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University of Southern California Dissertations and Theses
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Quantum information-theoretic aspects of chaos, localization, and scrambling
(USC Thesis Other)
Quantum information-theoretic aspects of chaos, localization, and scrambling
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Qu a ntum i nf o rma ti o n - th eo r e ti c a s pec ts o f c h a os , lo c a liz a ti o n , a nd sc r a mb li n g
b y
N a mit A n a nd
A Di s s e r t a t ion P r e s e n t e d t o the
F a c u l t y o f th e U S C Gr a d u a te Sc h o o l
Univ er sit y o f So u th er n C a li f o r ni a
I n P a r t i al F ul fi l lme n t of the
R e quir e me n ts for the D e gr e e
Do c t o r o f Ph i loso phy
( Phy si c s )
Au g us t 2022
C op y r i gh t 2022 N a mit A n a nd
T o Mom
Ackno w led g me n t s
They s a y I t T a k e s a V i ll a ge . I ca n ’t quit e c e r t i f y th a t , but I ca n s a y for s ur e , th a t I a m gr a t ef ul t o h a v e
be e n r ai s e d b y one . A s I s t a r t e d t o w r it e thi s, I r e al i z e d th a t it i s not quit e “ pr ofe s sion al , ” ( in th a t it i s not
c omp le t e ly de v o id of s e n t ime n t ) but I g ue s s the r e i s only s o m uch one ca n ca r e a bout , s o he r e it g oe s . F iv e
y e a r s i s a lo n g t ime , one w i th m a n y u ps a nd do w n s, a nd the s e fe w l ine s of th a nk y ous ca nnot do jus t ic e t o
the n ume r ous pe op le w ho h a v e done s o m uch for me . I w i l l tr y none the le s s .
F ir s t a nd for e mos t , I w ould l i k e t o sh a r e m y imme n s e gr a t itude t o m y a dv i s or , P a o lo Za n a r d i , w ho not
only t a u gh t me s o m uch P h ysic s, but mor e impor t a n tly , s e t u p a n ide al for w h a t it me a n s t o ch as e c ur iosit y .
A s a s tude n t , it i s r a r e t o h a v e the oppor tunit y t o s e e u p clos e the e v e r y d a y w ork in gs of a n exc e pt ion al
p h ysic i s t. Espe c i al ly one w ho i s in v e s t e d in t e a chin g y ou thos e thin gs i f y ou s o de sir e . I’ m v e r y gr a t ef ul for
the w onde r f ul hi k e s in D o lomit e s a nd Be n as que , w hich t oo k me out of m y c omfor t z one , but in r e tr ospe ct ,
left me w ith a de e p a ppr e c i a t ion for the moun t ain s . Y our gle e in t al k in g a bout cl imb in g a nd r unnin g h a v e
c e r t ainly be e n in s tr ume n t al in in sp ir in g me t o ch as e the s a me , a nd m y l i fe i s c e r t ainly mor e f ul fi l le d for
the m . I ca n s a y w ith c e r t ain t y th a t I’ m a be tt e r pe r s on t od a y th a n w he n I w al k e d in t o U S C , a nd I o w e s o
m uch of th a t t o y ou . On the r e s e a r ch side , y ou w e r e alw a ys v e r y w e lc omin g of ne w d ir e ct ion s, but not
in the “ do w h a t e v e r y ou w a n t , I don ’t r e al ly ca r e ” k ind of w a y , r a the r , y ou m a n tr a w as, ” i f y ou be l ie v e it ’ s
in t e r e s t in g , I’ m s ur e y ou ca n c on v inc e me , ” s ome thin g I de e p ly a ppr e c i a t e .
The r e ’ s a lon g l i s t of thin gs I’ m th a nk f ul for , but s ome honor a b le me n t ion s include , t h e b e s t s u m m er ev er
i i i
in I t aly , the c oun tle s s hour s spe n t in f r on t of the w hit e bo a r d , the m a n y d inne r s a t y our p l a c e , a nd a bo v e
al l y our f r ie nd ship a nd me n t or ship . A spe c i al th a nk y ou for the p a n dem ic y e a r s , it h as be e n a d i ffic ult t ime
for us al l , a nd y our p a t ie nc e , k indne s s, a nd flex i b i l it y m a de one of the w or s t t ime s in h um a n hi s t or y m uch
mor e be a r a b le .
T o T odd B r un, th a nk y ou for bein g a n a m a z in g me n t or . W he ne v e r I h a d a que s t ion, s c ie n t i fic or othe r -
w i s e , y our offic e door w as alw a ys ope n . Y ou w e r e alw a ys v e r y e nc our a g in g of ne w r e s e a r ch d ir e ct ion s a nd
I a m gr a t ef ul for the c our s e s y ou t a u gh t a nd our c oun tle s s d i s c us sion s in y our offic e . I hope th a t one d a y I
ca n al s o de v e lop the le v e l of p h ysical in si gh ts y ou h a v e a bout almos t a n y t op ic . Th a nk y ou al s o for s u ppor t
dur in g m y s e c ond y e a r , it i s de e p ly a ppr e c i a t e d!
T o D a nie l L id a r , th a nk y ou for the exc e pt ion al ly g ood c our s e s w hich not only t a u gh t me a lot of qua n tum
infor m a t ion the or y but al s o in sp ir e d m a n y r e s e a r ch ide as . I n m a n y w a ys, m y j our ney a t U S C be g a n w ith
y our he lp , a nd for th a t I ca nnot th a nk y ou e nou gh .
T o Kr z ys zt of P i lch a nd S t e p h a n H a as, m a n y th a nk s for bein g exc e pt ion al gr a dua t e a dv i s or s - y ou h a v e
m a de m y t ime he r e s e a mle s s .
T o A a r on L a ud a , th a nk y ou for y our s u ppor t a nd for t e a chin g us s o m a n y c oo l thin gs a bout ca t e g or y
the or y a nd t opo lo g y . Y ou a r e a n exc e pt ion al t e a che r!
T o R os a Di F e l ic e , th a nk y ou for a gr e ein g t o be on m y c ommitt e e , for the c oun tle s s d inne r s, a nd for the
s umme r s t a y in I t aly . Y ou h a v e alw a ys be e n v e r y k ind , p a t ie n t , a nd s u ppor t iv e , a nd I’ m gr a t ef ul for th a t.
T o L or e nz o C a mpos V e n ut i , th a nk y ou for bein g a w e s ome!
T o the U S C a dmini s tr a t iv e s t a ff , Be tt y , L i s a , C hr i s t in a , y ou al l h a v e be e n a m a z in g a nd m a de e v e r y thin g
in the l as t 5 y e a r s th a t m uch e asie r .
T o Ge or g ios S t y l i a r i s, I simp ly do not h a v e e nou gh w or d s t o th a nk y ou . S o I w i l l s a y thi s in s t e a d , y ou
us e d t o s a y th a t L or e nz o impr o v e d y our p h ysic s, y our cl imb in g , a nd y our l i fe! Y ou w e r e m y L or e nz o .
T o E v a n g e los V l a chos, it h as tr uly be e n a p le as ur e . F r om t ak in g me t o H im al a ya n H ous e t o t ak in g me
cl imb in g , y ou ’ v e be e n the r e thr ou gh the thick of it. I w i sh th a t one d a y , I ca n be the k i nd of f r ie nd th a t y ou
h a v e be e n t o me al l the s e y e a r s . Th a nk y ou for the F1 w a t ch p a r t ie s, c oun tle s s d inne r s, c y cl in g , Be n as que ,
t e a chin g me e v e r y thin g I kno w a bout “ r e al qub its ” a nd e v e r y thin g in be t w e e n . Y our f r ie nd ship h as e nr iche d
iv
m y l i fe a nd for th a t I’ m tr uly gr a t ef ul .
T o B i be k P o k h a r e l , th a nk y ou for bein g one of a k ind . I a m p a r t ic ul a rly gr a t ef ul t o y ou for in v it in g
H aime n g a nd I, on our fir s t d a y of cl as s, t o for m a s tudy gr ou p . I c ould ne v e r h a v e for e s e e n the m a n y y e a r s
of f r ie nd ship thi s w ould e n t ai l :-) The infinit e ly m a n y hour s of c on v e r s a t ion w e h a d o v e r t e a a r e pe rh a ps
ex a ctly ho w I w ould l i k e t o r e me mbe r m y P hD . O f the m a n y thin gs I a m gr a t ef ul t o y ou for , the r e i s one
I ca nnot th a nk y ou e nou gh for : E m a c s . I t ch a n g e d m y l i fe a nd it i s h a r d t o s a y i f I w ould h a v e e v e r g ott e n
in t o it , w as it not for y ou . I h a v e r a r e ly e v e r me t a n y one w ho i s s o o v e r the t op w i l l in g t o he lp s ome one e l s e ,
w ithout w a x . Th a nk y ou al s o for bein g the r e t o d i s c us s p h ysic s or l i fe , a n y t ime th a t I ne e de d t o .
T o H aime n g Z h a n g , th a nk y ou for y our w a r m f r ie nd ship . The m a n y hour s spe n t dr ink in g t e a a nd d i s -
c us sin g p h ysic s a nd p hi los op h y , the m a n y hi k e s, cl imb in g a nd c y cl in g , I’ m gr a t ef ul for the m al l . Y our one
of a k ind g e n uinit y i s a c on s t a n t r e minde r of w h a t , pe rh a ps, w e should al l s e e k t o be .
T o S ahi l G ul a ni a , th a nk y ou for y our f r ie nd ship , the s c ie n t i fic a nd me t as c ie n t i fic d i s c us sion s, a nd for
y our exc e pt ion al c oo k in g ! I h a v e no clue w h a t l i fe in L A w ould be l i k e i f y ou w e r e n ’t he r e .
T o Ad a m P e a r s on, y ou a r e one of the be s t h um a n bein gs I h a v e e v e r me t a nd y our f r ie nd ship i s tr uly a
g i ft. Th a nk y ou for the a m a z in g s umme r in L os A l a mos a nd the f r ie nd ship th a t fo l lo w e d . Y ou m ak e me a
be tt e r h um a n bein g.
T o A a r on W ir th w ein, y ou a r e a w e s ome . Y our c ommitme n t t o c y cl in g i s unl i k e a n y thin g I h a v e e v e r s e e n
befor e . Y ou in sp ir e me a nd y our f r ie nd ship i s a g i ft.
T o F r e d a nd V iv i a n, th a nk y ou for the d inne r s a nd for y our f r ie nd ship . Y our unique pe r s on al it ie s a r e jus t
a j o y t o h a v e .
T o B r i a n a nd F aidon, th a nk y ou for the c oun tle s s p h ysic s a nd mor e impor t a n tly , me t as c ie n t i fic d i s c us -
sion s, it h as be e n a p le as ur e .
T o fa mi ly , I w i sh I c ould ex p l ain t o y ou , w h y I m us t be s o fa r a w a y t o do the thin gs I do . B ut I’ m jus t gl a d
th a t d i s t a nc e h as ne v e r c ome in the w a y of y our s u ppor t for me . W ithout y our p a t ie nc e a nd s u ppor t , none
of thi s w ould e v e n be pos si b le a nd I don ’t kno w w h a t w or d s t o us e t o c on v ey th a t. S o I’ l l jus t hope y ou
alr e a dy kno w ho w m uch y ou al l me a n t o me . D a d , th a nk y ou for alw a ys bein g s u ppor t iv e w he n I ne e de d it ,
br othe r , th a nk y ou for bein g y ou , y ou h a v e r ai s e d me , a nd M om, I w i sh y ou w e r e he r e t o s e e ho w e v e r y thin g
v
tur ne d out t o be , I tr uly mi s s y ou . T o b h a b hi , y ou h a v e m a de a home a w a y f r om home for me . I a m de e p ly
inde bt e d t o al l of y ou .
T o Z oe , J eff , S alv a t or e , Ele a nor a nd the r e s t of the NA SA Qu A I L a nd KBR t e a m, th a nk y ou for the s e l as t
fe w mon th s, it h as be e n tr uly w onde r f ul w ork in g w ith y ou .
T o f r ie nd s b a ck home: N a nd i , M a nn a , S om ya , S om a , K alp a n a , B i kr a m, S a g a r , N e o g , A b hi she k , B a p i ,
H im a n sh u , U tt a m, al l of y ou h a v e he lpe d sh a pe the pe r s on I a m t od a y a nd y our f r ie nd ship i s a g i ft I do not
t ak e for gr a n t e d . Th a nk y ou for be in g the r e e v e n on the d a rk e s t of d a ys, it h as alw a ys me a n t a lot t o me ,
e v e n i f I m a y not s a y it e nou gh .
T o m y unde r gr a d pr ofe s s or s, A r un K P a t i , C o l in Be n j a min, a nd A n a mitr a M uk he r j e e; th a nk y ou for
be l ie v in g in me . W ithout y our c on s t a n t s u ppor t a nd e nc our a g e me n t , thi s j our ney w ould not h a v e be e n
pos si b le .
T o S hin y C houd h ur y , y e s, y ou ’ r e a t the e nd of thi s l i s t , but the r e i s oft e n a s y mme tr y be t w e e n the be-
g innin g a nd the e nd . N o a moun t of th a nk y ous w i l l e v e r s u ffic e , s o I w on ’t bothe r w ith the m . E v e r y sin gle
d a y of m y l i fe h as be e n m a de be tt e r b y y ou . Th a nk y ou for al l the p a t ie nc e y ou k e pt al l the s e y e a r s . A nd
th a nk y ou for th a t w onde r f ul , quirk y s e n s e of h umor , w hich h as m a de e v e n the d a rk e s t of d a ys th a t m uch
br i gh t e r . I t i s r a r e t o me e t s ome one w ho ca n he lp y ou g o bey ond y our s e l f , but w ith y ou , it ’ s alw a ys be e n
th a t w a y . A nd thi s i s jus t the be g innin g.
v i
T a ble of C on t e n t s
D ed i c a ti o n ii
A c kn o wled g ments iii
L is t o f T a b le s x
L is t o f Fi g u r e s x
L is t o f Pu b li c a ti o n s x v iii
A bs tr a c t x x
1 Qu a ntum c o h er en c e a s a si gn a tu r e o f c h a os 1
1.1 A bs tr a ct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 I n tr oduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 P r e l imin a r ie s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 A t the le v e l of s t a t e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 A t the le v e l of ch a nne l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.6 Di s c us sion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.7 L e v e l sp a c in g d i s tr i but ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.8 C o he r e nc e qua n t i fie r s for i n t e gr a b le a nd ch a ot ic ei g e n s t a t e s . . . . . . . . . . . . . . . . 34
v i i
1.9 P r oof s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2 Qu a ntum c o h er en c e a nd th e lo c a liz a ti o n tr a n siti o n 50
2.1 A bs tr a ct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.2 I n tr oduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3 Qua n tum c o he r e nc e of s t a t e s a nd ope r a t ion s . . . . . . . . . . . . . . . . . . . . . . . 52
2.4 C o he r e nc e- g e ne r a t in g po w e r a nd l ocal i za t ion in the 1- D A nde r s on mode l . . . . . . . . 64
2.5 C o he r e nc e- g e ne r a t in g po w e r a nd m a n y - body l ocal i za t ion . . . . . . . . . . . . . . . . . 67
2.6 Di ffe r e n t i al g e ome tr y of c o he r e nc e- g e ne r a t in g po w e r a nd MBL . . . . . . . . . . . . . . 71
2.7 C onclusion a nd outloo k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
A ppe nd ic e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3 Inf o rma ti o n Sc r a mb li n g o v er Bi p a r titi o n s: Eq u i li b r a ti o n , Entr o p y Pr o d uc ti o n ,
a nd T y pi c a lit y 85
3.1 A bs tr a ct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2 P r oof s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.3 H a a r me as ur e , unit a r y k - de si gn s a nd the b ip a r t it e O T O C . . . . . . . . . . . . . . . . . 116
3.4 Es t im a t in g the b ip a r t it e O T O C v i a l ine a r e n tr op y me as ur e me n ts of r a ndom pur e s t a t e s . 118
4 Inf o rma ti o n Sc r a mb li n g a nd Ch a os i n O pen Qu a ntum S y s te ms 120
4.1 A bs tr a ct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.2 I n tr oduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.3 Ge ne r al r e s ults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.4 S ome spe c i al ch a nne l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
4.5 Qua n tum S p in C h ain s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.6 C onclusion s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.7 R e v ie w of ope r a t or e n t a n gle me n t a nd e n t a n gl i n g po w e r . . . . . . . . . . . . . . . . . . 142
4.8 A pr ot oc o l for e s t im a t i n g the O pe n O T O C . . . . . . . . . . . . . . . . . . . . . . . . 143
4.9 P r oof s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
v i i i
5 BR O T O Cs a nd Qu a ntum Inf o rma ti o n Sc r a mb li n g a t Fi nite T e mper a tu r e 160
5.1 A bs tr a ct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.2 I n tr oduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.3 P r e l imin a r ie s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.4 M ain r e s ults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5.5 N ume r ical sim ul a t ion s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.6 C onclusion s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
Ou tlo o k 199
R efer en c e s 201
i x
Li st of T a bles
5.5.1 The de ca y r a t e γ for v a r ious H a mi lt oni a n mode l s a t β = 0; 1;1 , w ith r e spe ct t o the
A n s a tz G
( r)
β
( t) = α d
γ
. The pr efa ct or α i s n o n u n i v er sa l , the de t ai l s of w hich ca n be found
in the A ppe nd i x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.6.1 The log
2
( α) for v a r ious H a mi lt oni a n mode l s a t β = 0; 1;1 , g iv e n the A n s a tz G
( r)
β
( t) =
α d
γ
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
x
Li st of F i g ur es
1.4.1 R e l a t iv e e n tr op y of c o he r e nc e for ei g e n s t a t e s of the H a mi lt oni a n define d in e q . ( 1.13 ) as a
f unct ion of their e ne r g y , nor m al i z e d w i th the G OE pr e d ict ion usin g e q . ( 1.14 ),
⟨
c
( r e l)
B
⟩
GOE
10: 49 . R e s ults a r e r e por t e d for L = 15 w ith 5 sp in s u p a nd ω = 0 , ε
δ
= 0: 5 , J
xy
= 1 ,
J
z
= 0: 5 . The p lot m a rk e r s 1; 3; 5; 7 c or r e spond t o the v a r ious cho ic e s of the defe ct sit e ,
w ith δ = 1 a nd δ = 7 c or r e spond in g t o the in t e gr a b le a nd ch a ot ic l imits, r e spe ct iv e l y .
F i g ur e s ( a ) a nd ( b ) c or r e spond t o the t w o d i ffe r e n t b as e s, the sit e- b asi s a nd me a n- fie ld
b asi s, r e spe ct iv e ly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.2 2 - c o he r e nc e for ei g e n s t a t e s of the H a mi lt oni a n define d in e q . ( 1.13 ) as a f unct ion of their
e ne r g y , nor m al i z e d w ith the G OE pr e d ict ion o bt aine d n ume r ical ly ,
⟨
c
( 2)
B
⟩
GOE
0: 9991 .
R e s ults a r e r e por t e d for L = 15 w ith 5 sp in s u p a nd ω = 0 , ε
δ
= 0: 5 , J
xy
= 1 , J
z
= 0: 5 . The
p lot m a rk e r s 1; 3; 5; 7 c or r e spond t o the v a r ious cho ic e s of the defe ct sit e , w ith δ = 1 a nd
δ = 7 c or r e spond in g t o the in t e gr a b le a nd ch a ot ic l imits, r e spe ct iv e ly . F i g ur e s ( a ) a nd
( b ) c or r e spond t o the t w o d i ffe r e n t b as e s, the sit e- b asi s a nd m e a n- fie ld b asi s, r e spe ct iv e ly . 12
1.4.3 F r a ct ion of in t e gr a b le ei g e n s t a t e s th a t m a j or i z e ch a ot ic ei g e n s t a t e s for the H a mi lt oni a n
define d in e q . ( 1.13 ) s ys t e m si z e L . H e r e , δ = 1 for in t e gr a b le ei g e n s t a t e s, a nd δ =⌊ L= 2⌋
for ch a ot ic ei g e n s t a t e s . The p lot m a rk e r s c or r e spond t o the t w o d i ffe r e n t b as e s, the sit e-
b asi s a nd me a n- fie ld b asi s, r e spe ct iv e ly . F i g ur e s ( a ) a nd ( b ) c or r e spond t o the f ul l spe c -
tr um a nd 20% of ei g e n v e ct or s in the midd le of the spe ctr um, r e spe ct iv e ly . . . . . . . . . 14
x i
1.5.1 ( a ) L o g - L o g p lot of the v a r i a nc e of C GP a nd O T O C for n = 9 qub its . W e s tudy the
dy n a mic s of the H a mi lt oni a n g iv e n b y e q . ( 1.44 ) w ith g = 1; h = 0 as the in t e gr a b le l imit
a nd g = 1: 05; h = 0: 5 as the ch a ot ic one . W e s e t , V = σ
z
1
; W = σ
z
9
for the O T O C a nd
C GP in e q . ( 1.20 ). ( b ) F r a ct ion of the lon g -t ime a v e r a g e of the v a r i a nc e of ch a ot ic a nd
in t e gr a b le O T O C , C GP , th a t i s,
Var
integrable
Var
chaos
Var
integrable
for the H a mi lt oni a n g iv e n b y e q . ( 1.44 )
w ith g = 1; h = 0 as the in t e gr a b le l imit a nd g = 1: 05; h = 0: 5 as the ch a ot ic one . W e
s e t , V = σ
z
1
; W = σ
z
9
for the O T O C a nd C GP in e q . ( 1.20 ). . . . . . . . . . . . . . . . . 30
1.7.1 The tr a n sit ion in le v e l - sp a c in g d i s tr i but ion f r om P o i s s on t o the ( univ e r s al ) W i gne r - D ys on
d i s tr i but ion for the H a mi lt oni a n de s cr i be d in e q . ( 1.13 ) as w e mo v e the defe ctsit e t o the
midd le of the ch ain . F i g ur e s ( a ), ( b ), ( c ), a nd ( d ) c or r e spond t o the defe ct a t sit e s δ =
1; δ = 3; δ = 5; a nd δ = 7 , r e spe ct iv e ly . R e s ults a r e r e por t e d for L = 15 w ith 5 sp in s
u p a nd ω = 0; ε
δ
= 0: 5; J
xy
= 1; J
z
= 0: 5 . S imi l a r r e s ults w e r e o bt aine d for L = 15 a nd
δ = 1; 7 in R ef . [ 1 ] ( but not for in t e r me d i a t e posit ion s of the defe ct sit e ). . . . . . . . . . 34
1.8.1 I n v e r s e p a r t ic ip a t ion r a t io for ei g e n s t a t e s of the H a mi lt oni a n define d in e q . ( 1.13 ) as a
f unct ion of their e ne r g y . R e s ults a r e r e por t e d for L = 15 w ith 5 sp in s u p a nd ω = 0; ε
δ
=
0: 5; J
xy
= 1; J
z
= 0: 5 . The p lot m a rk e r s 1; 3; 5; 7 c or r e spond t o the v a r ious cho ic e s of the
defe ct sit e , w ith δ = 1; 7 c or r e spond in g t o the in t e gr a b le a nd ch a ot ic l imits, r e spe ct iv e ly .
F i g ur e s ( a ) a nd ( b ) c or r e spond t o the t w o d i ffe r e n t b as e s, the sit e- b asi s a nd the me a n- fie ld
b asi s, r e spe ct iv e ly . S imi l a r r e s ults w e r e o bt aine d for L = 18 a nd δ = 1; 9 in R ef . [ 1 ] ( but
not for in t e r me d i a t e posit ion s of the defe ct sit e ). . . . . . . . . . . . . . . . . . . . . . . 35
1.8.2 1 - c o he r e nc e for ei g e n s t a t e s of the H a mi lt oni a n define d in e q . ( 1.13 ) as a f unct ion of their
e ne r g y . R e s ults a r e r e por t e d for L = 15 w ith 5 sp in s u p a nd ω = 0; ε
δ
= 0: 5; J
xy
=
1; J
z
= 0: 5 . The p lot m a rk e r s 1; 3; 5; 7 c or r e spond t o the v a r ious cho ic e s of the defe ct sit e ,
w ith δ = 1; 7 c or r e spond in g t o the in t e gr a b le a nd ch a ot ic l imits, r e spe ct iv e ly . F i g ur e s
( a ) a nd ( b ) c or r e spond t o the t w o d i ffe r e n t b as e s, the sit e- b asi s a nd the me a n- fie ld b asi s,
r e spe ct iv e ly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
x i i
2.4.1 L o g -lo g p lot of the a v e r a g e r e tur n pr o b a b i l it y 1⟨ C
( 2)
B
(V
W
)⟩ as a f unct ion of the s ys t e m
si z e L for d i ffe r e n t v alue s of the d i s or de r s tr e n g th W . The s ys t e m i s in the local i z e d p h as e
for al l W > 0 , sinc e the as y mpt ot ic e s ca pe pr o b a b i l it y i s s tr ictly le s s th a t 1 for L!1 .
The n umbe r of r e al i za t ion s r a n g e f r om 30 000 for s m al l si z e s t o jus t 8 for the l a r g e s t si z e .
E r r or b a r s r e pr e s e n t one s t a nd a r d de v i a t ion . E n tr op y h as lo g a r ithm w ith b as e 2. . . . . . 67
2.4.2 L o g -l ine a r p lot of ⟨ C
( rel)
B
(V
W
)⟩ as a f unct ion of the s ys t e m si z e L for d i ffe r e n t v alue s of
the d i s or de r s tr e n g th W . The s ys t e m i s in the local i z e d p h as e for al l W > 0 , in w hich
the as y mpt ot ic v alue i s finit e . I n the e r g od ic p h as e ( W = 0 )⟨ C
( rel)
B
(V
W
)⟩ d iv e r g e s lo g a -
r ithmical ly . The n umbe r of r e al i za t ion s r a n g e f r om 30 000 for s m al l si z e s t o jus t 8 for the
l a r g e s t si z e . E r r or b a r s r e pr e s e n t one s t a nd a r d de v i a t ion . E n tr op y h as lo g a r ithm w ith b as e 2. 68
2.5.1 A s y mpt ot ic be h a v ior for the slope of the fo l lo w in g qua n t it ie s: log
2
(
1⟨ C
( 2)
B
(V
W
)⟩
)
=
log
2
( P
r e tur n
) ,⟨ C
( r e l)
B
(V
W
)⟩ , log
2
(
1⟨ f
( det)
B
( X
V W
)⟩
)
, log
2
(
1⟨ f
( t ime-a v g)
B
( X
V W
)⟩
)
, a nd
log
2
(
1⟨ f
(1)
B
( X
V W
)⟩
)
for l a r g e L as a f unct ion of the d i s or de r s tr e n g th W for the H a mi l -
t oni a n H
XXX
a t h
x
= 0: 3 . The slope w as ex tr a ct e d for si z e s L = 4; 6; ; 14 , w ith s a mp le
si z e s 20 000 , 20 000 , 20 000 , 8000 , 2000 , 800 ; exc e pt a t W = 3: 7 , w he r e the s a mp le si z e s
w e r e doub le d . The e r r or b a r s r e pr e s e n t the s t a nd a r d e r r or of the l ine a r fit. E n tr op y h as
lo g a r ithm w ith b as e 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.7.1 P lot of the e s ca pe pr o b a b i l it y ⟨ C
( 2)
B
(V
Γ
)⟩ as a f unct ion of the d i s or de r s tr e n g th Γ for the
L lo y d mode l H a mi lt oni a n H
Γ
, as pr e d ict e d a n aly t ical ly b y the he ur i s t ic E q . ( 2.28a ) ( s o l id
l ine ) a nd the n ume r ical sim ul a t ion s ( po in ts ). F or the cas e of the n ume r ical sim ul a t ion,
L ! 1 i s ex tr a po l a t e d b y a v e r a g in g o v e r d i s or de r for si z e s u p t o L = 2
12
. S t a nd a r d
de v i a t ion s a r e w ithin the po in t r a d ius . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
x i i i
2.7.2 P lot of the ( a ) a v e r a g e e s ca pe pr o b a b i l it y ⟨ C
( 2)
B
(V
W
)⟩ a nd ( b )⟨ C
( rel)
B
(V
W
)⟩ as a f unct ion
of the s ys t e m si z e L for d i ffe r e n t v alue s of the d i s or de r s tr e n g th W . The d i s or de r v alue s
d i sp l a y e d a r e W = 0: 4; 1: 0; 1: 4; 1: 8; 2: 5; 3: 1; 3: 7; 5: 0; 7: 0; 9: 0 ( monot onical ly f r om the t op
t o bott om in the p lots ) for L = 4; 6; ; 12 w ith s a mp le si z e s 20 000 , 20 000 , 20 000 ,
8000 , 2000 ; exc e pt a t the W = 3: 7 , w he r e the s a mp le si z e s w e r e doub le d . E r r or b a r s
r e pr e s e n t one s t a nd a r d de v i a t ion . E n tr op y h as lo g a r ithm w ith b as e 2. . . . . . . . . . . . 83
2.7.3 P lot of the g e ne r al i z e d - C GP me as ur e s, ( a )⟨ f
( t ime-a v g)
B
( X
V W
)⟩ , ( b )⟨ f
( de t)
B
( X
V W
)⟩ , a nd ( c )
⟨ f
(1)
B
( X
V W
)⟩ as a f unct ion of the s ys t e m si z e L for d i ffe r e n t v alue s of the d i s or de r s tr e n g th
W . The d i s or de r v alue s d i sp l a y e d a r e W = 0: 4; 1: 0; 1: 4; 1: 8; 2: 5; 3: 1; 3: 7; 5: 0; 7: 0; 9: 0 ( mono -
t onical ly f r om the t op t o bott om for ( a ) a nd bott om t o t op for ( b )) for L = 4; 6; ; 12
w ith s a mp le si z e s 20 000; 20 000; 20 000; 8000; 2000 ; exc e pt a t the W = 3: 7 , w he r e the
s a mp le si z e s w e r e doub le d . E r r or b a r s r e pr e s e n t one s t a nd a r d de v i a t ion . . . . . . . . . . 84
3.1.1 L o g a r ithmic p lot of v a r ious G e s t im a t e s, alon g w ith the ex a ct t ime-a v e r a g e , for fi xe d d
A
=
2 as a f unct ion of the t ot al n umbe r of sp in s n . G
Haar
1
= 3= 4 c or r e spond s t o the H a a r
e s t im a t e for n!1 . F or the c h a ot ic p h as e of the TFI M ( g = 1: 05 , h = 0: 5 ), the NR C
c on s t itut e s a s a t i sfa ct or y , thou gh impe r fe ct , a ppr o x im a t ion . The ch a ot ic a nd in t e gr a b le
p h as e s ( h = 0 ) ca n be cle a rly d i s t in g ui she d thr ou gh the e qui l i br a t ion be h a v ior of the
b ip a r t it e O T O C . F or the in t e gr a b le X X Z mode l (w e s e t J = 0: 4 , Δ = 2: 5 ), the NR C+
e s t im a t e c o inc ide s ( u p t o n ume r ical e r r or ) w ith the ex a ct t ime-a v e r a g e . I ne qual it y ( 3.11 )
ho ld s v al id in al l cas e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.4.1 P r ot oc o l t o for the e s t im a t ion of the pur it y 1 S
lin
[
Λ
( A)
t
(j ψ⟩⟨ ψj)
]
a c c or d in g t o E q . ( 3.13 ).
The r e s ult in g pur it y c on s t itut e s al s o a n e s t im a t e of the b ip a r t it e O T O C , u p t o a simp le
pr opor t ion al it y fa ct or . The fin al me as ur e me n t of the s w a p ope r a t or ca n be r e al i z e d , for
in s t a nc e , b y me as ur in g the ex pe ct a t ion v alue of A a n d A
′
o v e r a n y pr efe r r e d pr oduct b asi s
fj i⟩
j j⟩g
d A
i; j= 1
, w ithout the ne e d for c o he r e nc e s . . . . . . . . . . . . . . . . . . . . . . . 118
x iv
4.4.1 N on- unit a r y O T O C G( e
L t
) w ithL = iadS + λ(D
B
I) w he r e S i s the s w a p ope r a t or ,
B i s the Be l l b asi s ( d
A
= d
B
= 2 ). The d i ffe r e n t c ur v e s c or r e spond t o d i ffe r e n t cho ic e s
of the de p h asin g p a r a me t e r λ: O v e r the t ime s cale λ
1
on w hich de p h asin g be c ome s r e le-
v a n t the “ s cr a mb l in g e n tr op y ” ( fir s t t e r m in e q . ( 4.15 )) i s b al a nc e d , a nd e v e n tual ly o v e r -
w he lme d , b y the de c o he r e nc e-induc e d e n tr op y pr oduct ion ( s e c ond t e r m in e q . ( 4.15 )).
F or a n y fi xe d t ime t the O T O C s u ppr e s sion i s ex pone n t i al in the de p h asin g s tr e n g th λ:
M or e o v e r , in sh a r p c on tr as t w ith the unit a r y cas e , for a n y λ̸= 0; the infinit e t ime l imit of
the O T O C i s v a ni shin g. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.5.1 T e mpor al v a r i a t ion of the ope n O T O C G(E
t
) for the TFI M e q . ( 4.30 ) w ith L = 6 sp in s .
The thr e e c ur v e s c or r e spond t o v a r y in g cho ic e s of the d i s sip a t ion s tr e n g th α in the L ind -
b l a d ope r a t or s e q . ( 4.28 ). The ch a ot ic ( g = 1: 05; h = 0: 5 ) a nd in t e gr a b le ( g = 1; h =
0 ) p h as e s a r e cle a rly d i s t in g ui sh a b le for the α = 0 cas e ( clos e d s ys t e m ), ho w e v e r , incr e as -
in g the d i s sip a t ion s tr e n g th t o α = 0: 05 m ak e s the m fairly ind i s c e r ni b le a nd de s tr o ys the
r e v iv al s ( or fluctua t ion s ) ch a r a ct e r i s t ic of in t e gr a b le s ys t e m s . . . . . . . . . . . . . . . . 137
4.5.2 T e mpor al v a r i a t ion of the ope n O T O C G(E
t
) for the X X Z - NNN mode l e q . ( 4.31 ) w ith
L = 6 sp in s . The thr e e c ur v e s c or r e spond t o v a r y in g cho ic e s of the d i s sip a t ion s tr e n g th
α in the L ind b l a d ope r a t or s e q . ( 4.28 ) . The nonin t e gr a b le ( J = 1; Δ = 0: 5; J = 1; Δ
′
=
0: 5 ) a nd in t e gr a b le ( J = 1; Δ = 0 = J = Δ
′
) p h as e s a r e cle a rly d i s t in g ui sh a b le for the
α = 0 = γ cas e ( clos e d s ys t e m ). The in t e gr a b le mode l he r e ca n be m a ppe d on t o f r e e
fe r mion s a nd he nc e unl i k e the TFI M cas e , e v e n a ft e r incr e asin g the d i s sip a t ion s tr e n g th
( α = 0: 1 = γ ), the s ys t e m de mon s tr a t e s r e v iv al s ( or fluctua t ion s ) ch a r a ct e r i s t ic of in t e-
gr a b le s ys t e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
x v
4.5.3 T e mpor al v a r i a t ion of the ind iv idual t e r m s of the ope n O T O C G
( 1)
(E
t
) =
d B
d
2
Tr[ SE
2
( S
AA
′)]
a nd G
( 2)
(E
t
) =
1
d
2
Tr[ S
AA
′E
2
( S
AA
′)] w ith G(E
t
) = G
( 1)
(E
t
) G
( 2)
(E
t
) . The t w o fi g ur e s
c or r e spond t o the in t e gr a b le a nd ch a ot ic l imits as c on side r e d a bo v e for the ( a ) TFI M a nd
( b ) X X Z - NNN mode l w ith L = 6 sp in s, r e spe ct iv e ly . The d i s sip a t ion p a r a me t e r s a r e
α = 0: 01; γ = 0: 01 . The fir s t t e r m G
( 1)
(E
t
) or i g in a t e s f r om e n v ir onme n t al de c o he r e nc e
a nd i s simi l a r for both the in t e gr a b le a nd the ch a ot ic cas e . H o w e v e r , the s e c ond t e r m,
G
( 2)
(E
t
) i s cle a rly d i s t inct for the t w o p h as e s a nd ca n d i a gnos e qua n tum ch a os e v e n in the
pr e s e nc e of d i s sip a t ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5.4.1 A lo g -lo g p lot of the e qui l i br a t ion v alue ( lon g -t ime a v e r a g e ) of the G
( r)
β
( t) for the G UE
H a mi lt oni a n a t d = 100 for 10
10
β 10
3
c omp a r in g the n ume r ical e s t im a t e t o the
Be s s e l f unct ion for m a bo v e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
5.5.1 A lo g -lo g p lot of the e qui l i br a t ion v alue ( lon g -t ime a v e r a g e ) of the G
( r)
β
( t) for v a r ious
H a mi lt oni a n mode l s a t L = 6 as a f unct ion of the in v e r s e t e mpe r a tur e β a cr os s a s y mme t -
r ic b ip a r t it ion L= 2 : L= 2 . W e us e ex a ct t ime e v o lut ion for the in t e gr a b le TFI M a nd A n-
de r s on sinc e they do not s a t i sf y NR C . F or A nde r s on, MBL , a nd G UE, w e pe r for m ex a ct
d i a g on al i za t ion of the f ul l H a mi lt oni a n a nd us e the a n aly t ical ex pr e s sion in 5.4 . F or NR C -
PS a nd ME ( m a x im al ly e n t a n gle d mode l ), w e us e the a n aly t ical ex pr e s sion s in e q . ( 5.40 )
a nd e q . ( 5.48 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
5.5.2 A lo g p lot of the e qui l i br a t ion v alue ( lon g -t ime a v e r a g e ) of the G
( r)
β
( t) for v a r ious H a mi l -
t oni a n mode l s as a f unct ion of the s ys t e m si z e L a t β = 0 a cr os s a s y mme tr ic b ip a r t it ion
⌊ L= 2⌋ :⌈ L= 2⌉ . W e us e ex a ct t ime e v o lut ion for the in t e gr a b le TFI M a nd A nde r s on sinc e
they do not s a t i sf y NR C . F or A nde r s on, MBL , a nd G UE, w e pe r for m ex a ct d i a g on al i za -
t ion of the f ul l H a mi lt oni a n a nd us e the a n aly t ical ex pr e s sion in 5.4 . F or NR C - PS a nd ME
( m a x im al ly e n t a n gle d mode l ), w e us e the a n aly t ical ex pr e s sion s in e q . ( 5.40 ) a nd e q . ( 5.48 ). 187
x v i
5.5.3 A lo g p lot of the e qui l i br a t ion v alue ( lon g -t ime a v e r a g e ) of the G
( r)
β
( t) for v a r ious H a mi l -
t oni a n mode l s as a f unct ion of the s ys t e m si z e L a t β = 1 a cr os s a s y mme tr ic b ip a r t it ion
⌊ L= 2⌋ :⌈ L= 2⌉ . W e us e ex a ct t ime e v o lut ion for the in t e gr a b le TFI M a nd A nde r s on sinc e
they do not s a t i sf y NR C . F or A nde r s on, MBL , a nd G UE, w e pe r for m ex a ct d i a g on al i za -
t ion of the f ul l H a mi lt oni a n a nd us e the a n aly t ical ex pr e s sion in 5.4 . F or NR C - PS a nd ME
( m a x im al ly e n t a n gle d mode l ), w e us e the a n aly t ical ex pr e s sion s in e q . ( 5.40 ) a nd e q . ( 5.48 ). 187
5.5.4 A lo g p lot of the e qui l i br a t ion v alue ( lon g -t ime a v e r a g e ) of the G
( r)
β
( t) for v a r ious H a mi l -
t oni a n mode l s as a f unct ion of the s ys t e m si z e L a t β =1 a cr os s a s y mme tr ic b ip a r t it ion
⌊ L= 2⌋ :⌈ L= 2⌉ . The v a r ious d a t a po in ts h a v e c o ale s c e d in t o t w o c ur v e s, fir s t c on si s t in g of
al l the in t e gr a b le mode l s, w hos e gr ound s t a t e fo l lo w a r e a -l a w e n t a n gle me n t. A nd s e c ond ,
for the G UE a nd ME ( m a x im al ly e n t a n gle d mode l ), w hos e gr ound s t a t e s fo l lo w v o lume-
l a w e n t a n gle me n t. U sin g 5.3, w e simp ly c omput e the gr ound s t a t e pr oj e ct or for v a r ious
mode l s t o c omput e thi s n ume r ical ly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
x v i i
Li st of Public a tions
The w ork pr e s e n t e d in thi s the si s c on t ain s m a t e r i al f r om the fo l lo w in g pub l ica t ion s a nd pr e pr in ts:
• N a mit A n a nd , Ge or g ios S t y l i a r i s, M e e n u K um a r i , a nd P a o lo Za n a r d i . Qua n tum c o he r e nc e as a si g -
n a tur e of ch a os . Ph y s . R ev . R e se a r c h , 3, 023214 (2021).
doi:10.1103/PhysRevResearch.3.023214 .
• Ge or g ios S t y l i a r i s, N a mit A n a nd , L or e nz o C a mpos V e n ut i , a nd P a o lo Za n a r d i . Qua n tum c o he r e nc e
a nd the loc al i za t ion tr a n sit ion . Ph y s . R ev . B , 100:224204 (2019).
doi:10.1103/PhysRevB.100.224204 .
• Ge or g ios S t y l i a r i s, N a mit A n a nd , a nd P a o lo Za n a r d i . I nfor m a t ion S cr a mb l in g o v e r B ip a r t it ion s:
E qui l i br a t ion, E n tr op y P r oduct ion, a nd T y p ical it y . Ph y s . R ev . Lett . 126, 030601 (2021).
doi:10.1103/PhysRevLett.126.030601 .
• P a o lo Za n a r d i a nd N a mit A n a nd . I nfor m a t ion s cr a mb l in g a nd ch a os in ope n qua n tum s ys t e m s . Ph y s .
R ev . A , 103 , 062214 (2021).
doi:10.1103/PhysRevA.103.062214 .
• N a mit A n a nd a nd P a o lo Za n a r d i . BR O T O C s a nd Qu a n tum I nfor m a t ion S cr a mb l in g a t F init e T e m-
pe r a tur e . a r X i v:2111.07086 [ q u a n t-p h ] , (2021).
doi:10.48550/ARXIV.2111.07086 .
x v i i i
Othe r pub l ica t ion s a nd pr e pr in ts c omp le t e d dur in g the dur a t ion of the P hD but not include d in thi s the si s
a r e:
• B i be k P o k h a r e l , N a mit A n a nd , Be n j a min F or tm a n, a nd D a nie l A . L id a r . D e mon s tr a t ion of F ide l it y
I mpr o v e me n t U sin g D y n a mical D e c ou p l in g w ith S u pe r c onduct in g Qub its . Ph y s . R ev . Lett . , 121,
220502 (2018).
doi:10.1103/PhysRevLett.121.220502 .
• N a mit A n a nd a nd T odd A . B r un . Qua n t i f y in g non- M a rk o v i a nit y: a qua n tum r e s our c e-the or e t ic
a ppr o a ch . a r X i v:1903.03880 [ q u a n t-p h ] , (2019).
doi:10.48550/ARXIV.1903.03880 .
x i x
A b str a ct Thi s the si s ex p lor e s f r om the le n s of qua n tum infor m a t ion the or y , thr e e d i s t inct qua n tum m a n y - body
p he nome n a:( i) qua n tum ch a os, ( ii) A n de r s on a nd m a n y - body local i za t ion, a nd ( iii) infor m a t ion s cr a m-
b l in g.
Co h er en ce a n d c h a o s .— W e sho w th a t the qua n tum c o he r e nc e c on t e n t of both s t a t e s a nd dy n a mic s pr o -
v ide s a d i a g n o s t ic t oo l for the on s e t of qua n tum ch a os in m a n y - body s ys t e m s . I n p a r t ic ul a r , the a v e r a g e
c o he r e nc e of H a mi lt oni a n ei g e n s t a t e s in the midd le of the spe ctr um ca n d i s t in g ui sh in t e gr a b le a nd ch a ot ic
mode l s . M or e o v e r , ch a ot ic s ys t e m s a r e kno w n t o be f as t scr a m b ler s of infor m a t ion a nd w e sho w th a t the s o -
cal le d out - of -t ime- or de r e d c or r e l a t or ( O T O C ), a qua n t i fie r of infor m a t ion s cr a mb l in g , i s in fa ct in t im a t e ly
r e l a t e d t o the c o he r e nc e- g e ne r a t in g po w e r of the dy n a mic s .
Co h er en ce a n d lo c a l i za t io n.— The a bs e nc e of tr a n spor t in d i s or de r e d s ys t e m s, t e r me d lo c a l i za t io n , i s
us ual ly qua n t i fie d b y the ( as y mpt ot ic ) r e tur n pr o b a b i l it y of a p a r t icle . W e sho w th a t for non- de g e ne r a t e
H a mi l t oni a n s, thi s i s ex a ctly e qual t o the 2 - nor m c o he r e nc e of the qua n tum s t a t e , w ith r e spe ct t o the H a mi l -
t oni a n ei g e nb asi s . Thi s al lo w s us t o o bt ain a w e alth of r e s ults for v a r ious not ion s of c o he r e nc e- g e ne r a t in g
po w e r a nd e mp lo y the m as a si g n a t u r e of the e r g od ic -t o -local i za t io n tr a n sit ion .
Scr a m b l i n g a n d o p er a t o r en t a n g lem en t .— W e sho w th a t a spe c i al cl as s of a v e r a g e d O T O C s qua n t i f y ex -
a ctly the ope r a t or e n t a n gle me n t a nd e n t a n gl in g po w e r of unit a r y dy n a mic s, in tur n pr o v id in g ope r a t ion al
me a nin g t o O T O C s the m s e lv e s . A n e v o lut ion i s t e r me d scr a m b l i n g i f it quick ly d i s s e min a t e s local infor -
x x
m a t ion thr ou ghout the nonlocal de gr e e s of f r e e dom of the s ys t e m . F or unit a r y s cr a mb l in g e v o lut ion s, the
r e duc e d dy n a mic s i s ex pe ct e d t o be m i x i n g a nd w e qua n t i f y thi s b y ( i ) r e l a t in g the ( a v e r a g e ) local e n tr op y
pr oduct ion t o ( a v e r a g e d ) O T O C s a nd ( i i ) sho w in g th a t the ( a v e r a g e ) O T O C qua n t i fie s ho w fa r the r e-
duc e d dy n a mic s i s f r om a c omp le t e ly de po l a r i z in g ch a nne l .
Scr a m b l i n g i n o p en q u a n t u m sy s t em s .— A lmos t e v e r y p h ysical s ys t e m one e nc oun t e r s i s ult im a t e ly ope n
a nd he r e w e s tudy the r o le of the s e ope n s ys t e m effe cts on s cr a mb l in g. W e find th a t a v e r a g e d O T O C s
ca ptur e t w o d i s t inct c on tr i but ion s: one f r om glo b al e n v ir onme n t al de c o he r e nc e a nd the othe r f r om g e n-
uine infor m a t ion s cr a mb l in g w ithin the s ys t e m ’ s de gr e e s of f r e e dom . S ur pr i sin gly , the s e t w o c on tr i but ion s
co m p et e w ith e a ch othe r a nd the O T O C ca n v a ni sh e v e n for s ys t e m s w ith non tr iv i al a moun ts of s cr a mb l in g
a nd de c o he r e nc e . U sin g our a n aly t ical r e s ults, w e sho w ho w t o d isen t a n g le the s e c on tr i but ion s a nd r e c o v e r
si gn a t ur e s of s cr a mb l in g in the pr e s e nc e of e n v ir onme n t al de c o he r e nc e .
Scr a m b l i n g a t fi n it e t em p er a t u r e .— The no w fa mous, b o u n d o n c h a o s as in tr oduc e d b y M ald a c e n a a nd
S he nk a r s tud ie s infor m a t ion s cr a mb l in g in qua n tum s ys t e m s a t finit e t e mpe r a tur e , b y in tr oduc in g a ne w
cl as s of r e g u l a r i ze d O T O C s . W e sho w th a t s uit a b le a v e r a g e s of the s e r e g ul a r i z e d O T O C s a r e r e l a t e d t o
the pur it y of the t ime- e v o lv in g t h er m ofie l d do u b le s t a t e (w hich simp ly r efe r s t o a ca nonical pur i fica t ion of
the G i b bs s t a t e ). M or e o v e r , w e r e l a t e the s e r e g ul a r i z e d O T O C s t o a k ey qua n t i fie r of qua n tum ch a os, the
( a n aly t ical ly c on t in ue d ) spe ctr al for m fa ct or .
x x i
1
Qua n tum cohe r e nce a s a si g n a tur e of ch a o s
1.1 A bs tr a c t
W e e s t a b l i sh a r i g or ous c onne ct ion be t w e e n qua n tum c o he r e nc e a nd qua n tum ch a os b y e mp lo y in g c o -
he r e nc e me as ur e s or i g in a t in g f r om the r e s our c e the or y f r a me w ork as a d i a g n o s t ic t oo l for qua n tum ch a os .
W e qua n t i f y thi s c onne ct ion a t t w o d i ffe r e n t le v e l s: qua n tum s t a t e s a nd qua n tum c h a n n e ls . A t the le v e l of
s t a t e s, w e sho w ho w s e v e r al w e l l - s tud ie d qua n t i fie r s of ch a os a r e , in fa ct , qua n tum c o he r e nc e me as ur e s in
d i sg ui s e ( or clos e ly r e l a t e d t o the m ). W e f ur the r thi s c onne ct ion for a ll qua n tum c o he r e nc e me as ur e s b y
usin g t oo l s f r om m a j or i za t ion the or y . The n, w e n ume r ical ly s tudy the c o he r e nc e of ch a ot ic - v s -in t e gr a b le
ei g e n s t a t e s a nd find exc e l le n t a gr e e me n t w ith r a ndom m a tr i x the or y in the bul k of the spe ctr um . A t the
le v e l of ch a nne l s, w e sho w th a t the c o he r e nc e- g e ne r a t in g po w e r ( C GP ) — a me as ur e of ho w m uch c o -
1
he r e nc e a dy n a mical pr oc e s s g e ne r a t e s on a v e r a g e — e me r g e s as a s u bp a r t of the out - of -t ime- or de r e d c or -
r e l a t or ( O T O C ), a me as ur e of infor m a t ion s cr a mb l in g in m a n y - body s ys t e m s . V i a n ume r ical sim ul a t ion s
of the ( nonin t e gr a b le ) tr a n s v e r s e- fie ld I sin g mode l , w e sho w th a t the O T O C a nd C GP ca ptur e qua n tum
r e c ur r e nc e s in qua n t it a t iv e ly the s a me w a y . M or e o v e r , usi n g r a ndom m a tr i x the or y , w e a n aly t ical ly ch a r -
a ct e r i z e the O T O C - C GP c onne ct ion for the H a a r a nd G a us si a n e n s e mb le s . I n closin g , w e r e m a rk on ho w
our c o he r e nc e- b as e d si gn a tur e s of ch a os r e l a t e t o othe r d i a gnos t ic s, n a me ly the L os chmidt e cho , O T O C ,
a nd the S pe ctr al F or m F a ct or .
T ex t for thi s C h a pt e r i s a d a pt e d f r om [ 2 ].
1.2 In troduc tion
Qua n tum c o he r e nc e a nd qua n tum e n t a n gle me n t a r e a r g ua b ly the t w o ca r d in al a ttr i but e s of qua n tum the-
or y , or i g in a t in g f r om the s u pe r posit ion pr inc ip le a nd the t e n s or pr oduct s tr uctur e ( TPS ), r e spe ct iv e ly [ 3 –
5 ]. W hi le e n t a n gle me n t as a si gn a tur e of qua n tum ch a os h as be e n w e l l - s tud ie d in both the fe w - a nd m a n y -
body cas e [ 6 – 10 ], a r i g or ous c onne ct ion be t w e e n qua n tum c o he r e nc e a nd qua n tum ch a os s t i l l r e m ain s
e lusiv e . H e r e , w e cl a r i f y in a qua n t it a t iv e w a y the r o le th a t qua n tum c o he r e nc e p l a ys in the s tudy of ch a ot ic
qua n tum s ys t e m s . A p a r t f r om the found a t ion al r o le th a t the s u pe r posit ion pr inc ip le p l a ys in “ e v e r y thin g
qua n tum, ” the r e a r e ( a t le as t ) t w o d i s t inct w a ys in w hich qua n tum c o he r e nc e e n t e r s the s tudy of qua n tum
ch a ot ic s ys t e m s . The fir s t , a nd pe rh a ps the mor e c onc e ptual one , i s the E i g e n s t a t e The r m al i za t ion H y poth-
e si s ( E TH ) [ 11 – 13 ] a nd the d i a go n a l en sem b le as s oc i a t e d w ith it. The not ion of qua n tum c o he r e nc e i s a
b asis-dep en den t one a nd the d i a g on al e n s e mb le r e v e al s the H a mi lt oni a n ei g e nb asi s as the r e le v a n t p h ysical
b asi s, e spe c i al ly w he n s tudy in g the r m al i za t ion, e r g od ic it y , a nd othe r t e mpor al ch a r a ct e r i s t ic s . M or e o v e r ,
a n init i al s t a t e ’ s o v e rl a p w ith s u ffic ie n tly m a n y e ne r g y -le v e l s — w hich i s r e l a t e d t o c o he r e nc e in the e ne r g y -
ei g e nb asi s — i s a s u ffic ie n t c ond it ion for e qui l i br a t ion ( unde r s ome a dd it ion al as s umpt ion s ) [ 14 – 16 ]. S e c -
ond , the out - of -t ime- or de r e d c or r e l a t or ( O T O C ) [ 17 , 18 ] a qua n t i fie r of qua n tum ch a os ¹ a nd infor m a t ion
s cr a mb l in g , i s us ual ly s tud ie d v i a the input of t w o lo c a l unit a r ie s a nd gr o w s w he n they s t a r t nonc omm ut -
¹ The pr e c i s e r o le of the O T O C in ch a r a ct e r i z in g ch a ot ic it y i s n ua nc e d a nd w e r efe r the r e a de r t o s e ct ion 1 a nd R ef s . [ 19– 25 ]
for a de t ai le d d i s c us sion .
2
in g as one of the m spr e a d s unde r the H ei s e nbe r g t ime e v o lut ion . The local it y of the o bs e r v a b le s in the
O T O C “ pr o be s ” the e n t a n gle me n t s tr uctur e a nd its gr o w th [ 18 , 26 ]. A t the s a me t ime , it i s n a tur al t o ask ,
w h a t doe s the s tr en g t h of the nonc omm ut a t iv it y pr o be (w ithout r efe r e nc e t o a n y TPS )? W e a r g ue th a t thi s
i s pr e c i s e ly a me as ur e of qua n tum c o he r e nc e ( mor e spe c i fical ly , the inc omp a t i b i l it y of the b as e s as s oc i -
a t e d t o the unit a r ie s [ 27 ]). F or ex a mp le , g iv e n t w o ( non- de g e ne r a t e ) o bs e r v a b le s A; B a nd the as s oc i a t e d
ei g e nb as e s B
A
;B
B
, w e ca n ask , ho w c o he r e n t a r e the ei g e n s t a t e s of A w he n ex pr e s s e d in the ( ei g e n )b asi s
B
B
. C le a rly , i f [ A; B] = 0 the n the ei g e n s t a t e s of A a r e inc o he r e n t in B
B
. On the othe r h a nd , i f B
A
a nd
B
B
a r e m u t u a ll y u n bi ase d , the n the ei g e n s t a t e s of A a r e m ax i m a ll y co h er en t inB
B
, a nd v a r ious me as ur e s of
inc omp a t i b i l it y a r e m a x imi z e d [ 27 ]. F o l lo w in g thi s in tuit ion, w e w i l l sho w th a t the O T O C i s in t im a t e ly
r e l a t e d t o a me as ur e of inc omp a t i b i l it y cal le d the c o he r e nc e- g e ne r a t in g po w e r ( C GP ), as exe mp l i fie d b y
our 1.3 .
Q u a n t i f y i n g c h a o s .— S i gn a tur e s of qua n tum ch a os ca n be br o a d ly cl as si fie d in t o thr e e ca t e g or ie s: ( i )
spe ctr al pr ope r t ie s, s uch as le v e l - sp a c in g d i s tr i but ion [ 28 , 29 ], le v e l n umbe r v a r i a nc e [ 30 ], e t c ., ( i i ) ei g e n-
s t a t e s tr uctur e , s uch as ei g e n s t a t e e n t a n gle me n t ( define d as the a v e r a g e e n t a n gle me n t e n tr op y o v e r a ll
ei g e n s t a t e s ) a nd the as s oc i a t e d a r e a a nd v o lume l a w s [ 31 ], a nd ( i i i ) dy n a mical qua n t it ie s s uch as L os chmidt
e cho [ 32 – 35 ], e n t a n gl in g po w e r [ 6 , 36 – 39 ], qua n tum d i s c or d [ 40 ], O T O C s, e t c . ( s e e al s o R ef . [ 29 ] for
othe r ex a mp le s ), w hich, in g e ne r al a r e a pr ope r t y of both the ei g e n v alue s a nd ei g e n v e ct or s of the H a mi l -
t oni a n . I n thi s ch a pt e r , w e c onne ct qua n tum ch a os a nd qua n tum c o he r e nc e in the s e n s e of ( i i ) a nd ( i i i ),
b y ex a minin g the c o he r e nc e s tr uctur e of ch a ot ic - v s -in t e gr a b le ei g e n s t a t e s, a nd b y s tudy in g the c o he r e nc e-
g e ne r a t in g po w e r of ch a ot ic dy n a mic s .
1.3 Prel imin arie s .
R e so u r ce t h e o r y of q u a n t u m co h er en ce . — D e sp it e the f und a me n t al r o le th a t qua n tum c o he r e nc e p l a ys in
qua n tum the or y , a r i g or ous qua n t i fica t ion of c o he r e nc e as a p h y sic a l r e so u r ce w as only init i a t e d in r e c e n t
y e a r s [ 4 , 41 , 42 ]. W e br iefly r e v ie w the r e s our c e the or y of c o he r e nc e a nd the qua n t i fica t ion t oo l s it pr o -
v ide s . L e t H
=C
d
be the H i l be r t sp a c e as s oc i a t e d t o a d - d ime n sion al qua n tum s ys t e m a nd S(H) the s e t of
al l qua n tum s t a t e s . Qua n tum c o he r e nc e of s t a t e s i s qua n t i fie d w ith r e spe ct t o a pr ef er r e d or thonor m al b asi s
3
for the H i l be r t sp a c e , B =fj j⟩g
d
j= 1
. A l l s t a t e s th a t a r e d i a g on al in the b asi s B a r e de e me d i n co h er en t ( th a t
i s, de v o id of a n y r e s our c e ) w hi le othe r s co h er en t . Th a t i s, inc o he r e n t s t a t e s h a v e the for m, ρ =
∑
d
j= 1
p
j
Π
j
,
w he r e Π
j
j j⟩⟨ jj i s the r a nk - 1 pr oj e ct or as s oc i a t e d t o the b asi s s t a t e j j⟩ a nd p
j
0;
∑
d
j= 1
p
j
= 1 i s
a pr o b a b i l it y d i s tr i but ion . The c o l le ct ion of al l inc o he r e n t s t a t e s for m s a c on v ex s e t , I
B
( us ual ly cal le d
the “ f r e e s t a t e s ” of the r e s our c e the or y ) ² . A c ommon qua n t i fie r of the a moun t of r e s our c e in a s t a t e σ
i s t o me as ur e its ( minim um ) d i s t a nc e f r om the s e t I
B
, usin g a ppr opr i a t e ly chos e n d i s t a nc e me as ur e s, s a y
R
d
( σ) := min
δ2I
B
d( σ; δ) . w he r e d(;) i s a d i s t a nc e me as ur e on the s t a t e sp a c e a nd R
d
its as s oc i a t e d
r e s our c e qua n t i fie r ( us ual ly cal le d the “ r e s our c e me as ur e s ” of the r e s our c e the or y ). The c o he r e nc e qua n-
t i fi e r s th a t w e w i l l be w ork in g w ith in thi s ch a pt e r a r e the l
2
- nor m of c o he r e nc e ³ ( he r e a ft e r 2 - c o he r e nc e )
a nd the r e l a t iv e e n tr op y of c o he r e nc e , define d as [ 41 ],
c
( 2)
B
( ρ) := min
σ2I
B
∥ ρ σ∥
2
l 2
=∥ ρD
B
( ρ)∥
2
l 2
; (1.1)
c
( r e l)
B
( ρ) := min
σ2I
B
S
( r e l)
( ρjj σ) = S(D
B
( ρ)) S( ρ); (1.2)
w he r e ,D
B
( X) :=
∑
d
j= 1
Π
j
X Π
j
i s the de p h asin g s u pe r ope r a t or , S
( r e l)
( ρjj σ) i s the qua n tum r e l a t iv e e n tr op y ,
a nd S( ρ) i s the v on N e um a nn e n tr op y [ 41 ]. The 2 - c o he r e nc e ⁴ h as be e n ide n t i fie d as the e s ca pe pr o b a b i l it y ,
a k ey fi g ur e of me r it for fe w - a nd m a n y - body local i za t ion [ 43 ], w hi le the r e l a t iv e e n tr op y of c o he r e nc e h as
s e v e r al ope r a t ion al in t e r pr e t a t ion s, pr omine n t a mon gs t w hich a r e its r o le as the d i s t i l l a b le c o he r e nc e [ 44 ]
a nd as a me as ur e of de v i a t ion s f r om the r m al e qui l i br ium [ 45 ].
A fin al but k ey in gr e d ie n t of qua n tum r e s our c e the or ie s a r e the s o - cal le d “ f r e e ope r a t ion s, ” tr a n sfor m a -
t ion s th a t do not g e ne r a t e a n y r e s our c e , but m a y c on s ume it. F or the r e s our c e the or y of c o he r e nc e , w e
w i l l foc us on the cl as s of i n co h er en t o p er a t io n s ( IO ): c omp le t e ly -posit iv e ( C P ) m a ps s uch th a t the r e ex i s ts
a t le as t one Kr a us r e pr e s e n t a t ion w hich s a t i sfie s K
j
ρ K
y
j
= Tr
(
K
j
ρ K
y
j
)
2 I
B
8 ρ 2 I
B
; 8 j ⁵ . R e s our c e
² W e r e m a rk th a t t o qua n t i f y c o he r e nc e , inde e d a w e ak e r not ion th a n th a t of a b asi s i s r e quir e d , w hich t ak e s in t o a c c oun t the
f r e e dom in choosin g a r b itr a r y glo b al p h as e s a nd or de r in gs for the b asi s e le me n ts .
³ N ot e th a t althou gh the 2 - c o he r e nc e i s a monot one for al l unit al ch a nne l s (w hich include s unit a r y e v o lut ion ), it i s not mono -
t onic unde r the f ul l s e t of inc o he r e n t ope r a t ion s IO ( in tr oduc e d l a t e r ) [ 41 ]. H o w e v e r , thi s i s not a pr o b le m sinc e w e a r e only
c onc e r ne d w ith unit a r y e v o lut ion s in thi s w ork .
⁴ F or the pur pos e s of c omput in g the 2 - c o he r e nc e , r e cal l th a t the l
2
- nor m of a m a tr i x i s e qual t o its H i l be r t - S chmidt nor m .
⁵ One ca n al s o think of the m as g e ne r al i z e d me as ur e me n ts in s t e a d , sinc e th a t r e quir e s a spe c i fic Kr a us r e pr e s e n t a t ion [ 3 ]
4
me as ur e s th a t a r e n o n -i n cr e asi n g unde r the a ct ion of f r e e ope r a t ion s a r e cal le d r e s our c e m o n o t o n e s .
1.4 A t the l evel of s t a te s
1 Why s tu d y q u a ntum c o h er en c e ?
The s udde n de local i za t ion of ch a ot ic s ys t e m s fo l lo w in g a que nch h as be e n w e l l - s tud ie d for both cl as sical
a nd qua n tum s ys t e m s, s e e R ef s . [ 46 , 47 ] a nd the r efe r e nc e s the r ein . V a r ious qua n t i fie r s of thi s de local i za -
t ion h a v e be e n in tr oduc e d in the qua n tum ch a os l it e r a tur e t o ch a r a ct e r i z e in t e gr a b le a nd ch a ot ic qua n tum
s ys t e m s . H e r e , w e a r g ue th a t m a n y of the s e de local i za t ion me as ur e s a r e nothin g but qua n tum c o he r e nc e
me as ur e s in d i sg ui s e . W e a r g ue thi s in t w o w a ys: fir s t , w e c on side r s ome p a r a d i gm a t ic me as ur e s of de local -
i za t ion s uch as S h a nnon e n tr op y , p a r t ic ip a t ion r a t io , e t c ., [ 48 ] a nd c onne ct the m w ith me as ur e s of qua n tum
c o he r e nc e s tud ie d in the r e s our c e the or ie s f r a me w ork . M or e o v e r , thi s al s o r e v e al s th a t the not ion of de-
local i za t ion in the a v ai l a b le p h as e sp a c e , e ne r g y sp a c e , e t c ., i s pr e c i s e ly the not ion of qua n tum c o he r e nc e
in a n a ppr opr i a t e b asi s . S e c ond , w e sho w th a t the not ion of w he n one s t a t e i s mor e de lo c a l i ze d th a n the
othe r ( a nd me as ur e s t o qua n t i f y the m ) i s ca ptur e d in a v e r y g e ne r al w a y b y the m a the m a t ical for m al i s m
of m a j or i za t ion . Thi s f ur the r al lo w s us t o m ak e a pr e c i s e c onne ct ion t o the r e s our c e the or y of c o he r e nc e
sinc e s t a t e tr a n sfor m a t ion unde r inc o he r e n t ope r a t ion s i s c omp le t e ly ch a r a ct e r i z e d in t e r m s of m a j or i za -
t ion . F in al ly , usin g the m a j or i za t ion r e s ult f r om the r e s our c e the or e t ic f r a me w ork of c o he r e nc e , w e a r g ue
th a t qua n tum c o he r e nc e me as ur e s ca ptur e pr e c i s e ly w h a t de local i za t ion me as ur e s s e t out t o qua n t i f y: ho w
“ local i z e d ” or “ uni for mly spr e a d ” a qua n tum s t a t e i s a cr os s a b asi s . A lon g the w a y w e al s o r e m a rk on c o -
he r e nc e me as ur e s ’ a b i l it y t o pr o be e n t a n gle me n t me as ur e s, w hich h a v e lon g be e n us e d as qua n t i fie r s of
ch a os .
Co n n e c t io n w it h de lo c a l i za t io n m e as u r e s .— L e t us s t a r t w ith a simp le ex a mp le: G iv e n a s t a t e j ψ⟩ ex pr e s s e d
in s ome b asi s B =fj j⟩g ,j ψ⟩ =
∑
d
j= 1
c
j
j j⟩ , one ca n c on side r v a r ious w a ys t o qua n t i f y ho w u n i f o r m l y s pr e a d
the pr o b a b i l it y d i s tr i but ion g e ne r a t e d f r om f
c
j
2
g i s . F or in s t a nc e , a n inc o he r e n t s t a t e j j⟩ c or r e spond s t o
the ( ex tr e me ly ) non uni for m pr o b a b i l it y d i s tr i but ion p
j j⟩
=f 1; 0; ; 0g , th a t i s, it i s the mos t “ local i z e d ”
5
s t a t e; w hi le a hi ghly c o he r e n t s t a t e ⁶ of the for m j ψ⟩ =
1
p
d
d
∑
j= 1
e
iθ j
j j⟩ c or r e spond s t o the uni for m pr o b a -
b i l it y d i s tr i but ion p
j ψ⟩
=f
1
d
;
1
d
; ;
1
d
g , th a t i s, it i s m a x im al l y “ de local i z e d ” . The r efor e , i f w e qua n t i f y the
uni for mit y of the as s oc i a t e d pr o b a b i l it y d i s tr i but ion s b y e v alua t in g , for ex a mp le , their S h a nnon e n tr op y ,
w e s e e th a t the inc o he r e n t s t a t e c or r e spond s t o the minim um e n tr op y S(fj c
α
j
2
g) = 0 , w hi le the hi ghly
c o he r e n t s t a t e m a x imi z e s the S h a nnon e n tr op y , S(fj c
α
j
2
g) = log( d) . Thi s uni for mit y i s pr e c i s e ly w h a t
c o he r e nc e me as ur e s a nd de local i za t ion me as ur e s qua n t i f y .
W e no w d i s c us s s ome ex a mp le s w he r e the r e i s a pr e c i s e c onne ct ion be t w e e n the m . W e c on side r the s a me
not a t ion as a bo v e , a pur e s t a t e j ψ⟩ , a b asi sB = fj j⟩g , a ndf p
j
g
d
j= 1
, w he r e p
j
j⟨ kj ψ⟩j
2
i s the as s oc i a t e d
pr o b a b i l it y d i s tr i but ion .
1. The S h a nnon e n tr op y ( al s o kno w n as the infor m a t ion al e n tr op y in the qua n tum ch a os l it e r a tur e ) of
the pr o b a b i l it y d i s tr i but ion f p
j
g
d
j= 1
h as be e n us e d as a me as ur e of de local i za t ion [ 46 – 48 ]. W e not e th a t for
pur e s t a t e s, thi s i s e qual t o the r e l a t iv e e n tr op y of c o he r e nc e . Th a t i s,
c
( r e l)
B
( ρ) = S(f p
j
g) (1.3)
Thi s fo l lo w s f r om the definit ion in e q . ( 1.1 ) a nd the fa ct th a t the S h a nnon e n tr op y of pur e s t a t e s i s z e r o ,
th a t i s, S(j ψ⟩⟨ ψj) = 0 . I t i s w or th not in g th a t the S h a nnon e n tr op y i s the fir s t R é n y i e n tr op y [ 49 ], a fa mi ly
of e n tr op ie s w hich pr o v ide po w e r f ul c onne ct ion s w ith m a j or i za t ion the or y a nd s t a t e tr a n sfor m a t ion in
r e s our c e the or ie s [ 50 ].
2. The s e c ond p a r t ic ip a t ion r a t io ( al s o kno w n as the n umbe r of pr inc ip al c ompone n ts ) [ 46 – 48 ], define d
as
PR
2;B
(j ψ⟩) :=
∑
j
j⟨ jj ψ⟩j
4
: (1.4)
⁶ I n fa ct , thi s fa mi ly of s t a t e s a r e m a x im al ly c o he r e n t in the r e s our c e the or y of c o he r e nc e w ith inc o he r e n t ope r a t ion s; a n alo -
g ous t o ho w B e l l s t a t e s a r e m a x im al ly e n t a n gle d in the r e s our c e the or y of pur e b ip a r t it e e n t a n gle me n t.
6
N ot e th a t for pur e s t a t e s a nd a n y g iv e n b asi s B , the PR
2;B
i s e qual t o one min us the 2 - c o he r e nc e , th a t i s ⁷,
PR
2;B
(j ψ⟩) = 1c
( 2)
B
(j ψ⟩⟨ ψj): (1.5)
M or e o v e r , the ne g a t iv e lo g a r ithm of PR
2
i s e qual t o the s e c ond R é n y i e n tr op y [ 49 ] of the pr o b a b i l it y
d i s tr i but ion f p
j
g . A nd both the fir s t a nd s e c ond R é n y i e n tr op ie s a r e me as ur e s of qua n tum c o he r e nc e [ 4 ].
3. W e no w r e v ie w thr e e qua n t it ie s, the L os chmidt e cho , the e s ca pe pr o b a b i l it y a nd the effe ct iv e d i -
me n sion, w hich find a m ult itude of a pp l ica t ion s in qua n tum ch a os, the r m al i za t ion, a nd local i za t ion . The
L os chmidt e cho i s define d as the o v e rl a p be t w e e n the init i al s t a t e j ψ⟩ a nd the s t a t e a ft e r t ime t [ 32 , 33 , 35 ],
L
t
(j ψ⟩) :=
⟨ ψj e
iHt
j ψ⟩
2
: (1.6)
The effe ct iv e d ime n sion of a qua n tum s t a t e i s define d as its in v e r s e pur it y [ 14 , 15 ],
d
eff
( ρ) =
1
Tr[ ρ
2
]
; (1.7)
w hich in tuit iv e ly c or r e spond s t o the n umbe r of pur e s t a t e s th a t c on tr i but e t o the ( in g e ne r al ) mi xe d s t a t e
ρ . I n R ef s . [ 14 , 15 ], d
eff
( ρ) w as us e d t o pr o v ide a s u ffic ie n t c ond it ion for e qui l i br a t ion in clos e d qua n tum
s ys t e m s . A nd fin al ly , w e r e cal l th a t the infinit e-t ime a v e r a g e of a qua n t it y A i s define d as
A := lim
T!1
1
T
T
∫
0
A( t) dt: (1.8)
I nfinit e-t ime a v e r a g in g c onne cts the s e v a r ious qua n t it ie s as fo l lo w s (w ith ρ =j ψ⟩⟨ ψj )
L
t
(j ψ⟩) = PR
2;B H
( ρ) =
1
d
eff
( ρ)
= 1P
ψ
; (1.9)
w he r eB
H
i s the H a mi lt oni a n ei g e nb asi s a nd P
ψ
:= 1j⟨ ψj e
iHt
j ψ⟩j
2
i s the e s ca pe pr o b a b i l it y of the s t a t e
⁷ A pr oof of thi s fo l lo w s imme d i a t e ly b y ex p a nd in g the for m ul a for 2 - c o he r e nc e of pur e s t a t e s, c
( 2)
B
( ρ) = 1⟨ ρ;D
B
( ρ)⟩ .
7
j ψ⟩ ; w hich usin g P r oposit ion 4 of R ef . [ 27 ] i s al s o e qual t o th e 2 - c o he r e nc e in the H a mi lt oni a n ei g e nb asi s .
N ot e th a t , the pr oof of P r oposit ion 4 in R ef . [ 27 ] ca n pot e n t i al ly r e v e al m a n y mor e c onne ct ion s sinc e
the r e it w as o bs e r v e d th a t the infinit e t ime-a v e r a g e of the t ime e v o lut ion ope r a t or ( for a non- de g e ne r a t e
H a mi lt oni a n ) U
t
:= U() U
y
i s e quiv ale n t t o de p h asin g in the H a mi lt oni a n ei g e nb asi s, th a t i s, U
t
=D
B H
.
The a ct ion ofD
B H
r e v e al s the “ d i a g on al e n s e mb le , ” f und a me n t al t o the s tudy of the r m al i za t ion in clos e d
qua n tum s ys t e m s [ 13 ].
A r bitr a r y co h er en ce m e as u r e s a n d m aj o r i za t io n.— G iv e n t w o v e ct or s ⃗ v;⃗ w 2 R
n
, w e s a y th a t “ ⃗ v i s m a -
j or i z e d b y ⃗ w , ” ( e quiv ale n tly ⃗ w m a j or i z e s ⃗ v ) w r itt e n as⃗ v≺ ⃗ w , i f [ 51 ]
k
∑
j= 1
v
[ j]
k
∑
j= 1
w
[ j]
;8 k = 1; ; n 1
n
∑
j= 1
v
[ j]
=
n
∑
j= 1
w
[ j]
;
(1.10)
w he r e v
[ j]
i s the j th e le me n t of ⃗ v w he n s or t e d in a nonincr e asin g or de r . M a j or i za t ion induc e s a pr e-
or de r ⁸ on the v e ct or s in R
n
a nd it i s n a tur al t o ask w h a t f unct ion s pr e s e r v e thi s pr e or de r? A l l f unct ion s
f : R
n
! R s uch th a t ⃗ v ≺ ⃗ w =) f(⃗ v) f(⃗ w) a r e cal le d S ch ur - c on v ex ( e quiv ale n tly , S ch ur - c onca v e
i f⃗ v ≺ ⃗ w =) f(⃗ v) f(⃗ w) ). M a n y f unct ion al s e mp lo y e d in the s tudy of qua n tum ch a os l i k e S h a nnon
e n tr op y , the fa mi ly of R é n y i e n tr op ie s, a nd othe r s, a r e a n ex a mp le of S ch ur - c onca v e f unct ion s th a t pr e s e r v e
the or de r in g impos e d f r om m a j or i za t ion . U sin g a the or e m of H a r dy - L ittle w ood - P o lya [ 51 ], w e h a v e the
fo l lo w in g
⃗ v≺ ⃗ w ()
n
∑
j= 1
g
(
v
j
)
n
∑
j= 1
g
(
w
j
)
; (1.11)
for al l c on t in uous c on v ex f unct ion s g :R!R .
Th a t i s, s tudy in g m a j or i za t ion i s e quiv ale n t t o s tudy in g the or de r in g induc e d f r om a ll c on t in uous c on v ex
f unct ion s o bey in g a n or de r in g. I t i s in thi s spe c i fic s e n s e th a t m a j or i za t ion al lo w s us t o g o bey ond a n y
spe c i fic qua n tum c o he r e nc e me as ur e a nd al lo w s us t o d i s c us s the be h a v ior of a ll c o he r e nc e me as ur e s .
⁸ A pr e or de r i s a b in a r y r e l a t ion th a t i s r eflex iv e a nd tr a n sit i v e but not ne c e s s a r i ly a n t i s y mme tr ic .
8
T o m ak e the c onne ct ion t o qua n tum c o he r e nc e , w e not e th a t g iv e n t w o s t a t e s ρ; σ a nd a c o he r e nc e me a -
s ur ec
B
() , i fc
B
( ρ) > c
B
( σ) the n σ ca nnot be tr a n sfor me d in t o ρ v i a inc o he r e n t ope r a t ion s ( sinc e IO
ca n only n o n i n cr e a se the a moun t of c o he r e nc e in a s t a t e ). On the othe r h a nd , c
B
( ρ) c
B
( σ) pr o v ide s
a ne c e s s a r y ( but not s u ffic ie n t ) c ond it ion on the s t a t e tr a n sfor m a t ion σ 7! ρ usin g IO . A ne c e s s a r y a nd
s u ffic ie n t c ond it ion w as o bt aine d in R ef . [ 52 ] in t e r m s of m a j or i za t ion ( the the or e m h as be e n r e p hr as e d
for simp l ic it y ). I n the fo l lo w in g , D
B
i s the de p h asin g s u pe r ope r a t or in the b asi s B ; a nd the not ion of m a -
tr i x m a j or i za t ion h as be e n us e d , w ith A≺ B () spec( A)≺ spec( B) , w he r e spec( A) i s the v e ct or of
ei g e n v a lue s of A .
P r opositio n 1.1: [ 52 ]
A qua n tum s t a t e j ψ⟩ ca n be tr a n sfor me d t o a nothe r s t a t e j φ⟩ v i a inc o he r e n t ope r a t ion s i f a nd only i f
D
B
(j ψ⟩⟨ ψj)≺D
B
(j φ⟩⟨ φj) .
R em a r k : F ir s t , not e th a t D
B
(j ψ⟩⟨ ψj) p
ψ
i s i s omor p hic t o the pr o b a b i l it y v e ct or o bt aine d f r om the
s t a t e j ψ⟩ ex pr e s s e d in the b asi s B . The r efor e , the c ond it ion D
B
(j ψ⟩⟨ ψj) ≺ D
B
(j φ⟩⟨ φj) = p
ψ
≺ p
φ
,
th a t i s, it i s e quiv ale n t t o the s t a t e j ψ⟩ bein g mor e uni for mly spr e a d in the b asi s B th a n the s t a t e j φ⟩ , in
the s e n s e of m a j or i za t ion . N o w , sinc e the m a j or i za t ion c ond it ion i s e quiv ale n t ⁹ t o tr a n sfor min g j ψ⟩ 7!
j φ⟩ v i a a n inc o he r e n t ope r a t ion, the a moun t of c o he r e nc e in j ψ⟩ i s gr e a t e r th a n or e qual t o the a moun t of
c o he r e nc e in j φ⟩ , for ev er y qua n tum c o he r e nc e me as ur e . F or m al ly , R
c
(j ψ⟩⟨ ψj)R
c
(j φ⟩⟨ φj) , for e v e r y
c o he r e nc e monot one R
c
: S(H) ! R
+
0
. The r efor e , qua n tum c o he r e nc e me as ur e s ca ptur e in a pr e c i s e
s e n s e w h a t tr a d it ion al de local i za t ion me as ur e s s e t out t o qua n t i f y: ho w uni for mly spr e a d i s a qua n tum
s t a t e w ith r e spe ct t o a b asi s B ; in fa ct , the a bo v e the or e m qua n t it a t iv e ly sho w s th a t the s e t w o not ion s a r e
e quiv ale n t.
H a v in g e s t a b l i she d a w e b of c onne ct ion s be t w e e n s e v e r al k ey qua n t it ie s us e d in the s tudy of qua n tum
ch a os a nd e qui l i br a t ion, w e no w d i s c us s ho w qua n tum c o he r e nc e me as ur e s ca n inhe r it their a b i l it y t o d i -
a gnos e qua n tum ch a os f r om their in t e r p l a y w ith e n t a n gle me n t me as ur e s .
⁹ The c ond it ion i s only s u ffic ie n t but be c ome s ne c e s s a r y for the g e ne r ic cas e of f ul l - r a nk pur e s t a t e s (w hich ca n be o bt aine d
9
-4 -2 0 2
0.85
0.90
0.95
1.00
(a)
-4 -2 0 2
0.0
0.2
0.4
0.6
0.8
(b)
Figure 1.4.1: Relative entrop y of coherence fo r eigenstates of the Hamiltonian defined in eq. ( 1.13 )
as a function of their energy , no rmalized with the GOE p rediction using eq. ( 1.14 ),
⟨
c
( rel)
B
⟩
GOE
10: 49 . Results a re rep o rted fo r L = 15 with 5 spins up and ω = 0 , ε
δ
= 0: 5 , J
xy
= 1 , J
z
= 0: 5 . The
plot ma rk ers 1; 3; 5; 7 co rresp ond to the va rious choices of the defect si te, with δ = 1 and δ = 7 co rre-
sp onding to the integrable and chaotic limits, resp ectively . Figures (a) and (b) co rresp ond to the t w o
different bases, the site-basis and mean-field basis, resp ectively .
Co h er en ce a n d its i n t er p l a y w it h en t a n g lem en t .— The s tudy of qua n tum c o he r e nc e pe r s e , m ak e s no r ef -
e r e nc e t o the lo c a l it y ( or TPS ) of a qua n tum s ys t e m . H o w e v e r , m a n y - body s ys t e m s a r e oft e n e ndo w e d
w ith a n a tur al TPS a nd t o s tudy the in t e r p l a y be t w e e n c o he r e nc e a nd e n t a n gle me n t , it i s oft e n c on v e nie n t
t o choos e inc o he r e n t s t a t e s th a t a r e co m p a t i b le w ith the TPS , n a me ly , the inc o he r e n t s t a t e s a r e al s o pr od -
uct s t a t e s [ 53 , 54 ]. C on side r , for ex a mp le , a t w o - qub it s ys t e m, H
= C
2
C
2
, w ith a n inc o he r e n t b asi s
B =fj 00⟩;j 01⟩;j 10⟩;j 11⟩g th a t i s al s o s e p a r a b le ¹⁰ . The n, not ic e th a t a n y e n t a n gle d s t a t e i s a ut om a t ical ly
c o he r e n t , sinc e j Ψ
AB
⟩ i s e n t a n gle d i f a nd only i f j Ψ
AB
⟩̸=j φ⟩
A
j φ⟩
B
for a n y j φ⟩
A( B)
2H
A( B)
. The r efor e ,
w he n ex pr e s s e d as a l ine a r c omb in a t ion of the b asi s e le me n ts in B , w e not e th a t , for e v e r y e n t a n gle d s t a t e ,
j Ψ
AB
⟩ =
∑
1
j; k= 0
c
jk
j j⟩
A
j k⟩
B
, w e h a v e a t le as t t w o non- z e r o c oeffic ie n ts c
jk
— th a t i s, they a r e c o he r e n t as
w e l l . C le a rly , not e v e r y c o he r e n t s t a t e i s e n t a n gle d , for ex a mp le , c on side r the s t a t e j 0⟩
j+⟩ . Thi s c on-
s tr uct ion ca n be g e ne r al i z e d t o the ( simp le s t ) m ult ip a r t it e ¹¹ cas e as fo l lo w s: L e t H
=H
1
H
2
H
n
be a n -p a r t it e H i l be r t sp a c e w ith F
e
bein g the s e t of f ul ly s e p a r a b le s t a t e s ( th a t i s, they a r e c on v ex c om-
b in a t ion s of s t a t e s th a t fa ct or i z e o v e r a n y t e n s or fa ct or ) a nd F
c
bein g the s e t of inc o he r e n t s t a t e s th a t a r e
b y a n a r b itr a r i ly s m al l pe r tur b a t ion ) a nd ho ld s tr ue for p h ysical ly r e le v a n t s c e n a r ios .
¹⁰ A n ex a mp le of “ inc omp a t i b le ” qua n tum c o he r e nc e w ould be , for in s t a nc e , i f the inc o he r e n t b asi s for a 2 - qub it s ys t e m i s
chos e n t o be the Be l l - b asi s .
¹¹ I n g e ne r al , m ult ip a r t it e e n t a n gle me n t i s m uch r iche r a nd le s s tr a ct a b le th a n b ip a r t it e e n t a n gle me n t a nd th a t i s w h y w e
c o n side r the simp le s t s c e n a r io he r e [ 5 ].
10
al s o f ul ly s e p a r a b le . The n, it i s e as y t o s e e th a t F
c
F
e
( sinc e the F
c
i s co m p a t i b le w ith the TPS ). A s
a n imme d i a t e c on s e que nc e , not e th a t , i f R(;) : S(H) ! R
+
0
i s a c on tr a ct iv e d i s t a nc e ( unde r the as -
s oc i a t e d f r e e ope r a t ion s, th a t le a v e the s e t of f r e e s t a t e s in v a r i a n t ), the n, one ca n define a “ d i s t a nc e- b as e d
me as ur e , ”R
α
( ρ) := min
σ2F α
R( ρ; σ) , w he r e α =f c; eg . The n, usin g the s e t inclusion of F
c
;F
e
, w e h a v e ,
8 ρ;R
c
( ρ) R
e
( ρ) , th a t i s, the a moun t of c o he r e nc e i s lo w e r - bounde d b y the e n t a n gle me n t ; or , the
a moun t of c o he r e nc e i s a n u ppe r bound on the a moun t of e n t a n gle me n t ¹² .
I n l i gh t of the a bo v e o bs e r v a t ion, it i s w or th not in g th a t the r e i s a s e m a n t ical i s s ue in cal l in g the s e f unc -
t ion al s de lo c a l i za t io n me as ur e s sinc e the r e i s, pe r s e , no lo c a l it y in their definit ion . A t thi s po in t , it i s mor e
a ppr opr i a t e t o think of the m as qua n t i f y in g the c o he r e nc e of a s t a t e in s ome b asi s, B ; in fa ct , their definit ion
r e v e al s th a t thi s i s pr e c i s e ly w h a t they do . T o c onne ct qua n tum c o he r e nc e w ith e n t a n gle me n t in a qua n t i -
t a t iv e w a y ( a p a r t f r om the bound s r e al i z e d f r om the d i s c us sion a bo v e ), as a fir s t s t e p , one ne e d s t o define a
qua n t it y th a t r e mo v e s the b asi s - de pe nde nc e of c o he r e nc e ( sinc e e n t a n gle me n t i s b asis-i n dep en den t ), w hich
ca n be o bt aine d b y opt imi z in g o v e r v a r ious cho ic e s of b as e s . H e r e , w e pr o v e one s uch r e s ult b y minimi z in g
the a moun t of c o he r e nc e o v e r al l local b as e s: G iv e n pur e s t a t e s in H
=H
a
H
b
, w e h a v e ,
P r opositio n 1.2
min
B a;B
b
c
( 2)
B a
B
b
(j Ψ⟩⟨ Ψj) = 1
ρ
a
2
2
=: S
lin
( ρ
a
); (1.12)
w he r e ρ
a
= Tr
b
(j Ψ⟩⟨ Ψj) i s the r e duc e d de n sit y m a tr i x a nd S
lin
() i s the l ine a r e n tr op y , a qua n t i fie r
of e n t a n g le m e n t.
Th a t i s, b y minimi z in g the a moun t of c o he r e nc e o v e r al l local b as e s, w e ca n ( ind ir e ctly ) c omput e a me a -
s ur e of e n t a n gle me n t. A nothe r qua n t it a t iv e c onne ct ion w as o bt aine d in R ef . [ 56 ], w he r e , b y m a x imi z in g
the a moun t of c o he r e nc e o v e r al l b as e s, the a moun t of c o he r e nc e in a s t a t e w as c onne ct e d w ith its pur it y .
¹² Thi s c on s tr uct ion ho ld s not only for c on tr a ct i v e d i s t a nc e s but the g e ne r al cl as s of f unct ion al s cal le d g a u g e f unct ion s [ 55].
11
-4 -2 0 2
0.992
0.994
0.996
0.998
1.000
(a)
-4 -2 0 2
0.0
0.2
0.4
0.6
0.8
1.0
(b)
Figure 1.4.2: 2 -coherence fo r eigenstates of the Hamilt onian defined in eq. ( 1.13 ) as a function of
their energy , no rmalized with the GOE p rediction obtained numerically ,
⟨
c
( 2)
B
⟩
GOE
0: 9991 . Results
a re rep o rted fo r L = 15 with 5 spins up and ω = 0 , ε
δ
= 0: 5 , J
xy
= 1 , J
z
= 0: 5 . The plot ma rk ers
1; 3; 5; 7 co rresp ond to the va rious c hoices of the defect site, with δ = 1 and δ = 7 co rresp onding to the
integrable and chaotic limits, resp ectively . Figures (a) and (b) co rresp ond to the t w o different bases,
the site-basis and mean-field basis, resp ectively .
I n s umm a r y , qua n tum c o he r e nc e me as ur e s pr o v ide both u ppe r bound s a nd in s ome cas e s pr e c i s e c onne c -
t ion s w ith e n t a n gle me n t me as ur e s . S inc e e n t a n gle me n t me as ur e s h a v e be e n w ide ly us e d t o de t e ct qua n tum
ch a os, w e no w tur n t o s tudy in g qua n tum c o he r e nc e in ch a ot ic s ys t e m s .
2 Co h er en c e o f ma ny - bo d y ei gen s t a te s: X X Z spi n - c h a i n with d efec t
The e n t a n gle me n t s tr uctur e of exc it e d s t a t e s h as be e n sho w n t o be a s uc c e s sf ul d i a gnos t ic of qua n tum
ch a os [ 57 – 59 ]. H e r e , w e n ume r ical ly s tudy the c o he r e nc e s tr uctur e of H a mi lt oni a n ei g e n s t a t e s, usin g a n
ope n X X Z sp in- ch ain w ith a n on sit e defe ct ¹³ , de s cr i be d v i a a H a mi lt oni a n of the for m [ 1 , 61 ]
H =
1
4
L 1
∑
j= 1
(
J
xy
(
σ
x
j
σ
x
j+ 1
+ σ
y
j
σ
y
j+ 1
)
+ J
z
σ
z
j
σ
z
j+ 1
)
| {z }
H
XXZ
+
1
2
0
@
L
∑
j= 1
ω σ
z
j
+ ε
δ
σ
z
δ
1
A
| {z }
H z
;
(1.13)
¹³ S e e R ef . [ 60 ] for othe r H a mi lt oni a n s ys t e m s th a t be c ome qua n tum ch a ot ic in the pr e s e nc e of defe cts .
12
w he r e δ 2 f 1; 2; ; Lg i s the l a be l for the defe ct sit e . W e s e t ℏ = 1 a nd al l sit e s h a v e the s a me e ne r g y
sp l itt in g , exc e pt the sit e δ , w hich h as a sp l itt in g of ω+ ε
δ
( the defe ct c or r e spond s t o a d i ffe r e n t v alue of the
Z e e m a n sp l itt in g ). W e as s ume ope n bound a r y c ond it ion s a nd s e t the v a r ious p a r a me t e r s t o the fo l lo w in g
v a lue s: ω = 0; ε
δ
= 0: 5; J
xy
= 1; J
z
= 0: 5 ; for a de t ai le d d i s c us sion of the p h ysic s s ur r ound in g the
cho i c e of p a r a me t e r s a nd ho w thi s le a d s t o the on s e t of ch a os, s e e S e c . I I of R ef . [ 1 , 61 ]. I t i s e as y t o s e e
th a t the t ot al sp in in z - d ir e ct ion i s c on s e r v e d , th a t i s, [ H; σ
z
total
] , w he r e σ
z
total
∑
L
j= 1
σ
z
j
. The H a mi lt oni a n
in e q . ( 1.13 ) i s in t e gr a b le w he n the defe ct i s on the e d g e s of the ch ain, th a t i s, δ = 1 or L , w hi le it i s
non-in t e gr a b le for the defe ct in the midd le of the ch ain δ = ⌊ L= 2⌋ [ 1 , 61 ]. One w a y t o o bs e r v e thi s
tr a n sit ion t o non-in t e gr a b i l it y i s v i a the le v e l - sp a c in g d i s tr i but ion of the H a mi lt oni a n, as s tud ie d in R ef . [ 1 ,
61 ] a nd r e pr oduc e d inde pe nde n tly in fi g. 1.7.1 . The le v e l - sp a c in g d i s tr i but ion tr a n sit ion s f r om a P o i s s on
t o a ( univ e r s al ) W i gne r - D ys on for m, a c ommon si gn a tur e of qua n tum ch a os . N ot e th a t , in g e ne r al , t o
o bt ain a W i gne r - D ys on le v e l - sp a c in g d i s tr i but ion for ch a ot ic s ys t e m s, one ne e d s t o m ak e s ur e th a t al l the
s y mme tr ie s h a v e be e n r e mo v e d , th a t i s, w e a r e w ork in g in a spe c i fic s y mme tr y s e ct or of the s ys t e m . F or
the s ys t e m in e q . ( 1.13 ), w e c on side r the sp in s ubsp a c e c or r e spond in g t o
⌊
L
3
⌋
sp in s u p; onc e w e a r e in
thi s s ubsp a c e , the r e a r e no de g e ne r a c ie s in the H a mi lt oni a n, s e e R ef s . [ 1 , 61 ] for mor e de t ai l s . M or e o v e r ,
for the X X Z mode l w ith defe ct , R ef s . [ 62 – 64 ] v e r i fie d othe r si gn a tur e s of qua n tum ch a os s uch as local
o bs e r v a b le s s a t i sf y in g d i a g on al E TH [ 46 , 47 ], a nd the lon g -t ime dy n a mic s de v e lop in g spe ctr al c or r e l a t ion s .
F ur the r mor e , in r e c e n t y e a r s, thi s mode l h as al s o be e n e mp lo y e d in the s tudy of m a n y - body ch a os [ 65 ],
the r m al i za t ion [ 66 , 67 ], a nd qua n tum tr a n spor t [ 68 ].
I n R ef . [ 69 ], the p a r t ic ip a t ion r a t io as a n ind ica t or of ch a os w as s tud ie d a nd r e s ults simi l a r t o fi g. 1.8.1
w e r e o bt aine d . U sin g the r e l a t iv e e n tr op y of c o he r e nc e , 2 - c o he r e nc e , in v e r s e p a r t ic ip a t ion r a t io ( I PR ), a nd
1 - nor m c o he r e nc e , w e s tudy the on s e t of ch a os, as the defe ct sit e i s mo v e d t o the midd le of the ch ain . W e
s t udy c o he r e nc e in t w o d i ffe r e n t b as e s, the “ sit e b asi s ” a nd the “ me a n- fie ld b asi s ” . The sit e b asi s i s simp ly the
local σ
z
b asi s a t e a ch sit e a nd c o he r e nc e in thi s b asi s i s a me as ur e of ho w uni for mly spr e a d i s the ei g e n s t a t e
w i th r e spe ct t o the local s ubs ys t e m s . T o define the me a n- fie ld b asi s, w e s t a r t b y ex pr e s sin g the t ot al H a mi l -
t oni a n as H
total
= H
0
+ V , w he r e H
0
i s the H a mi lt oni a n of nonin t e r a ct in g p a r t icle s ( or , mor e g e ne r al ly ,
de gr e e s of f r e e dom ) a nd V the in t e r a ct ion be t w e e n the m [ 70 , 71 ]. The me a n- fie ld b asi s i s the n the ei g e n-
13
▲
▲ ▲
▲
▲
▲
▲
▲ ▲
▲
▲
▲
■
■
■
■
■
■ ■
■
■
■
■
■
6 8 10 12 14 16
0.0
0.1
0.2
0.3
0.4
0.5
0.6
▲
■
(a)
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
▲
■
■
■
■
■
■
■
■
■
■
■
■
6 8 10 12 14 16
0.0
0.1
0.2
0.3
0.4
0.5
0.6
(b)
Figure 1.4.3: F raction of integrable eigenstates that majo rize chaotic eigenstates fo r the Hamiltonian
defined in eq. ( 1.13 ) system size L. Here, δ = 1 fo r integrable eigenstates, and δ = ⌊ L= 2⌋ fo r chaotic
eigenstates. The plot ma rk ers co rresp ond to the t w o different bases, the site-basis and mean-field
basis, r esp ectively . Figures (a) and (b) co rresp ond to the full sp ectrum and 20% of eigenvecto rs in the
middle of the sp ectrum, resp ectively .
b asi s of the “ me a n- fie ld H a mi lt oni a n, ” H
0
. Thi s i s, in fa ct , quit e simi l a r t o the me a n- fie ld a ppr o a ch us e d in
a t omic a nd n ucle a r p h ysic s ( a nd he nc e the t e r mino lo g y ). I t i s imme d i a t e ly a pp a r e n t th a t s uch a de c ompo -
sit ion of the t ot al H a mi lt oni a n i s n o t unique , ho w e v e r , in m a n y p h ysical s c e n a r ios, the r e i s a n a tur al cho ic e
of the me a n- fie ld b asi s . The in tuit ion he r e i s th a t as the in t e r a ct ion s tr e n g th incr e as e s, the ei g e n s t a t e s of the
t ot al H a mi lt oni a n w i l l be c ome mor e uni for mly spr e a d w he n ex pr e s s e d in the me a n- fie ld b asi s . F o l lo w in g
R ef s . [ 1 , 61 ], w e t ak e the me a n- fie ld H a mi lt oni a n t o be J
xy
̸= 0; ε
δ
̸= 0; J
z
= 0 . N ot ic e th a t thi s i s n o t the
s a me a s the in t e gr a b le l imit a bo v e .
R a n do m m a tr i x t h e o r y .— Befor e g o in g in t o the de t ai l s of our n ume r ical s tud ie s, le t us br iefly r e cal l s ome
k ey ide as f r om r a ndom m a tr i x the or y ( R MT ) a nd its pr e d ict ion s for qua n tum ch a ot ic s ys t e m s . F ir s t in tr o -
duc e d b y W i gne r [ 72 – 74 ] a nd l a t e r de v e lope d b y D ys on [ 75 ], R MT h as be e n w ide ly us e d t o s tudy c omp lex
s ys t e m s a nd in p a r t ic ul a r , qua n tum ch a ot ic s ys t e m s ( s e e R ef s . [ 46 , 47 ] for a pe d a g o g ical r e v ie w ). M a n y of
the or i g in al ly in tr oduc e d me as ur e s ( l i k e le v e l - sp a c in g d i s tr i but ion ) w e r e pur e ly spe ctr al pr ope r t ie s, but in
r e c e n t y e a r s, the r e h as be e n mor e in t e r e s t in g o in g bey ond the spe ctr al pr ope r t ie s t o unde r s t a nd the ei g e n-
s t a t e s tr uctur e of ch a ot ic s ys t e m s [ 46 , 47 ]. F or in s t a nc e , i f qua n tum ch a ot ic s ys t e m s ca n be w e l l - de s cr i be d
b y R MT , the n their ei g e n s t a t e pr ope r t ie s a r e ex pe ct e d t o r e s e mb le thos e of r a n do m v e ct or s in the H i l be r t
sp a c e ( n a me ly , the ei g e n v e ct or s of R MT H a mi lt oni a n s ). H o w e v e r , thi s i s not the c omp le t e p ictur e . M a n y
of the tr a d it ion al G a us si a n e n s e mb le s l i k e the G a us si a n Or tho g on al E n s e mb le ( G OE ), G a us si a n U nit a r y
14
E n s e mb le ( G UE ), e t c . a r e e n s e mb le s of m a n y- b o d y in t e r a ct ion s a nd not 2 - a nd 3 - body in t e r a ct ion s ( r e m-
ini s c e n t of p h ysical H a mi lt oni a n s ), a nd the pr ope r t ie s of fe w - body H a mi lt oni a n s ca n be mode l le d mor e
a c c ur a t e ly b y the us e of the s o - cal le d em b e d de d en sem b le s [ 48 ]. M or e o v e r , n ume r ical s tud ie s h a v e r e v e ale d
th a t g e ne r ical ly , only ei g e n s t a t e s in the midd le of the spe ctr um c or r e spond w e l l t o the ( us ual ) R MT pr e-
d ict ion ( as w i l l al s o be r e le v a n t for our n ume r ical s tud ie s ) [ 48 , 69 , 70 ].
W e al s o not e th a t usin g the c onne ct ion be t w e e n S h a nnon e n tr op y a nd r e l a t iv e e n tr op y of c o he r e nc e as
d i s c us s e d in s e ct ion 1 , w e ca n infe r a n aly t ical ly the e n s e mb le a v e r a g e d r e l a t iv e e n tr op y of c o he r e nc e for
G OE ei g e n s t a t e s ( s e e S e c . 2.3.2 of R ef . [ 48 ])
⟨
c
( r e l)
B
⟩
GOE
= ln( 0: 48d)+ O
(
1
d
)
; (1.14)
w he r e d i s the H i l be r t sp a c e d ime n sion . S inc e G OE ei g e n v e ct or s a r e ( H a a r ) uni for mly d i s tr i but e d , the
b asi sB i s a gen er ic b asi s, th a t i s, the e s t im a t e for the e n s e mb le a v e r a g e ho ld s tr ue for a n y b asi s [ 46 , 47 ]. W e
us e thi s a n aly t ical ex pr e s sion for nor m al i z in g the qua n t it ie s s tud ie d in fi gs . 1.4.1 a nd 1.4.2 .
The H a mi lt oni a n in e q . ( 1.13 ) i s r e al a nd s y mme tr ic a nd be lon gs t o the G a us si a n Or tho g on al E n s e mb le
( G OE ) univ e r s al it y cl as s . I n fi gs . 1.4.1 , 1.4.2 , 1.8.1 a nd 1.8.2 , w e s tudy the a for e me n t ione d c o he r e nc e me a -
s ur e s nor m al i z e d b y the G OE pr e d ict ion a nd find th a t , in the midd le of the spe ctr um, the ch a ot ic mode l
doe s r e pr oduc e the G OE pr e d ict ion ; w hich i s c on si s t e n t w ith pr e v iously kno w n r e s ults ( th a t the ei g e n-
s t a t e s of s ys t e m s w ith fe w - body in t e r a ct ion s de local i z e in the midd le of the spe ctr um ) [ 1 , 48 , 61 , 69 , 70 ].
Th us, thi s v ind ica t e s the v a r ious c o he r e nc e me as ur e s as a si gn a tur e of the tr a n sit ion t o ch a os .
W h a t a b o u t o t h er q u a n t u m co h er en ce m e as u r e s? — A p a r t f r om the spe c i fic qua n tum c o he r e nc e me as ur e s
s tud ie d a bo v e , w h a t , i f a n y thin g , ca n be s aid a bout a n a r bitr a r y c o he r e nc e me as ur e s ’ a b i l it y t o pr o be qua n-
tum ch a os in a simi l a r w a y ? T o a n s w e r thi s que s t ion, w e tur n t o the po w e r f ul m a the m a t ical for m al i s m of
m a j or i za t ion the or y [ 51 ] as d i s c us s e d in s e ct ion 1 . W e n ume r ical ly s tudy the m a j or i za t ion c ond it ion in
1.1 for the in t e gr a b le a nd ch a ot ic ei g e n s t a t e s of the X X Z sp in- ch ain in e q . ( 1.13 ) a nd a n aly z e the ex t e n t t o
w hich the induc e d pr e or de r or de r ho ld s tr ue . S pe c i fical ly , for a g iv e n s ys t e m si z e L , w e c on side r the s e t
of in t e gr a b le a nd ch a ot ic ei g e n s t a t e s or de r e d r e spe ct iv e ly b y the e ne r g ie s of the c or r e spond in g H a mi lt oni -
15
a n s . The n, w e n ume r ical ly che ck for the m a j or i za t ion c ond it ion in 1.1 be t w e e n the k th ch a ot ic ei g e n s t a t e
a nd the k th in t e gr a b le ei g e n s t a t e (w he r e the index k i s or de r e d w ith r e spe ct t o the e ne r g y ). W e find th a t
the m a j or i za t ion c ond it ion doe s not ho ld for al l p air s of ei g e n s t a t e s ( or de r e d b y e ne r g y ). F or thi s r e as on,
w e in tr oduc e a w e ak e r not ion of “ m a j or i za t ion f r a ct ion, ” w hich i s the f r a ct ion of ei g e n s t a t e s for w hich the
m a j or i za t ion c ond it ion i s tr ue . L e t η be the n umbe r of ch a ot ic ei g e n s t a t e s th a t a r e m a j or i z e d b y the c or -
r e spond in g in t e gr a b le ei g e n s t a t e s a nd d the t ot al n umbe r of ei g e n s t a t e s, the n, the m a j or i za t ion f r a ct ion i s
simp ly the r a t io
η
d
. I n fi g. 1.4.3 , w e p lot the m a j or i za t ion f r a ct ion as a f unct ion of the s ys t e m si z e L , for both
the sit e- b asi s a nd the me a n- fie ld b asi s . W e s e e th a t , for l a r g e r s ys t e m si z e s, a ch a ot ic ei g e n s t a t e p ick e d a t r a n-
dom ( uni for mly ) i s, w ith r e l a t iv e ly hi gh pr o b a b i l it y , m a j or i z e d b y its in t e gr a b le c oun t e r p a r t a nd th us w i l l
h a v e a l a r g e r v alue for a n y c o he r e nc e me as ur e; for ex a mp le , as d i sp l a y e d b y the r e l a t iv e e n tr op y of c o he r -
e nc e a nd the 2 - c o he r e nc e in fi gs . 1.4.1 a nd 1.4.2 . S inc e p h ysical ei g e n s t a t e s r e s e mb le r a ndom v e ct or s in the
midd le of the spe ctr um, w e f ur the r c on side r the m a j or i za t ion f r a ct ion for 20% of ei g e n s t a t e s in the midd le
of the spe ctr um, a nd find a simi l a r incr e as e w ith s ys t e m si z e ( a nd a non- monot onic it y a t s m al l si z e s ).
1.5 A t the l evel of c h an nel s
H a v in g de mon s tr a t e d the a b i l it y of qua n tum c o he r e nc e me as ur e s t o d i s t in g ui sh ch a ot ic - v s -in t e gr a b le ei g e n-
s t a t e s a nd a flur r y of c onne ct ion s w ith de local i za t ion me as ur e s, w e no w tur n t o ch a os a t the le v e l of qua n-
tum dy n a mic s ( or mor e g e ne r al ly qua n tum ch a nne l s ¹⁴ ). I n p a r t ic ul a r , the a b i l it y of ch a ot ic dy n a mic s t o
g e ne r a t e qua n tum c or r e l a t ion s h as pr o v e n t o be a r ich f r a me w ork [ 6 , 38 , 40 ] a nd he r e w e e s t a b l i sh r i g or ous
c onne ct ion s w ith their a b i l it y t o g e ne r a t e qua n tum c o he r e nc e .
1 T h e O T O C , q u a ntum c h a os , a nd its c o nnec ti o n with C GP
I n r e c e n t y e a r s, the out - of -t ime- or de r e d c or r e l a t or ( O T O C ) h as e me r g e d as a pr omine n t d i a gnos t ic for
qua n tum ch a os a t the le v e l of dy n a mic s [ 17 , 18 , 76 – 80 ]. The pr e c i s e r o le th a t the O T O C p l a ys in ch a r a c -
t e r i z in g qua n tum ch a os v i a its shor t -t ime ex pone n t i al gr o w th i s be tt e r unde r s t ood in s ys t e m s w ith eithe r a
¹⁴ W e r e m a rk th a t qua n tum ch a nne l s [ 3 ] pr o v ide a g e ne r al f r a me w ork th a t e nca ps ul a t e s the not ion s of unit a r y dy n a mic s as
w e l l as ope n s ys t e m effe cts, a nd the r efor e w e r efe r t o the c onne ct ion s he nc efor th as “ a t the le v e l of ch a nne l s, ” for its g e ne r al it y .
16
s e micl as sical l imit or s ys t e m s w ith a l a r g e n umbe r of local de gr e e s of f r e e dom [ 18 , 76 ]. On the othe r h a nd ,
the shor t -t ime gr o w th doe s not s e e m t o p l a y a n y r o le for qua n tum ch a os in finit e s ys t e m s s uch as sp in- ch ain s
(w ithout a s e micl as sical a n alo g ) [ 19 – 24 ]. H o w e v e r , the lon g -t ime l imit of O T O C s m a y be ex pe ct e d t o p l a y
a mor e cle a r r o le , s e e R ef s . [ 81 , 82 ]. M or e o v e r , t o f ur the r our unde r s t a nd in g of the O T O C , s e v e r al w ork s
h a v e tr ie d t o e s t a b l i sh a c onne ct ion t o w e l l - s tud ie d si gn a tur e s of ch a os s uch as L os chmidt e cho [ 83 ] a nd e n-
t a n gl in g po w e r [ 84 ], w hich s u gg e s t th a t a n infor m a t ion-the or e t ic in v e s t i g a t ion of the O T O C’ s pr ope r t ie s
mi gh t pr o v ide a f r uitf ul d ir e ct ion .
The O T O C qua n t i fie s the r a p id de local i za t ion of qua n tum infor m a t ion init i al i z e d in local s ubs ys t e m s,
w hich h as be e n t e r me d “ infor m a t ion s cr a mb l in g ” . One w a y t o qua n t i f y thi s spr e a d i s t o c on side r the gr o w th
of local ope r a t or s unde r H ei s e nbe r g t ime e v o lut ion, ca ptur e d b y the fo l lo w in g qua n t it y ( he r e a ft e r r efe r r e d
t o as the “ s qua r e d c omm ut a t or ” for br e v it y )
C
( β)
V; W
( t) := Tr
(
[ V; W( t)]
y
[ V; W( t)] ρ
β
)
=∥[ V; W( t)]∥
2
β
;
(1.15)
w he r e W( t) = U
y
t
( W) i s the H ei s e nbe r g - e v o lv e d ope r a t or , ρ
β
e
β H
= Tr
[
e
β H
]
i s the G i b bs s t a t e a t
in v e r s e t e mpe r a tur e β , a nd∥∥
β
be the nor m induc e d f r om the inne r pr oduct ⟨ X; Y⟩
β
:= Tr
(
X
y
Y ρ
β
)
. R e-
ex pr e s sin g C
( β)
V; W
( t) in the c omm ut a t or for m r e s e mb le s a ( s t a t e- de pe nde n t ) v a r i a n t of the L ie b - R o b in s on
c on s tr uct ion, w hich in tur n impos e s f und a me n t al l imits on the spe e d of infor m a t ion pr op a g a t ion in non-
r e l a t iv i s t ic s ys t e m s [ 26 , 85 – 87 ]. I n thi s w a y , C
( β)
V; W
( t) ca ptur e s the spr e a d of infor m a t ion thr ou gh nonlocal
de gr e e s of f r e e dom of a s ys t e m .
The c onne ct ion be t w e e n the s qua r e d c omm ut a t or a nd the O T O C i s r e v e ale d w he n w e choos e V; W t o
be u nit a r y [ 17 , 18 ]
C
( β)
V; W
( t) = 2
(
1Re
{
F
( β)
V; W
( t)
})
;
w he r e , F
( β)
V; W
( t) Tr
(
W( t)
y
V
y
W( t) V ρ
β
)
;
(1.16)
i s a four -po in t f unct ion (w ith un us ual t ime- or de r in g ) cal le d the O T O C . S inc e the s qua r e d c omm ut a t or
17
a bo v e a nd the O T O C a r e r e l a t e d v i a a simp le a ffine f unct ion, w e w i l l foc us he r e on the s qua r e d c omm ut a t or
a nd r efe r t o it in t e r ch a n g e a b ly as the O T O C ( the d i s t inct ion should be cle a r f r om the c on t ex t ). I n thi s
ch a pt e r , w e w i l l foc us on the infinit e-t e mpe r a tur e ( β = 0 ) cas e , th a t i s, ρ
β
=
I
d
. H e r e a ft e r , w e define ,
C
( β= 0)
V; W
( t) C
V; W
( t) a nd F
( β= 0)
V; W
( t) F
V; W
( t) . I n the fo l lo w in g , w e w i l l c onne ct the out - of -t ime- or de r e d
c or r e l a t or w ith the c o he r e nc e- g e ne r a t in g po w e r , w hich w e a r e no w r e a dy t o in tr oduc e .
Co h er en ce-gen er a t i n g p o w er .— H o w m uch c o he r e nc e doe s a n e v o lut ion g e ne r a t e on a v e r a g e? M ot iv a t e d
f r om the r e s our c e the or y of c o he r e nc e , s e v e r al me a nin g f ul qua n t i fie r s for thi s w e r e o bt aine d in R ef s . [ 88 –
90 ]. H e r e , w e w i l l c on side r the “ ex tr e m al C GP , ” define d as [ 43 ]
C
B
(U) =
1
d
d
∑
j= 1
c
B
(U( Π
j
)); (1.17)
w he r eU() = U() U
y
i s a unit a r y ch a nne l , c
B
() i s a c o he r e nc e me as ur e , a nd B =f Π
j
g
d
j= 1
i s a n or thonor -
m al b asi s for the d - d ime n sion al H i l be r t sp a c e ( s e e the s e ct ion 4.3 for mor e de t ai l s ). The C GP me as ur e s
the a v e r a g e c o he r e nc e g e ne r a t e d unde r t ime e v o lut ion U b y its a ct ion on the pur e s t a t e s in B . F or the r e s t
of the ch a pt e r w e choos e c
( 2)
B
() in the a bo v e e qua t ion, th a t i s, C
B
(U) =
1
d
d
∑
j= 1
c
( 2)
B
(U( Π
j
)) , w hich h as a
clos e d for m ex pr e s sion as [ 43 ]
C
B
(U) = 1
1
d
Tr
(
X
T
U
X
U
)
;
w he r e[ X
U
]
j; k
= Tr
(
Π
j
U( Π
k
)
)
:
(1.18)
H e r e a ft e r , w e w i l l r efe r t o the a bo v e qua n t it y simp ly as C GP for br e v it y . I t i s w or th me n t ionin g th a t the for -
m al i s m in tr oduc e d in R ef s . [ 43 , 88 – 90 ] i s m uch mor e g e ne r al th a n the definit ion e q . ( 1.17 ). I n p a r t ic ul a r ,
one ca n c on side r v a r ious cho ic e s of c o he r e nc e me as ur e s a nd d i s tr i but ion s o v e r inc o he r e n t s t a t e s .
The C GP define d a bo v e h as m a n y in t e r e s t in g pr ope r t ie s, s ome of w hich w e r e v ie w no w . F ir s t , in the
c on t ex t of A nde r s on local i za t ion a nd m a n y - body local i za t ion, it w as sho w n th a t the C GP a cts as a n “ or de r
p a r a me t e r ” for the e r g od ic -t o -local i za t ion tr a n sit ion [ 43 ]. S e c ond , in the r e s our c e-the or e t ic s tudy of in-
c omp a t i b i l it y of qua n tum me as ur e me n ts, the C GP a r i s e s n a tur al ly as a n inc omp a t i b i l it y me as ur e [ 27 ]. A nd
thir d , the C GP le nd s its e l f t o a po w e r g e ome tr ic c onne ct ion : the C
B
(U) i s pr opor t ion al t o the ( s qua r e of
18
the ) Gr as mm a nni a n d i s t a nc e be t w e e n t w o m a x im al ly a be l i a n s ub al g e br as, the one g e ne r a t e d b y al l bounde d
o bs e r v a b le s d i a g on al in B a nd thos e d i a g on al in U(B) [ 90 ]. U sin g thi s c onne ct ion, a clos e d for m ex pr e s -
sion for C GP in a c omm ut a t or for m ca n be o bt aine d as fo l lo w s ¹⁵
C
B
(U) =
1
2d
d
∑
j; k= 1
[
Π
j
;U( Π
k
)
]
2
2
=
1
2d
d
∑
j; k= 1
Tr
(
[
Π
j
;U( Π
k
)
]
y
[
Π
j
;U( Π
k
)
]
)
:
(1.19)
W ith the C GP ex pr e s s e d in the c omm ut a t or for m in e q . ( 1.19 ), w e a r e no w r e a dy t o in tr oduc e its c on-
ne ct ion t o the O T O C C
V; W
( t) . I n a n t ic ip a t ion of the the or e m be lo w , w e define the fo l lo w in g : L e t V; W
be t w o nonde g e ne r a t e unit a r ie s w ith a spe ctr al de c omposit ion V =
d
∑
j= 1
v
j
Π
j
; W =
d
∑
j= 1
w
j
e
Π
j
. L e tB
V
=
f Π
j
g;B
W
=f
e
Π
j
g be the c or r e spond in g ei g e nb as e s, the n, V
B V!B W
i s a unit a r y in t e r t w ine r c onne ct in g B
V
t oB
W
, w ho s e a ct ion i s V
B V!B W
(
Π
j
)
=
e
Π
j
8 j2f 1; 2; ; dg .
P r opositio n 1.3
G iv e n a unit a r y e v o lut ion ope r a t or U
t
, a nd t w o nonde g e ne r a t e unit a r y ope r a t or s V a nd W , the i nfinit e-
t e mpe r a tur e out - of -t ime- or de r e d c or r e l a t or ( C
V; W
( t) ) a nd the C GP ( C
B
() ) a r e r e l a t e d as
C
V; W
( t) = 2C
B V
(U
t
◦V
B V!B W
)
2
d
Re
8
<
:
∑
j̸= l; k̸= m
v
j
w
k
v
l
w
m
Tr
(
e
Π
k
( t) Π
j
e
Π
m
( t) Π
l
)
9
=
;
: (1.20)
R em a r k s .— ( a ) W hi le qua n tum c o he r e nc e ( a nd he nc e the C GP ) i s a b asi s - de pe nde n t qua n t it y , the
a bo v e the or e m r e l a t e s the O T O C t o a C GP n a t u r a ll y . I n tuit iv e ly , the O T O C me as ur e s the gr o w th of
the nonc omm ut a t iv it y be t w e e n the ope r a t or s W( t) a nd V , a nd thi s in tuit ion i s m a de pr e c i s e b y the C GP
¹⁵ N ot e th a t thi s for m ul a us e s the ex tr e m al pr o b a b i l it y d i s tr i but ion o v e r the inc o he r e n t s t a t e s, in s t e a d of the uni for m d i s tr i -
but ion, w hich a c c oun ts for the d i ffe r in g fa ct or s of d( d+ 1) .
19
C
B V
(U
t
) , w hich me as ur e s the i n co m p a t i bi l it y [ 27 ] be t w e e n th e b as e s B
V
a ndB
U t
.
( b ) I n 1.3 it i s impor t a n t t o e mp h asi z e th a t the C GP e me r g e s as a s u bp a r t of the O T O C . B y p lu gg in g in
the spe ctr al de c omposit ion of the ope r a t or s V a nd W , w e o bt ain a s umm a t ion o v e r four ind ic e s a nd b y c on-
side r in g a s ubs e t of the s e t e r m s, w e o b ain the C GP . The “ ex tr a ” t e r m i s of the for m Tr
(
e
Π
k
( t) Π
j
e
Π
m
( t) Π
l
)
(w hich i s the s e c ond t e r m on the R HS of e q . ( 1.20 )) a nd w e r efe r t o thi s as the “ off - d i a g on al ” t e r m . Th a t i s,
the C GP i s “ c on t aine d ” in the O T O C . W e r efe r the r e a de r t o the pr oof in the A ppe nd i x for mor e de t ai l s .
( c ) T o he lp unde r s t a nd 1.3 , le t us c on side r a simp le cas e: as s ume th a t the t w o ope r a t or s c omm ut e a t
t ime t = 0 , th a t i s, [ V; W] = 0 . Thi s i s a c ommon as s umpt ion w he n s tudy in g the O T O C’ s dy n a mical
fe a tur e s, for ex a mp le , b y choosin g local ope r a t or s on d i ffe r e n t sit e s ( or , i f they a r e on the s a me sit e , b y
choosin g the m t o be the s a me ope r a t or ), the n, V
B V!B W
=I , th a t i s, the in t e r t w ine r ca n be chos e n t o be
the ( tr iv i al ) ide n t it y s u pe r ope r a t or . T o f ul fi l l the nonde g e ne r a c y cr it e r i a (w hich w e as s ume d init i al ly ), w e
ca n choos e V a nd W t o be quasi local . N o w , sinc e [ V; W] = 0 , the fir s t t e r m be c ome s e qual t o 2C
B
(U
t
) ,
w i thB
V
=B
W
B . Th a t i s, simp ly ( t w ic e ) the C GP of the t ime e v o lut ion unit a r y w he n me as ur e d in the
b asi s of the ope r a t or s V a nd W . U sin g the for thc omin g d i s c us sion, s e e E qua t ion 1.26 , le t the ei g e n v alue s
of V a nd W be uni for mly d i s tr i but e d o v e r [ 0; 2 π) , the n w e h a v e , ⟨ C
V; W
( t)⟩
V; W
= 2C
B
(U
t
) , w he r e⟨⟩
V; W
de not e s a v e r a g in g o v e r V; W . Th a t i s, the “ ex tr a t e r m ” v a ni she s a nd the a v e r a g e d O T O C i s e xa c t l y e qual t o
t w ic e the C GP .
P r o j e c t io n O T O Cs .— H e r e w e e s t a b l i sh a nothe r c onne ct ion be t w e e n the O T O C a nd the C GP b y choos -
in g V a nd W t o be pr oj e ct ion ope r a t or s in the O T O C . S imi l a r c on s tr uct ion s h a v e be e n c on side r e d befor e ,
for ex a mp le , in R ef . [ 91 ], the a uthor s us e d “ pr oj e ct ion O T O C s ” t o c onne ct w ith the p a r t ic ip a t ion r a t io . I n
p a r t ic ul a r , simi l a r a qua n t it y kno w n as “ fide l it y O T O C s ” w as pr opos e d in R ef . [ 92 ] as a n ex pe r ime n t al ly
pr omi sin g a ppr o a ch t o me as ur e O T O C s a nd , in tur n, t o the s tudy of s cr a mb l in g a nd the r m al i za t ion . L e t
B
V
= f Π
α
g a ndB
W
= f
e
Π
β
g , w e s t a r t b y p lu gg in g in V = Π
α
; W =
e
Π
β
in t o the O T O C t o o bt ain
C
Π α;
e
Π
β
( t) =
1
d
[
Π
α
;
e
Π
β
( t)
]
2
2
. The n, b y s ummin g o v e r α , w e h a v e ,
d
∑
α= 1
C
Π α;
e
Π
β
( t) =
1
d
d
∑
α= 1
[
Π
α
;
e
Π
β
( t)
]
2
2
=
2
d
c
( 2)
B V
(
e
Π
β
( t)); (1.21)
20
w he r e c
( 2)
B V
() i s the 2 - nor m c o he r e nc e . The n, i f w e s um o v e r β ,w e h a v e ,
d
∑
α= 1; β= 1
C
Π α;
e
Π
β
( t) =
2
d
d
∑
β= 1
c
( 2)
B V
(
e
Π
β
( t))
= 2C
B V
(U
t
◦V
B V!B W
):
(1.22)
The r efor e , g iv e n t w o b as e s, B
V
;B
W
, w e h a v e th a t the O T O C “ a v e r a g e d ” o v e r the s e b as e s i s e qual t o ( t w ic e )
the c o he r e nc e- g e ne r a t in g po w e r of the unit a r y e v o lut ion ( a nd the in t e r t w ine r c onne ct in g the b as e s ). M or e-
o v e r , i f B
V
=B
W
, w e h a v e ,
d
∑
α; β
C
Π α; Π
β
( t) = 2C
B V
(U
t
): (1.23)
Th a t i s, the O T O C a v e r a g e d o v e r v a r ious pr oj e ct or s i s e qual t o the C GP of the t ime e v o lut ion unit a r y . N ot e
th a t for a non- de g e ne r a t e H a mi lt oni a n, the C GP i s e qual t o the a v e r a g e e s ca pe pr o b a b i l it y [ 27 ], w hich i s
in t im a t e ly c onne ct e d t o qua n t it ie s l i k e the L os chmidt e cho , p a r t ic ip a t ion r a t io , a nd othe r s, as d i s c us s e d in
s e ct ion 1 .
A v er a ge O T O C , co h er en ce, a n d ge o m etr y .— I n the fo l lo w in g w e e s t a b l i sh a c onne ct ion be t w e e n the a v e r -
a g e O T O C a nd the g e ome tr y of the s e t of m a x im al ly a be l i a n s ub al g e br as of the ope r a t or sp a c e ( as s oc i a t e d
t o the qua n tum s ys t e m ). F or thi s, le t us br iefly in tr oduc e the g e ome tr ic r e s ults o bt aine d in R ef . [ 90 ] c on-
c e r nin g C GP a nd 2 - c o he r e nc e . G iv e n a b asi s B , le tA
B
be the a be l i a n al g e br a g e ne r a t e d b y its e le me n ts .
The n, A
B
i s a s ubsp a c e of the ope r a t or al g e br a B(H) v ie w e d as a H i l be r t sp a c e H
HS
, e ndo w e d w ith the
H i l be r t - S chmidt inne r pr oduct , ⟨ A; B⟩
HS
:= Tr
(
A
y
B
)
, w hich induc e s the nor m, ∥ A∥
HS
=
√
⟨ A; A⟩
HS
=
√
Tr( A
y
A) . I fB i s o bt aine d v i a a m a x im al or tho g on al r e s o lut ion of the ide n t it y in B(H) , the n,A
B
i s a
m a x i m al a be l i a n s ub al g e br a ( M A SA) [ 90 , 93 ]. The s e t of al l M A SA s i s a t opo lo g ical ly non tr iv i al s ubs e t of
the Gr as s m a nni a n of d - d ime n sion al s ubsp a c e s of H
HS
a nd w e ca n define a d i s t a nc e be t w e e n t w o M A SA s,
A
B
a ndA
e
B
as [ 90 ],
D
(
A
B
;A
e
B
)
:=
D
B
D
e
B
HS
; (1.24)
21
w he r e for s u pe r ope r a t or s, w e h a v e , Tr
HS
(E) :=
∑
d
j; k= 1
⟨j j⟩⟨ kj;E (j j⟩⟨ kj)⟩ . I n fa ct , the C GP tur n s out t o
be pr opor t ion al t o the ( s qua r e d ) d i s t a nc e be t w e e n the al g e br as A
B
a ndU(A
B
) , th a t i s [ 90 ],
C
B
(U) =
1
2d
D
2
(A
B
;U (A
B
)): (1.25)
W e a r e no w r e a dy t o in tr oduc e the m ain r e s ult of thi s s e ct ion, the de t ai le d pr oof s of w hich ca n be found
in s e ct ion 1.9. L e tB
V
=f Π
α
g a ndB
W
=f
e
Π
β
g be t w o b as e s . C on side r unit a r ie s d i a g on al in the r e spe ct iv e
b as e s, V =
∑
α
e
iθ α
Π
α
a nd W =
∑
β
e
i
e
θ
βe
Π
β
, w ithf θ
α
g a ndf
e
θ
β
g inde pe nde n t a nd ide n t ical ly d i s tr i but e d
uni for mly on the in t e r v al [ 0; 2 π) . The n,
⟨∥[ V; W( t)]∥
2
2
⟩
θ
= 2dC
B V
(U
t
◦V
B V!B W
): (1.26)
Th a t i s, the O T O C a v e r a g e d o v e r d i a g on al unit a r ie s w ith p h as e s d i s tr i but e d uni for mly r e v e al s the C GP of
the dy n a mic s . M or e o v e r , i f B
V
= B
W
, the n, the r e l a t ion simp l i fie s t o , ⟨∥[ V; W( t)]∥
2
2
⟩
θ
= 2dC
B V
(U
t
):
U sin g the c onne ct ion w ith d i s t a nc e in the Gr as s m a nni a n, w e h a v e ,
⟨∥[ V; W( t)]∥
2
2
⟩
θ
= D
2
(A
B V
;U
t
(A
B V
)): (1.27)
The r efor e , thi s a v e r a g e O T O C qua n t i fie s ex a ctly the d i s t a nc e ( s qua r e d ) in the Gr as s m a nni a n be t w e e n
M A SA sA
B V
a ndU
t
(A
B V
) . Thi s i s y e t a nothe r w a y t o unde r s t a nd the O T O C as me as ur in g the inc om-
p a t i b i l it y be t w e e n the ope r a t or s V a nd U
t
a nd the b as e s as s oc i a t e d t o the m .
F ur the r mor e , w e ca n al s o us e a v e r a g e O T O C s t o e s t im a t e the c o he r e nc e of a s t a t e . F or thi s, w e fir s t
pr o v e the fo l lo w in g r e s ult : g iv e n a s t a t e ρ a nd a unit a r y V , w e h a v e
⟨
∥[D
B
( V); ρ]∥
2
2
⟩
V2 Haar
=
2
d
c
( 2)
B
( ρ): (1.28)
The n, as a c or o l l a r y , w e c on side r the fo l lo w in g o p en sy s t em O T O C ,
⟨
∥[E
t
( V); ρ]∥
2
2
⟩
V2 Haar
, w he r efE
t
g
t
i s a fa mi ly of qua n tum ch a nne l s [ 94 ]. I ffE
t
g
t
i s s uch th a t E
t
t!1
! D
B
, th a t i s, in the lon g -t ime l imit ,
22
E
t
c on v e r g e s t o the de p h asin g ch a nne l in the b asi s B [ 94 ], the n, the e qui l i br a t ion v alue of thi s a v e r a g e d
O T O C r e v e al s the 2 - c o he r e nc e of the s t a t e ρ . Th a t i s,
⟨
∥[E
t
( V); ρ]∥
2
2
⟩
V2 Haar
t!1
!
2
d
c
( 2)
B
( ρ): (1.29)
One ca n al s o c on side r in s t e a d of the qua n tum ch a nne l E
t
, unit a r y dy n a mic s unde r a ( t ime-inde pe nde n t )
non- de g e ne r a t e H a mi lt oni a n . H o w e v e r , in thi s cas e , the l imit lim
t!1
U
t
doe s not ex i s t ( as oppos e d t o lim
t!1
E
t
,
w hi ch doe s ), a nd s o w e c on side r the infinit e-t ime a v e r a g e d v alue of the O T O C , w hich ca n be us e d t o ex tr a ct
e qui l i br a t ion v alue s of p h ysical qua n t it ie s for unit a r y dy n a mic s th a t doe s e qui l i br a t e [ 14 , 15 ].
The a bo v e r e s ult ca n al s o be g e ne r al i z e d t o the fo l lo w in g s c e n a r io: c on side r t w o unit a r ie s V; W a nd t w o
b as e sB;
e
B . The n, the fo l lo w in g H a a r -a v e r a g e d O T O C i s pr opor t ion al t o the ( s qua r e d ) d i s t a nc e in the
Gr as s m a nni a n be t w e e n the M A SA s as s oc i a t e d t o the b as e s B;
e
B . Th a t i s,
⟨
[
D
B
( V);D
e
B
( W)
]
2
2
⟩
V; W2 Haar
=
1
d
2
D
2
(A
B
;A
e
B
) (1.30)
F o l lo w in g a simi l a r c or o l l a r y as a bo v e , c on side r t w o ch a nne l s E
t
a ndN
τ
, w hos e lon g -t ime l imit a r e the
de p h asin g ch a nne l s D
B
a ndD
e
B
, r e spe ct iv e ly . The n, the e qui l i br a t ion v alue of the fo l lo w in g O T O C r e v e al s
the Gr as s m a nni a n d i s t a nc e ( s qua r e d ) be t w e e n the M A SA s A
B
a ndA
e
B
,
⟨∥[E
t
( V);N
τ
( W)]∥
2
2
⟩
V; W2 Haar
t; τ!1
!
1
d
2
D
2
(A
B
;A
e
B
) (1.31)
Th a t i s, a v e r a g e O T O C s of the a bo v e for m ca n be us e d t o pr o be g e ome tr ical d i s t a nc e in the Gr as s m a nni a n
be t w e e n the t w o M A SA s a bo v e .
N ot e th a t the H a a r a v e r a g e s d i s c us s e d a bo v e c on si s t of a sin gle a d j o in t a ct ion of V ( or W ) a nd the r e-
for e , the s a me e s t im a t e ca n be o bt aine d b y simp ly a v e r a g in g o v e r e le me n ts of a 1 - de si gn in s t e a d [ 95 – 98 ].
F or qub it s ys t e m s, P a ul i m a tr ic e s for m a 1 - de si gn a nd s o the s e a v e r a g e s ca n be a c c e s s e d in a r e l a t iv e ly sim-
p le r w a y . The s a me al s o ho ld s tr ue for the H a a r -a v e r a g e d 4 -po in t O T O C s as they do not pr o be the f ul l
H a a r r a ndomne s s eithe r , w hich w ould (g e ne r al ly ) r e quir e c on side r in g e v e n hi ghe r -po in t f unct ion s [ 99 ].
23
I n s umm a r y , s uit a b ly a v e r a g e d O T O C s ca n pr o be 2 - c o he r e nc e of a s t a t e , the C GP of the dy n a mic s, a nd
the Gr as s m a nni a n d i s t a nc e ( s qua r e d ) be t w e e n M A SA s; a nd in thi s s e n s e qua n t it a t iv e ly c onne ct w ith the
not ion of c o he r e nc e a nd inc omp a t i b i l it y . A nd fin al ly , it i s w or th e mp h asi z in g th a t althou gh the C GP i s
r e l a t e d t o qua n t it ie s s uch as the L os chmidt e cho ( or s ur v iv al pr o b a b i l it y ) a nd effe ct iv e d ime n sion ; s e e the
d i s c us sion in R ef . [ 43 ], it r e m ain s uncle a r i f the O T O C - C GP c onne ct ion h as a n y d ir e ct imp l ica t ion s for
ch a r a ct e r i z in g qua n tum ch a os .
2 C GP , r a nd om ma tr i c e s , a nd sh o r t - time gr o w th
The un us ual effe ct iv e ne s s of R MT in pr e d ict in g the p h ysic s of qua n tum ch a ot ic s ys t e m s i s quit e as t oni sh-
in g , e spe c i al ly sinc e p h ysical H a mi lt oni a n s ( a nd their ei g e n s t a t e s ) a r e fa r f r om r a ndom . I n s e ct ion 1 w e
s a w th a t the c o he r e nc e of ei g e n s t a t e s in the midd le of the spe ctr um i s clos e t o the e n s e mb le a v e r a g e s o b -
t aine d f r om R MT . W e no w tur n t o dy n a mical fe a tur e s w hich a r e r e le v a n t for ex pe r ime n t al s ys t e m s s uch
as c o ld a t om s a nd ion tr a ps [ 9 , 100 ] w hich foc us on t ime e v o lut ion ; as oppos e d t o spe ctr al fe a tur e s, us e-
f ul in othe r s e tu ps s uch as n ucle a r s ca tt e r in g ex pe r ime n ts [ 72 , 73 ]. H e r e , w e pr o v ide a n a n aly t ical u ppe r
bound on the C GP a v e r a g e d o v e r G UE H a mi lt oni a n s a nd unr a v e l a c onne ct ion w ith the S pe ctr al F or m
F a ct or ( S FF ) [ 29 , 48 , 99 , 101 , 102 ], a pr omine n t me as ur e of spe ctr al c or r e l a t ion s for qua n tum ch a os . W e
be g in b y r e cal l in g th a t the G UE i s define d v i a the fo l lo w in g pr o b a b i l it y d i s tr i but ion o v e r d d H e r mit i a n
m a tr ic e s,
P( H)/ exp
(
d
2
Tr( H
2
)
)
: (1.32)
I t i s e as y t o s e e th a t tr a n sfor m a t ion s of the for m H 7! UHU
y
le a v e the e n s e mb le in v a r i a n t ( th a t i s, it i s
unit a r i ly in v a r i a n t ). The pr o b a b i l it y me as ur e ca n al s o be w r itt e n in t e r m s of the ei g e n v alue s f λ
j
g
d
j= 1
as the
fo l l o w in g j o in t pr o b a b i l it y d i s tr i but ion
P( λ
1
; λ
2
:::; λ
d
) = exp
(
d
2
∑
i
λ
2
i
)
∏
i< j
(
λ
i
λ
j
)
2
: (1.33)
24
The n, definin g the j o in t pr o b a b i l it y d i s tr i but ion of n ei g e n v alue s, th a t i s, the spe ctr al n -po in t c or r e l a t ion
f unct ion ( for n < d ) as
ρ
( n)
( λ
1
;:::; λ
n
) =
∫
d λ
n+ 1
::: d λ
d
P( λ
1
;:::; λ
d
); (1.34)
w he r e w e in t e gr a t e al l ei g e n v alue s f r om n+ 1 t o d . W e a r e no w r e a dy t o define the S FF , w hich i s the F our ie r
tr a n sfor m of the n -po in t c or r e l a t ion f unct ion, [ 29 , 48 , 99 , 102 ]
R
2k
( t) =
∑
i 1; i 2;; i
k
j 1; j 2;; j
k
∫
d λ ρ
( 2k)
( λ
1
;:::; λ
2k
)
e
i( λ i 1
++ λ i
k
λ j 1
λ j
k
) t
;
(1.35)
w he r e k i s a n y posit iv e in t e g e r . I n p a r t ic ul a r , the four -po in t S FF i s
R
4
( t) =
∑
k; l; m; n
∫
d λ ρ
( 4)
( λ
k
; λ
l
; λ
m
; λ
n
)
e
i( λ
k
+ λ
l
λ m λ n) t
:
(1.36)
B y c on side r in g the H a mi lt oni a n in the C GP C
B
( e
iHt
) as a r a ndom v a r i a b le o v e r the G UE, w e pr o v ide
a n a n aly t ical u ppe r bound on its a v e r a g e v alue in t e r m s of the four -po in t S FF .
P r opositio n 1.4
The c o he r e nc e- g e ne r a t in g po w e r a v e r a g e d o v e r the G a us si a n U nit a r y E n s e mb le ( G UE ) i s u ppe r bounde d
b y the four -po in t spe ctr al for m fa ct or as
⟨
C
B
(
e
iHt
)⟩
GUE
1
1
d( d+ 1)( d+ 2)( d+ 3)
∑
k; l; m; n
∫
d λ ρ
( 4)
( λ
k
; λ
l
; λ
m
; λ
n
) e
i( λ
k
+ λ
l
λ m λ n) t
| {z }
R 4
:
(1.37)
25
M or e o v e r , the bound i s t i gh t for shor t t ime s .
1.3 a nd 1.4 e s t a b l i sh a thr e e-w a y c onne ct ion be t w e e n C GP , O T O C s, a nd S FF; w ith the C GP a s ubp a r t of
the O T O C a nd its G UE a v e r a g e u ppe r bounde d b y the S FF . The S FF as a f unct ion of t ime h as a ch a r a ct e r -
i s t ic qual it a t iv e fe a tur e s for qua n tum ch a ot ic s ys t e m s r e s e mb l in g a slope , d ip , r a mp , a nd p l a t e a u [ 99 , 103 ].
A s a f utur e w ork , it w ould be in t e r e s t in g t o s e e w he the r the C GP — w hich i s c onne ct e d t o the S FF v i a 1.4
— ca n ca ptur e simi l a r fe a tur e s, a nd in tur n be us e d t o de t e ct as s oc i a t e d qua n tum si gn a tur e s of ch a os .
I n a simi l a r sp ir it t o the R MT a v e r a g e a bo v e , one ca n tr e a t the t ime e v o lut ion unit a r y U its e l f as a r a ndom
v a r i a b le . Thi s al lo w s us t o a ddr e s s a n impor t a n t que s t ion : H o w w e l l ca n ch a ot ic dy n a mic s be a ppr o x im a t e d
b y r a ndom unit a r ie s? The pur s uit of thi s que s t ion h as r e v e ale d m a n y p h ysical in si gh ts in t o the n a tur e of
s tr on gly -in t e r a ct in g s ys t e m s, f r om c onde n s e d m a tt e r s ys t e m s t o b l a ck ho le s a nd h as in sp ir e d a m ult itude
of qua n t it a t iv e c onne ct ion s be t w e e n ch a os a nd r a ndom unit a r ie s; s e e for ex a mp le R ef s . [ 99 , 103 , 104 ]. T o
e s t a b l i sh simi l a r c onne ct ion s, w e no w c omput e the H a a r a v e r a g e of the O T O C - C GP r e l a t ion usin g 1.3 .
P r opositio n 1.5
The H a a r -a v e r a g e d O T O C i s g iv e n b y
⟨ C
V; W
⟩
U Haar
=
2( d 1)
( d+ 1)
+
2
d
2
( d
2
1)
Re
8
<
:
∑
j̸= l a nd k̸= m
v
j
w
k
v
l
w
m
9
=
;
(1.38)
2
d( d+ 1)
Re
8
<
:
∑
j̸= l
v
j
v
l
+
∑
k̸= m
w
k
w
m
9
=
;
; (1.39)
w he r e U Haar r e pr e s e n ts H a a r -a v e r a g in g o v e r the t ime- e v o lut ion unit a r y .
The fir s t t e r m in thi s ex pr e s sion i s o bt aine d f r om the H a a r -a v e r a g e of the 2 - C GP , w hi le the othe r t w o
t e r m s or i g in a t e f r om the off - d i a g on al c on tr i but ion . W e br iefly r e m a rk th a t sinc e the f unct ion C
V; W
( t) i s
L ips chitz c on t in uous, usin g t oo l s f r om me as ur e c onc e n tr a t ion a nd L e v y ’ s le mm a [ 105 ], w e h a v e th a t the
pr o b a b i l it y of a r a ndom in s t a nc e of C
V; W
( t) de v i a t in g f r om its H a a r a v e r a g e ⟨ C
V; W
⟩
U Haar
i s ex pone n t i al ly
26
s u ppr e s s e d . Th a t i s, the H a a r -a v e r a g e i s r e pr e s e n t a t iv e of a l m o s t a ll in s t a nc e s of the O T O C . F ur the r mor e ,
simi l a r t o the d i s c us sion fo l lo w in g 1.3 , th a t i s, usin g the r e s ult of E qua t ion 1.26 , w e not e th a t a v e r a g in g
o v e r c omm ut in g unit a r ie s w ith their p h as e s d i s tr i but e d uni for mly on [ 0; 2 π) , the ex tr a t e r m s v a ni sh for the
a v e r a g e d O T O C s . M or e o v e r , for g e ne r ic ope r a t or s V a nd W , the m ain c on tr i but ion c ome s f r om the H a a r -
a v e r a g e of the C GP , w hich g e ts ex pone n t i al ly clos e t o 2 in the d ime n sion ( i f d s cale s as 2
n
for n qub its ).
The r efor e , the t y p ical O T O C for H a a r - r a ndom e v o lut ion s i s ex pone n t i al ly w e l l -a ppr o x im a t e d b y the C GP
v a lue .
Sh o r t-t i m e g r o w t h.— T o f ur the r e s t a b l i sh dy n a mical fe a tur e s of the C GP , w e foc us on its shor t -t ime be-
h a v i or . W hi le the O T O C’ s shor t -t ime gr o w th h as be e n us e d as a d i a gnos t ic of ch a os for s ys t e m s w ith a
s e micl as sical or l a r g e- N l imits, its be h a v ior for g e ne r al m a n y - body s ys t e m s w ith local in t e r a ct ion s a nd fi -
nit e de gr e e s of f r e e dom ca n simp ly be unde r s t ood v i a L ie b - R o b in s on bound s [ 21 , 106 – 108 ] a nd doe s not
ne c e s s a r i ly ch a r a ct e r i z e ch a os [ 19 – 25 , 109 – 112 ]. T o pr o v ide infor m a t ion-the or e t ic me a nin g t o a s ubp a r t
of the O T O C ( th a t i s, the C GP ), w e c onne ct it t o the not ion of qua n tum fluctua t ion s a nd inc omp a t i b i l -
it y . I nc omp a t i b i l it y of o bs e r v a b le s in qua n tum the or y i s pe rh a ps mos t c ommonly unde r s t ood in t e r m s of
a non- v a ni shin g c omm ut a t or ( for ex a mp le , the ca nonical [^ x;^ p] c omm ut a t or ) a nd the r e l a t e d H ei s e nbe r g
unc e r t ain t y r e l a t ion s . I n r e c e n t y e a r s, ho w e v e r , e n tr op ic unc e r t ain t y r e l a t ion s h a v e e me r g e d as a g e ne r al -
i z e d a nd mor e r o bus t w a y t o qua n t i f y the inc omp a t i b i l it y of o bs e r v a b le s [ 113 ]. I n R ef . [ 27 ], the a uthor s
in tr oduc e d a for m al i s m th a t e nc omp as s e s both a nd qua n t i fie d the not ion of inc omp a t i b i l it y be t w e e n b as e s
B
0
a ndB
1
( a nd not jus t o bs e r v a b le s ). A mon g m a n y in t e r e s t in g c onne ct ion s, it w as sho w n ho w thi s inc om-
p a t i b i l it y m a ni fe s ts its e l f as the c o he r e nc e of s t a t e s j ψ⟩ 2 B
0
w he n ex pr e s s e d as a l ine a r c omb in a t ion of
e le me n ts f r om B
1
. M or e o v e r , usin g t oo l s f r om m a tr i x m a j or i za t ion, a p a r t i al or de r on b as e s w as un v ei le d ,
w ith the or de r qua n t i f y in g inc omp a t i b i l it y . I n p a r t ic ul a r , the C GP w as e s t a b l i she d as a me as ur e of inc om-
p a t i b i l it y be t w e e n d i ffe r e n t b as e s a nd its c onne ct ion t o e n tr op ic unc e r t ain t y r e l a t ion s w as d i s c us s e d . I n the
the or e m be lo w , w e find th a t the shor t -t ime gr o w th of the C GP ca ptur e s inc omp a t i b i l it y be t w e e n the b asi s
B in w hich w e me as ur e c o he r e nc e a nd the b asi s of the H a mi lt oni a n B
H
.
27
P r opositio n 1.6
The shor t -t ime gr o w th of the C GP i s c onne ct e d t o the v a r i a nc e of the H a mi lt oni a n as
1
2
d
2
C
B
(U
t
)
dt
2
t= 0
=
1
d
d
∑
j= 1
var
j
( H); (1.40)
w he r e v a r
j
( H)⟨ H
2
⟩
Π j
⟨ H⟩
2
Π j
i s the v a r i a nc e of the H a mi lt oni a n in the b asi s s t a t e Π
j
. M or e o v e r ,
the fo l lo w in g bound s ho ld :
1
d
d
∑
j= 1
var
j
( H)
∥ H∥
2
2
d
1 X
T
(B;B
H
) X(B;B
H
)
1
∥ H∥
2
1
q(B;B
H
)
(1.41)
w he r e , [ X(B;B
H
)]
j; k
Tr
(
Π
j
P
k
)
a nd q(B
H
;B
0
)∥ 1 X
T
(B;B
H
) X(B;B
H
)∥
1
.
T o unde r s t a nd the u ppe r bound ,
1
2
d
2
C
B
(U t)
dt
2
t= 0
∥ H∥
2
1
q(B;B
H
) , fir s t not e th a t the m a tr i x X(B;B
H
)
i s b i s t och as t ic . A nd , s o i s X
T
(B;B
H
) X(B;B
H
) , usin g the fa ct th a t the s e t of b i s t och as t ic m a tr ic e s i s clos e d
unde r tr a n sposit ion a nd m ult ip l ica t ion [ 51 ]. U sin g thi s, it i s e as y t o s e e th a t q(B;B
H
) 1 , the r efor e ,
w e h a v e the fo l lo w in g bound
1
∥ H a∥
2
1
1
2
d
2
C
B
(U t)
dt
2
t= 0
1 . N ot e th a t thi s qua n t it y al s o pr o v ide s a p h ysical ly
me a nin g f ul nor m al i za t ion on the shor t -t ime gr o w th of the C GP : w he n c omp a r in g the t ime s cale s g e ne r a t e d
b y H a mi lt oni a n dy n a mic s, U
t
= e
iHt
, one ca n incr e as e/de cr e as e the as s oc i a t e d t ime s cale s b y s cal in g the
H a mi lt oni a n H7! α H . T o fi x thi s a r b itr a r ine s s, w he n c omp a r in g t w o d i ffe r e n t dy n a mic s, it m ak e s s e n s e t o
nor m al i z e the nor m of v a r ious H a mi lt oni a n s, w hich, in thi s cas e h a ppe n s n a tur al ly v i a the ope r a t or nor m .
T o f ur the r e luc id a t e the the or e m a bo v e a nd the as s oc i a t e d bound s, w e in tr oduc e a fa mi ly of c omm ut in g
k -local H a mi lt oni a n s of the for m,
H
( k)
:=
L( k 1)
∑
j= 1
(
σ
x
j
σ
x
j+ 1
σ
x
j+( k 1)
)
: (1.42)
R e cal l th a t a H a mi lt oni a n i s cal le d k -local ( k L ) i f it ca n be w r itt e n as a s um o v e r t e r m s w hich a ct on a t
28
m o s t k s ubs ys t e m s [ 114 ]. L e tB be the c omput a t ion al b asi s ( th a t i s, the local σ
z
b asi s ), the n, w e pr o v e the
fo l lo w in g ,
1
∥ H
( k)
∥
2
1
1
2
d
2
C
B
(U
t
)
dt
2
t= 0
=
1
L( k 1)
: (1.43)
W e pr o v ide a br ief sk e t ch of the pr oof in s e ct ion 1.9. F or k = 1 thi s g e ne r a t e s a 1 -local H a mi lt oni a n, th a t
i s, c ompos e d of pur e ly local in t e r a ct ion s, H
( 1)
=
L
∑
j= 1
σ
x
j
, w hich doe s not g e ne r a t e e n t a n gle me n t or c or r e l a -
t ion s . A nd , for k = L , w e h a v e , a hi ghly nonlocal H a mi lt oni a n, H
( L)
=
L
j= 1
σ
x
j
, w hich ca n g e ne r a t e a n L -
qub it Gr e e nbe r g e r – H or ne–Z ei l in g e r ( GH Z ) s t a t e s t a r t in g f r om pr oduct s t a t e s ¹⁶ [ 3 , 115 ]. U sin g the g e n-
e r al r e s ult a bo v e , w e not ic e th a t i f k = O( 1) , the n, the nor m al i z e d shor t -t ime gr o w th,
1
∥ H
( k= O( 1))
∥
2
1
1
2
d
2
C
B
(U t)
dt
2
t= 0
O( 1= L) t o le a d in g or de r , w hi le it i s s a tur a t e d for the nonlocal H a mi lt oni a n
1
∥ H
( k= L)
∥
2
1
1
2
d
2
C
B
(U t)
dt
2
t= 0
= 1 .
F in al ly , w e not e th a t the v a r i a nc e of the H a mi lt oni a n th a t sho w s u p in the the or e m a bo v e i s in t im a t e ly r e-
l a t e d t o ( i ) qua n tum spe e d l imits a nd the r e s our c e the or y of as y mme tr y ( s e e [ 116 , 117 ] a nd the r efe r e nc e s
the r ein ) a nd ( i i ) the “ s tr e n g th f unct ion, ” w ide ly us e d in qua n tum ch a os l it e r a tur e ( s e e S e c . 3 of R ef . [ 47 ]
for mor e de t ai l s ). I t w ould be a n in t e r e s t in g f utur e d ir e ct ion t o qua n t it a t iv e ly e s t a b l i sh the s e c onne ct ion s
f ur the r .
3 Qu a nti f y i n g c h a os with r ec u r r en c e s: n umer i c a l sim u l a ti o n s
O T O C s ca ptur e the s cr a mb l in g of qua n tum infor m a t ion . A s local i z e d infor m a t ion spr e a d s thr ou gh the
nonlocal de gr e e s of f r e e dom of a s ys t e m, it be c ome s in a c c e s si b le t o local o bs e r v a b le s a nd their ex pe ct a -
t ion v alue s r e v e al a n e qui l i br a t ion of the s ubs ys t e m s t a t e . Thi s a pp a r e n t ir r e v e r si b le los s of infor m a t ion
unde r unit a r y dy n a mic s (w hich i s r e v e r si b le ) h as be e n t e r me d s cr a mb l in g. S i gn a tur e s of s cr a mb l in g ca n be
o bs e r v e d in the lon g -t ime a v e r a g e s of both simp le p h ysical qua n t it ie s l i k e local ex pe ct a t ion v alue s a nd in
“ c omp lex ” qua n t it ie s s uch as the O T O C a nd C GP . H o w e v e r , in finit e s ys t e m s, s uch lon g -t ime a v e r a g e s do
not c on v e r g e in the l imit t!1 , in s t e a d they t y p ical ly os c i l l a t e a r ound s ome e qui l i br ium v alue . Thi s e qui -
¹⁶ Thi s fo l lo w s imme d i a t e ly b y ex p a nd in g exp[ iH
b
t] = cos( t)I i sin( t) H
b
, le tt in g t = π= 4 , a nd choosin g the init i al s t a t e
t o bej 0⟩
n
, usin g w hich, w e g e t , j ψ( t = π= 4)⟩ =
1
p
2
(j 0⟩
n
ij 1⟩
n
) .
29
1 10 100 1000
10
-5
10
-4
0.001
0.010
0.100
(a)
2 3 4 5 6 7 8
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
(b)
Figure 1.5.1: (a) Log-Log plot of the va riance of CGP and OTOC fo r n = 9 qubits. W e study the
dynamics of the Hamiltonian given b y eq. ( 1.44 ) wi th g = 1; h = 0 as the integrable limit and g =
1: 05; h = 0: 5 as the chaotic one. W e set, V = σ
z
1
; W = σ
z
9
fo r the OTOC and CGP in eq. ( 1.20 ).
(b) F raction of the long-time average of the va riance of chaotic and integrable OTOC, CGP , that is,
Var
integrable
Var
chaos
Var
integrable
fo r the Hamiltonian given b y eq. ( 1.44 ) with g = 1; h = 0 as the integrable limit and
g = 1: 05; h = 0: 5 as the chaotic one. W e set, V = σ
z
1
; W = σ
z
9
fo r the OTOC and CGP in eq. ( 1.20 ).
l i br ium v alue ca n be o bt aine d f r om the infinit e-t ime a v e r a g e , A := lim
T!1
1
T
T
∫
0
A( t) dt . I n R ef . [ 118 ], the
infinit e-t ime a v e r a g e of the a v e r a g e d O T O C (w ith a b ip a r t it ion in the s ys t e m H i l be r t sp a c e ) w as s tud ie d for
both in t e gr a b le a nd ch a ot ic mode l s a nd its e qui l i br a t ion v alue w as us e d t o s uc c e s sf ul ly d i s t in g ui sh the t w o
p h as e s; s e e al s o r e l a t e d w ork s tudy in g the lon g -t ime l imit of O T O C s for the in t e gr a b i l it y -t o - ch a os tr a n si -
t ion [ 81 , 82 ]. A lon g the w a y , c onne ct ion s w ith e n tr op y pr oduct ion, ope r a t or e n t a n gle me n t , a nd c h a nne l
d i s t in g ui sh a b i l it y w e r e al s o d i s c us s e d .
I t w as pr e v iously sho w n th a t in the lon g -t ime l imit , the s tr e n g th of r e c ur r e nc e s ca n d i s t in g ui sh ch a ot ic
a nd in t e gr a b le s ys t e m s [ 119 , 120 ]. L e t n be the n umbe r of qub its ( or mor e g e ne r al ly , the s ys t e m si z e ), the n
in t e gr a b le s ys t e m s t y p ical ly h a v e a qua n tum r e c ur r e nc e t ime th a t i s a po ly nomi al in n , w hi le ch a ot ic s ys t e m s
t y p ical ly h a v e r e c ur r e nc e t ime s th a t a r e doub ly ex pone n t i al in n , th a t i s, O( e
e
n
) . The r efor e , w he n s tudy in g
r e c ur r e nc e s in the ex pe ct a t ion v alue s of o bs e r v a b le s for a finit e ( but l a r g e ) t ime , one ex pe cts in t e gr a b le
s ys t e m s t o sho w l a r g e r r e c ur r e nc e s th a n ch a ot ic s ys t e m s . B ui ld in g on the w ork of R ef s . [ 99 , 104 ], w e sho w
th a t b y c on side r in g the O T O C a nd the C GP as “ c omp lex o bs e r v a b le s ” a nd qua n t i f y in g their r e c ur r e nc e s
v i a their t e mpor al v a r i a nc e , one ca n d i s t in g ui sh in t e gr a b le a nd ch a ot ic r e g ime s . W e al s o a r g ue th a t for the
pur pos e s of d i s t in g ui shin g the s e t w o p h as e s v i a the s tr e n g th of their r e c ur r e nc e s, the O T O C a nd C GP
ca ptur e effe ct iv e ly the s a me be h a v ior , v ind ica t in g our 1.3 .
30
The p h ysical s ys t e m w e us e t o s tudy thi s t e mpor al v a r i a nc e i s the p a r a d i gm a t ic tr a n s v e r s e- fie ld I sin g
mode l w ith ope n bound a r y c ond it ion s,
H
TFIM
=
0
@
L 1
∑
j= 1
σ
z
j
σ
z
j+ 1
+
L
∑
j= 1
g σ
x
j
+ h σ
z
j
1
A
: (1.44)
The s ys t e m h as a n in t e gr a b le l imit for h = 0 , w he r e the H a mi lt oni a n ca n be m a ppe d on t o f r e e fe r mion s;
w e s e t g = 1; h = 0 as the in t e gr a b le po in t. The s ys t e m i s qua n tum ch a ot ic for the p a r a me t e r cho ic e s g =
1: 05; h = 0: 5 w hich ca n be s e e n, for ex a mp le , b y s tudy in g the le v e l sp a c in g d i s tr i but ion . I n R ef . [ 104 ],
the O T O C a v e r a g e d o v e r local o bs e r v a b le s w as us e d t o d i s t in g ui sh the t w o p h as e s a nd it w as o bs e r v e d
th a t in the ch a ot ic l imit , the s ys t e m quick ly as y mpt ot e s t o jus t be lo w the H a a r -a v e r a g e d v alue , w hi le in the
in t e gr a b le r e g ime , the s ys t e m s d i sp l a ys l a r g e r e c ur r e nc e s a nd doe s not sho w a n y fe a tur e s of s cr a mb l in g. A
simi l a r be h a v ior w as o bs e r v e d for the m utual infor m a t ion be t w e e n d i ffe r e n t s ubs ys t e m s . H e r e , w e c omp a r e
the dy n a mical be h a v ior of the O T O C a nd the C GP for V = σ
z
1
; W = σ
z
L
for a n L - s it e s ys t e m . N ot ic e
th a t our n ume r ical sim ul a t ion s us e ex a ct dy n a mic s but a r e l imit e d t o t ime s cale s fa r be lo w the ex pe ct e d
r e c u r r e nc e t ime for the ch a ot ic l imit. H o w e v e r , w e a r e a b le t o o bs e r v e a nd qua n t i f y r e c ur r e nc e s for the
in t e gr a b le c as e in a w a y th a t i s s u ffic ie n t t o d i s t in g ui sh the t w o p h as e s .
F or s ys t e m s s a t i sf y in g the E TH A n s a tz [ 11 – 13 ], fluctua t ion s a r ound the lon g -t ime a v e r a g e s of ex pe c -
t a t ion v alue s of o bs e r v a b le s w i l l be ex pone n t i al ly s m al l in the s ys t e m si z e [ 46 , 47 ]. W hi le the C GP a nd
O T O C a r e “ c omp lex ” qua n t it ie s, their be h a v ior ca n be ex pe ct e d t o r e s e mb le th a t of simp le r o bs e r v a b le s,
e spe c i al ly for finit e s ys t e m s a nd simp le local ope r a t or s s uch as P a ul i m a tr ic e s . S inc e qua n tum ch a ot ic s ys -
t e m s t y p ical ly o bey the E TH A n s a tz ( a ft e r r e mo v in g tr iv i al s y mme tr ie s ), the fluctua t ion s in the O T O C
a nd C GP a r ound their lon g -t ime a v e r a g e m a y be ex pe ct e d t o be c ome ex pone n t i al ly s m al l in the s ys t e m
si z e . Our n ume r ical find in gs s umm a r i z e d in fi g. 1.5.1 v ind ica t e thi s in tuit ion : w e c on side r the lon g -t ime
a v e r a g e of the O T O C a nd the C GP in the in t e gr a b le a nd ch a ot ic r e g ime s . I n the ch a ot ic r e g ime the v a r i -
a nc e of the C GP a nd the O T O C a r e e qual u p t o n ume r ical e r r or ( 10
10
in d ime n sionle s s units ), w hi le
in the in t e gr a b le r e g ime the v a r i a nc e s e e m s t o as y mpt ot e t o d i ffe r e n t v alue s for the C GP a nd the O T O C
– w hich i s simp ly a c on s e que nc e of the d i ffe r e n t t ime s cale s of r e c ur r e nc e s in the s e t w o qua n t it ie s . A mor e
31
me a nin g f ul c omp a r i s on ca n be o bt aine d b y c omput in g the r e l a t i v e fluctua t ion s in the in t e gr a b le a nd ch a ot ic
r e g ime s, for w hich w e c omput e the r a t io
Var
integrable
Var
chaos
Var
integrable
;
w he r e Var
integrable
i s the lon g -t ime a v e r a g e of the t e mpor al v a r i a nc e of the C GP/O T O C in the in t e gr a b le
r e g ime , pe r for me d n ume r ical ly . W e find th a t for both the O T O C a nd C GP , thi s qua n t it y be c ome s ex po -
ne n t i al ly clos e t o one as a f unct ion of the s ys t e m si z e . The r efor e , the fluctua t ion s a r ound the a v e r a g e in the
ch a ot ic r e g ime a r e ex pone n t i al ly s m al le r th a n th a t in the in t e gr a b le cas e , as ex pe ct e d , a nd both the O T O C
a nd its s u bp a r t , the C GP ca n d i a gnos e ch a ot ic it y in thi s w a y .
1.6 D iscu ssion
W hi le the r o le of qua n tum e n t a n gle me n t in ch a r a ct e r i z in g qua n tum ch a os h as be e n w ide ly ex p lor e d , it
r e m aine d uncle a r w h a t pr e c i s e r o le qua n tum c o he r e nc e p l a ys, i f a n y , in d i a gnosin g qua n tum ch a os . Our
w ork a ffir m a t iv e ly a n s w e r s thi s que s t ion b y e s t a b l i shin g r i g or ous c onne ct ion s be t w e e n me as ur e s of qua n-
tum c o he r e nc e a nd si gn a tur e s of qua n tum ch a os . C o he r e nc e of H a mi lt oni a n ei g e n s t a t e s i s sho w n t o be a n
“ or de r p a r a me t e r ” for the in t e gr a b le-t o - ch a ot ic tr a n sit ion a nd w e n ume r ical ly de mon s tr a t e thi s b y s tudy in g
qua n tum ch a os in a n X X Z sp in- ch ain w ith defe ct a nd find exc e l le n t a gr e e me n t w ith r a ndom m a tr i x the-
or y ( R MT ) in the bul k of the spe ctr um, as ex pe ct e d . F ur the r mor e , usin g the m a the m a t ical for m al i s m of
m a j or i za t ion the or y a nd f und a me n t al r e s ults f r om the r e s our c e the or y of c o he r e nc e , w e a r g ue w h y ev er y
qua n tum c o he r e nc e me as ur e i s a “ de local i za t ion ” me as ur e — a cl as s of si gn a tur e s of qua n tum ch a os th a t
qua n t i f y spr e a d , in s a y , the posit ion ei g e nb asi s, e ne r g y ei g e nb asi s, a nd othe r s . M or e o v e r , our 1.2 sho w s th a t
for pur e s t a t e s in a b ip a r t it e s ys t e m, the 2 - c o he r e nc e minimi z e d o v e r pr oduct b as e s i s e qual t o the l ine a r e n-
tr op y of the r e duc e d s t a t e . Th a t i s, qua n tum c o he r e nc e me as ur e s ca n be us e d t o de t e ct the e n t a n gle me n t
in a qua n tum s t a t e , as h as al s o be e n de mon s tr a t e d pr e v iously [ 53 ].
F or dy n a mical si gn a tur e s of ch a os, our 1.3 e s t a b l i she s the c o he r e nc e- g e ne r a t in g po w e r ( C GP ) as a s u b -
p a r t of the O T O C , a pr omine n t me as ur e of infor m a t ion s cr a mb l in g in qua n tum s ys t e m s . I n p a r t ic ul a r , the
32
( as s oc i a t e d ) s qua r e d - c omm ut a t or ’ s gr o w th si gn al s the incr e asin g inc omp a t i b i l it y of the ope r a t or s unde r
t ime- e v o lut ion . Our the or e m p a v e s a w a y t o m ak e thi s in tuit ion pr e c i s e as the C GP qua n t i fie s inc omp a t i b i l -
it y be t w e e n the b as e s as s oc i a t e d t o the t ime- e v o lv in g ope r a t or in the O T O C a nd the fi xe d one . M or e o v e r ,
w e a n aly t ical ly sho w , in m a n y d i ffe r e n t w a ys, ho w the O T O C , s uit a b ly a v e r a g e d , c onne cts w ith 2 - c o he r e nc e
of a s t a t e , the C GP of dy n a mic s, a nd the g e ome tr ic d i s t a nc e be t w e e n the M A SA s as s oc i a t e d t o the b as e s of
the ope r a t or s in the O T O C . A mon g a p le thor a of othe r r e as on s, the C GP i s p a r t ic ul a rly w e l l - s uit e d t o qua n-
t i f y thi s inc omp a t i b i l it y sinc e it al s o h a ppe n s t o be a for m al me as ur e in the r e s our c e the or y of me as ur e me n t
inc omp a t i b i l it y [ 27 ].
F ur the r mor e , usin g R MT w e pr o v ide a n u ppe r bound on the a v e r a g e C GP for G UE H a mi lt oni a n s in
t e r m s of the S pe ctr al F or m F a ct or , a w e l l - e s t a b l i she d me as ur e of qua n tum ch a os . W e al s o find a n a n aly t -
ical ex pr e s sion for the H a a r -a v e r a g e d O T O C - C GP r e l a t ion, w hich al lo w s us t o a r g ue th a t unde r c e r t ain
as s umpt ion s, the O T O C i s a ppr o x im a t e d ex pone n t i al ly -w e l l ( in the s ys t e m si z e ) b y the C GP .
The shor t -t ime be h a v ior of the O T O C h as r e c eiv e d c on side r a b le a tt e n t ion in r e c e n t y e a r s a nd s o w e
a n aly z e the shor t -t ime gr o w th of the C GP ( a s ubp a r t of the O T O C ) w hich, t o le a d in g or de r , i s ch a r a ct e r i z e d
b y the v a r i a nc e of the H a mi lt oni a n w ith r e spe ct t o a b asi s; for the O T O C thi s b asi s i s inhe r it e d f r om the
cho ic e of the O T O C ope r a t or s . W e r e m a rk th a t thi s v a r i a nc e of the H a mi lt oni a n ( for pur e s t a t e s ) i s r e l a t e d
t o qua n tum spe e d l imits a nd the r e s our c e the or y of as y mme tr y [ 116 ]. A nd fin al ly , w e n ume r ical ly s tudy
the lon g -t ime be h a v ior of the O T O C a nd C GP in a tr a n s v e r s e- fie ld I sin g mode l a nd find th a t their t e mpor al
v a r i a nc e s qua n t i f y ch a os in effe ct iv e ly the s a me w a y .
I n closin g , our r e s ults e s t a b l i sh qua n tum c o he r e nc e as a si gn a tur e of qua n tum ch a os, both a t the le v e l
of s t a t e s a nd dy n a mic s . A s a f utur e w ork , it w ould be in t e r e s t in g t o s e e ho w w e l l s uit e d me as ur e s of qua n-
tum c o he r e nc e a r e t o the s tudy fe w - body ch a os, in p a r t ic ul a r , usin g p a r a d i gm a t ic s ys t e m s l i k e the qua n tum
k ick e d t op [ 6 ]. F e w - body s ys t e m s pr o v ide a po w e r f ul ex pe r ime n t al t e s t be d for s tudy in g si gn a tur e s of the r -
m al i za t ion a nd s cr a mb l in g , w hich a r e in t im a t e ly l ink e d w ith qua n tum c o he r e nc e me as ur e s . Qua n t it a t iv e ly
e s t a b l i shin g the s e c onne ct ion s w i l l al s o be a pr omi sin g f utur e d ir e ct ion .
33
A ppendic e s
1.7 L evel sp a c ing dis trib u tion
△ △
△ △
△ △
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△ △
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△ △
△ △
△ △
△ △ △ △ △ △
△ △ △ △
△ △ △ △ △ △
△ △ △ △ △ △ △ △
△ △
△ △ △ △ △ △ △ △ △ △ △ △
△ △ △ △
△ △ △ △ △ △ △ △ △ △ △ △ △
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
(a)
△ △
△ △
△ △
△ △
△ △
△ △
△ △
△ △
△ △
△ △
△ △
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△ △
△ △ △ △
△ △ △ △
△ △
△ △
△ △ △ △ △ △
△ △ △ △ △ △ △ △
△ △ △ △ △ △
△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
(b)
△ △
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△ △ △ △
△ △
△ △
△ △ △ △ △ △
△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
(c)
△ △
△ △
△ △
△ △
△ △ △ △
△ △
△ △
△ △
△ △
△ △
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△ △
△ △
△ △
△ △ △ △
△ △
△ △
△ △ △ △
△ △ △ △ △ △ △ △ △ △
△ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △ △
0 1 2 3 4 5
0.0
0.2
0.4
0.6
0.8
1.0
△
(d)
Figure 1.7.1: The transition in level-spacing distribution from P oisson to the (universal) Wigner-
Dyson distribution fo r the Hamiltonian describ ed in eq. ( 1.13 ) as w e move the defectsite to the mid-
dle of the chain. Figures (a), (b), (c), and (d) co rresp ond to the defect at sites δ = 1; δ = 3; δ = 5;
and δ = 7 , resp ectively . Results a re rep o rted fo r L = 15 with 5 spins up and ω = 0; ε
δ
= 0: 5; J
xy
=
1; J
z
= 0: 5 . Simil a r results w ere obtained fo r L = 15 and δ = 1; 7 in Ref. [ 1 ] (but not fo r intermediate
p ositions of the defect site).
1.8 Coherenc e qu an tifier s for in teg r ab l e and c h a otic eig en s t a te s
34
-4 -2 0 2
0
200
400
600
800
1000
1200
(a)
-4 -2 0 2
0
100
200
300
400
500
(b)
Figure 1.8.1: Inverse pa rticipation ratio fo r eigenstates of the Hamiltonian defined in eq. ( 1.13 ) as a
function of their energy . Results a re rep o rted fo r L = 15 with 5 spins up and ω = 0; ε
δ
= 0: 5; J
xy
=
1; J
z
= 0: 5 . The plot ma rk ers 1; 3; 5; 7 co rresp ond to the va rious choices of the defect site, with δ = 1; 7
co rresp onding to the integrable and chaotic limits, resp ectively . Figures (a) and (b) co rresp ond to the
t w o different bases, the site-basis and the mean-field basis, resp ectively . Simila r results w ere obtained
fo r L = 18 and δ = 1; 9 in Ref. [ 1 ] (but not fo r intermediate p ositions of the defect site).
-4 -2 0 2
0
500
1000
1500
2000
(a)
-4 -2 0 2
0
200
400
600
800
1000
1200
(b)
Figure 1.8.2: 1 -coherence fo r eigenstates of the Hamiltonian defined in eq. ( 1.13 ) as a function of
their energy . Results a re rep o rted fo r L = 15 with 5 spins up and ω = 0; ε
δ
= 0: 5; J
xy
= 1; J
z
= 0: 5 . The
plot ma rk ers 1; 3; 5; 7 co rresp ond to the va rious choices of the defect site, w ith δ = 1; 7 co rresp onding
to the integrable and chaotic limits, resp ectively . Figures (a) and (b) co rresp ond to the t w o different
bases, the site-basis and the mean-field basis, resp ectively .
1.9 Pro of s
H e r e w e r e s t a t e the P r oposit ion s, The or e m s, as w e l l as othe r m a the m a t ical cl aim s a ppe a r in g in the m ain
t ex t , a nd g iv e their pr oof s .
35
Pr o o f o f 1.2
P r o of . W e s t a r t b y c o l le ct i n g a fe w simp le r e s ults . F ir s t , r e cal l th a t the 2 - c o he r e n c e i s
c
( 2)
B
( ρ) =∥ ρD
B
( ρ)∥
2
2
(1.45)
=⟨ ρD
B
( ρ); ρD
B
( ρ)⟩ =⟨ ρ; ρ⟩⟨ ρ;D
B
( ρ)⟩⟨D
B
( ρ); ρ⟩+⟨D
B
( ρ);D
B
( ρ)⟩
(1.46)
=⟨ ρ; ρ⟩⟨ ρ;D
B
( ρ)⟩; (1.47)
w he r e in the s e c ond l ine , w e h a v e us e d ⟨ ρ;D
B
( ρ)⟩ =⟨D
B
( ρ); ρ⟩ sinc eD
B
i s a s e l f -a d j o in t s u pe r ope r -
a t or a nd ⟨D
B
( ρ);D
B
( ρ)⟩ =⟨ ρ;D
B
( ρ)⟩ sinc eD
B
i s a pr oj e ct ion s u pe r ope r a t or , th a t i s, (D
B
)
2
=D
B
.
F or pur e s t a t e s, w e h a v e , ⟨ ρ; ρ⟩ = 1 a nd the r efor e , the 2 - c o he r e nc e for pur e s t a t e s i s e qual t o
c
( 2)
B
( ρ) = 1⟨ ρ;D
B
( ρ)⟩ .
S e c ond , a pur e b ip a r t it e s t a t e , j Ψ⟩
AB
2H
=H
A
H
B
ca n be w r itt e n in the S chmidt for m ( th a t
i s, usin g S chmidt d e c omposit ion the or e m ) [ 3 ],
j Ψ⟩
AB
=
minf d A; d Bg
∑
j= 1
λ
j
j j⟩
A
j
e
j⟩
B
; (1.48)
w he r efj j⟩
A
g;fj
e
j⟩
B
g i s a n or thonor m al b asi s for s ubs ys t e m s A , B , r e spe ct iv e ly , a nd f λ
j
g a r e non-
ne g a t iv e c oeffic ie n ts s a t i sf y in g
∑
j
λ
2
j
= 1 . The c oeffic ie n ts λ
2
j
a r e the ei g e n v alue s of the r e duc e d de n sit y
m a tr i x ρ
A
; r e cal l al s o th a t ρ
A
a nd ρ
B
a r e i s ospe ctr al . The n, r e- ex pr e s sin g the s t a t e in thi s for m, w e h a v e
( dr opp in g the s ubs cr ipts for the s ubs y t e m s A; B ),
j Ψ⟩⟨ Ψj =
∑
j; k
λ
j
λ
k
j j⟩⟨ kj
j
e
j⟩⟨
e
kj: (1.49)
36
A nd thir d , the de p h asin g s u pe r ope r a t or fa ct or i z e s, th a t i s,
D
B a
B
b
=D
B a
D
B
b
: (1.50)
T o s e e thi s, le t B
a
=f Π
( a)
j
g
d
j= 1
a ndB
b
=f Π
( b)
k
g
d
k= 1
, the n, the a ct ion of D
B a
B
b
i s
D
B a
B
b
( X) =
d
∑
j; k= 1
(
Π
( a)
j
Π
( b)
k
)
X
(
Π
( a)
j
Π
( b)
k
)
(1.51)
a nd the a ct ion of D
B a
D
B
b
i s,
D
B a
D
B
b
( X) =D
B a
(
d
∑
k= 1
(
I
Π
( b)
k
)
X
(
I
Π
( b)
k
)
)
(1.52)
=
d
∑
j; k= 1
(
Π
( a)
j
I
)(
I
Π
( b)
k
)
X
(
I
Π
( b)
k
)(
Π
( a)
j
I
)
(1.53)
=
d
∑
j; k= 1
(
Π
( a)
j
Π
( b)
k
)
X
(
Π
( a)
j
Π
( b)
k
)
=D
B a
B
b
( X): (1.54)
W e a r e no w r e a dy t o pr o v e the m ain r e s ult.
min
B a;B
b
c
( 2)
B a
B
b
(j Ψ⟩⟨ Ψj) = min
B a;B
b
f 1⟨j Ψ⟩⟨ Ψj;D
B a
B
b
(j Ψ⟩⟨ Ψj)⟩g = 1 max
B a;B
b
f⟨j Ψ⟩⟨ Ψj;D
B a
B
b
(j Ψ⟩⟨ Ψj)⟩g:
(1.55)
L e t us c on side r the t e r m in side the m a x imi za t ion, ⟨j Ψ⟩⟨ Ψj;D
B a
B
b
(j Ψ⟩⟨ Ψj)⟩ . W e us e the S chmidt
37
for m of j Ψ⟩ a nd s ubs t itut e D
B a
B
b
b yD
B a
D
B
b
t o g e t
⟨j Ψ⟩⟨ Ψj;D
B a
B
b
(j Ψ⟩⟨ Ψj)⟩ =
d
∑
j; k; l; m
λ
j
λ
k
λ
l
λ
m
Tr
(
j l⟩⟨ mj
j
e
l⟩⟨e mjD
B a
(j j⟩⟨ kj)
D
B
b
(
j
e
j⟩⟨
e
kj
))
(1.56)
=
d
∑
j; k; l; m
λ
j
λ
k
λ
l
λ
m
Tr(j l⟩⟨ mjD
B a
(j j⟩⟨ kj)) Tr
(
j
e
l⟩⟨e mjD
B
b
(
j
e
j⟩⟨
e
kj
))
:
(1.57)
I t i s e as y t o s e e th a t t o m a x imi z e the s e inne r pr oducts, w e ne e d t o choos e the de p h asin g b asi s t o be
the s a me as the local b asi s fj j⟩g;fj
e
j⟩g , r e spe ct iv e ly . T o s e e thi s, le t B
a
= fj φ
j
⟩⟨ φ
j
jg , the n, the t e r m
Tr(j l⟩⟨ mjD
B a
(j j⟩⟨ kj)) be c ome s,
d
∑
j= 1
⟨ φ
j
j l⟩⟨ mj φ
j
⟩⟨ φ
j
j j⟩⟨ kj φ
j
⟩; (1.58)
a n u ppe r bound on w hich ca n be o bt aine d usin g C a uch y - S ch w a r z ine qual it y r e pe a t e d ly t o s e e th a t it i s
m a x imi z e d w he n j φ
j
⟩ =j j⟩ 8 j . Th a t i s, the local b asi s in the S chmidt de c omposit i on of the s t a t e a nd
the de p h asin g b asi s a r e the s a me . The r efor e , D
B a
(j j⟩⟨ kj) =j j⟩⟨ kj δ
j; k
a ndD
B
b
(
j
e
j⟩⟨
e
kj
)
=j
e
j⟩⟨
e
kj δ
j; k
.
P lu gg in g it b a ck , w e h a v e ,
max
B a;B
b
f⟨j Ψ⟩⟨ Ψj;D
B a
B
b
(j Ψ⟩⟨ Ψj)⟩g =
d
∑
j; k; l; m
λ
j
λ
k
λ
l
λ
m
δ
jk
δ
m; j
δ
l; j
=
d
∑
j= 1
λ
4
j
=
ρ
a
2
2
: (1.59)
The r efor e , putt in g e v e r y thin g t o g e the r , w e h a v e ,
min
B a;B
b
c
( 2)
B a
B
b
(j Ψ⟩⟨ Ψj) = 1
ρ
a
2
2
=: S
l in
( ρ
a
): (1.60)
■
38
Pr o o f o f 1.3
P r o of . C on side r the infinit e-t e mpe r a tur e O T O C , C
( β= 0)
V; W
( t) =
1
d
Tr
(
[ V; W( t)]
y
[ V; W( t)]
)
. The n,
p lu gg in g in the spe ctr al de c omposit ion of V; W , th a t i s, V =
∑
j
v
j
Π
j
; W( t) =
∑
j
w
j
e
Π
j
( t) , w e h a v e
C
V; W
( t) =
1
d
∑
j; k; l; m
v
j
w
k
v
l
w
m
Tr
(
[
Π
j
;
e
Π
k
( t)
]
y
[
Π
l
;
e
Π
m
( t)
]
)
:
The n , ex tr a ct in g the j = l a nd k = m t e r m s, w e h a v e ,
C
V; W
( t) =
1
d
∑
j; k
v
j
2
j w
k
j
2
Tr
(
[
Π
j
;
e
Π
k
( t)
]
y
[
Π
j
;
e
Π
k
( t)
]
)
(1.61)
+
1
d
∑
j̸= l; k̸= m
v
j
w
k
v
l
w
m
Tr
(
[
Π
j
;
e
Π
k
( t)
]
y
[
Π
l
;
e
Π
m
( t)
]
)
: (1.62)
S inc e , V; W a r e unit a r y , w e h a v e ,
v
j
2
= 1 =
w
j
2
8 j2f 1; 2; ; dg . The r efor e ,
C
V; W
( t) =
1
d
∑
j; k
Tr
(
[
Π
j
;
e
Π
k
( t)
]
y
[
Π
j
;
e
Π
k
( t)
]
)
+
1
d
∑
j̸= l; k̸= m
v
j
w
k
v
l
w
m
Tr
(
[
Π
j
;
e
Π
k
( t)
]
y
[
Π
l
;
e
Π
m
( t)
]
)
:
(1.63)
The n, r e cal l in g e q . ( 1.19 ), w e h a v e ,
C
V; W
( t) = 2C
B V
(U
t
◦V
B V!B W
)+
1
d
∑
j̸= l; k̸= m
v
j
w
k
v
l
w
m
Tr
(
[
Π
j
;
e
Π
k
( t)
]
y
[
Π
l
;
e
Π
m
( t)
]
)
; (1.64)
w he r eV
B V!B W
i s the in t e r t w ine r c onne ct in g the b as e s B
V
t oB
W
asV
B V!B W
(
Π
j
)
=
e
Π
j
8 j2f 1; 2; ; dg .
39
N ex t , w e w ould l i k e t o simp l i f y the s e c ond t e r m of the s umm a t ion . F or thi s, not e th a t
Tr
(
[
Π
j
;
e
Π
k
( t)
]
y
[
Π
l
;
e
Π
m
( t)
]
)
= Tr
({
e
Π
k
Π
j
Π
j
e
Π
k
}{
Π
l
e
Π
m
e
Π
m
Π
l
})
(1.65)
= Tr
(
e
Π
k
Π
j
Π
l
e
Π
m
)
Tr
(
e
Π
k
Π
j
e
Π
m
Π
l
)
Tr
(
Π
j
e
Π
k
Π
l
e
Π
m
)
+ Tr
(
Π
j
e
Π
k
e
Π
m
Π
l
)
(1.66)
= δ
km
δ
jl
Tr
(
Π
j
e
Π
k
)
Tr
(
e
Π
k
Π
j
e
Π
m
Π
l
)
Tr
(
Π
j
e
Π
k
Π
l
e
Π
m
)
+ δ
jl
δ
km
Tr
(
Π
j
e
Π
k
)
(1.67)
= 2 δ
km
δ
jl
Tr
(
Π
j
e
Π
k
)
2Re
{
Tr
(
e
Π
k
Π
j
e
Π
m
Π
l
)}
: (1.68)
The s umm a t ion ind ic e s for the s e c ond t e r m a r e j ̸= l OR k ̸= m , w hich h as thr e e pos si b i l it ie s:
j ̸= l a nd k ̸= m , j ̸= l but k = m , a nd fin al ly , j = l but k ̸= m . I n e a ch cas e , the pr oduct of de lt a
f unct ion s, δ
km
δ
jl
v a ni she s a nd w e a r e left w ith the s e c ond t e r m only . The r efor e ,
C
V; W
( t) = 2C
B V
(U
t
◦V
B V!B W
)
2
d
Re
8
<
:
∑
j̸= l; k̸= m
v
j
w
k
v
l
w
m
Tr
(
e
Π
k
( t) Π
j
e
Π
m
( t) Π
l
)
9
=
;
; (1.69)
w he r e w e e mp h asi z e th a t the ind ic e s of the s umm a t ion h a v e the thr e e pos si b i l it ie s l i s t e d a bo v e .
The r e l a t ion be t w e e n F
V; W
( t) a ndC
B V
i s o bt aine d simp ly b y usin g C
V; W
( t) = 2( 1Ref F
V; W
( t)g) .
Thi s c omp le t e s the pr oof . ■
Pr o o f o f Eq u a ti o n 1.26 , Eq u a ti o n 1.28 , a nd Eq u a ti o n 1.30
L e t ’ s s t a r t w ith E qua t ion 1.30 . W e h a v e t w o unit a r ie s V; W a nd t w o b as e s B;
e
B . The n, the fo l lo w in g H a a r -
a v e r a g e d s qua r e d c omm ut a t or i s pr opor t ion al t o the ( s qua r e d ) d i s t a nc e in the Gr as s m a nni a n be t w e e n the
M A SA s as s oc i a t e d t o the b as e s B;
e
B . Th a t i s,
⟨
[
D
B
( V);D
e
B
( W)
]
2
2
⟩
V; W2 Haar
=
1
d
2
D
2
(A
B
;A
e
B
) (1.70)
40
W e s t a r t b y ex p a nd in g
[
D
B
( V);D
e
B
( W)
]
2
2
w ith v
α
:= Tr( Π
α
V); w
β
:= Tr
(
e
Π
β
W
)
. The n,
[
D
B
( V);D
e
B
( W)
]
2
2
=
d
∑
α; β
v
α
w
β
[
Π
α
;
e
Π
β
]
2
2
=
d
∑
α; β; γ; η
v
α
w
β
v
γ
w
η
Tr
(
[
Π
α
;
e
Π
β
]
y
[
Π
γ
;
e
Π
η
]
)
:
(1.71)
N o w , v
α
v
γ
= Tr
(
Π
α
V
y
)
Tr
(
Π
γ
V
)
= Tr
((
Π
α
Π
γ
)(
V
y
V
))
a nd usin g the le mm a ,
∫
Haar
dAA
y
A =
S
d
; w he r e S i s the S W A P ope r a t or , (1.72)
w e h a v e
⟨
v
α
v
γ
⟩
V2 Haar
=
∫
Haar
dV Tr
((
Π
α
Π
γ
)(
V
y
V
))
=
1
d
Tr
((
Π
α
Π
γ
)
S
)
=
1
d
Tr
(
Π
α
Π
γ
)
=
1
d
δ
α; γ
:
(1.73)
S imi l a rly , for
⟨
w
β
w
η
⟩
W2 Haar
=
1
d
δ
β; η
. P utt in g e v e r y thin g t o g e the r , w e h a v e ,
⟨
[
D
B
( V);D
e
B
( W)
]
2
2
⟩
V; W2 Haar
=
1
d
2
d
∑
α; β; γ; η
δ
α; γ
δ
β; η
Tr
(
[
Π
α
;
e
Π
β
]
y
[
Π
γ
;
e
Π
η
]
)
(1.74)
=
1
d
2
d
∑
α; β
[
Π
α
;
e
Π
β
]
2
2
=
1
d
2
D
2
(A
B
;A
e
B
): (1.75)
T o pr o v e E qua t ion 1.28 ,
⟨
∥[D
B
( V); ρ]∥
2
2
⟩
V2 Haar
=
2
d
c
( 2)
B
( ρ) , w e fo l lo w a simi l a r s e que nc e of a r g ume n ts
as a bo v e . W e pr o v e a sl i gh tly g e ne r al v e r sion of the r e s ult he r e , w he r e V i s a unit a r y a nd X a n a r b itr a r y
ope r a t or ( a nd not ne c e s s a r i ly a qua n tum s t a t e )
∥[D
B
( V); X]∥
2
2
=
d
∑
α; β
v
α
v
β
Tr
(
[ Π
α
; X]
y
[
Π
β
; X
]
)
: (1.76)
41
A s a bo v e ,
⟨
v
α
v
β
⟩
V2 Haar
=
1
d
δ
α; β
. The r efor e ,
⟨∥[D
B
( V); X]∥
2
2
⟩
V2 Haar
=
1
d
d
∑
α
∥[ Π
α
; X]∥
2
2
=
2
d
c
( 2)
B
( X): (1.77)
A nd fin al ly , t o pr o v e E qua t ion 1.26 ,⟨∥[ V; W( t)]∥
2
2
⟩
θ
= 2dC
B V
(U
t
◦V
B V!B W
) , w e pr oc e e d as a bo v e a nd
not e th a t the k ey s t e p i s
⟨
v
α
v
γ
⟩
θ
=
⟨
e
i( θ γ θ α)
⟩
θ
=
1
2 π
2 π
∫
0
e
i( θ γ θ α)
dθ = δ
α; γ
. A nd simi l a rly for
⟨
w
β
w
η
⟩
θ
=
δ
β; η
. P utt in g e v e r y thin g t o g e the r , w e the n h a v e the de sir e d r e s ult.
Pr o o f o f 1.4
P r o of . L e t
^
S be the S W A P ope r a t or define d in e q . ( 1.87 ). The n,
C
B
(
e
iHt
)
1
1
d
∑
j
[
Tr
(
P
j
UP
j
U
y
)]
2
(1.78)
= 1
1
d
∑
j
Tr
(
P
j
2
U
2
P
j
2
U
y
2
)
(1.79)
= 1
1
d
∑
j
Tr
(
^
SP
4
j
(
U
2
U
y
2
))
(1.80)
= 1
1
d
∑
j
∑
k; l; m; n
Tr
(
P
4
j
V
4
( P
k
P
l
P
m
P
n
)
)
e
i( E
k
+ E
l
E m E n) t
; (1.81)
w he r e in the fir s t ine qual it y , w e h a v e dr oppe d the off - d i a g on al t e r m s in the C GP , th a t i s, usin g C
B
( U) =
1 1= d
∑
j; k
[
Tr
(
Π
j
U Π
k
U
y
)]
2
( s e e e q . ( 1.18 )) a nd only k e e p in g the t e r m s w ith j = k . I n the s e c ond
l ine , w e h a v e simp ly r e- ex pr e s s e d the tr a c e b y usin g [ Tr( A)]
2
= Tr( A
A) . A nd , in the l as t l ine w e
h a v e p lu gg e d in U =
∑
k
e
iE
k
t
VP
k
V
y
, w he r eV() = V() V
y
i s the unit a r y in t e r t w ine r c onne ct in g the
H a mi lt oni a n ei g e nb asi s w ith the b asi s B .
I n the fo l lo w in g w e w i l l m ak e a simp le ch a n g e of not a t ion for both c on v e nie nc e a nd c on si s t e nc y
42
w ith othe r w ork s: E
j
7! λ
j
. N o w , r e cal l th a t for G UE, w e h a v e , P( H)/ exp
(
d
2
Tr( H
2
)
)
, the r efor e ,
∫
dHP( H) =
∫
d λ P( λ)
∫
dV; (1.82)
w he r e the a v e r a g e de c ompos e s in t o the ei g e n v alue s a nd the ei g e n v e ct or s . R e cal l th a t V i s H a a r - d i s tr i but e d .
The n,
P( λ) = cj Δ( λ)j
2
e
d
2
∑
j
λ
2
j
; w he r e Δ( λ)
∏
1 j< k d
(
λ
j
λ
k
)
i s the V a nde r monde m a tr i x . (1.83)
The n,
⟨
C
B
( e
iHt
)
⟩
GUE
1
1
d
∑
j
∑
k; l; m; n
(∫
d λ P( λ) e
i( λ
k
+ λ
l
λ m λ
l
) t
∫
dV Tr
(
P
4
j
V
4
( P
k
P
l
P
m
P
n
)
)
)
:
(1.84)
N ot ic e th a t i f λ
j
= λ
k
for a n y j; k the n Δ( λ) = 0 . The r efor e , in the s umm a t ion
∑
k; l; m; n
, w e only ne e d t o
c on side r λ
k
̸= λ
l
̸= λ
m
̸= λ
n
. The n, one ca n sho w th a t ,
∫
dV Tr
(
P
4
j
V
4
( P
k
P
l
P
m
P
n
)
)
=
1
d( d+ 1)( d+ 2)( d+ 3)
; s e e R ef . [ 121 ] for in t e gr al s of thi s for m . The r efor e ,
⟨
C
B
(
e
iHt
)⟩
GUE
1
1
d( d+ 1)( d+ 2)( d+ 3)
∑
k; l; m; n
∫
e
i( λ
k
+ λ
l
λ m λ n) t
P( λ) d λ
| {z }
R 4
: (1.85)
T o s e e th a t the bound i s t i gh t for shor t t ime s, not ic e th a t in or de r t o e s t a b l i sh the c onne ct ion t o the
spe ctr al for m fa ct or , the fir s t s t e p in the pr oof i s the ine qual it y , C
B
( e
iHt
) 1
1
d
∑
j
[
Tr
(
P
j
UP
j
U
y
)]
2
w hich i s o bt aine d b y i gnor in g the off - d i a g on al t e r m s in the C GP , C
B
( U) = 1 1= d
∑
j; k
[
Tr
(
Π
j
U Π
k
U
y
)]
2
(w e dr op the j ̸= k t e r m s ). N o w , for shor t t ime s, le t t = O( ε) , the n, the c on tr i but ion f r om the off -
d i a g on al t e r m s s cale s as O( ε
4
)≪ 1 , m ak in g the bound t i gh t. ■
Pr o o f o f 1.5
43
P r o of . F ir s t , not e th a t usin g 1.3 , w e ca n H a a r -a v e r a g e the C GP a nd the “ off - d i a g on al ” t e r m s inde pe n-
de n tly . F o l lo w in g R ef s . [ 43 , 88 ], w e h a v e th a t
⟨C
B
(U)⟩
Haar
=
( d 1)
( d+ 1)
: (1.86)
N o w , for the “ off - d i a g on al ” t e r m, le t us loo k a t t e r m s of the for m Tr
(
e
Π
k
( t) Π
j
e
Π
m
( t) Π
l
)
. L e tH =
H
A
H
A
′ , w he r eH
A
= H
A
′ , th a t i s, w e t ak e t w o c op ie s of the H i l be r t sp a c e . The S W A P ope r a t or
a ct in g on thi s doub le d sp a c e i s define d as,
^
S =
∑
i; j
j i⟩
A
⟨ jj
j j⟩
A
′⟨ ij: (1.87)
I t i s e as y t o sho w th a t Tr( XY) = Tr
(
^
SX
Y
)
, w hich w e us e in the fo l lo w in g ( a n d v a r i a n ts the r e of ).
The n,
Tr
(
e
Π
k
( t) Π
j
e
Π
m
( t) Π
l
)
= Tr
(
Π
l
e
Π
k
( t)
Π
j
e
Π
m
( t)
^
S
)
(1.88)
= Tr
((
Π
l
Π
j
)(
e
Π
k
( t)
e
Π
m
( t)
)
^
S
)
= Tr
((
Π
l
Π
j
)
U
2
t
(
e
Π
k
e
Π
m
)
^
S
)
: (1.89)
The n, t o H a a r -a v e r a g e the a bo v e t e r m, w e c o l le ct a fe w r e s ults,
⟨
U
2
( X)
⟩
Haar
=
1
2
(
I+
^
S
)
d( d+ 1)
Tr
((
I+
^
S
)
X
)
+
1
2
(
I
^
S
)
d( d 1)
Tr
((
I
^
S
)
X
)
: (1.90)
N o w , t ak in g X =
e
Π
k
e
Π
m
, w e h a v e ,
Tr
((
I
^
S
)
e
Π
k
e
Π
m
)
= 1 δ
km
: (1.91)
44
The n,
⟨
Tr
(
e
Π
k
( t) Π
j
e
Π
m
( t) Π
l
)⟩
Haar
(1.92)
= Tr
((
Π
l
Π
j
)⟨
U
2
t
(
e
Π
k
e
Π
m
)⟩
Haar
^
S
)
: (1.93)
U sin g ,
(
I
^
S
)
^
S =
(
^
S I
)
, w e h a v e , Tr
((
Π
l
Π
j
)(
^
SI
))
= δ
lj
1 .
P utt in g e v e r y thin g t o g e the r , a nd r e cal l in g th a t the “ off - d i a g on al ” t e r m h as the for m,
∑
j̸= l; k̸= m
v
j
w
k
v
l
w
m
Tr
(
e
Π
k
( t) Π
j
e
Π
m
( t) Π
l
)
;
w he r e , w e r e cal l th a t the ind ic e s h a v e the for m j̸= l OR k̸= m .
The n, for d i ffe r e n t cho ic e s of ind ic e s, w e h a v e ,
F or j̸= l a nd k̸= m :
1
2d( d+ 1)
1
2d( d 1)
=
1
d( d
2
1)
; (1.94)
F or j̸= l a nd k = m :
1
d( d+ 1)
; (1.95)
F or j = l a nd k̸= m :
1
d( d+ 1)
: (1.96)
C omb inin g w ith the p h as e s, w e h a v e ,
2
d( d
2
1)
Re
8
<
:
∑
j̸= l a nd k̸= m
v
j
w
k
v
l
w
m
9
=
;
2
d( d+ 1)
Re
8
<
:
∑
j̸= l; k= m
j w
k
j
2
v
j
v
l
9
=
;
2
d( d+ 1)
Re
8
<
:
∑
j= l; k̸= m
j v
k
j
2
w
k
w
m
9
=
;
(1.97)
The n, sinc e V; W a r e unit a r ie s, j w
k
j
2
= 1 =j v
k
j
2
8 k2f 1; 2; ; dg . The r efor e , the a bo v e t e r m
be c ome s,
2
d( d
2
1)
Re
8
<
:
∑
j̸= l a nd k̸= m
v
j
w
k
v
l
w
m
9
=
;
2
d( d+ 1)
Re
8
<
:
∑
j̸= l
v
j
v
l
+
∑
k̸= m
w
k
w
m
9
=
;
(1.98)
45
C o l le ct in g the C GP a nd “ off - d i a g on al ” t e r m s t o g e the r , w e h a v e the de sir e d r e s ult.
■
Pr o o f o f 1.6
P r o of . U sin g P r oposit io n 1 of R ef . [ 43 ], w e h a v e ,
C
B
(U
t
) = 1
1
d
∑
j; k
Tr
(
Π
j
Π
k
( t) Π
j
Π
k
( t)
)
; w he r e Π
j
( t)U
t
( Π
j
); (1.99)
= 1
1
d
∑
j; k
Tr
((
Π
j
Π
j
)
U
2
t
( Π
k
Π
k
)
^
S
)
; (1.100)
w he r e
^
S i s the S W A P ope r a t or on the doub le d H i l be r t sp a c e define d in e q . ( 1.87 ).
N o w , r e cal l th a t the t ime e v o lut ion s u pe r ope r a t or ca n be ex p a nde d a t shor t t ime s as,
U
t
I iH t
1
2
H
2
t
2
+ (1.101)
w he r eI i s the I de n t it y s u pe r ope r a t or a nd H( X) [ H; X] . The r efor e ,
U
2
t
I
I it(H
I +I
H)
t
2
2
(H
I +I
H)
2
+ (1.102)
L e t us c on side r the v a r ious t e r m s in the shor t -t ime ex p a n sion of the doub le d e v o lut ion .
Z e r oth or de r :
Tr
((
Π
j
Π
j
)
I
2
( Π
k
Π
k
)
^
S
)
= Tr
(
Π
j
Π
k
Π
j
Π
k
^
S
)
= δ
jk
δ
jk
: (1.103)
=)
1
d
∑
j; k
Tr() = 1: (1.104)
The r efor e , the z e r oth or de r t e r m i s one .
46
F ir s t or de r :
Tr
(
Π
2
j
(H
I +I
H) Π
2
k
^
S
)
(1.105)
= Tr
(
Π
2
j
(H
I) Π
2
k
^
S
)
+ Tr
(
Π
2
j
(I
H) Π
2
k
^
S
)
(1.106)
L e t us c on side r th e fir s t t e r m in the s umm a t ion : = Tr
(
Π
j
H( Π
k
)
Π
j
Π
k
^
S
)
(1.107)
= Tr
(
Π
j
H( Π
k
) Π
j
Π
k
)
= δ
jk
Tr
(
Π
j
H( Π
k
)
)
= Tr
(
Π
j
H( Π
j
)
)
= 0: (1.108)
The s a me ho ld s for the s e c ond t e r m in the s umm a t ion a bo v e . The r efor e , the l ine a r t e r m i s z e r o .
S e c ond or de r :
Tr
(
Π
2
j
(H
I +I
H)
2
Π
2
k
^
S
)
= Tr
(
Π
2
j
(H
2
I +I
H
2
+ 2H
H) Π
2
k
^
S
)
(1.109)
= 2 Tr
(
Π
2
j
(H
I +H
H) Π
2
k
^
S
)
; (1.110)
w he r e , the l as t e qual it y fo l lo w s f r om a simp le s y mme tr y a r g ume n t.
N ot e th a t H
2
I ( X
Y) = [ H;[ H; X]]
Y a ndH
H( X
Y) = [ H; X]
[ H; Y] . The r efor e ,
H
2
I( Π
k
Π
k
) = [ H;[ H; Π
k
]]
Π
k
=f H( H Π
k
Π
k
H)( H Π
k
Π
k
H) Hg
Π
k
(1.111)
= ( H
2
Π
k
2H Π
k
H+ Π
k
H
2
)
Π
k
: (1.112)
47
P lu gg in g thi s b a ck in t o the tr a c e , w e h a v e ,
Tr
(
Π
2
j
(H
2
I) Π
2
k
^
S
)
(1.113)
= Tr
(
Π
2
j
( H
2
Π
k
2H Π
k
H+ Π
k
H
2
)
Π
k
^
S
)
(1.114)
= δ
jk
Tr
(
Π
j
( H
2
Π
k
2H Π
k
H+ Π
k
H
2
)
)
(1.115)
= Tr
(
Π
j
( H
2
Π
k
2H Π
k
H+ Π
k
H
2
)
)
(1.116)
= 2
(
Tr
(
Π
j
H
2
Π
j
)
(
Tr
(
Π
j
H
))
2
)
(1.117)
= 2var
j
( H); w he r e var
j
( H)⟨ H
2
⟩
Π j
⟨ H⟩
2
Π j
: (1.118)
N o w , w e ne e d t o loo k a t the H
H t e r m .
Tr
(
Π
2
j
[ H; Π
k
]
[ H; Π
k
]
^
S
)
= Tr
(
Π
j
[ H; Π
k
] Π
j
[ H; Π
k
]
)
(1.119)
= Tr
((
Π
j
H Π
k
Π
j
Π
k
H
)(
Π
j
H Π
k
Π
j
Π
k
H
))
= 0: (1.120)
Th a t i s, the H
H t e r m i s z e r o .
The r efor e , putt in g e v e r y thin g t o g e the r , w e h a v e ,
1
2
d
2
C
B
(U
t
)
dt
2
t= 0
=
1
d
d
∑
j= 1
var
j
( H): (1.121)
■
Pr o o f o f sh o r t - time gr o w th o f k - lo c a l c om m u ti n g Ha mi l t o ni a n s , Eq u a ti o n 1.43
T o pr o v e thi s w e ne e d thr e e in gr e d ie n ts . F ir s t , not ic e th a t sinc e e a ch t e r m in the H a mi lt oni a n H
( k)
c om-
m ut e s, w e h a v e ,
H
( k)
1
=
L( k 1)
∑
j= 1
σ
x
j
σ
x
j+ 1
σ
x
j+( k 1)
1
= L ( k 1) w he r e in the s e c ond
e qual it y , w e h a v e us e d the fa ct th a t σ
x
j
σ
x
j+ 1
σ
x
j+( k 1)
i s a unit a r y for e a ch j a nd∥ U∥
1
= 1 for al l
unit a r ie s . S e c ond , t o c omput e
1
2
d
2
C
B
(U t)
dt
2
t= 0
, w e ca n us e its e qual it y w ith
1
d
d
∑
j= 1
var
j
( H) .
48
The n, w e not e th a t ,
1
d
d
∑
j= 1
var
j
( H) =
1
d
0
@
d
∑
j= 1
Tr
(
H
2
Π
j
)
d
∑
j= 1
(
Tr
(
H Π
j
))
2
1
A
=
1
d
0
@
Tr( H
2
)
d
∑
j= 1
H
2
jj
1
A
; (1.122)
w he r e H
jj
=⟨ jj Hj j⟩ .
Thir d , for e a ch H
( k)
not ic e th a t ,
Tr
[
(
H
( k)
)
2
]
=
L( k 1)
∑
α; β= 1
Tr
[
(
σ
x
α
σ
x
α+ 1
σ
x
α+( k 1)
)
(
σ
x
β
σ
x
β+ 1
σ
x
β+( k 1)
)]
: (1.123)
I t i s e as y t o s e e th a t sinc e Tr[ σ
x
] = 0 the a bo v e tr a c e i s only nonz e r o i f α = β a nd the r efor e , w e h a v e , usin g
Tr[I] = d ,
Tr
[
(
H
( k)
)
2
]
= d( L( k 1)) (1.124)
M or e o v e r , not ic e th a t , H
jj
= 0 8 j ( sinc e σ
x
’ s fl ip the sp in s ). The r efor e ,
1
d
d
∑
j= 1
var
j
( H) = L( k 1) (1.125)
A nd , fin al ly , nor m al i z in g thi s w ith the ( s qua r e d ) ope r a t or nor m of the H a mi lt oni a n, w e h a v e the de sir e d
r e s ult.
49
2
Qua n tum cohe r e nce a nd the lo c ali z a tion tr a nsition
2.1 A bs tr a c t
A dy n a mical si gn a tur e of local i za t ion in qua n tum s ys t e m s i s the a bs e nc e of tr a n spor t w hich i s g o v e r ne d b y
the a moun t of c o he r e nc e th a t c onfi g ur a t ion sp a c e s t a t e s pos s e s s w ith r e spe ct t o the H a mi lt oni a n ei g e nb asi s .
T o m ak e thi s o bs e r v a t ion pr e c i s e , w e s tudy the local i za t ion tr a n sit ion v i a qua n tum c o he r e nc e me as ur e s
a r i sin g f r om the r e s our c e the or y of c o he r e nc e . W e sho w th a t the e s ca pe pr o b a b i l it y , w hich i s kno w n t o
sho w d i s t inct be h a v ior in the e r g od ic a nd local i z e d p h as e s, a r i s e s n a tur al ly as the a v e r a g e of a c o he r e nc e
me as ur e . M or e o v e r , usin g the the or y of m a j or i za t ion, w e a r g ue th a t br o a d fa mi l ie s of c o he r e nc e me as ur e s
ca n de t e ct the uni for mit y of the tr a n sit ion m a tr i x ( be t w e e n the H a mi lt oni a n a nd c onfi g ur a t ion b as e s ) a nd
he nc e a ct as pr o be s t o local i za t ion . W e pr o v ide s u ppor t in g n ume r ical e v ide nc e for A nde r s on a nd m a n y -
50
body local i za t ion ( MBL ).
F or infinit e sim al pe r tur b a t ion s of the H a mi lt oni a n, the d i ffe r e n t i al c o he r e nc e define s a n as s oc i a t e d R ie-
m a nni a n me tr ic . W e sho w th a t the l a tt e r i s ex a ctly g iv e n b y the dy n a mical c onduct iv it y , a qua n t it y of ex -
pe r ime n t al r e le v a nc e w hich i s kno w n t o h a v e a d i s t inct iv e ly d i ffe r e n t be h a v ior in the e r g od ic a nd in the
m a n y - body local i z e d p h as e s .
T ex t for thi s C h a pt e r i s a d a pt e d f r om [ 122 ].
2.2 In troduc tion
One of the c onc e ptual p i l l a r s of qua n tum the or y i s the s u pe r posit ion pr inc ip le a nd , d ir e ctly a r i sin g f r om it ,
the not ion of q u a n t u m co h er en ce [ 123 ]. A qua n tum s t a t e i s de e me d t o be c o he r e n t w ith r e spe ct t o a c om-
p le t e s e t of s t a t e s i f it ca n be ex pr e s s e d as a non tr iv i al l ine a r s u pe r posit ion of the s e s t a t e s . R e c e n tly , the r e
h as be e n a n effor t t o for m ul a t e a r e s our c e the or y of qua n tum c o he r e nc e [ 124 – 126 ]. The foc us of thi s the-
or y h as be e n qua n tum infor m a t ion pr oc e s sin g t ask s, sinc e g e ne r a t in g a nd pr e s e r v in g qua n tum c o he r e nc e
c on s t itut e s one of the e s s e n t i al pr e r e qui sit e s .
I n thi s w ork , w e ut i l i z e the po w e r f ul t oo l s th a t a r os e f r om thi s infor m a t ion-the or e t ic pe r spe ct iv e on c o -
he r e nc e t o s tudy p h as e tr a n sit ion s in qua n tum one- a nd m a n y - body s ys t e m s . M or e spe c i fical ly , w e foc us
on A nde r s on [ 127 , 128 ] a nd m a n y - body local i za t ion ( MBL ) tr a n sit ion s [ 129 – 131 ]. The s e “ infinit e t e m-
pe r a tur e ” or “ ei g e n s t a t e ” p h as e tr a n sit ion s a r e ch a r a ct e r i z e d b y a n a br u pt ch a n g e oc c ur r in g a t the le v e l of
w ho le H a mi lt oni a n ei g e n s t a t e s as oppos e d , e . g., t o the gr ound s t a t e only .
A c onne ct ion be t w e e n qua n tum c o he r e nc e a nd the tr a n sit ion of a qua n tum s ys t e m f r om a n e r g od ic
p h as e t o a local i z e d one ca n be c onc e ptual ly for m al i z e d as fo l lo w s . One of the si gn a tur e s of local i za t ion i s
the a bs e nc e of tr a n spor t , w ith r e spe ct t o s ome pr ope rly define d posit ion al de gr e e of f r e e dom . On the othe r
h a nd , tr a n spor t pr ope r t ie s a r e g o v e r ne d b y the c o he r e nc e be t w e e n the H a mi lt oni a n ei g e nb asi s a nd the
posit ion al one . H e nc e one should ex pe ct a n a br u pt ch a n g e in the c o he r e nc e pr ope r t ie s of the H a mi lt oni a n
ei g e n v e ct or s a t the tr a n sit ion po in t.
H e r e w e m ak e the a bo v e in tuit ion qua n t it a t iv e ly pr e c i s e b y in v e s t i g a t in g the a moun t of c o he r e nc e th a t
ca n be g e ne r a t e d on a v e r a g e b y the qua n tum dy n a mic s s t a r t in g f r om inc o he r e n t s t a t e s, the co h er en ce-gen er a t i n g
51
p o w er ( C GP ) of a qua n tum e v o lut ion . S uch qua n t it ie s e s s e n t i al ly ca ptur e the d i ffe r e nc e be t w e e n t w o c om-
p le t e or thonor m al s e ts of ei g e n s t a t e s as s oc i a t e d w ith t w o he r mit i a n ope r a t or s [ 132 ] ( s e e al s o C h a pt e r 3 ).
W e fir s t sho w th a t a w e l l - s tud ie d qua n t it y in local i za t ion, the e s c a p e pr o b a bi l it y ( or , e quiv ale n tly , the se co n d
p a r t icip a t io n r a t io ) ca n be ex pr e s s e d as a c o he r e nc e a v e r a g e . W e the n a r g ue th a t br o a d fa mi l ie s of c o he r -
e n c e me as ur e s, a r i sin g f r om the r e s our c e-the or e t ic pe r spe ct iv e , ca n be us e d t o define a n “ or de r p a r a me-
t e r ” for loca l i za t ion . W e pr o v ide s u ppor t in g n ume r ical e v ide nc e for both A nde r s on a nd MBL tr a n sit ion s .
M or e o v e r , w e sho w th a t the d i ffe r e n t i al - g e ome tr ic v e r sion of our a v e r a g e c o he r e nc e i s ex a ctly g iv e n b y a n
infinit e t e mpe r a tur e dy n a mical c onduct iv it y , a n ex pe r ime n t al ly a c c e s si b le qua n t it y , w hich i s kno w n t o be-
h a v e d i ffe r e n tly in the e r g od ic a nd MBL p h as e s [ 133 ]. The s e find in gs ope n the pos si b i l it y of o bs e r v in g
ex pe r ime n t al ly the c o he r e nc e- g e ne r a t in g po w e r of qua n tum dy n a mic s .
2.3 Qu an tum c oherenc e of s t a te s and oper a tion s
Co h er en c e o f s t a te s
C on side r a qua n tum s ys t e m, de s cr i be d b y a finit e d ime n sion al H i l be r t sp a c e H
=C
d
. A s t a t e j ψ⟩2H i s
de e me d co h er en t w ith r e spe ct t o a fiduc i al or thonor m al b asi s
{
j φ
i
⟩
}
d
i= 1
i f the ex p a n sion j ψ⟩ =
∑
i
a
i
j φ
i
⟩
c on t ain s mor e th a n one non v a ni shin g t e r m, othe r w i s e it i s cal le d i n co h er en t . Thi s not ion ex t e nd s s tr ai gh t -
for w a r d ly t o the s e t of de n sit y ope r a t or s S(H) . A n y ρ2S(H) i s r e g a r de d as co h er en t w ith r e spe ct t o the
pr efe r r e d b asi s i f the c or r e spond in g m a tr i x ρ
ij
h as nonz e r o off - d i a g on al e le me n ts, othe r w i s e it i s t e r me d
i n co h er en t .
Qua n tum c o he r e nc e i s us ual ly define d r e l a t iv e t o a r efe r e nc e b asi s . I n fa ct , one ne e d s a w e ak e r not ion
th a n th a t of a b asi s, sinc e p h as e de gr e e s of f r e e dom a nd or de r in g of a n or thonor m al b asi s
{
j φ
i
⟩
}
d
i= 1
a r e
p h ysical ly r e dund a n t. I n othe r t e r m s, b as e s d i ffe r in g b y tr a n sfor m a t ion s of the for m j φ
j
⟩ 7! e
iθ j
j φ
π( j)
⟩
( π2 S
d
i s a pe r m ut a t ion ) a r e e quiv ale n t as fa r as c o he r e nc e i s c onc e r ne d . The r e le v a n t o bj e ct , t ak in g in t o
a c c oun t thi s f r e e dom, i s a c omp le t e s e t of or tho g on al , r a nk -1 pr oj e ct ion ope r a t or s B = f Π
i
g
d
i= 1
, w he r e
Π
i
:=j φ
i
⟩⟨ φ
i
j . I n the r e s t of thi s w ork , w e w i l l r efe r for c on v e nie nc e t o the s e t B its e l f as a “ b asi s . ”
W hi le al l s t a t e s nond i a g on al in B ca r r y c o he r e nc e , s ome of the m mi gh t r e s e mb le inc o he r e n t s t a t e s mor e
52
th a n othe r s . Thi s not ion i s m a de pr e c i s e b y the in tr oduct ion of ( B - de pe nde n t ) f unct ion al s, c
B
:S(H)!
R
+
0
th a t a r e s aid t o qua n t i f y c o he r e nc e [ 125 ]. Qua n t i fie r s of c o he r e nc e ( al s o cal le d co h er en ce m o n o t o n e s )
s a t i sf y c
B
( ρ
inc
) = 0 for al l s t a t e s d i a g on al in B a nd , in a dd it ion, a r e nonincr e asin g unde r the f r e e ope r a t ion s
of the r e s our c e the or y ¹ . I n thi s w ork , w e m ak e us e of the 2- co h er en ce a nd the r e l a t i v e en tr o p y of co h er en ce ,
define d r e spe ct iv e ly b y
c
( 2)
B
( ρ) :=∥(ID
B
) ρ∥
2
2
=
∑
i̸= j
ρ
ij
2
(2.1a)
c
( rel)
B
( ρ) := S[D
B
( ρ)] S( ρ) ; (2.1b)
w he r e w e h a v e in tr oduc e d the B - de p h asin g s u pe r ope r a t or
D
B
( X) :=
d
∑
i= 1
Π
i
X Π
i
; (2.2)
S a bo v e de not e s the us ual v on- N e um a nn e n tr op y S( ρ) := Tr( ρ log( ρ)) a nd the ( S ch a tt e n ) 2- nor m of
a n ope r a t or X i s define d as∥ X∥
2
:=
√
Tr( X
y
X) . R e l a t iv e e n tr op y of c o he r e nc e i s a c e n tr al me as ur e in
the r e s our c e the or ie s of c o he r e nc e a nd a dmits a n ope r a t ion al in t e r pr e t a t ion, e . g., as a c on v e r sion r a t e of
infor m a t ion-the or e t ic pr ot oc o l s [ 135 , 137 ]. The 2- c o he r e nc e a dmits a n in t e r pr e t a t ion as a n e s ca pe pr o b a -
b i l it y , as w i l l be sho w n mome n t a r i ly ² .
Co h er en c e o f u nit a r y q u a ntum pr o c e sse s v i a pr o b a b i lis ti c a v er a ge s
I n thi s s e ct ion w e d i s c us s ho w , g iv e n a c o he r e nc e me as ur e c
B
a nd a unit a r y s u pe r ope r a t or U , one ca n ca ptur e
the a b i l it y of the unit a r y U t o g e ne r a t e c o he r e nc e b y c omput in g the a v e r a g e a moun t of c o he r e nc e th a t ca n
be g e ne r a t e d s t a r t in g f r om inc o he r e n t s t a t e s . Thi s i s the co h er en ce-gen er a t i n g p o w er ( C GP ) of the qua n tum
ope r a t ion U . S inc e s ome of our c on v e n t ion s w i l l be d i ffe r e n t f r om the one s in C h a pt e r 3 , w e in tr oduc e the
¹ W e not e th a t the r e ex i s t v a r ious pr opos al s for the f r e e ope r a t ion s in the r e s our c e the or ie s of c o he r e nc e ( s e e [ 134 ] for mor e
de t ai l s ). I n the fo l lo w in g , w e w i l l us e the t e r m I nc o he r e n t O pe r a t ion s for the f r e e ope r a t ion s but , in fa ct , al l r e s ults ho ld for a n y
cl as s th a t c on t ain s S tr ictly I nc o he r e n t O pe r a t ion s [ 135 , 136 ].
² W e not e , ho w e v e r , th a t the 2- c o he r e nc e mi gh t fai l t o s a t i sf y the monot onic it y pr ope r t y unde r s ome cl as s e s of f r e e ope r a -
t ion s .
53
r e le v a n t definit ion s ex p l ic itly .
C on side r a b asi s B = f Π
i
g
d
i= 1
a nd define a pr o b a b i l i s t ic e n s e mb le of inc o he r e n t s t a t e s, i . e ., a r a ndom
v a r i a b le ρ
inc
( p) =
∑
i
p
i
Π
i
, w he r ef p
i
g
i
( p
i
0 ,
∑
i
p
i
= 1) a r e r a ndom a nd d i s tr i but e d a c c or d in g t o a
pr e s cr i be d me as ur e μ( p) . The n, the c or r e spond in g C GP
C(U; c
B
; μ) :=
∫
d μ( p) c
B
[
U
(
ρ
inc
( p)
)]
(2.3)
ch a r a ct e r i z e s the a v e r a g e effe ct iv e ne s s of the qua n tum pr oc e s s U t o g e ne r a t e c o he r e nc e out of r a ndom
inc o he r e n t s t a t e s in B . S inc e the unit a r y U( X) = UXU
y
ca n be thou gh t of as c onne ct in g the b as e s B a nd
B
′
=fU ( Π
i
)g
i
, one ca n al s o in t e r pr e t C(U; c
B
; μ) as the a v e r a g e c o he r e nc e w ith r e spe ct t o B of a r a ndom
s t a t e w hich i s inc o he r e n t in B
′
.
W ithout a n y a dd it ion al s tr uctur e , it i s a n a tur al cho ic e t o c on side r a v e r a g in g only o v e r pur e s t a t e s w ith
e qual w ei gh t o v e r e a ch of the m , i . e ., t ak e
μ
unif
( p) :=
1
d
∑
i
δ( p e
i
) (2.4)
w he r e ( e
i
)
j
:= δ
ij
. Thi s cho ic e d ir e ctly le a d s t o the ex pr e s sion
C
(
U; c
B
; μ
unif
)
=
1
d
d
∑
i= 1
c
B
[U ( Π
i
)] : (2.5)
W e no w simp l i f y E q . ( 2.5 ) w he n the c o he r e nc e me as ur e i s the 2- c o he r e nc e or the r e l a t iv e e n tr op y of
c o he r e nc e , n a me ly for
C
( 2)
B
(U) := C
(
U; c
( 2)
B
; μ
unif
)
; (2.6a)
C
( rel)
B
(U) := C
(
U; c
( rel)
B
; μ
unif
)
: (2.6b)
54
P r opositio n 2.1
L e t B = f Π
i
g
d
i= 1
be a b asi s,U a unit a r y qua n tum pr oc e s s a nd X
U
de not e the ( b i s t och as t ic ) m a tr i x
w ith e le me n ts ( X
U
)
ij
:= Tr
(
Π
i
U( Π
j
)
)
. The n,
C
( 2)
B
(U) = 1
1
d
Tr
(
X
T
U
X
U
)
: (2.7)
a nd
C
( rel)
B
(U) = H( X
U
) ; (2.8)
w he r e H( X) :=
1
d
∑
i; j
X
ij
log( X
ij
) de not e s the g e ne r al i za t ion of the S h a nnon e n tr op y o v e r b i s -
t och as t ic m a tr ic e s .
P r o of . ( i ) W e fo l lo w a pr oc e dur e simi l a r t o the one in C h a pt e r 3 . W e m ak e us e of the H i l be r t - S chmidt
inne r pr oduct ⟨ A; B⟩ := Tr
(
A
y
B
)
o v e r the sp a c e B(H) of bounde d l ine a r ope r a t or s o v e r H . S t a r t in g
f r om E q . ( 2.5 ) w ith c
B
= c
( 2)
B
, w e g e t
C
( 2)
B
(U) =
1
d
∑
i
∥(ID
B
)U Π
i
∥
2
2
=
1
d
∑
i
⟨(ID
B
)U Π
i
;(ID
B
)U Π
i
⟩
=
1
d
∑
i
(∥U Π
i
∥
2
2
∥D
B
U Π
i
∥
2
2
) ;
w he r e w e h a v e us e d the fa ct th a t the de p h asin g s u pe r ope r a t or D
B
2 B(B(H)) i s s e l f -a d j o in t D
y
B
=
D
B
w ith r e spe ct t o the H i l be r t - S chmidt inne r pr oduct , as w e l l as a pr oj e ct ion D
2
B
= D
B
. U nit a r y
in v a r i a nc e of the 2- nor m imp l ie s ∥U Π
i
∥
2
2
= 1 . U sin g the definit ion E q . ( 2.2 ), a s tr ai gh tfor w a r d calc u-
55
l a t ion g iv e s
C
( 2)
B
(U) = 1
1
d
∑
ij
( X
U
)
2
ji
(2.9)
w hich r e duc e s t o the cl aime d r e s ult.
( ii ) L e t us de not e the S h a nnon e n tr op y of a pr o b a b i l it y v e ct or as H( p) :=
∑
i
p
i
log( p
i
) . S inc e
S(U Π
i
) = S( Π
i
) = 0 , E q . ( 2.5 ) w ith c
B
= c
( rel)
B
g iv e s
C
( rel)
B
(U) =
1
d
∑
i
S(D
B
U Π
i
)
=
1
d
∑
i
S
0
@
∑
j
( X
U
)
ji
Π
j
1
A
=
1
d
∑
i
H
(
{
( X
U
)
ji
}
j
)
= H( X
U
) :
■
The t w o C GP qua n t it ie s a r e r e l a t e d as
C
( rel)
B
log
(
1 C
( 2)
B
)
: (2.10)
The ine qual it y fo l lo w s f r om the a bo v e pr oposit ion, t o g e the r w ith the c onca v it y of the lo g a r ithmic f unct ion .
Gener a l pr o per ti e s o f c o h er en c e - gener a ti n g po wer me a s u r e s
Both qua n t it ie s C
( 2)
B
(U) a nd C
( rel)
B
(U) in tr oduc e d e a rl ie r ca n be c on side r e d as f unct ion s of the ( tr a n sit ion )
m a tr i x X
U
, in s t e a d of U its e l f . I n othe r w or d s, the p h as e s as s oc i a t e d w ith U
ij
( tr e a t e d as a m a tr i x in the B
b asi s ) a r e ir r e le v a n t. I n fa ct , as w e w i l l sho w mome n t a r i ly , thi s i s a g e ne r al fe a tur e of a n y C GP me as ur e
56
C
(
U; c
B
; μ
unif
)
a r i sin g f r om a c o he r e nc e monot one c
B
.
M ot iv a t e d b y the a bo v e o bs e r v a t ion, w e define as a gen er a l i ze d C GP m e as u r e a n y f unct ion f
B
m a pp in g
b i s t och as t ic m a tr ic e s t o non- ne g a t iv e r e al n umbe r s s uch th a t :
( i ) f
B
( Π) = 0 i f Π2 S
d
i s a pe r m ut a t ion .
( i i ) f
B
( Π X Π
′
) = f
B
( X) , w he r e Π; Π
′
2 S
d
a r e pe r m ut a t ion s .
( i i i ) f
B
( MX) f
B
( X) for a n y b i s t och as t ic m a tr i x M .
P r opositio n 2.2
L e t c
B
be a c o he r e nc e me as ur e . The n, the c or r e spond in g c o he r e nc e- g e ne r a t in g po w e r f
B
( X
U
) :=
C
(
U; c
B
; μ
unif
)
s a t i sfie s( i)( iii) a bo v e .
P r o of . W e fir s t sho w th a t , for a fi xe d c o he r e nc e me as ur e c
B
, the qua n t it y C
(
U; c
B
; μ
unif
)
( ex p l ic itl y
g iv e n in E q . ( 2.5)) ca n be ex pr e s s e d as a f unct ion of X
U
. Thi s imp l ie s th a t the p h as e s of U ( c on side r e d
as a m a tr i x in the B =f Π
i
g
i
=fj φ
i
⟩⟨ φ
i
jg
i
b asi s, w he r e U( X) = UXU
y
) a r e ir r e le v a n t.
C on side r a pur e s t a t e j ψ⟩ . The v alue of c
B
(j ψ⟩⟨ ψj) ca n only de pe nd on the modulus of the c oeffi -
c ie n ts
{
⟨ φ
i
j ψ⟩
}
d
i= 1
. Thi s fo l lo w s f r om the fa ct th a t the unit a r y tr a n sfor m a t ion s V( ρ) = V ρ V
y
, s uch
th a t Vj ψ⟩ alt e r s the p h as e s or pe r m ut e s the c oeffic ie n ts f⟨ φ
i
j ψ⟩g
d
i= 1
, for m a s ub gr ou p of the I nc o he r -
e n t O pe r a t ion s . H e nc e al l c o he r e nc e monot one s should m ain t ain a c on s t a n t v alue o v e r a gr ou p or b it.
A s a r e s ult , c
B
(U( Π
j
)) ca n be ex pr e s s e d as a f unct ion of f( X
U
)
ij
g
d
i= 1
( r e cal l ( X
U
)
ij
=
⟨ φ
i
j Uj φ
j
⟩
2
).
H e nc e , al s o C
(
U; c
B
; μ
unif
)
ca n be ex pr e s s e d as a f unct ion of the w ho le m a tr i x X
U
( in fa ct , a n a dd it iv e
one o v e r the c o lumn s ).
P r ope r t y ( i ) fo l lo w s d ir e ctly f r om the fa ct th a t c o he r e nc e me as ur e s v a ni sh o v e r inc o he r e n t s t a t e s .
F or pr ope r t y ( i i ), in v a r i a nc e unde r pr e-pr oc e s sin g b y a pe r m ut a t ion Π
′
ho ld s sinc e the a v e r a g in g o v e r
the s t a t e s i s uni for m . I n v a r i a nc e unde r pos t -pr oc e s sin g b y Π ho ld s sinc e unit a r y tr a n sfor m a t ion s th a t
pe r m ut e the e le me n ts of B be lon g t o I nc o he r e n t O p e r a t or s .
57
W e no w pr o v e pr ope r t y ( i i i ). F ir s t not ic e th a t , sinc e the v alue of c
B
(j ψ⟩⟨ ψj) ca n only de pe nd on the
modul i of the c oeffic ie n ts
{
⟨ φ
i
j ψ⟩
}
d
i= 1
, the f unct ion f
B
( X) i s in fa ct w e l l - define d o v e r al l b i s t och as t ic
m a tr ic e s ( a nd not jus t uni s t och as t ic a one s ).
C on side r a c o l le ct ion of pur e s t a t e s fj ψ
j
⟩⟨ ψ
j
jg
d
j= 1
s uch th a t
j ψ
j
⟩ =
∑
i
√
( MX
U
)
ij
j φ
i
⟩ : (2.11)
The n, one h as th a t
Tr
(
Π
i
j ψ
j
⟩⟨ ψ
j
j
)
=
∑
k
M
ik
Tr
(
Π
k
U( Π
j
)
)
8 i; j : (2.12)
T o pr o v e the de sir e d ine qual it y of ( i i i ), w e w i l l sho w th a t c
B
(
j ψ
j
⟩⟨ ψ
j
j
)
c
B
(U( Π
j
))8 j . I nde e d ,
the pr e v ious ho ld s tr ue for al l c o he r e nc e me as ur e s c
B
i f for e v e r y j the r e ex i s ts a n I nc o he r e n t O pe r a t or E
s uch th a t E
(
j ψ
j
⟩⟨ ψ
j
j
)
=U( Π
j
) . The l as t i s g ua r a n t e e d ( in fa ct , w ithin S tr ictly I nc o he r e n t O pe r a t or s )
b y the m ain r e s ult of [ 139 ] w hich ca n be a pp l ie d sinc e , b y the b i s t och as t ic it y of M , E q . ( 2.12 ) imp l ie s
th a t D
B
(
U( Π
j
)
)
≻D
B
(
j ψ
j
⟩⟨ ψ
j
j
)
.
■
a A b i s t och as t ic m a tr i x M
ij
i s cal le d uni s t och as t ic i f the r e ex i s ts a unit a r y m a tr i x U
ij
s uch th a t M
ij
=
U
ij
2
( s e e [ 138 ]
for mor e de t ai l s ).
On p h ysical gr ound s, al l qua n t it ie s C
(
U; c
B
; μ
unif
)
a r e ex pe ct e d t o qua n t i f y ho w “ uni for m ” or “ spr e a d ”
i s the tr a n sit ion m a tr i x X
U
be t w e e n the b as e s B a nd B
′
= U( B) . Thi s in tuit ion i s r efle ct e d in p a r t ( i i i )
of P r oposit ion 2.2: “ pos t -pr oc e s sin g ” the tr a n sit ion m a tr i x X 7! MX b y a n y b i s t och as t ic m a tr i x M w i l l
c e r t ainly incr e as e a n y C GP me as ur e C
(
U; c
B
; μ
unif
)
, w he r e c
B
ca n be a n y c o he r e nc e monot one .
Ge ne r al i z e d C GP me as ur e s ca n be thou gh t of as f unct ion s th a t ch a r a ct e r i z e the uni for mit y of a ( b i s -
t och as t ic ) m a tr i x . They alw a ys a chie v e their m a x im um v alue o v e r the tr a n sit ion m a tr i x ( X
V
)
ij
= 1= d , i . e .,
w he nV c onne cts t w o unb i as e d b as e s, as fo l lo w s b y c omb inin g pr ope r t ie s ( ii) a nd( iii) . I n a simi l a r m a n-
ne r , the minim um v alue i s a chie v e d o v e r pe r m ut a t ion m a tr ic e s a nd i s s e t t o z e r o ( as a nor m al i za t ion ) b y
58
( i) . F or in s t a nc e , a n y c onca v e f unct ion th a t s a t i sfie s pr ope r t ie s ( i) a nd( ii) a ut om a t ical ly s a t i sfie s ( iii) , i . e .,
i s a g e ne r al i z e d C GP me as ur e .
E x a mp le s of g e ne r al i z e d me as ur e s a r i sin g f r om pr e v ious w ork s on C GP [ 140 ] a r e
f
( det)
B
( X
V
) := 1j det( X
V
)j
1
d
(2.13)
f
(1)
B
( X
V
) :=
I X
T
V
X
V
1
; (2.14)
w he r e∥()∥
1
de not e s the ope r a t or nor m . N ot ic e th a t f
( det)
B
( X
V
) = 1(
∏
i
s
i
)
1
d
a nd al s o 0 f
( det)
B
( X
V
)
1 , w hi le f
(1)
B
( X
V
) = 1 s
2
d
( he r e s
i
a r e the sin g ul a r v alue s of X
V
s or t e d in de cr e asin g or de r ).
A s ys t e m a t ic w a y t o ca ptur e the a moun t of uni for mit y of a m a tr i x i s pr o v ide d b y the not ion of m ul -
t iv a r i a t e m a j or i za t ion [ 141 ]. A n ex a mp le i s co l u m n m aj o r i za t io n , in w hich a s t och as t ic m a tr i x X c o lumn
m a j or i z e s a nothe r s t och as t ic m a tr i x Y , de not e d as X≻
c
Y , i f X
c
i
≻ Y
c
i
8 i ; he r e X
c
i
a nd Y
c
i
s t a nd for the i th
c o lumn v e ct or of X a nd Y , r e spe ct iv e ly , a nd “ ≻ ” de not e s or d in a r y m a j or i za t ion of pr o b a b i l it y v e ct or s .
I t i s the n n a tur al t o ask w he the r the C GP qua n t it ie s C
(
U; c
B
; μ
unif
)
a r i sin g f r om d i ffe r e n t c o he r e nc e
me as ur e s c
B
j o in tly ca ptur e s ome not ion of uni for mit y of the tr a n sit ion m a tr i x X
U
, as de s cr i be d b y m ult i -
v a r i a t e m a j or i za t ion . W e a n s w e r thi s in the a ffir m a t iv e v i a the pr oposit ion be lo w .
P r opositio n 2.3
L e t c
B
be a c o he r e nc e me as ur e . The n, the c or r e spond in g c o he r e nc e- g e ne r a t in g po w e r f
B
( X
U
) :=
C
(
U; c
B
; μ
unif
)
c on side r e d o v e r b i s t och as t ic m a tr ic e s i s a monot one of c o lumn m a j or i za t ion, i . e .,
X ≻
c
Y) f
B
( X) f
B
( Y) . C on v e r s e ly , i f f
B
( X) f
B
( Y) for al l f
B
a r i sin g f r om c on t in uous c o he r -
e nc e monot one s o v e r pur e s t a t e s, the n X≻
c
Y .
P r o of . The fir s t p a r t fo l lo w s b y g e ne r al i z in g the pr oof of p a r t ( i i i ) of P r oposit ion 2.2 . One ca n d ir e ctly
ex t e nd the c on s tr uct ion b y c on side r in g t w o s e ts of pur e s t a t e s fj ψ
j
⟩⟨ ψ
j
jg
d
j= 1
a ndfj ψ
′
j
⟩⟨ ψ
′
j
jg
d
j= 1
s uch
59
th a t
j ψ
j
⟩ =
∑
i
√
Y
ij
j φ
i
⟩ (2.15a)
j ψ
′
j
⟩ =
∑
i
√
X
ij
j φ
i
⟩ : (2.15b)
The n the c on v e r t i b i l it y a r g ume n t j ψ
j
⟩⟨ ψ
j
j 7! j ψ
′
j
⟩⟨ ψ
′
j
j v i a s tr ictly inc o he r e n t ope r a t ion s a pp l ie s due
t o t he m a j or i za t ion c ond it ion, g iv in g the de sir e d r e s ult.
F or the c on v e r s e , w e w i l l fir s t sho w th a t the f unct ion s o v e r pur e s t a t e s c
B
(j ψ⟩⟨ ψj) =
∑
i
φ( Tr( Π
i
j ψ⟩⟨ ψj))
a r e monot one s, w he r e φ i s a n y c on t in uous c onca v e f unct ion . I nde e d , f r om the m ain r e s ult of R ef . [ 139 ],
a c on v e r sion j ψ⟩⟨ ψj7!j ψ
′
⟩⟨ ψ
′
j v i a S tr ictly I nc o he r e n t O pe r a t ion s i s pos si b le i f a nd only i f D
B
(j ψ
′
⟩⟨ ψ
′
j)≻
D
B
(j ψ⟩⟨ ψj) .
H e r e w e not e th a t , s tr ictly , the m a j or i za t ion c ond it ion i s only s u ffic ie n t for c on v e r t i b i l it y . I t be-
c ome s al s o ne c e s s a r y i f a n a dd it ion al c ond it ion a bout the r a nk of the de p h as e d s t a t e s i s s a t i sfie d ( s e e
R ef . [ 139 ] for mor e de t ai l s ). N e v e r the le s s, i f one c on side r s c on v e r t i b i l it y w ith s ome e r r or ( a r b itr a r i ly
s m al l ), w hich i s the r e le v a n t not ion in al l p h ysical s c e n a r ios, the r a nk c ond it ion s be c ome s ir r e le v a n t.
A s t a nd a r d r e s ult b y H a r dy , L ittle w ood a nd P ó lya s t a t e s th a t for t w o pr o b a b i l it y v e ct or s it ho ld s
th a t p ≻ q i f a nd only i f
∑
i
φ( p
i
)
∑
i
φ( q
i
) for al l c on t in uous c onca v e φ [ 141 ]. A s a r e s ult ,
D
B
(j ψ
′
⟩⟨ ψ
′
j)≻D
B
(j ψ⟩⟨ ψj) i s e quiv ale n t t o
∑
i
φ( Tr( Π
i
j ψ
′
⟩⟨ ψ
′
j))
∑
i
φ( Tr( Π
i
j ψ⟩⟨ ψj)) , i . e .,
the a for e me n t i one d f unct ion s c
B
a r e monot one s o v e r pur e s t a t e s .
B y as s umpt i on, the f unct ion s f
B
a r i s e f r om c on t in uous c o he r e nc e monot one s o v e r pur e s t a t e s . F r om
the s t a t e me n t in the pr e v ious p a r a gr a p h it the n fo l lo w s th a t , in fa ct , al l f
B
( X) =
∑
ij
φ
(
X
ij
)
for c on-
t in uous c onca v e φ a r e s uch f unct ion s . H e nc e ,
∑
ij
φ
(
X
ij
)
∑
ij
φ
(
Y
ij
)
. F in al ly , the a for e me n t ione d
r e s ult b y H a r dy , L i ttle w ood , a nd P ó lya [ 141 ] in the c on t ex t of c o lumn m a j or i za t ion imp l ie s X≻
c
Y .
■
The l as t p a r t of the a bo v e pr oposit ion e s t a b l i she s the fa ct th a t the r e a r e e nou gh c o he r e nc e monot one s
o v e r pur e s t a t e s one ca n c on side r s uch th a t , i f al l c or r e spond in g me as ur e s f
B
a r e monot onic, the n c o lumn
60
m a j or i za t ion i s g ua r a n t e e d . I n othe r w or d s, the s e f unct ion s for m a c omp le t e s e t of monot one s . I n th a t
s e n s e , the define d fa mi ly of C GP me as ur e s j o in tly ca ptur e s a not ion of uni for mit y for the tr a n sit ion m a tr i x
th a t i s a t le as t as s tr ict as c o lumn m a j or i za t ion .
Co h er en c e a nd e sc a pe pr o b a b i lit y
L e t us c on side r a finit e- d ime n sion al qua n tum s ys t e m w hos e dy n a mic s i s spe c i fie d b y a H a mi lt oni a n H .
S u ppos e the s ys t e m i s init i al i z e d in a s t a t e j ψ⟩ a nd one i s in t e r e s t e d in the e sc a p e pr o b a bi l it y
P
ψ
:= 1j⟨ ψj e
iHt
j ψ⟩j
2
; (2.16)
w he r e the o v e rl ine de not e s the infinit e t ime a v e r a g e
f( t) := lim
T!1
1
T
∫
T
0
dt f( t) : (2.17)
F or in s t a nc e , in the cas e of a p a r t icle hopp in g on a l a tt ic e w hich i s init i al i z e d o v e r a sin gle sit e j ,P
j
c or r e-
spond s t o the a v e r a g e pr o b a b i l it y of the p a r t icle e s ca p in g the init i al sit e .
A t thi s po in t , le t us not e th a t in finit e d ime n sion s o bs e r v a b le qua n t it ie s s uch as ⟨ A( t)⟩ := Tr[ A( t) ρ
0
] =
Tr[ A ρ( t)] do not c on v e r g e t o a n y l imit as t!1 . I n s t e a d they s t a r t f r om a n init i al v alue a nd the n os c i l l a t e
a r ound a v alue g iv e n b y ⟨ A( t)⟩ [ 142 – 145 ]. S inc e i f a f unct ion f( t) h as a l imit for t ! 1 , thi s l imit m us t
c o inc ide w ith f( t) , the infinit e t ime a v e r a g e pr o v ide s a w a y t o ex tr a ct the infinit e t ime l imit e v e n w he n the
l a tt e r s tr ictly spe ak in g doe s not ex i s t.
I f the H a mi lt oni a n in c on side r a t ion h as nonde g e ne r a t e e ne r g y g a ps, i . e ., i f the e ne r g y d i ffe r e nc e s s a t i sf y
E
i
E
i
′ = E
j
E
j
′ =)( i = i
′
^ j = j
′
)_( i = j^ i
′
= j
′
) ( al s o kno w n as the nonr e s on a nc e c ond it ion ),
the eff e c t i v e d i m en sio n d
eff
:= ( 1P
ψ
)
1
d ict a t e s the e qui l i br a t ion pr ope r t ie s of the s ys t e m : the l a r g e r d
eff
the s m al le r a r e the t e mpor al fluctua t ion s of the o bs e r v a b le s a r ound their me a n v alue s [ 142 , 143 ], i . e ., e qui -
l i br a t ion i s s tr on g e r . S inc e m a n y - body local i za t ion i s a me ch a ni s m b y w hich qua n tum s ys t e m s ca n e s ca pe
e qui l i br a t ion, it i s pe rh a ps no s ur pr i s e th a t the effe ct iv e d ime n sion i s r e l a t e d t o the local i za t ion tr a n sit ion
( s e e A ppe nd i x B for mor e de t ai l s on r e l a t e d qua n t it ie s ).
61
A ft e r in tr oduc in g the b asic f r a me w ork , w e a r e no w r e a dy t o pr e s e n t our fir s t r e s ult. The fo l lo w in g P r opo -
sit ion e s t a b l i she s the fa ct th a t the 2- c o he r e nc e of a s t a t e , qua n t i fie d w ith r e spe ct t o the H a mi lt oni a n ei g e n-
b asi s, i s the t ime-a v e r a g e d e s ca pe pr o b a b i l it y of the s t a t e .
P r opositio n 2.4
L e t H =
∑
i
E
i
j φ
i
⟩⟨ φ
i
j be a nonde g e ne r a t e H a mi lt oni a n .
( i ) F or a n y s t a t e j ψ⟩ ,
P
ψ
= c
( 2)
B
(j ψ⟩⟨ ψj) ; (2.18)
w he r e B =
{
j φ
i
⟩⟨ φ
i
j
}
i
i s the ei g e nb asi s of the H a mi lt oni a n .
( i i ) D e not e the e s ca p e p r o b a b i l it y a v e r a g e d o v e r a s e t of or thonor m al s t a t e s B
′
=fj i⟩⟨ ijg
d
i= 1
as
P
B
′ :=
1
d
d
∑
i= 1
P
i
: (2.19)
The n,
P
B
′ = C
( 2)
B
(V) = C
( 2)
B
′
(V
y
) ; (2.20)
w he r e B =
{
j φ
i
⟩⟨ φ
i
j
}
i
i s the ei g e nb asi s of the H a mi lt oni a n a nd V() := V() V
y
, w he r e
V =
∑
i
j i⟩⟨ φ
i
j i s the in t e r t w ine r be t w e e n B a nd B
′
.
P r o of . ( i ) The k ey o bs e r v a t ion i s th a t the de p h asin g s u pe r ope r a t or D
B
a r i s e s as the ( infinit e ) t ime
a v e r a g e of the S chr öd in g e r e v o lut ion U
t
() = e
itH
() e
itH
, n a me lyU
t
= D
B
. U sin g the H i l be r t -
62
S chmidt inne r pr oduct o v e r B(H) ( s e e pr oof of P r op . 2.1) a nd s e tt in g Π
ψ
=j ψ⟩⟨ ψj , w e g e t
P
ψ
= 1 Tr
(
Π
ψ
U
t
( Π
ψ
)
)
= 1 Tr
(
Π
ψ
D
B
( Π
ψ
)
)
= 1⟨ Π
ψ
;D
B
Π
ψ
⟩ = 1⟨D
B
Π
ψ
;D
B
Π
ψ
⟩
=⟨(ID
B
) Π
ψ
;(ID
B
) Π
ψ
⟩
=
(ID
B
) Π
ψ
2
2
= c
( 2)
B
( Π
ψ
) :
( ii ) The fir s t e qual it y of E q . ( 2.20 ) fo l lo w s b y c omb in g p a r t ( i) of the P r oposit ion w ith E q . ( 2.5 ). F or
the s e c ond e qual it y , f r om the unit a r y in v a r i a nc e of the 2- nor m, w e h a v e
C
( 2)
B
(V) =
1
d
∑
i
∥(ID
B
)j i⟩⟨ ij∥
2
2
=
1
d
∑
i
V
y
(ID
B
′)V (j i⟩⟨ ij)
2
2
=
1
d
∑
i
∥(ID
B
′)V (j i⟩⟨ ij)∥
2
2
= C
( 2)
B
′
(V) :
H o w e v e r , not ic e th a t X
V
y = X
T
V
w hich f r om E q . ( 2.7 ) imp l i e s C
( 2)
B
′
(V
y
) = C
( 2)
B
′
(V) . ■
The l as t e qua t ion a bo v e de mon s tr a t e s th a t the r o le of the b as e s B a nd B
′
ca n be in t e r ch a n g e d . F or in-
s t a nc e , one ca n e quiv ale n tly think in t e r m s of the a v e r a g e c o he r e nc e o v e r H a mi lt oni a n ei g e n s t a t e s, qua n t i -
fie d w ith r e spe ct t o the posit ion b asi s .
A p h ysical ly r e le v a n t fa mi ly of unit a r y tr a n sfor m a t ion s U
t
i s the t ime e v o lut ion g e ne r a t e d b y the H a mi l -
t oni a n of a s ys t e m . One ca n, for in s t a nc e , c on side r the t ime-a v e r a g e of C
( 2)
B
′
(U
t
) . F or a H a mi lt oni a n w ith
nonde g e ne r a t e e ne r g y g a ps, the a for e me n t ione d qua n t it y a dmits the clos e d for m ex pr e s sion
C
( 2)
B
′
(U
t
) = 1
2
d
∑
ij
⟨ X
c
i
; X
c
j
⟩
2
+
1
d
∑
i
⟨ X
c
i
; X
c
i
⟩
2
; (2.21)
63
he r e X
c
i
s t a nd s for the c o lumn v e ct or of the tr a n sit ion m a tr i x X
V
, w hi le V =
∑
i
j i⟩⟨ φ
i
j i s the in t e r t w ine r
be t w e e n the H a mi lt oni a n ei g e nb asi s B = fj φ
i
⟩⟨ φ
i
jg a nd B
′
= fj i⟩⟨ ijg
i
. I n fa ct , the r e s ult in g qua n t it y
f
( t ime-a v g)
B
( X
V
) := C
( 2)
B
′
(U
t
) fai l s t o be a g e ne r al i z e d C GP me as ur e . The de t ai l s ca n be found in A ppe nd i x A .
The ide n t i fica t ion be t w e e n e s ca pe pr o b a b i l it y a nd 2- c o he r e nc e g iv e s a p h ysical in t e r pr e t a t ion t o the l a t -
t e r a nd its as s oc i a t e d C GP . M or e impor t a n tly , the e s ca pe pr o b a b i l it y ( or the r e tur n pr o b a b i l it y , P
r e tur n
:=
1P
B
′ ) i s a w e l l -kno w n me as ur e in the the or y of local i za t ion [ 127 , 146 ] a nd the fa ct th a t it ca n be thou gh t
of as c o he r e nc e g iv e s r i s e t o the que s t ion : C a n othe r me as ur e s a r i sin g f r om the r e s our c e the or e t ic f r a me-
w ork of c o he r e nc e g iv e r i s e t o pr o be s of local i za t ion in a simi l a r m a nne r?
I n v ie w of P r oposit ion 2.2 , C GP me as ur e s r e v e al infor m a t ion r e g a r d in g the uni for mit y of the tr a n sit ion
m a tr i x X . H e nc e w he n the l a tt e r i s chos e n t o be be t w e e n the H a mi lt oni a n a nd posit ion ei g e nb as e s, a n y
a br u pt ch a n g e in the o v e rl a p of the t w o b as e s, as for in s t a nc e in the local i za t ion tr a n sit ion, i s ex pe ct e d t o be
de t e ct a b le v i a C GP me as ur e s . I n w h a t fo l lo w s, w e de mon s tr a t e th a t thi s i s inde e d the cas e , b y c on side r in g
A nde r s on a nd MBL .
2.4 Coherenc e - g ener a ting po wer and l o c al iz a tion in the 1- D A nder son m odel
The A nde r s on mode l [ 127 ] in one d ime n sion i s de s cr i be d b y the H a mi lt oni a n
H
W
=
L
∑
i= 1
(j i⟩⟨ i+ 1j+j i+ 1⟩⟨ ij)+
L
∑
i= 1
ε
i
j i⟩⟨ ij (2.22)
o v e r L sit e s ( i . e ., d = L ) w ith pe r iod ic bound a r y c ond it ion s, w he r e the on- sit e e ne r g ie s ε
i
a r e inde pe nde n t
a n d ide n t ical ly d i s tr i but e d ( i .i . d .) r a ndom v a r i a b le s a nd fo l lo w a uni for m d i s tr i but ion of w idth 2W . I t i s
kno w n th a t th e mode l i s local i z e d for a n y de gr e e of d i s or de r W > 0 [ 147 ].
L ocal i za t ion ca n be dy n a mical ly ch a r a ct e r i z e d b y the a bs e nc e of tr a n spor t , a not ion r efe r r in g t o the in-
t e r p l a y be t w e e n the “ posit ion ” b asi s B
′
=fj i⟩⟨ ijg
L
i= 1
in E q . ( 2.22 ) a nd the H a mi lt oni a n ei g e nb asi s B . H e r e ,
w e c on side r c o he r e nc e qua n t i fie d w ith r e spe ct t o the l a tt e r b asi s . L e t us no w ex a mine the be h a v ior of f unc -
t ion al s C
B
(V
W
) , w he r e the unit a r y V
W
i s the in t e r t w ine r be t w e e n H a mi lt oni a n a nd posit ion ei g e nb as e s . I n
fa ct , P r oposit ion 2.4 imme d i a t e ly imp l ie s th a t ⟨ C
( 2)
B
(V
W
)⟩ i s a pr o be t o local i za t ion ( ⟨⟩ de not e s a v e r a g in g
64
o v e r d i s or de r ). M or e spe c i fical ly , local i za t ion imp l ie s th a t in the the r mody n a mic l imit the r e tur n pr o b a b i l -
it y ( a v e r a g e d o v e r d i s or de r ) in the local i z e d p h as e i s non v a ni shin g , i . e ., lim
L!1
⟨j⟨ jj e
iH W t
j j⟩j
2
⟩ > 0 for
a n y W > 0 . I n tur n, thi s i s e quiv ale n t t o P
j
< 1 ( in the the r mody n a mic l imit ) for al l sit e s j , he nc e al s o
lim
L!1
⟨ C
( 2)
B
(V
W
)⟩ < 1 (2.23)
b y E q . ( 2.19 ). N ot ic e th a t H
W
for W > 0 i s g e ne r ical ly nonde g e ne r a t e s o P r op . 2.4 a pp l ie s . W e v e r i f y thi s
cl aim b y n ume r ical sim ul a t ion s ( s e e F i g u r e 2.4.1 a nd F i g ur e 2.4.2 ).
The H a mi lt oni a n H
W= 0
i s de g e ne r a t e in the e r g od ic p h as e , he nc e the in t e r t w ine r V
H W= 0
i s not w e l l de-
fine d . N e v e r the le s s, as w e sho w in A ppe nd i x C , for a n y cho ic e of ei g e nb asi s of H
W
it ho ld s th a t
lim
L!1
C
( 2)
B
(V
W= 0
) = 1 ; (2.24)
he nc e the a v e r a g e c o he r e nc e ⟨ C
( 2)
B
(V
W
)⟩ un a mb i g uously d i s t in g ui she s the t w o be h a v ior s .
The r o le of the qua n t it y C
( 2)
B
(V
W
) mi gh t s e e m spe c i al as a pr o be t o local i za t ion due t o its in t e r pr e t a t ion
as a v e r a g e e s ca pe pr o b a b i l it y . I n fa ct , othe r me as ur e s, a r i sin g f r om a n infor m a t ion-the or e t ic v ie w po in t of
c o he r e nc e , h a v e a n alo g ous pr ope r t ie s . L e t ’ s no w c on side r the r e l a t iv e e n tr op y C GP of the in t e r t w ine r ,
n a me ly C
( rel)
B
(V
W
) . I ts v alue as a f unct ion of the s ys t e m si z e L for d i ffe r e n t v alue s of the d i s or de r s tr e n g th
W i s p lott e d in F i g ur e 2.4.1 . I n the e r g od ic p h as e W = 0 it d iv e r g e s lo g a r ithmical ly
C
( rel)
B
(V
W= 0
) log( L) : (2.25)
Thi s ca n be e asi ly v e r i fie d a n aly t ical ly for a n in t e r t w ine r c onne ct in g t w o m utual ly unb i as e d b as e s, i . e ., for
⟨ ij φ
j
⟩
= 1=
p
L for al l i; j . I n th a t cas e E q . ( 2.25 ) ho ld s w ith e qual it y , as it d ir e ctly fo l lo w s f r om P r op . 2.1.
I n A ppe nd i x C w e sho w th a t the r e s ult a g ain ho ld s in the the r mody n a mic l imit inde pe nde n tly of the spe c i fic
cho ic e for the in t e r t w ine r .
W e no w pr o v ide a nonr i g or ous a r g ume n t t o r e l a t e the a v e r a g e s ⟨ C
( 2)
B
(V
W> 0
)⟩ a nd⟨ C
( rel)
B
(V
W> 0
)⟩ t o the
c or r e spond in g local i za t ion le n g th s ξ
j
. I n the local i z e d p h as e , the ei g e n v e ct or s t y p ical ly de ca y ex pone n t i al ly ,
65
i . e .,
⟨ ij φ
j
⟩
2
c
j
exp
(
j i α
j
j= ξ
j
)
; (2.26)
w he r e α
j
i s the sit e a r ound w hich j φ
j
⟩ i s local i z e d , w hi le due t o the pe r iod ic bound a r y c ond it ion s
i α
j
a bo v e should be unde r s t ood as min(
i α
j
;
i α
j
L
) ). I f one us e s the a n s a tz
( X
V W
)
ji
=
⟨ ij φ
j
⟩
2
= c
j
exp
(
j i α
j
j= ξ
j
)
; (2.27)
the n for L≫ 1
⟨ C
( 2)
B
(V
W> 0
)⟩
= 1
1
L
∑
j
tanh
2
[( 2 ξ
j
)
1
]
tanh( ξ
1
j
)
(2.28a)
a nd
⟨ C
( rel)
B
(V
W> 0
)⟩
=
1
L
L
∑
j= 1
(
[
ξ
j
sinh( 1= ξ
j
)
]
1
ln
(
tanh
[
( 2 ξ
j
)
1
])
)
(2.28b)
( e n tr op y he r e h as n a tur al lo g a r ithm ). A de t ai le d de r iv a t ion ca n be found in A ppe nd i x D .
The ex pr e s sion ( 2.28b ) for ξ
j
≫ 1 ca n be ex p a nde d as
⟨ C
( rel)
B
(V
Γ
)⟩ =
1
L
L
∑
j= 1
(
1+ ln
(
2 ξ
j
)
+ O( ξ
2
j
)
)
;
w hich i s c on si s t e n t w ith the n ume r ical ly o bs e r v e d be h a v ior th a t it r e m ain s finit e in the local i z e d p h as e
w hi le it d iv e r g e s lo g a r ithmical ly as a f unct ion of L in the e r g od ic one .
The a c c ur a c y of e qua t ion s ( 2.28 ) ca n be as s e s s e d b y c omp a r in g w ith cas e s for w hich a n a n aly t ical ex -
pr e s sion ca n be o bt aine d for the local i za t ion le n g th s ξ
j
as a f unct ion of the d i s or de r s tr e n g th . W e no w
c on side r s uch a cas e , de s cr i be d b y a H a mi lt oni a n as in E q . ( 2.22 ), but w ith on- sit e e ne r g ie s th a t fo l lo w a
C a uch y d i s tr i but ion w ith p a r a me t e r Γ a nd v a ni shin g me a n ( al s o kno w n as L lo y d mode l [ 148 ]). W e foc us
66
5 10 50 100 500 1000
0.005
0.010
0.050
0.100
0.500
1
L
1- C
B
(2)
(
W
)
W
10
2
10
1
10
-1
Figure 2.4.1: Log-log plot of the average return p robabilit y 1⟨ C
( 2)
B
(V
W
)⟩ as a function of the sys-
tem size L fo r different values of the diso rder strength W . The system is in the lo calized phase fo r all
W > 0 , since the asymptotic escap e p robabilit y is strictly less that 1 fo r L ! 1 . The numb er of
realizations range from 30 000 fo r small sizes to just 8 fo r the la rgest size . Erro r ba rs rep resent one
standa rd deviation. Entrop y has loga rithm with base 2.
for c oncr e t e ne s s on E q . ( 2.28a ) a nd w e de not e the c or r e spond in g H a mi lt oni a n a nd in t e r t w ine r as H
Γ
a nd
V
Γ
, r e spe ct iv e ly . U t i l i z in g a w e l l -kno w n r e s ult f r om Thoule s s [ 149 ] th a t c onne cts the local i za t ion le n g th
w ith the e ne r g y spe ctr um, one ca n ex pr e s s the R HS of E q . ( 2.28a ) as a f unct ion of the d i s or de r s tr e n g th Γ .
Thi s al lo w s for a d ir e ct c omp a r i s on w ith n ume r ical e v alua t ion s of the me a n ⟨ C
( 2)
B
(V
Γ
)⟩ , y ie ld in g a s ound
a gr e e me n t for s m al l d i s or de r ( Γ < 1 ). W e pr e s e n t the de t ai l s in A ppe nd i x E.
2.5 Coherenc e - g ener a ting po wer an d many - bod y lo c al iz a tion
W e no w tur n t o a d i s or de r e d qua n tum m a n y - body s ys t e m a dmitt in g a p h as e d i a gr a m w ith a n e r g od ic p h as e
a t lo w e nou gh d i s or de r a nd a n MBL p h as e a t s tr on g d i s or de r . F or thi s pur pos e , w e c on side r a tr a n s v e r s e-
fie ld H ei s e nbe r g sp in-1/2 ch ain in a r a ndom m a gne t ic fie ld ( alon g the ^ z a x i s ) o v e r L sit e s ( d = 2
L
) w ith
pe r iod ic bound a r y c ond it ion s, de s cr i be d b y the H a mi lt oni a n
67
5 10 50 100 500 1000
0
2
4
6
8
L
C
B
(rel)
(
W
)
W
5
2
1
1/2
0
Figure 2.4.2: Log-linea r plot of⟨ C
( rel)
B
(V
W
)⟩ as a function of the system size L fo r different value s of
the diso rder strength W . The sys tem is in the lo calized phase fo r all W > 0 , in which the asymptotic
value is finite. In the ergo dic phase ( W = 0 )⟨ C
( rel)
B
(V
W
)⟩ diverges loga rithmically . The numb er of
realizations range from 30 000 fo r small sizes to just 8 fo r the la rgest si ze. Erro r ba rs rep resent one
standa rd deviation. Entrop y has loga rithm with base 2.
68
H
XXX
=
1
2
L
∑
i= 1
[
σ
x
i
σ
x
i+ 1
+ σ
y
i
σ
y
i+ 1
+ σ
z
i
σ
z
i+ 1
]
+ h
x
L
∑
i= 1
σ
x
i
+
L
∑
i= 1
w
i
σ
z
i
; (2.29)
w he r e the h
x
i s the s tr e n g th of the tr a n s v e r s e fie ld a nd the local fie ld s tr e n g th s a r e i .i . d . r a ndom v a r i a b le s w ith
uni for m d i s tr i but ion w
i
2 [ W; W] . N ot ic e th a t the tr a n s v e r s e fie ld br e ak s the r ot a t ion al s y mme tr y of the
H a mi l t oni a n . The mode l h as be e n ex t e n siv e ly s tud ie d n ume r ical ly a nd i s kno w n t o ex hi b it a tr a n sit ion f r om
a n e r g od ic t o a n MBL p h as e a t d i s or de r s tr e n g th W
C
3: 7 ( in the a bs e nc e of the tr a n s v e r s e fie ld t e r m ), s e e
R ef s . [ 146 , 150 ] a nd r efe r e nc e s the r ein . W e no w tur n t o n ume r ical ly v e r i f y th a t the local i za t ion tr a n sit ion
ca n be de t e ct e d thr ou gh the s cal in g of v a r ious of the C GP qua n t it ie s in tr oduc e d e a rl ie r .
S imi l a r t o the A nde r s on H a mi lt oni a n, w e fir s t s tudy the be h a v ior of the C GP ⟨ C
( 2)
B
(V
W
)⟩ a nd⟨ C
( rel)
B
(V
W
)⟩ ,
w he r eV
W
i s the in t e r t w ine r be t w e e n the H a mi lt oni a n a nd the c onfi g ur a t ion sp a c e b asi s, w hich he r e i s
t ak e n t o be the pr oduct
⊗
i
σ
z
i
ei g e nb asi s . W e find a d i s t inct be h a v ior of the qua n t it ie s ⟨ C
( 2)
B
(V
W
)⟩ a nd
⟨ C
( rel)
B
(V
W
)⟩ be t w e e n the e r g od ic a nd MBL p h as e s of the mode l , as al s o hin t e d f r om the n ume r ical r e s ults
in R ef s . [ 151 – 156 ].
F or si z e s u p t o L = 14 , none of the s tud ie d C GP qua n t it ie s s e e m s t o r e a ch a c on s t a n t as y mpt ot ic v alue
as in the A nde r s on cas e . N one the le s s, the ( a v e r a g e ) r e tur n pr o b a b i l it y P
r e tur n
as a f unct ion of the n umbe r
of sp in s L i s c on s i s t e n t w i th a n ex pone n t i al de ca y
P
r e tur n
/ 2
λ
( 2)
W
L
= d
λ
( 2)
W
: (2.30)
The ex tr a po l a t e d r a t e s λ
( 2)
W
, p lott e d in F i g ur e 2.5.1 , a r e clos e t o 1 in the e r g od ic p h as e , w hi le they dr op a t the
tr a n sit ion po in t , o bt ainin g a si gni fica n tly r e duc e d v alue a t the MBL p h as e . On the othe r h a nd , the r e l a t iv e
e n tr op y C GP i s c on si s t e n t w ith a s cal in g
⟨ C
( rel)
B
(V
W
)⟩ = λ
( rel)
W
L+ c on s t; (2.31)
69
w ith a r a t e λ
( rel)
W
th a t i s c los e t o 1 for s m al l d i s or de r a nd dr ops si gni fica n tly in the MBL p h as e .
W e no w tur n t o the g e ne r al i z e d C GP me as ur e s f
( det)
B
a nd f
(1)
B
, w hos e be h a v ior i s al s o c on si s t e n t w ith a
s cal in g
1⟨ f
( x)
B
( X
V W
)⟩/ 2
λ
( x)
W
L
= d
λ
( x)
W
; (2.32)
a nd a r a t e λ
( x)
W
sho w in g d i s t inct be h a v ior in the d i ffe r e n t p h as e s ( x = det or x = 1 ). I n A ppe nd i x F w e
sho w th a t
λ
( det)
W
1
2
λ
( 2)
W
; (2.33)
w hich i s s a tur a t e d for s m al l d i s or de r v alue s a nd i s v e r i fie d b y the o bs e r v e d n ume r ical sim ul a t ion s . E x po -
ne n t i al de ca y i s al s o e nc oun t e r e d for the t ime-a v e r a g e ⟨ f
( t ime-a v g)
B
( X
V W
)⟩ , al s o p lott e d in F i g ur e 2.5.1 . N ot ic e
th a t , althou gh the l a tt e r fai l s t o be a g e ne r al i z e d C GP me as ur e , it ca n s t i l l be e mp lo y e d t o de t e ct the tr a n si -
t ion .
F or w h a t r e g a r d s ⟨ C
( 2)
B
(V
W
)⟩ , w e ca n o bt ain its be h a v ior in the l imit of infinit e d i s or de r a nd in the e r -
g od ic p h as e . F ir s t , w e w r it e the r et u r n pr o b a bi l it y P
r e tur n
:= 1P
B
′ = 1⟨ C
( 2)
B
(V
W
)⟩ as
P
r e tur n
=
1
d
d
∑
i= 1
⟨ ijE(j i⟩⟨ ij)j i⟩ ; (2.34)
w he r eE := ⟨U
t
⟩ i s the a v e r a g e of the qua n tum ( s u pe r ope r a t or ) e v o lut ion U
t
() = e
itH XXX
() e
itH XXX
a nd
j i⟩ de not e s the p r oduct
⊗
i
σ
z
i
( I sin g ) b asi s .
F in al ly , w e c omme n t on our find in gs f r om the t y p ical it y po in t of v ie w . I n C h a pt e r 3 it w as sho w n th a t
i f the in t e r t w ine r i s chos e n a t r a ndom f r om the unit a r y gr ou p V 2 U( d) a c c or d in g t o the H a a r me as ur e ,
the n C
( 2)
B
i s c onc e n tr a t e d ne a r its me a n
⟨ C
( 2)
B
(V)⟩
H a a r
= 1
2
d+ 1
(2.35)
70
λ
W
(time-avg)
λ
W
(rel)
λ
W
(2)
λ
W
(det)
λ
W
( ∞)
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Disorder strength (W)
Figure 2.5.1: Asymptotic b ehavio r fo r the slop e of the follo wing quantities: log
2
(
1⟨ C
( 2)
B
(V
W
)⟩
)
= log
2
( P
return
) ,⟨ C
( rel)
B
(V
W
)⟩ , log
2
(
1⟨ f
( det)
B
( X
V W
)⟩
)
, log
2
(
1⟨ f
( time-avg)
B
( X
V W
)⟩
)
, and
log
2
(
1⟨ f
(1)
B
( X
V W
)⟩
)
fo r la rge L as a function of the diso rder s trength W fo r the Hamiltonian H
XXX
at h
x
= 0: 3 . The slop e w as extracted fo r sizes L = 4; 6; ; 14 , with sample sizes 20 000 , 20 000 ,
20 000 , 8000 , 2000 , 800 ; except at W = 3: 7 , whe re the sample sizes w ere doubled. The erro r ba rs rep re-
sent the standa rd erro r of the linea r fit. Entrop y has loga rithm with base 2.
(⟨⟩
H a a r
de not e s the H a a r a v e r a g e o v e r the in t e r t w ine r ), w ith o v e r w he lmin g pr o b a b i l it y for l a r g e H i l be r t
sp a c e d ime n sion d ( he r e B ca n be a n y fi xe d b asi s ). I n othe r w or d s, the t y p ical r a t e for P
r e tur n
i s λ
( 2)
H a a r
1 .
F r om th a t pe r spe ct iv e , a n e r g od ic be h a v ior i s the t y p ical one , w hi le the MBL cas e ca n be s e e n as a hi ghly
a t y p ical outl ie r .
2.6 D ifferen ti al g eome tr y of c oherenc e - g ener a ting po wer and M BL
I n thi s s e ct ion w e s tudy the be h a v ior of the C GP C
( 2)
B
( δV) w he n the in t e r t w ine r δV c onne cts t w o b as e s
th a t a r e “ infinit e sim al ly clos e ” t o e a ch othe r . Thi s r e s ults in a d i ffe r e n t i al - g e ome tr ic c on s tr uct ion w hos e
71
c e n tr al qua n t it y i s a R ie m a nni a n me tr ic . A s w e w i l l sho w , the r e s ult in g me tr ic
( i ) i s d ir e ctly c onne ct e d t o the dy n a mical c onduct iv it y , w hich i s a qua n t it y of ex pe r ime n t al r e le v a nc e
a nd
( i i ) be h a v e s d i s t inc tly in the MBL a nd e r g od ic p h as e s .
The de t ai le d m a the m a t ical s tr uctur e i s pr e s e n t e d in A ppe nd i x G .
C on side r a c omp le t e or thonor m al fa mi ly of s t a t e s
{
j φ
i
( λ)⟩
}
d
i= 1
, p a r a me tr i z e d b y a s e t of p a r a me t e r s
λ . Thi s i s the r e le v a n t cas e , for in s t a nc e , w he n one s tud ie s the ei g e n v e ct or s as s oc i a t e d w ith a fa mi ly of
H a mi lt oni a n s H( λ) . The infinit e sim al a d i a b a t ic in t e r t w ine r δV i s a unit a r y m a p define d b y
δV
(
j φ
i
( λ)⟩⟨ φ
i
( λ)j
)
=j φ
i
( λ + d λ)⟩⟨ φ
i
( λ + d λ)j ; (2.36)
w he r e H( λ)j φ
i
( λ)⟩ = E
i
( λ)j φ
i
( λ)⟩ .
I t ca n be sho w n th a t the C GP of δV h as the for m C
( 2)
B
( δV) = 2gd λ
2
, w he r e g i s a me tr ic g iv e n b y
g :=
1
d
d
∑
i= 1
χ
i
; (2.37a)
χ
i
:=
⟨
@ φ
i
@ λ
@ φ
i
@ λ
⟩
⟨
φ
i
@ φ
i
@ λ
⟩⟨
@ φ
i
φ
i
φ
i
⟩
; (2.37b)
i . e ., it i s its e l f a me a n of the me tr ic s χ
i
w hich a r e as s oc i a t e d t o the v e ct or s j φ
i
⟩ . W he n the l a tt e r a r e H a mi l -
t oni a n ei g e n s t a t e s, χ
i
a r e kno w n as fi de l it y s uscep t i bi l it ie s [ 157 – 159 ] a nd the gr ound s t a t e s us c e pt i b i l it y χ
0
p l a ys a k ey r o le in the d i ffe r e n t i al g e ome tr ic a ppr o a ch t o qua n tum p h as e tr a n sit ion s [ 160 ].
I n or de r t o c onne ct w ith qua n t it ie s of ex pe r ime n t al r e le v a nc e , le t us no w c on side r the the r m al a n alo g of
the me tr ic g . W e de not e g
T
=
∑
i
p
i
χ
i
, w he r e p
i
= exp( E
i
= T)= Z a r e the the r m al w ei gh ts a nd Z de not e s
the p a r t it ion f unct ion . The qua n t it y g
T
, define d in R ef . [ 161 ] as a g e ne r al i za t ion of the fide l it y s us c e pt i b i l it y
a t finit e t e mpe r a tur e ( g = g
T=1
), ca n be thou gh t of as the me tr ic as s oc i a t e d w ith the the r m al a n alo g of the
C GP C(V; c
( 2)
B
; μ
T
) , w he r e the me as ur e μ
T
w ei gh ts the H a mi lt oni a n ei g e n s t a t e s w ith the as s oc i a t e d G i b bs
w ei gh ts . The qua n t it y g
T
ca n be ex pr e s s e d v i a the ( im a g in a r y p a r t of the ) dy n a mical s us c e pt i b i l it y χ
VV
( ω) ,
72
w he r e V = @
λ
H( λ) . M or e pr e c i s e ly ( s e e R ef . [ 161 ]),
g
T
=
∫
1
0
d ω
π
χ
′′
VV
( ω)
ω
2
coth
(
ω
2T
)
: (2.38)
The a bo v e for m ul a i s r e m a rk a b le , as it de mon s tr a t e s th a t the , a pp a r e n tly a bs tr a ct , qua n t it y C
( 2)
B
( δV) i s sim-
p ly c onne ct e d w ith a qua n t it y me as ur a b le in ex pe r ime n t al s e tu ps [ 162 – 164 ]. W e al s o not e th a t , althou gh
E q . ( 2.38 ) i s not s tr ai gh tfor w a r d ly a pp l ica b le in the infinit e t e mpe r a tur e l imit , in thi s l imit one o bt ain s
g = g
T=1
=
1
π
∫
1
1
σ
VV
( ω)
ω
2
d ω ; (2.39)
w he r e σ
VV
( ω) i s the h i gh-t e mpe r a tur e dy n a mical c onduct iv it y ³ g iv e n b y
σ
VV
( ω) =
2 π
d
∑
n̸= m
j V
n; m
j
2
δ[ ω( E
m
E
n
)]: (2.40)
I n thi s cas e , the r o le of g i s p l a y e d b y the d . c . d ie le ctr ic po l a r i za b i l it y [ 133 , 165 ].
The qua n t it ie s g
T
a nd g not only al lo w t o m ak e c on t a ct w ith ex pe r ime n ts but h a v e al s o be e n s tud ie d in
the c on t ex t of the r m al i za t ion a nd MBL . I n p a r t ic ul a r , it i s be l ie v e d th a t g ! 1 in the the r mody n a mic
l imit , both for the e r g od ic a nd the s ubd i ff usiv e p h as e . I n s t e a d , in the MBL p h as e g ! constant <1
[ 133 ]. I n the l i gh t of E q . ( 2.37 ), the s e r e s ults me a n th a t the C GP of the a d i a b a t ic in t e r t w ine r be t w e e n
ne a r b y H a mi lt oni a n s h as d i s t inct iv e ly d i ffe r e n t be h a v ior s in the e r g od ic a nd in the MBL p h as e s .
2.7 Conc l u sion and ou t lo ok
I n thi s C h a pt e r w e h a v e br ou gh t t o g e the r ide as f r om qua n tum infor m a t ion a nd g e ome tr y , on one h a nd ,
a nd the p h ysic s of d i s or de r e d s ys t e m s on the othe r . W e e s t a b l i she d a c onne ct ion be t w e e n the qua n t it a t iv e
a ppr o a ch t o c o he r e nc e , or i g in a t in g f r om the pe r spe ct iv e of qua n tum r e s our c e the or ie s [ 125 , 126 ], a nd
local i za t ion [ 127 , 129 – 131 ].
³ The n a me dy n a mical c onduct iv it y c ome s f r om its us e w he n χ
VV
i s the ( ch a r g e ) c ur r e n t - c ur r e n t c or r e l a t ion .
73
M or e spe c i fical ly , w e s tud ie d the be h a v ior of the e r g od ic, A nde r s on, a nd m a n y - body local i z e d p h as e s in
t e r m s of the s cal in g pr ope r t ie s of c o he r e nc e a v e r a g e s th a t a r e as s oc i a t e d t o the in t e r t w ine r c onne ct in g the
H a mi lt oni a n ei g e n v e ct or s w ith the c onfi g ur a t ion sp a c e b asi s . The in tr oduc e d qua n t it ie s a r e a b le t o de t e ct
the uni for mit y of the tr a n sit ion m a tr i x c onne ct in g the t w o b as e s, he nc e they ca n s e n s e a br u pt ch a n g e s in
the e n t ir e s e t of e ne r g y ei g e n s t a t e s, si gn al in g the local i za t ion tr a n sit ion . The l a tt e r pr ope r t y i s g ua r a n t e e d
b y the s tr uctur e of c o he r e nc e monot one s .
F ur the r mor e , w e bui lt a n as s oc i a t e d d i ffe r e n t i al - g e ome tr ic v e r sion for infinit e sim al pe r tur b a t ion s of the
H a mi lt oni a n, a nd sho w e d th a t the r e s ult in g R ie m a nni a n me tr ic ca n be m a ppe d on t o kno w n p h ysical qua n-
t it ie s w hich h a v e a sh a r p ly d i s t inct be h a v ior in the e r g od ic a nd in the MBL p h as e s .
I n v e s t i g a t in g ho w d i ffe r e n t r e pr e s e n t a t iv e s of the in tr oduc e d fa mi ly of me as ur e s ca n ex tr a ct v a r ious p h ys -
ical fe a tur e s r e g a r d in g the n a tur e of the local i za t ion tr a n sit ion r e m ain s a d ir e ct ion for f utur e r e s e a r ch .
A ppendic e s
A T ime - a v er a ged C GP
I n thi s s e ct ion w e s tudy the t ime-a v e r a g e of the C GP C
( 2)
B
′
(U
t
) , w he r eU
t
( X) = exp( iHt) X exp( iHt) i s
the t ime e v o lut ion ope r a t or . F or the fo l lo w in g , w e w i l l as s ume th a t the H a mi lt oni a n H =
∑
i
E
i
j φ
i
⟩⟨ φ
i
j
s a t i sfie s the n o n r e so n a n ce co n d it io n , i . e ., its e ne r g y g a ps a r e nonde g e ne r a t e . U nde r thi s as s umpt ion, w e w i l l
sho w th a t
C
( 2)
B
′
(U
t
) = 1
2
d
∑
ij
⟨ X
c
i
; X
c
j
⟩
2
+
1
d
∑
i
⟨ X
c
i
; X
c
i
⟩
2
(2.41)
w he r e V =
∑
i
j i⟩⟨ φ
i
j i s the in t e r t w i ne r be t w e e n B =f Π
i
:=j φ
i
⟩⟨ φ
i
jg
i
a nd B
′
=f P
i
:=j i⟩⟨ ijg
i
.
74
W e h a v e ,
C
( 2)
B
′
(U
t
) = 1
1
d
∑
i
⟨D
B
′U
t
( P
i
);D
B
′U
t
( P
i
)⟩
= 1
1
d
∑
i
⟨ P
i
;U
y
t
D
B
′U
t
( P
i
)⟩
= 1
1
d
∑
ijkk
′
ll
′
[
exp[ i( E
k
E
k
′ + E
l
E
l
′) t]
Tr
(
P
i
Π
k
P
j
Π
k
′ P
i
Π
l
P
j
Π
l
′
)
]
:
The nonr e s on a nc e c ond it ion imp l ie s th a t
exp[ i( E
k
E
k
′ + E
l
E
l
′) t] = δ
kk
′ δ
ll
′ + δ
kl
′ δ
k
′
l
δ
kk
′ δ
k
′
l
δ
ll
′ :
A s tr ai g h tfor w a r d calc ul a t ion g iv e s
C
( 2)
B
′
(U
t
) = 1
1
d
(
2
∑
ijkl
( X
V
)
ki
( X
V
)
kj
( X
V
)
li
( X
V
)
lj
∑
ijk
( X
V
)
2
ki
( X
V
)
2
kj
)
w hich r e duc e s t o E q . ( 2.41 ).
A n e as y calc ul a t ion for a sin gle qub it r e v e al s th a t f
( t ime-a v g)
B
( X) i s not a g e ne r e l i z e d C GP me as ur e , sinc e
its m a x im um v alue i s not a tt aine d o v e r the tr a n sit ion m a tr i x w ith e le me n ts X
ij
= 1= 2 .
B Inv er se p a r ti c i p a ti o n r a ti o , effec tiv e d imen si o n , a nd L osc hmi d t ec h o
F or a nonde g e ne r a t e H a mi lt oni a n H =
∑
i
E
i
j φ
i
⟩⟨ φ
i
j , the e s ca pe pr o b a b i l it y P
ψ
i s d ir e ctly c onne ct e d
w ith the s e c ond P a r t ic ip a t ion R a t io of j ψ⟩ o v e r the H a mi lt oni a n ei g e nb asi s PR
2
:=
∑
i
⟨ φ
i
j ψ⟩
4
asP
ψ
=
1 PR
2
.
The s e c ond P a r t ic ip a t ion r a t io , in tur n, i s in t im a t e ly c onne ct e d t o t w o othe r qua n t it ie s of p h ysical in t e r e s t
in the s tudy of e qui l i br a t ion a nd the r m al i za t ion, n a me ly the eff e c t i v e d i m en sio n a nd the Lo sc h m i d t e c h o [ 142 ,
75
143 ]. The effe ct iv e d ime n sion of a qua n tum s t a t e i s define d as its in v e r s e pur it y ,
d
eff
( ρ) =
1
Tr[ ρ
2
]
; (2.42)
w hich in tuit iv e ly c or r e spond s t o the n umbe r of pur e s t a t e s th a t c on tr i but e t o the ( in g e ne r al ) mi xe d s t a t e
ρ . G iv e n a nonde g e ne r a t e H a mi lt oni a n, it i s e as y t o sho w th a t the effe ct iv e d ime n sion of the ( infinit e )
t i me-a v e r a g e d s t a t e i s e qual t o the in v e r s e of the s e c ond P a r t ic ip a t ion r a t io , th a t i s,
d
eff
( ρ) =
1
Tr( ρ
2
)
=
1
∑
i
⟨
φ
i
j ψ
⟩
4
=
1
PR
2
; (2.43)
w he r e ρ =j ψ⟩⟨ ψj .
R e cal l th a t the L os chmidt e cho i s define d as the o v e rl a p be t w e e n the init i al s t a t e j ψ⟩ a nd the s t a t e a ft e r
t ime t ,
L
t
:=
⟨ ψj e
iHt
j ψ⟩
2
: (2.44)
the infinit e t ime a v e r a g e of w hich ca n be ide n t i fie d w ith the r et u r n pr o b a bi l it y of the s t a t e j ψ⟩ . The n, in the
nonde g e ne r a t e cas e , the t ime-a v e r a g e d L os chmidt e cho i s r e l a t e d t o the s e c ond P a r t ic ip a t ion r a t io a nd the
effe ct iv e d ime n sion as
L
t
= PR
2
=
1
d
eff
( ρ)
: (2.45)
W e al s o not e th a t the L os chmidt e cho a ppe a r s n a tur al ly in the s tudy of the w ork d i s tr i but ion [ 166 ], a qua n-
t it y of the r mody n a mic impor t a nc e .
C C GP i n th e A nd er so n mo d el f o r th e d egener a te c a se W = 0
The spe ctr um of A nde r s on H a mi lt oni a n E q . ( 2.22 ) for the d i s or de r - f r e e cas e i s de g e ne r a t e , he nc e the
in t e r t w ine r V
W= 0
be t w e e n the posit ion a nd H a mi lt oni a n ei g e nb as e s i s not unique ly define d . N e v e r the-
le s s, w e sho w he r e th a t the be h a v ior of the qua n t it ie s C
( 2)
B
(V
W= 0
) a nd C
( rel)
B
(V
W= 0
) in the the r mody n a mic
76
l imit i s inde pe nde n t of the spe c i fic cho ic e of the H a mi lt oni a n ei g e nb asi s, n a me ly C
( 2)
B
(V
W= 0
) ! 1 w hi le
C
( rel)
B
(V
W= 0
) log( L) for L!1 .
The spe ctr um of the H a mi lt oni a n i s
{
2 cos
(
2 π j
L
)}
L 1
j= 0
, he nc e the r e a r e n
L
d i s t inct t w o - d ime n sion al de-
g e ne r a t e s ubsp a c e s, w he r e n
L
= ( L 2)= 2 for L e v e n a nd n
L
= ( L 1)= 2 for L odd . I n v o k in g the F our ie r
ei g e nb asi s
j φ
k
⟩ =
1
p
L
L 1
∑
j= 0
exp
(
i
2 π jk
L
)
j j⟩ (2.46)
as r efe r e nc e , the g e ne r al ei g e nb asi s of H
W= 0
m a y d i ffe r f r om b asi s ( 2.46 ) as
j φ
′
k
⟩ = e
i γ
k
(
e
i α
k
cos( θ
k
)j φ
k
⟩+ e
i β
k
sin( θ
k
)j φ
L k
⟩
)
(2.47a)
j φ
′
L k
⟩ = e
i γ
k
(
e
i β
k
sin( θ
k
)j φ
k
⟩+ e
i α
k
cos( θ
k
)j φ
L k
⟩
)
(2.47b)
for k = 1;:::; n
L
, w he r e the a n gle s
{
α
k
; β
k
; γ
k
; θ
k
}
spe c i f y the ( unit a r y ) tr a n sfor m a t ion w ithin the k th
t w ofo ld de g e ne r a t e s ubsp a c e .
A s tr ai gh tfor w a r d calc ul a t ion g iv e s
⟨ lj φ
′
k
⟩
2
=
⟨ lj φ
′
L k
⟩
2
=
1
L
[
1+ cos
(
2( L 2k) l π
L
+ α
k
β
k
)
sin( 2θ
k
)
]
: (2.48)
f r om w hich one ca n d ir e ctly s e e th a t the pos si b le H a mi lt oni a n ei g e nb as e s d i ffe r in the s um
∑
i; j
( X
U
)
2
ji
a t mos t of a n or de r 1 t e r m . H e nc e , f r om E q . ( 2.9 ) it fo l lo w s th a t a n y s uch c on tr i but ion v a ni she s a t the
the r mody n a mic l imit , y ie ld in g C
( 2)
B
(V
W= 0
)! 1 .
F or C
( rel)
B
(V
W= 0
) , w e fir s t in v o k e the s t a nd a r d ine qual it y be t w e e n the S h a nnon e n tr op y a nd the pur it y
H(f p
i
g) log(
∑
i
p
2
i
) ( fo l lo w in g f r om the monot onic it y of the R é n y i e n tr op ie s [ 167 ]). B y the us e of
E q . ( 2.48 ), the pur it y of the pr o b a b i l it y d i s tr i but ion
{
⟨ lj φ
′
k
⟩
2
}
L
l= 1
i s
L
∑
l= 1
⟨ lj φ
′
k
⟩
4
=
2+ sin
2
( 2θ
k
)
2L
;
77
the r efor e the pr e v ious ine qual it y imp l ie s
H
({
⟨ lj φ
′
k
⟩
2
}
l
)
log L log
(
2+ sin
2
( 2θ
k
)
2
)
:
F in al ly , thi s imp l ie s b y E q . ( 2.8 ) th a t C
( rel)
B
(V
W= 0
) d iv e r g e s lo g a r ithmical ly w ith L for a n y cho ic e of the
H a mi lt oni a n ei g e nb asi s .
D D er iv a ti o n o f Eqs . ( 2.28 )
I n thi s s e ct ion w e sho w ho w usin g the a n s a tz ( X
V W
)
ji
= c
j
exp
(
j i α
j
j= ξ
j
)
, one ca n de r iv e E qs . ( 2.28 ).
A s s umin g pe r iod ic bound a r y c ond it ion s as in the m ain t ex t , a nd sinc e
∑
i
( X
V W
)
ji
= 1 , the c oeffic ie n ts
c
j
ca n be ex pr e s s e d for L≫ 1 as
( c
j
)
1
2
1
∑
x= 0
e
x= ξ
j
1
the r efor e
c
j
= tanh[( 2 ξ
j
)
1
] : (2.49)
F r om E q . ( 2.7 ),
C
( 2)
B
= 1
1
L
∑
ij
( X
V W
)
2
ij
= 1
1
L
∑
j
tanh
2
[( 2 ξ
j
)
1
]
tanh( ξ
1
j
)
;
w hich i s ( 2.28a ).
S imi l a rly , f r om E q . ( 2.8 ) w e h a v e
H( X
V W> 0
) =
1
L
L
∑
i; j= 1
c
j
e
j i α jj= ξ
j
ln
[
c
j
e
j i α jj= ξ
j
]
=
1
L
∑
j
(
ln c
j
c
j
∑
i
e
j i α jj= ξ
j
i α
j
ξ
j
)
:
78
The s um
∑
i
for L≫ 1 i s
L
∑
i= 1
e
j i α jj= ξ
j
i α
j
ξ
j
2
1
∑
x= 1
e
x= ξ
j
x
ξ
j
=
2
ξ
j
d
d( ξ
j
)
1
1
∑
x= 1
e
x= ξ
j
= 2
e
1= ξ
j
(
e
1= ξ
j
1
)
2
ξ
j
:
U s in g E q . ( 2.49 ) t o g e the r w ith the a bo v e , w e g e t t o t he de sir e d for m ( 2.28b ).
E E v a l u a ti o n o f Eq . ( 2.28a ) f o r o n - site ener gi e s f o llo wi n g C a uc hy d is tr i b u ti o n
W e c on side r the H a mi lt oni a n ( 2.22 ) w ith i .i . d . on- sit e e ne r g ie s ε
i
, d i s tr i but e d a c c or d in g t o the C a uch y
d i s tr i but ion
f
Γ
( ε) =
1
π Γ
[
Γ
2
ε
2
+ Γ
2
]
: (2.50)
The local i za t ion le n g th ξ( E; Γ) ca n be calc ul a t e d b y in v o k in g the for m ul a due t o Thoule s s [ 149 ], w hich in
our not a t ion i s
cosh
(
1
2 ξ( E; Γ)
)
=
√
( 2+ E)
2
+ Γ
2
+
√
( 2 E)
2
+ Γ
2
4
: (2.51)
T o e v alua t e E q . ( 2.28a ) for thi s mode l in the the r mody n a mic l imit , w e tr a n sit ion t o the c on t in uum l imit
1
L
∑
j
g( E
j
)7!
∫
dE ρ
Γ
( E) g( E) . The de n sit y of s t a t e s ρ
Γ
( E) ca n be o bt aine d e asi ly f r om the c or r e spond in g
r e s o lv e n t , calc ul a t e d for the L lo y d mode l in R ef . [ 148 ], a nd E q . ( 2.51 ). The r e s ult in g in t e gr al i s n ume r ical ly
e v alua t e d a nd y ie ld s the d a t a p lott e d in F i g ur e 2.7.1 .
F Comp a r iso n o f t w o me a s u r e s
I n thi s s e ct ion, w e w i l l sho w th a t
P
r e tur n
= 1 C
( 2)
B
(V)
(
1 f
( det)
B
( X
V
)
)
2
: (2.52)
79
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Γ
C
B
(2)
(
Γ
)
Figure 2.7.1: Plot of the escap e p robabilit y⟨ C
( 2)
B
(V
Γ
)⟩ as a function of the diso rder strength Γ fo r
the Llo yd mo del Hamiltonian H
Γ
, as p redicted analytically b y the heuristic Eq. ( 2.28a ) (solid line) and
the numerical simulations (p oints). F o r the case of the numerical simulation, L ! 1 is extrap olated
b y averaging over diso rder fo r sizes up to L = 2
12
. Standa rd deviations a re within the p oint radius.
I nde e d ,
1 C
( 2)
B
(V) =
1
d
∥ X
V
∥
2
2
=
1
d
∑
i
s
2
i
(
1
d
∑
i
s
i
)
2
2
4
(
∏
i
s
i
) 1
d
3
5
2
=
(
1 f
( det)
B
( X
V
)
)
2
;
w he r e s
i
de not e s the sin g ul a r v alue s of X
V
. The fir s t e qual it y fo l lo w s f r om the c on v ex it y of the me a n a nd
the s e c ond one f r om the s t a nd a r d ine qual it y be t w e e n the a r ithme t ic a nd g e ome tr ic me a n .
The ine qual it y for the r a t e s E q . ( 2.33 ) fo l lo w s b y p lu gg in g in t o the ine qual it y ( 2.52 ) the for m s ( 2.30 )
a nd ( 2.32 ).
G Co h er en c e - Gener a ti n g P o wer a nd d is t a n c e i n th e Gr a ssma nni a n
H e r e w e pr e s e n t in mor e de t ai l the unde rly in g d i ffe r e n t i al - g e ome tr ic s tr uctur e th a t i s in tr oduc e d in s e c -
t ion 2.6 .
80
L e tH de not e the finit e d ime n sion al H i l be r t sp a c e of the qua n tum s ys t e m a nd B(H) the as s oc i a t e ope r -
a t or al g e br a . The s e t B(H) e quippe d w ith the H i l be r t - S chmidt s cal a r pr oduct ⟨ X; Y⟩ := Tr
(
X
y
Y
)
tur n s
in t o a H i l be r t sp a c e ( the sp a c e of H i l be r t - S chmidt ope r a t or s ) th a t w e w i l l de not e b y H
HS
. S u pe r ope r a t or s
O m a pp in gH
HS
in t o its e l f ca n be the n e ndo w e d w ith the fo l lo w in g nor m
∥O∥
HS
:=
√
Tr
HS
(O
y
O) ; (2.53)
w he r e
( a ) O
y
de not e s the H i l be r t - S chmidt c on ju g a t e of O , i . e .,⟨O( X); Y⟩ =⟨ X;O
y
( Y)⟩8 X; Y2H
HS
.
( b ) I ffj i⟩g
d
i= 1
i s a n y or thonor m al b asi s of H , one define s Tr
HS
O :=
∑
d
i; j= 1
⟨j i⟩⟨ jj;O(j i⟩⟨ jj)⟩ .
A s w e d i s c us s e d in the m ain t ex t , in s t e a d of in v o k in g or thonor m al s e que nc e s of k e ts fj i⟩g
d
i= 1
, it i s mor e
c on v e nie n t t o w ork w ith s e ts of or tho g on al , r a nk -1 pr oj e ct ion ope r a t or s B = f P
i
:=j i⟩⟨ ijg
d
i= 1
. L e t us
in tr oduc e the sp a c e of al l s uch s e ts o v e r the H i l be r t sp a c e , w hich w e de not e as M(H) . Thi s i s e s s e n t i al ly
the s e t of al l pos si b le or thonor m al b as e s o v e r the H i l be r t sp a c e onc e the p h as e de gr e e s of f r e e dom a nd
or de r in g h a v e be e n modde d out [ 132 ]. The e le me n ts B2M(H) a r e in one-t o - one c or r e sponde nc e w ith
the s e t of de p h asin g s u pe r - ope r a t or s, i . e ., the m a p B 7! D
B
( define d in E q . ( 2.2 )) i s in j e ct iv e . G iv e n a
B2M
d
, the c or r e spond in g s e t of B - d i a g on al ope r a t or s i s
A
B
:= Spanf P
i
g
d
i= 1
H
HS
; (2.54)
w hich i s al s o the r a n g e of the B - de p h asin g s u pe r ope r a t or D
B
: One ca n s e e th a t E q . ( 2.54 ) a ctual ly define s
a m a x im al ly a be l i a n s ub al g e br a ( M A SA) of H
HS
; mor e o v e r it ca n be pr o v e n th a t the s e t of M A SA s of H
HS
ca n be ide n t i fie d w ith M(H) ( s e e R ef . [ 132 ] for a pr oof ). I n thi s w a y , the s e t M(H) ca n be no w s e e n as a
s ubs e t of the Gr as s m a nni a n m a ni fo ld of d - d ime n sion al s ubsp a c e s of H
HS
. The a dv a n t a g e of thi s a ppr o a ch
i s th a t M(H) d ir e ctly inhe r its the n a tur al me tr ic s tr uctur e of the Gr as s m a nni a n
D(A
B
;A
B
′) :=∥D
B
D
B
′∥
HS
: (2.55)
81
W e w i l l no w c onne ct the s e c onc e pts t o the 2- C GP of unit a r y qua n tum m a ps .
F r om its definit ion, C
( 2)
B
(U) s e e m s t o ca ptur e s ome not ion of s e p a r a t ion be t w e e n the s e ts B =f P
i
g
d
i= 1
a nd B
′
= fU ( P
i
)g
d
i= 1
. I n fa ct , the B - c o he r e nc e g e ne r a t in g po w e r of a unit a r y m a p U i s pr opor t ion al t o
the ( s qua r e of the ) Gr as s m a nni a n d i s t a nc e be t w e e n the input B - d i a g on al al g e br a A
B
a nd its im a g e unde r
U [ 132 ]. F or m al ly:
C
( 2)
B
(U) =
1
2d
D(A
B
;U(A
B
))
2
: (2.56)
w he r e the d i s t a nc e f unct ion D i s g iv e n b y ( 2.55 ). The m a x im um of thi s f unct ion i . e ., max
U
C
( 2)
B
(U) =
1 1= d i s a chie v e d for unit a r y ope r a t or s U th a t c onne ct e d m utual ly unb i as e d b as e s, n a me ly j⟨ ij Uj j⟩j = 1= d
(8 i; j ), a nd c or r e spond s t o a m a x im um d i s t a nc e o v e r M(H) g iv e n b y D
max
=
√
2( d 1):
I t i s impor t a n t t o s tr e s s th a t , in the l i gh t of P r oposit ion 2.4, the Gr as s m a nni a n d i s t a nc e be t w e e n M A SA s
i s e ndo w e d w ith a p h ysical me a nin g in the c on t ex t of qua n tum me ch a nic s .
W e no w tur n t o e s t a b l i sh a c onne ct ion be t w e e n the d i ffe r e n t i al s tr uctur e of M(H) , as induc e d b y the
d i s t a nc e f unct ion ( 2.55 ), a nd MBL . One h as the n a tur al R ie m a nni a n me tr ic o v e r the Gr as s m a nni a n
ds
2
= D( Π; Π + d Π)
2
= Tr( d Π
2
) (2.57)
( Π de not e the pr oj e ct or s o v e r the d - d ime n sion al s ubsp a c e s c ompr i sin g the Gr as s m a nni a n ). The l a tt e r ,
in v ie w of E q . ( 2.56 ), h as in tur n the p h ysical in t e r pr e t a t ion as the C
( 2)
B
of the unit a r y as s oc i a t e d w ith a n
infinit e sim al tr a n sfor m a t ion
{
j φ
i
( λ)⟩
}
d
i= 1
7!
{
j φ
i
( λ + d λ)⟩
}
d
i= 1
. The for m of the me tr ic ( 2.37 ) fo l lo w s
d ir e ctly b y the calc ul a t ion of P r oposit ion 6 in R ef . [ 132 ].
H D e t a i l s o f th e n umer i c a l c a lc u l a ti o n s f o r MBL
I n thi s s e ct ion, w e l i s t f ur the r de t ai l s of the qua n t it ie s s tud ie d a cr os s the e r g od ic - MBL tr a n sit ion, n a me ly 1
⟨ C
( 2)
B
(V
W
)⟩ ,⟨ C
( rel)
B
(V
W
)⟩ ,⟨ f
( t ime-a v g)
B
( X
V W
)⟩ , a nd⟨ f
( de t)
B
( X
V W
)⟩ . I n F i g ur e 2.5.1 , w e p lot the ex tr a po l a t e d
r a t e s ( for l a r g e L ) as a f unct ion of the d i s or de r s tr e n g th W for the H a mi lt oni a n H
XXX
a t h
x
= 0: 3 . F or thi s
82
4 6 8 10 12
0.0
0.2
0.4
0.6
0.8
1.0
Lattice size (L)
C
B
(2)
(
W
)
(a)
4 6 8 10 12
2
4
6
8
10
Lattice size (L)
C
B
(rel)
(
W
)
(b)
Figure 2.7.2: Plot of the (a) average escap e p robabilit y⟨ C
( 2)
B
(V
W
)⟩ and (b)⟨ C
( rel)
B
(V
W
)⟩ as a func-
tion of the system size L fo r different values of the diso rder strength W . The diso rder values displa y ed
a re W = 0: 4; 1: 0; 1: 4; 1: 8; 2: 5; 3: 1; 3: 7; 5: 0; 7: 0; 9: 0 (monotonically from the top to b ottom in the plots) fo r
L = 4; 6; ; 12 with sample sizes 20 000 , 20 000 , 20 000 , 8000 , 2000 ; except at the W = 3: 7 , where the
sample si zes w ere doubled. Erro r ba rs rep resent one standa rd deviation. Entrop y has loga rithm with
base 2.
pur pos e w e c on side r , e . g., for the r e tur n pr o b a b i l it y 1⟨ C
( 2)
B
(V
W
)⟩ a n a n s a tz of the for m
g( L) = α + 2
λ L
; (2.58)
w he r e α i s the as y mpt ot ic v alue a nd λ i s the r a t e of de ca y w ith s ys t e m si z e L . B y pe r for min g a nonl ine a r
fit a t d i ffe r e n t d i s or de r v alue s for the v a r ious qua n t it ie s l i s t e d a bo v e , w e found th a t the α 0 (w ithin the
unc e r t ain t y of the fitt in g p a r a me t e r s ), e v e n for the l a r g e s t d i s or de r th a t w e c on side r ( W = 9: 0 ). The r efor e ,
w e simp l i f y our a n s a tz t o the for m g( L)/ 2
λ L
a nd ex tr a ct the as y mpt ot ic r a t e s b y t ak in g the lo g a r ithm of
the d e sir e d qua n t it ie s .
I n F i g ur e s 2.7.2 a nd 2.7.3 w e p lot our d a t a for a s a mp le of d i s or de r v alue s a nd for s ys t e m si z e s L =
4; ; 12 . E r r or b a r s r e pr e s e n t one s t a nd a r d de v i a t ion .
83
4 6 8 10 12
0.0
0.2
0.4
0.6
0.8
1.0
Lattice size (L)
f
B
(time-avg)
X
W
(a)
4 6 8 10 12
0.0
0.2
0.4
0.6
0.8
1.0
Lattice size (L)
f
B
(det)
X
W
(b)
4 6 8 10 12
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Lattice size (L)
f
B
( ∞)
X
W
(c)
Figure 2.7.3: Plot of the generalized-CGP measures, (a)⟨ f
( time-avg)
B
( X
V W
)⟩ , (b)⟨ f
( det)
B
( X
V W
)⟩ ,
and (c)⟨ f
(1)
B
( X
V W
)⟩ as a funct ion of the system size L fo r different values of the diso rder strength
W . The diso rder values displa y ed a re W = 0: 4; 1: 0; 1: 4; 1: 8; 2: 5; 3: 1; 3: 7; 5: 0; 7: 0; 9: 0 (monotonically
from the top to b ottom fo r (a) and b ottom to top fo r (b)) fo r L = 4; 6; ; 12 with sample sizes
20 000; 20 000; 20 000; 8000; 2000 ; except at the W = 3: 7 , where the sample sizes w ere doubled. Erro r
ba rs rep resent one standa rd deviation.
84
3
I nfor m a tion S cr a mblin g o v e r B i p a r titions:
E qui li br a tion, E n tr op y P r o duction, a nd T y pic alit y
3.1 A bs tr a c t
I n r e c e n t y e a r s, the out - of -t ime- or de r c or r e l a t or ( O T O C ) h as e me r g e d as a d i a gnos t ic t oo l for infor m a t ion
s cr a mb l in g in qua n tum m a n y - body s ys t e m s . H e r e , w e pr e s e n t ex a ct a n aly t ical r e s ults for the O T O C for a
t y p ical p air of r a ndom local ope r a t or s s u ppor t e d o v e r t w o r e g ion s of a b ip a r t it ion . Quit e r e m a rk a b ly , w e
sho w th a t thi s “ b ip a r t it e O T O C” i s e qual t o the ope r a t or e n t a n gle me n t of the e v o lut ion a nd w e de t e r mine
its in t e r p l a y w ith e n t a n gl in g po w e r . F ur the r mor e , w e c omput e lon g -t ime a v e r a g e s of the O T O C a nd r e-
85
v e al their c onne ct ion w ith ei g e n s t a t e e n t a n gle me n t. F or H a mi lt oni a n s ys t e m s, w e unc o v e r a hie r a r ch y of
c on s tr ain ts o v e r the s tr uctur e of the spe ctr um a nd e luc id a t e ho w thi s a ffe cts the e qui l i br a t ion v alue of the
O T O C . F in al ly , w e pr o v ide ope r a t ion al si gni fica nc e t o thi s b ip a r t it e O T O C b y unr a v e l in g in t im a t e c on-
ne ct ion s w ith a v e r a g e e n tr op y pr oduct ion a nd s cr a mb l in g of infor m a t ion a t the le v e l of qua n tum ch a nne l s .
T ex t for thi s C h a pt e r i s a d a pt e d f r om [ 168 ].
I n tr o d u c t io n.— A ch a r a ct e r i s t ic fe a tur e of c e r t ain qua n tum m a n y - body s ys t e m s i s their a b i l it y t o quick ly
spr e a d “ local i z e d ” infor m a t ion o v e r s ubs ys t e m s, the r e b y m ak in g it in a c c e s si b le t o local o bs e r v a b le s . A l -
thou gh unit a r y e v o lut ion r e t ain s al l infor m a t ion, thi s local in a c c e s si b i l it y m a ni fe s ts its e l f as e qui l i br a t ion
in clos e d s ys t e m s, a nd h as be e n t e r me d “ infor m a t ion s cr a mb l in g ” [ 169 – 173 ].
F or H a mi lt oni a n qua n tum dy n a mic s, s cr a mb l in g ca n be pr o be d b y ex a minin g the o v e rl a p of a t ime-
e v o lv e d local ope r a t or V( t) := U
y
t
VU
t
w ith a s e c ond s t a t ic ope r a t or W . Thi s o v e rl a p i s c ommonly qua n t i -
fie d v i a the s tr e n g th of the c omm ut a t or ¹
C
V; W
( t) :=
1
2
Tr
(
[ V( t); W]
y
[ V( t); W] ρ
β
)
(3.1)
w he r e ρ
β
de not e s the the r m al s t a t e a t in v e r s e-t e mpe r a tur e β . F r om the pe r spe ct iv e of infor m a t ion spr e a d -
in g , C
V; W
( t) i s a n a tur al qua n t it y t o c on side r sinc e it c on s t itut e s a s t a t e- de pe nde n t v a r i a n t of the L ie b -
R o b in s on s che me; the l a tt e r e nfor c e s a f und a me n t al r e s tr ict ion on the spe e d of c or r e l a t ion s spr e a d in g in
non- r e l a t iv i s t ic qua n tum s ys t e m s [ 174 – 177 ]. I n E q . ( 3.1 ), it i s c on v e nie n t t o c on side r p air s of ope r a t or s
V; W w hich a t t = 0 a ct non tr iv i al ly on d i ffe r e n t s ubs ys t e m s, th us c omm ut e; w e fo l lo w thi s c on v e n t ion
he r e .
The c omm ut a t or C
V; W
( t) i s in t im a t e ly l ink e d t o the out - of -t ime- or de r c or r e l a t or ( O T O C ) [ 178 , 179 ]
w hich i s a 4-po in t f unct ion w ith a n unc on v e n t ion al t ime- or de r in g
F
V; W
( t) := Tr
(
V
y
( t) W
y
V( t) W ρ
β
)
: (3.2)
The c onne ct ion be t w e e n the t w o a r i s e s w he n V; W a r e unit a r y; E q . ( 3.1) the n imme d i a t e ly r e duc e s t o
¹ I n fa ct , C
V; W
( t) =
1
2
[
V( t); W
]
2
for the nor m as s oc i a t e d w ith the inne r pr oduct ⟨ X; Y⟩
β
= T r
(
X
y
Y ρ
β
)
, β <1 .
86
C
V; W
( t) = 1 Re[ F
V; W
( t)] . I n thi s ch a pt e r w e foc us on the infinit e t e mpe r a tur e , β = 0 cas e .
Thr ou gh the y e a r s, s e v e r al k ey si gn a tur e s of qua n tum ch a os [ 180 – 183 ] h a v e be e n in tr oduc e d . The init i al
ex pone n t i al gr o w th of the O T O C w as pr opos e d as a d i a gnos t ic of qua n tum ch a os [ 184 – 191 ]. H o w e v e r , a
ca r ef ul a n alysi s h as r e v e ale d th a t infor m a t ion s cr a mb l in g doe s not alw a ys ne c e s sit a t e ch a os [ 19 – 24 ].
P e r s e , the O T O C’ s a b i l it y t o pr o be dy n a mical fe a tur e s cle a rly de pe nd s on the cho ic e of ope r a t or s V; W .
H o w e v e r , it i s de sir a b le t o be a b le t o ca ptur e the s e fe a tur e s as inde pe nde n tly as pos si b le f r om the spe c i fic
cho ic e of ope r a t or s . Thi s in s e n sit iv it y ca n be a chie v e d b y a v e r a g in g o v e r a s e t of ope r a t or s, a s tr a t e g y al s o
c on side r e d in R ef s . [ 190 , 192 – 197 ]. I t i s cr uc i al t o r e m a rk th a t for the a v e r a g e d O T O C t o faithf ul ly ca ptur e
infor m a t ion spr e a d in g , the a v e r a g in g pr oc e s s m us t pr e ser v e t h e i n it i a l lo c a l it y of t h e sy s t em , i . e ., w hich s ub -
s ys t e m s V; W init i al ly a ct u pon — a n o bs e r v a t ion th a t w as quin t e s s e n t i al in r e v e al in g the c or r e ct be h a v ior
of the O T O C a nd its c onne ct ion w ith L os chmidt e cho [ 197 ].
G iv e n a b ip a r t it ion of a finit e- d ime n sion al H i l be r t sp a c e H =H
A
H
B
=C
d A
C
d B
, w e w i l l he nc efor th
foc us on a v e r a g in g C
V A; W B
( t) o v e r the ( inde pe nde n t ) unit a r y ope r a t or s V
A
a nd W
B
, w hos e s u ppor t i s o v e r
s ubs ys t e m s A a nd B , r e spe ct iv e ly . The r e s ult in g qua n t it y
G( t) := 1
1
d
Re
∫
dVdW Tr
(
V
y
A
( t) W
y
B
V
A
( t) W
B
)
; (3.3)
de pe nd s only on the dy n a mic s a nd the H i l be r t sp a c e c ut , w he r e w e de not e V
A
= V
I
B
, W
B
= I
A
W
a n d the a v e r a g in g i s pe r for me d a c c or d in g t o the H a a r me as ur e [ 198 ]. W e w i l l r efe r t o G( t) for br e v it y as
the bip a r t it e O T O C , a nd a n aly z in g its pr ope r t ie s w i l l be the foc us of the pr e s e n t ch a pt e r .
I t w as r e c e n tly sho w n in R ef . [ 197 ], w he r e G( t) w as fir s t in tr oduc e d , th a t unde r the as s umpt ion s of
( i ) w e ak c ou p l in g be t w e e n A a nd B , a nd ( i i ) M a rk o v i a nit y , th a t G( t) ex hi b its a clos e c onne ct ion w ith the
L o s chmidt e cho [ 32 , 199 ]; the l a tt e r h as be e n w ide ly e mp lo y e d t o ch a r a ct e r i z e ch a os [ 200 , 201 ]. H e r e ,
w e fir s t sho w , w ithout a n y of the pr e v ious as s umpt ion s, th a t G( t) i s, in fa ct , a me n a b le t o ex a ct a n aly t ical
tr e a tme n t , a nd w e unc o v e r its d ir e ct r e l a t ion w ith e n tr op y pr oduct ion, infor m a t ion spr e a d in g , a nd e n t a n-
gle me n t. W e al s o r i g or ously pr o v e th a t the a v e r a g e cas e i s al s o the t y p ical one , he nc e jus t i f y in g the a v e r a g in g
pr oc e s s . Our m ain r e s ults a r e s t a t e d in the the or e m s th a t fo l lo w . A l l pr oof s of the cl aim s a ppe a r in g in the
87
t ex t ca n be found in A ppe nd i x 3.2 .
Th e bip a r t it e O T O C .— W e be g in b y br in g in g G( t) in a mor e ex p l ic it for m w hich w i l l be the s t a r t in g
po in t for a s e que nc e of r e s ults . Thi s ca n be a chie v e d b y w ork in g on the doub le d sp a c e H
H
′
, w he r e
H
′
=H
A
′
H
B
′ i s a r e p l ica of the or i g in al H i l be r t sp a c e .
P r opositio n 3.1
L e t S
AA
′ be the ope r a t or o v e r H
H
′
th a t s w a ps A w ith its r e p l ica A
′
a nd d = dim(H) . The n
G( t) = 1
1
d
2
Tr
(
S
AA
′ U
2
t
S
AA
′ U
y
2
t
)
: (3.4)
The a n alo g ous ex pr e s sion for BB
′
al s o ho ld s .
The a bo v e for m ul a imme d i a t e ly ex pos e s a c onne ct ion be t w e e n the b ip a r t it e O T O C a nd the o p er a t o r
en t a n g lem en t of the e v o lut ion E
op
( U
t
) , as define d in R ef . [ 202 ] ( s e e al s o A ppe nd i x 3.2 for the r e le v a n t def -
init ion s ). The t w o qua n t it ie s, r e m a rk a b ly , c o inc ide ex a ctly . Thi s o bs e r v a t ion al s o al lo w s one t o ex pr e s s
the en t a n g l i n g p o w er [ 203 ] e
P
( U
t
) as a f unct ion of the b ip a r t it e O T O C for the s y mme tr ic cas e d
A
= d
B
.
The for me r qua n t i fie s the a v e r a g e e n t a n gle me n t pr oduc e d b y the e v o lut ion a nd h as be e n e s t a b l i she d as a n
ind ica t or of glo b al ch a os in fe w - body s ys t e m s [ 204 – 207 ].
P r opositio n 3.2
L e t G
U
de not e the b ip a r t it e O T O C for the e v o lut ion U . The n, ( i ) E
op
( U
t
) = G
U t
, a nd ( i i ) for a
s y mme tr ic b ip a r t it ion d
A
= d
B
,
e
P
( U
t
) =
d
(
p
d+ 1)
2
( G
U t
+ G
U t S AB
G
S AB
): (3.5)
88
F or the finit e t e mpe r a tur e cas e , E q . ( 3.4) a dmits a s tr ai gh tfor w a r d g e ne r al i za t ion w hich w e r e por t in
A ppe nd i x 3.2 . H o w e v e r , a d ir e ct c onne ct ion w ith ope r a t or e n t a n gle me n t a nd e n t a n gl in g po w e r m a y not
be s o simp le .
H o w i nf o r m a t i v e is t h e a v er a ge G( t) ? — U s ual ly , one i s in t e r e s t e d in be h a v ior of the O T O C for a t y p ical
cho ic e of r a ndom unit a r y ope r a t or s . Due t o me as ur e c onc e n tr a t ion [ 208 ], w e pr o v e th a t the t w o e s s e n t i al ly
c o inc ide , i . e ., the pr o b a b i l it y th a t a r a ndom in s t a nc e de v i a t e s si gni fica n tly f r om the me a n i s ex pone n t i al ly
s u ppr e s s e d as the d ime n sion of eithe r of the s ubs ys t e m s A a nd B gr o w s l a r g e .
P r opositio n 3.3
L e t P( ε) be the pr o b a b i l it y th a t a r a ndom in s t a nc e of C
V A; W B
( t) de v i a t e s f r om its H a a r a v e r a g e G( t)
mor e th a n ε . The n,
P( ε) 2 exp
(
ε
2
d
max
64
)
; (3.6)
w he r e d
max
= maxf d
A
; d
B
g .
I n the definit ion of the b ip a r t it e O T O C a nd t o o bt ain the r e p l ica for m ul a E q . ( 3.4), w e h a v e s o fa r c on-
side r e d a v e r a g in g o v e r the uni for m ( H a a r ) e n s e mb le w hich c on t in uously ex t e nd s o v e r the w ho le unit a r y
gr ou p . A lthou gh n a tur al f r om a m a the m a t ical v ie w po in t , thi s cho ic e ca n tur n out t o be r a the r c omp l ica t e d
on p h ysical a nd n ume r ical gr ound s [ 209 ]. N one the le s s, w e sho w in A ppe nd i x 3.3 th a t H a a r a v e r a g in g ca n
be r e p l a c e d b y a n y unit a r y e n s e mb le th a t for m s a 1- de si gn [ 95 – 98 ] w ithout alt e r in g G( t) . S uch e n s e m-
b le s mimic the H a a r r a ndomne s s only u p t o the fir s t mome n t , w hich i s the de pth of r a ndomne s s th a t the
O T O C ca n pr o be [ 190 ]. The l a tt e r as s umpt ion i s th us m uch w e ak e r th a n H a a r r a ndomn s e s s . F or in s t a nc e ,
c on side r the cas e of a sp in- 1= 2 m a n y - body s ys t e m sp l it in t o t w o p a r ts, A a nd B . I n s t e a d of a v e r a g in g o v e r
H a a r r a ndom unit a r ie s V
A
a nd W
B
, th a t t y p ical ly do not fa ct or , the 1- de si gn ( e quiv ale n t ) p ictur e pr e s cr i be s
t o in s t e a d c on side r only f ul ly fa ct or i z e d unit a r ie s w ith s u ppor t o v e r A a nd B , e . g., pr oducts of local P a ul i
m a tr ic e s .
89
T i m e-a v er a g i n g t h e bip a r t it e O T O C .— I n finit e d ime n sion al qua n tum s ys t e m s, non tr iv i al qua n tum ex -
pe ct a t ion v alue s or qua n t it ie s s uch as C
V; W
( t) do not c on v e r g e t o a l imit for t!1 . I n s t e a d , a ft e r a lon g
t ime they t y p ical ly os c i l l a t e a r ound a n e qui l i br ium v alue [ 142 , 143 , 210 – 212 , 212 , 213 ] w hich ca n be ex -
tr a ct e d b y t ime-a v e r a g in g X( t) := lim
T!1
1
T
∫
T
0
dt X( t) . W e no w tur n t o ex a mine thi s lon g -t ime be h a v ior
G( t) of the b ip a r t it e O T O C as a f unct ion of the H a mi lt oni a n a nd the H i l be r t sp a c e c ut.
L e t us be g in w ith the cas e of a ch a ot ic dy n a mic s, w hich e n t ai l s le v e l r e pul sion s t a t i s t ic s [ 183 ] a nd a n “ in-
c omme n s ur a b le ” r e l a t ion a mon g the e ne r g y le v e l s . A s s uch, ch a ot ic H a mi lt oni a n s s a t i sf y ( eithe r ex a ctly or
t o v e r y g ood a ppr o x im a t ion ) the no - r e s on a nc e c ond it ion ( NR C ): The e ne r g y le v e l s a nd e ne r g y g a ps fe a -
tur e nonde g e ne r a c y . Thi s h as impor t a n t imp l ica t ion s for the lon g -t ime be h a v ior of their b ip a r t it e O T O C ,
as w e w i l l s e e s oon .
L e t us spe ctr al ly de c ompos e H =
∑
k
E
k
j φ
k
⟩⟨ φ
k
j a nd us e ρ
( χ)
k
:= T r
χ
(
j φ
k
⟩⟨ φ
k
j
)
t o de not e the r e duc e d
de n sit y ope r a t or o v e r χ = A; B c or r e spond in g t o the kth H a mi lt oni a n ei g e n s t a t e ( χ c or r e spond s t o the
c omp le me n t ). Be lo w , ⟨ X; Y⟩ := T r( X
y
Y) de not e s the H i l be r t - S chmidt inne r pr oduct [ 214 ], w hich g iv e s
r i s e t o the ope r a t or 2- nor m ∥ X∥
2
:=
√
⟨ X; X⟩ .
P r opositio n 3.4
C on side r a H a mi lt oni a n s a t i sf y in g the NR C . The n
G( t)
NRC
= 1
1
d
2
∑
χ2f A; Bg
(
R
( χ)
2
2
1
2
R
( χ)
D
2
2
)
(3.7)
w he r e R
( χ)
i s the Gr a m m a tr i x of the r e duc e d H a mi lt oni a n ei g e n s t a t e s f ρ
( χ)
k
g
d
k= 1
, i . e .,
R
( χ)
kl
:=⟨ ρ
( χ)
k
; ρ
( χ)
l
⟩ (3.8)
w hi le
(
R
( χ)
D
)
kl
:= R
( χ)
kl
δ
kl
.
L e t us fir s t po in t out s ome b asic, y e t impor t a n t pr ope r t ie s of the a bo v e for m ul a . The m a tr i x R
( χ)
i s
90
r e al a nd s y mme tr ic, w hi le R
( χ)
D
i s posit iv e- s e midefinit e a nd d i a g on al . M or e o v e r , the c omp le t e ne s s of the
H a mi lt oni a n ei g e n v e ct or s impos e s
∑
k
ρ
( χ)
k
= d
χ
I , th us the r e s cale d
~
R
( χ)
:= R
( χ)
= d
χ
a r e doub ly s t och as t ic,
i . e .,
∑
i
~
R
( χ)
ij
=
∑
i
~
R
( χ)
ji
= 18 j . A s
~
R
( χ)
i s a ( r e s cale d ) Gr a m m a tr i x , its ei g e v alue s a r e nonne g a t iv e , u ppe r
bounde d b y 1, a nd a t mos t d
2
χ
of the m a r e nonz e r o [ 214 ]. Thi s l as t pr ope r t y fo l lo w s f r om the fa ct th a t
Rank
~
R
( χ)
= dim Spanf ρ
( χ)
k
g
k
d
2
χ
. Obs e r v e al s o th a t
R
( A)
D
2
2
=
R
( B)
D
2
2
as t w o s t a t e s ρ
( A)
k
a nd ρ
( B)
k
alw a ys h a v e the s a me spe ctr um ( u p t o ir r e le v a n t z e r oe s ).
B ip a r t it e O T O C a n d en t a n g lem en t .— E qua t ion 3.4 m ak e s it pos si b le t o br id g e the lon g -t ime be h a v ior of
the b ip a r t it e O T O C w ith the e n t a n gle me n t s tr uctur e of the H a mi lt oni a n ei g e n s t a t e s . L e t us be g in w ith the
s y mme tr ic cas e w he r e d
A
= d
B
a nd al l j φ
k
⟩ a r e m a x im al ly e n t a n gle d w ith r e spe ct t o the A - B H i l be r t sp a c e
c ut. Thi s l imit unique ly de t e r mine s the t ime-a v e r a g e for the NR C cas e , r e g a r d le s s of the ex a ct H a mi lt oni a n
ei g e nb asi s . I n g e ne r al , ho w e v e r , kno w le d g e of the e n t a n gle me n t i s not e nou gh t o unique ly de t e r mine the
e qui l i br a t ion v alue; the inne r pr oducts R
( χ)
kl
g o bey ond pr o b in g jus t the spe ctr um of the r e duc e d s t a t e s . A
simp le s ubs t itut ion in E q . ( 3.7) g iv e s for the m a x im al ly e n t a n gle d cas e G
ME
( t)
NRC
= ( 1 1= d)
2
. W e w i l l
l a t e r sho w the u ppe r bound G( t) 1 1= d
2
min
, the r efor e the e qui l i br ium v alue for the b ip a r t it e O T O C in
thi s c as e i s ne a rly m a x im al , as ex pe ct e d for hi ghly e n t a n gle d mode l s ( e . g., [ 215 , 216 ]).
H o w r o bus t i s thi s c onclusion for ch a ot ic H a mi lt oni a n s w ith a pos si b ly as y mme tr ic b ip a r t it ion? T y p ical
ei g e n s t a t e s of ch a ot ic H a mi lt oni a n s, as al s o pr e d ict e d b y the ei g e n s t a t e the r m al i za t ion h y pothe si s [ 217 –
219 ], a r e be l ie v e d t o o bey a v o lume l a w for the e n t a n gle me n t e n tr op y . M or e o v e r , their e n t a n gle me n t pr op -
e r t ie s in the bul k r e s e mb le thos e of H a a r r a ndom pur e s t a t e s [ 59 , 220 , 221 ]. W e w i l l no w sho w th a t hi gh
e n t a n g le m e n t for the H a mi lt oni a n ei g e n s t a t e s ne c e s s a r i ly imp l ie s th a t the de v i a t ion of the a ctual e qui l i br a -
t ion v alue f r om G
ME
( t)
NRC
i s s m al l .
I t i s c on v e nie n t for thi s pur pos e t o qua n t i f y the a moun t of e n t a n gle me n t v i a the l ine a r e n tr op y [ 222 , 223 ]
of the r e duc e d s t a t e E(j ψ
AB
⟩) := S
lin
(
T r
χ
j ψ
AB
⟩⟨ ψ
AB
j
)
, w he r e S
lin
( ρ) := 1 T r( ρ
2
) . The l a tt e r w i l l al s o
e me r g e n a tur al ly l a t e r w he n w e ex pr e s s the b ip a r t it e O T O C in t e r m s of e n tr op y pr oduct ion . N ot ic e th a t
E 1 1= d
max
:= E
max
, w hich i s a chie v a b le only for d
A
= d
B
.
91
P r opositio n 3.5
I f E
max
E(j φ
k
⟩) ε ho ld s for a t le as t a f r a ct ion α of the H a mi lt oni a n ei g e n s t a t e s, the n
G
ME
( t)
NRC
G( t)
NRC
α J+( 1 α) K , w he r e
J :=
6 ε
d
min
+
5 ε
2
2
+ 2
λ
2
1
d
2
max
(3.9a)
K :=
(
1+
2
d
min
)
( 1 α)+
2
d
+ 4( ε +
p
ε) (3.9b)
a nd λ = d
max
= d
min
.
The a bo v e bound pr o v ide s a s u ffic ie n t c ond it ion s uch th a t the b ip a r t it e O T O C e qui l i br a t e s a r ound
G
ME
( t)
NRC
. I t i s ex pr e s s e d in t e r m s of the f r a ct ion α of the hi ghly e n t a n gle d ei g e n s t a t e s, their e n t a n gle-
me n t a nd the as y mme tr y of the A - B b ip a r t it ion . N ot ic e th a t the bound simp l i fie s c on side r a b ly for the cas e
α = 1 a nd d
min
= d
max
=
p
d , th a t i s,
G
ME
( t)
NRC
G( t)
NRC
ε( 6=
p
d + 5 ε= 2) w hich should ho ld
t o a g ood a ppr o x im a t ion for H a mi lt oni a n s w ith hi gh e n t a n gle me n t in the bul k of the e ne r g ie s . A pp l ie d t o
ch a ot ic H a mi lt oni a n s ² , the bound of E qua t ion 3.5 ind ica t e s th a t the b ip a r t it e O T O C w i l l e qui l i br a t e ne a r
G
ME
( t)
NRC
, w ith de v i a t ion s u p t o O( 1= d
2
min
) . F or a fi xe d r a t io λ a nd as d gr o w s, G( t)
NRC
he nc e c on v e r g e s
t o G
ME
( t)
NRC
for al l ch a ot ic s ys t e m s . S inc e G( t) 1 1= d
2
min
, fluctua t ion s a r ound the t ime-a v e r a g e a r e
ne c e s s a r i ly in si gni fica n t , jus t i f y in g the t e r m e qui l i br a t ion .
Be y o n d c h a o t ic H a m i l t o n i a n s .— W e no w r e l a x the “ s tr on g ” le v e l r e pul sion, i . e ., NR C , cr it e r ion a nd un-
c o v e r ho w a hie r a r ch y of c on s tr ain ts, e a ch imp ly in g a d i ffe r e n t s tr e n g th of ch a os, i s r efle ct e d in the e qui l i -
br a t ion v alue of the b ip a r t it e O T O C .
I n t e gr a b le mode l s, w hich pos s e s s a s tr uctur e d spe ctr um, a r e ex pe ct e d t o v io l a t e the NR C . N e v e r the le s s,
not ic e th a t E q . ( 3.7 ), althou gh de r iv e d unde r the NR C , ca n s t i l l be e v alua t e d for a n ( a r b itr a r y ) cho ic e of
or thonor m al ei g e n v e ct or s of the H a mi lt oni a n . W e w i l l r efe r t o the r e s ult in g v alue as the N R C e s t i m a t e
² H e r e ch a ot ic it y c oncr e t e ly me a n s th a t the H a mi lt oni a n spe ctr um s a t i sfie s the NR C a nd th a t the e n t a n gle me n t of the t y p -
ical ei g e n v e ct or s in the bul k , w hich de t e r mine the e qui l i br a t ion v alue , r e s e mb le s th a t of H a a r r a ndom v e ct or s [ 224 , 225 ], i . e .,
Tr
(
ρ
2
χ
)
( d
A
+ d
B
)=( d+ 1) th us ε = O( 1= d
min
) a nd α 1 .
92
of the t ime-a v e r a g e a nd w e w i l l shor tly sho w th a t thi s e s t im a t e alw a ys c on s t itut e s a n u ppe r bound of the
a ctual e qui l i br a t ion v alue ( a nd c o inc ide s w ith it for ch a ot ic H a mi lt oni a n s ). Thi s i s both of c onc e ptual a nd
pr a ct ical impor t a nc e , as e v alua t in g the NR C e s t im a t e i s c on side r a b ly le s s in t e n siv e th a n calc ul a t in g the
ex a ct v alue .
I n fa ct , one ca n m ak e a br o a de r cl aim . F or th a t , w e fir s t sk e t ch thr e e t y pe s of a v e r a g in g pr oc e s s e s o v e r
G , incr e asin gly shi ft in g a w a y f r om the s tr on g ch a ot ic it y l imit. Ea ch of the m g iv e s r i s e t o a c or r e spond in g
e s t im a t e for the ( ex a ct ) e qui l i br a t ion t ime-a v e r a g e v alue G( t) . ( i ) G
Haar
: A v e r a g in g o v e r (glo b al ) H a a r
r a ndom unit a r y ope r a t or s U 2 U( d) in p l a c e of the t ime- e v o lut ion . Thi s a v e r a g in g pr oc e s s i s “ bey ond
ch a os ” , in the s e n s e th a t it doe s not c on s e r v e e ne r g y , in c on tr as t w ith t ime-a v e r a g in g o v e r a n y H a mi lt oni a n
e v o lut ion s . I ts e s t im a t e ( only a f unct ion of the d ime n sion ) i s g iv e n l a t e r in E q . ( 3.10 ). ( i i ) G( t)
NRC
: T ime-
a v e r a g e , as s umin g the H a mi lt oni a n h as nonde g e ne r a t e e ne r g y le v e l s a nd nonde g e ne r a t e e ne r g y g a ps . The
c or r e spond in g e s t im a t e i s E q . ( 3.7). ( i i i ) G( t)
NRC
+
: A s befor e , but as s umin g the H a mi lt oni a n m a y h a v e
de g e ne r a t e spe ctr um, but the e ne r g y g a ps ( be t w e e n the d i ffe r e n t le v e l s ) a r e nonde g e ne r a t e . I ts e s t im a t e
de pe nd s only on the ei g e npr oj e ct or s of the H a mi lt oni a n a nd ca n be found in A ppe nd i x ( 3.2).
The v alue of the H a a r a v e r a g e ca n be pe r for me d ex a ctly , w ith r e s ult
G
Haar
=
( d
2
A
1)( d
2
B
1)
d
2
1
: (3.10)
The fo l lo w in g or de r in g ho ld s .
P r opositio n 3.6
F or a n y g iv e n H a mi lt oni a n, the c or r e spond in g e s t im a t e s a r e r e l a t e d w ith the ex a ct t ime-a v e r a g e G( t)
as
G
Haar
G( t)
NRC
G( t)
NRC
+
G( t) : (3.11)
93
▲
▲
▲
▲
▲
▲
▲
▲
▲
△
△
△
△
△
△
△
△ △
■
■
■
■
■
■
■
■
■
◆
◆
◆
◆
◆
◆
◆
◆
◆
2 4 6 8 10
10
-6
10
-5
10
-4
0.001
0.010
0.100
▲
△
■
◆
Figure 3.1.1: Loga rithmic plot of va rious G estimates, along with the exact time-average, fo r fixed
d
A
= 2 as a function of the total numb er of spins n . G
Haar
1
= 3= 4 co rresp onds to the Haa r estimate
fo r n ! 1 . F o r the chaotic phase of the TFIM ( g = 1: 05 , h = 0: 5 ), the NRC constitutes a satis-
facto ry , though imp erfect, app ro ximation. The chaotic and integrable phases ( h = 0 ) can b e clea rly
distinguished through the equilib ration b ehavio r of the bipa rtite OTOC. F o r the integrable XXZ mo del
(w e set J = 0: 4 , Δ = 2: 5 ), the NRC
+
estimate coincides (up to numerical e rro r) with the exact time-
average. Inequalit y ( 3.11 ) holds valid in all cases.
The a bo v e c on s t itut e s a pr oof th a t c o inc ide nc e s in the spe ctr um of a H a mi lt oni a n u p t o the “ g a ps of g a ps ”
( i . e ., de g e ne r a c y o v e r the e ne r g y le v e l s a nd their g a ps ) alw a ys r e d u ce the e qui l i br a t ion v alue of the b ip a r t it e
O T O C .
L e t us no w n ume r ical ly c omp a r e e a ch of the e s t im a t e s for t w o mode l s of sp in-1/2 ch ain s w ith ope n-
bound a r y c ond it ion s: ( i ) tr a n s v e r s e- fie ld I sin g mode l ( TFI M ) w ith ne a r e s t nei ghbour in t e r a ct ion, H
I
=
∑
i
( σ
z
i
σ
z
i+ 1
+ g σ
x
i
+ h σ
z
i
) ( i i ) ne a r e s t - nei ghbor X X Z in t e r a ct ion H
XXZ
= J
∑
i
( σ
x
i
σ
x
i+ 1
+ σ
y
i
σ
y
i+ 1
+
Δ σ
z
i
σ
z
i+ 1
) . R e cal l th a t H
I
for h = 0 i s in t e gr a b le in t e r m s of f r e e- fe r mion s, w hi le H
XXZ
b y Be the A n s a tz
t e chnique s . The t w o t y pe s of s o lut ion s y ie ld qual it a t iv e ly d i ffe r e n t spe ctr a; f r e e fe r mion s o lut ion s ne c e s -
s a r i ly v io l a t e nonde g e ne r a c y of the g a ps . Thi s i s r efle ct e d in the a c c ur a c y of the e s t im a t e s ( s e e F i g ur e 3.1.1 ).
A lthou gh the NR C e s t im a t e pr o v ide s e s s e n t i al ly the ex a ct e qui l i br a t ion v alue s for the ch a ot ic p h as e of the
TFI M , it o v e r e s t im a t e s the m in the in t e gr a b le p h as e . On the othe r h a nd , NR C+ i s e s s e n t i al ly ex a ct for the
in t e gr a b le cas e of the H
XXZ
due t o the l a ck of c o inc ide nc e s in the g a ps . The r e s ults o bt aine d he r e c or r o bo -
r a t e ex i s t in g s tud ie s in the l it e r a tur e , w he r e the ( shor t - a nd ) lon g -t ime be h a v ior of the O T O C w as s tud ie d
for v a r ious m a n y - body s ys t e m s, s e e R ef s . [ 81 , 82 , 226 ].
B ip a r t it e O T O C a n d s u b sy s t em ev o l u t io n.— W e h a v e s o fa r foc us e d on ex a minin g the be h a v ior of the
94
b ip a r t it e O T O C f r om the pe r spe ct iv e of clos e d s ys t e m s, i . e ., o v e r the f ul l b ip a r t it e H i l be r t sp a c e H
A
H
B
.
One ca n in s t e a d ex pr e s s G( t) as a f unct ion of the r e duc e d t ime- dy n a mic s o v e r only eithe r H
A
orH
B
( a nd
the c or r e spond in g du p l ica t e ), a t the ex pe n s e of g iv in g u p unit a r it y . Thi s ca n be e asi ly r e al i z e d b y for m al ly
pe r for min g a p a r t i al tr a c e in E q . ( 3.4 ), w hich imme d i a t e ly r e s ults in the fo l lo w in g e quiv ale n t ex pr e s sion for
the b ip a r t it e O T O C .
P r opositio n 3.7
L e t Λ
( A)
t
( ρ
A
):= T r
B
[
U
t
(
ρ
A
I
B
d
B
)
U
y
t
]
be the r e duc e d dy n a mic s o v e r A w he n the e n v ir onme n t B
i s init i al i z e d in a m a x im al ly mi xe d s t a t e . The n,
G( t) = 1
1
d
2
A
T r
[
S
AA
′
(
Λ
( A)
t
)
2
( S
AA
′)
]
: (3.12)
The a n alo g ous ex pr e s sion for BB
′
al s o ho ld s .
The qua n tum m a p Λ
( χ)
t
i s unit al , i . e ., the m a x im al ly mi xe d s t a t e i s a fi xe d po in t. A s s uch, the tr a n sfor m a -
t ion ρ
χ
7! Λ
( χ)
t
( ρ
χ
) r e s ults alw a ys in a n out put s t a t e w hos e spe ctr um i s mor e d i s or de r e d th a n the input
one [ 138 ]. A s a r e s ult , w he n ρ
χ
i s pur e , the effe ct of the r e duc e d t ime- dy n a mic s i s t o s cr a mb le a nd he nc e
pr oduc e e n tr op y . L e t us no w tur n t o ex a mine thi s c onne ct ion mor e clos e ly .
B ip a r t it e O T O C as en tr o p y pr o d u c t io n.— W e no w sho w th a t the b ip a r t it e O T O C G( t) i s nothin g but a
me as ur e of the a v e r a g e e n tr op y pr oduct ion o v e r pur e s t a t e s, w ith the l a tt e r qua n t i fie d b y l ine a r e n tr op y S
lin
.
P r opositio n 3.8
G( t) =
d
χ
+ 1
d
χ
∫
dU S
lin
[
Λ
( χ)
t
(j ψ
U
⟩⟨ ψ
U
j)
]
(3.13)
95
w he r e χ = A; B a ndj ψ
U
⟩ := Uj ψ
0
⟩ c or r e spond s t o H a a r r a ndom pur e s t a t e s o v e r H
χ
.
I n thi s m a nne r , the b ip a r t it e O T O C ca n be f ul ly ch a r a ct e r i z e d b y l ine a r e n tr op y me as ur e me n ts o v e r a n y
of the A; B s ubs ys t e m s . T o o bt ain a s a t i sfa ct or y e s t im a t e of the me a n in the R HS of E q . ( 3.13 ), one doe s
not , in pr a ct ic e , ne e d t o s a mp le o v e r the f ul l H a a r e n s e mb le . A n a de qua t e e s t im a t e ca n be o bt aine d w ith
a r a p id ly de cr e asin g n umbe r of ne c e s s a r y s a mp le s, as the d ime n sion d
χ
gr o w s . M or e pr e c i s e ly , le t
~
P( ε) be
the pr o b a b i l it y of the e n tr op y S
lin
[
Λ
( χ)
t
(
j ψ⟩⟨ ψj
)]
de v i a t in g f r om
d χ
d χ+ 1
G( t) mor e th a n ε for a n in s t a nc e of
a r a ndom s t a t e . W e sho w in A ppe nd i x 3.2 th a t
~
P( ε) exp
(
d
χ
ε
2
64
)
: (3.14)
The l ine a r e n tr op y , althou gh, pe r s e , a nonl ine a r f unct ion al , ca n be tur ne d in t o a n or d in a r y ex pe ct a t ion
v alue i f t w o ( unc or r e l a t e d ) c op ie s of the qua n tum s t a t e a r e sim ult a ne ously a v ai l a b le , 1 S
lin
= T r( S ρ
2
)
for S = S
AA
′ S
BB
′ . Thi s fa ct ca n be ex p lo it e d simp l i f y its ex pe r ime n t al a c c e s si b i l it y [ 227 – 231 ]. M or e r e-
c e n tly , pr ot oc o l s b as e d on c or r e l a t in g me as ur e me n ts o v e r r a ndom b as e s h a v e al s o be e n de v e lope d t o me a -
s ur e e n tr op ie s [ 232 – 235 ], as w e l l as O T O C s [ 236 , 237 ]. A s a r e s ult , E qua t ion 3.8 a nd the t y p ical it y r e s ult
E q . ( 4.17 ) s u gg e s t th a t the b ip a r t it e O T O C i s, in tur n, tr a ct a b le v i a l ine a r e n tr op y me as ur e me n ts . W e pr o -
v ide mor e de t ai l s in A ppe nd i x 3.4 .
F r om E q . ( 3.13 ) one ca n al s o infe r the u ppe r bound G( t) 1 1= d
2
χ
:= G
( χ)
max
a nnounc e d e a rl ie r th a t
fo l lo w s f r om the r a n g e of the l ine a r e n tr op y f unct ion . The bound i s th us a chie v a b le only w he n Λ
( χ)
t
i s e qual
t o the c omp le t e ly de po l a r i z in g m a p T
( χ)
() := Tr()
I
χ
d
χ
.
F in al ly , w e r e m a rk th a t l ine a r e n tr op y oc c ur s r a the r n a tur al ly in r e l a t ion w ith the b ip a r t it e O T O C , as
de mon s tr a t e d b y E qua t ion 3.2 (w he r e it l ie s imp l ic itly in the definit ion of ope r a t or e n t a n gle me n t a nd e n-
t a n gl in g po w e r ) a nd E qua t ion 3.8 . Thi s fa ct h as its r oots in the definit ion of the O T O C , w hich i s in t i -
m a t e ly r e l a t e d t o the F r o be nious nor m . R e le v a n t r e l a t ion s for the l ine a r e n tr op y h a v e be e n al s o r e por t e d
in [ 193 ]. S t a r t in g f r om the ine qual it y S
lin
( ρ) S( ρ) be t w e e n the l ine a r a nd v on N e um a nn e n tr op ie s
( S( ρ) := T r[ ρ log( ρ)] ), one ca n al s o o bt ain th e c or r e spond in g e s t im a t e s for the l a tt e r .
B ip a r t it e O T O C a n d i nf o r m a t io n s pr e a d i n g .— The b ip a r t it e O T O C me as ur e s the a v e r a g e a b i l it y of the
r e duc e d t ime- e v o lut ion t o e r as e infor m a t ion, as ca ptur e d b y the e n tr op y pr oduct ion o v e r a r a ndom pur e
96
s t a t e . Thi s n a tur al ly r ai s e s the que s t ion as t o w he the r G( t) ca n al s o be unde r s t ood as a me as ur e of d i s t a nc e
be t w e e n Λ
( χ)
t
a nd the de po l a r i z in g m a p T
( χ)
, th a t i s, in the sp a c e of qua n tum ch a nne l s ( i . e ., C omp le t e ly
P osit iv e a nd T r a c e P r e s e r v in g ( C P TP ) m a ps [ 238 ]).
A s tr ai gh tfor w a r d a n s w e r ca n be o bt aine d b y r e s or t in g t o the dual it y be t w e e n qua n tum s t a t e s a nd ope r -
a t ion s [ 238 ]. L e t ρ
E
:= E
I(j φ
+
⟩⟨ φ
+
j) de not e the ( C ho i ) s t a t e c or r e spond in g t o the C P TP m a p E ,
w he r ej φ
+
⟩ := d
1= 2
∑
d
i= 1
j ii⟩ i s a m a x im al ly e n t a n gle d s t a t e .
P r opositio n 3.9
The b ip a r t it e O T O C i s a me as ur e of the d i s t a nc e be t w e e n the r e duc e d t ime- e v o lut ion a nd the de po -
l a r i z in g m a p:
G( t) = G
( χ)
max
ρ
Λ
( χ)
t
ρ
T
( χ)
2
2
: (3.15)
A s a n a pp l ica t ion, the pr oposit ion a bo v e ca n be ut i l i z e d t o bound the d i s t a nc e
Λ
( χ)
t
T
( χ)
♢
g iv e n b y
the d i a mond nor m [ 114 , 239 ]; the l a tt e r i s a w e l l - e s t a b l i she d me as ur e of d i s t a nc e be t w e e n qua n tum ch a n-
ne l s ³ sinc e it a dmits a n ope r a t ion al in t e r pr e t a t ion in t e r m s of d i s cr imin a t ion on the le v e l of qua n tum pr o -
c e s s e s [ 240 ]. The d i s t in g ui sh a b i l it y of the t w o ope r a t ion s s a t i sfie s
Λ
( χ)
t
T
( χ)
♢
d
3= 2
χ
√
G
( χ)
max
G( t)
( s e e A ppe nd i x 3.2 ), the r efor e i f G
( χ)
max
G( t) de ca ys fas t e r th a n d
3
χ
, the n as y mpt ot ical ly the t w o ch a nne l s
a r e e s s e n t i al ly ind i s t in g ui sh a b le .
Su m m a r y .— W e sho w e d th a t the b ip a r t it e O T O C i s a me n a b le t o ex a ct a n aly t ical tr e a tme n t a nd , quit e r e-
m a rk a b ly , i s e qual t o the ope r a t or e n t a n gle me n t of the dy n a mic s . Thi s ide n t it y al lo w s one t o e s t a b l i sh a r i g -
or ous qua n t it a t iv e c onne ct ion be t w e e n the O T O C a nd the not ion of e n t a n gl in g po w e r , a w e l l - e s t a b l i she d
qua n t i fie r of fe w - body ch a os . Thi s m a y pr o v ide in si gh ts in t o r e c e n t w ork in v o lv in g “ dual - unit a r ie s ” a nd
m a n y - body ch a os [ 241 – 244 ]; the l a tt e r m a x imi z e ope r a t or e n t a n gle me n t [ 244 , 245 ]. W e the n tur ne d t o
l a t e-t ime a v e r a g e s of the b ip a r t it e O T O C a nd pr o v ide d a hie r a r ch y of e s t im a t e s for s ys t e m s th a t v io l a t e
³ Bound in g the d i ffe r e nc e in t e r m s of the qua n tum pr oc e s s e s al s o c on s tr ain ts the d i s t in g ui sh a b i l it y in t e r m s of s t a t e s:
E
1
( ρ)E
2
( ρ)
1
E
1
E
2
♢
for al l s t a t e s a nd qua n tum pr oc e s s e s .
97
the c ond it ion s of a “ g e ne r ic spe ctr um ” . F in al ly , w e unr a v e le d the ope r a t ion al si gni fica nc e of the O T O C
b y e s t a b l i shin g in t im a t e c onne ct ion s w ith e n tr op y pr oduct ion a nd infor m a t ion s cr a mb l in g a t the le v e l of
qua n tum ch a nne l s . P os si b le f utur e d ir e ct ion s include a pp ly in g f ur the r the s e the or e t ical t oo l s t o c oncr e t e
m a n y - body s ys t e m s a nd unc o v e r in g r e l a t ion s w ith the r m al i za t ion, local i za t ion, a nd othe r m a n y - body p he-
nome n a .
A ppendic e s
3.2 Pro of s
H e r e w e r e s t a t e the The or e m s a nd P r oposit ion s, as w e l l as othe r m a the m a t ical cl aim s a ppe a r in g in the m ain
t ex t , a nd g iv e their pr oof .
T h eo r e m 3.1
P r o of . L e t S be the ope r a t or o v e r H
H
′
th a t s w a ps H w ith its r e p l ica H
′
. The n for a n y ope r a t or s
X; Y a ct in g o v e r H it ho ld s th a t
Tr( XY) = Tr[ S( X
Y)]; (3.16)
as it ca n be e asi ly v e r i fie d b y ex pr e s sin g both side s in a b asi s . N ot ic e th a t in our cas e , w he r e H ca r r ie s
a b ip a r t it ion, one ca n f ur the r de c ompos e S = S
AA
′ S
BB
′ .
U sin g the a bo v e ide n t it y the O T O C a v e r a g in g in E q . ( 3.3 ) ca n be w r itt e n as
G( t) = 1
1
d
Re
∫
dVdW Tr
(
S V
y
A
( t) W
y
B
V
A
( t) W
B
)
= 1
1
d
Re
∫
dVdW Tr
(
SU
y
2
t
( V
y
A
V
A
) U
2
t
( W
y
B
W
B
)
)
= 1
1
d
Re Tr
[
SU
y
2
t
(∫
dVV
y
A
V
A
)
U
2
t
(∫
dWW
y
B
W
B
)]
:
98
N o w the t w o inde pe nde n t a v e r a g e s ca n be e asi ly pe r for me d sinc e for unit a r y ope r a t or s o v e r H
=C
d
the c or r e spond in g H a a r in t e gr al s e v alua t e t o
∫
dUU
U
y
=
S
d
(3.17)
w he r e S i s a g ain the s w a p ope r a t or o v e r the doub le d sp a c e .
A quick w a y t o pr o v e the w e l l -kno w n ide n t it y ( 3.17 ) i s b y usin g E q . ( 3.16 ) t o w r it e
UXU
y
= T r
H
′
[
( U
U
y
)( X
I) S
]
a n d t he n usin g the fa ct th a t
∫
dUUXU
y
=
Tr( X)
d
(3.18)
w hich fo l lo w s d ir e ctly f r om the left/r i gh t in v a r i a nc e of the H a a r me as ur e [ 198 ].
U sin g E q . ( 3.17 ) t w ic e , w e g e t
G( t) = 1
1
d
Re Tr
(
SU
y
2
t
S
AA
′
d
A
U
2
t
S
BB
′
d
B
)
= 1
1
d
2
Tr
(
S
AA
′ U
2
t
S
AA
′ U
y
2
t
)
:
S inc e[ S; X
2
] = 0 for al l ope r a t or s X , the a n alo g ous ex pr e s sion for BB
′
ho ld s, i . e .,
G( t) = 1
1
d
2
Tr
(
S
BB
′ U
2
t
S
BB
′ U
y
2
t
)
: (3.19)
■
N ot ic e th a t the s y mme tr y of the H a a r me as ur e for c e s the b ip a r t it e O T O C t o be t ime- r e v e r s al in v a r i a n t ,
i . e ., G( t) = G( t) .
F in al ly , w e al s o not e th a t th a t the r e i s a s tr ai gh tfor w a r d g e ne r al i za t ion of E qua t ion 3.1 t o a n y finit e t e m-
pe r a tur e the r m al s t a t e . F o l lo w in g simi l a r s t e ps as a bo v e , one g e ts for for the the r m al v e r sion of the b ip a r t it e
99
O T O C
G( t) = 1
1
d
Re Tr
(
( ρ
β
I
A
′
B
′) U
y
2
t
S
AA
′ U
2
t
S
AA
′
)
: (3.20)
T h eo r e m 3.2
Befor e g iv in g the pr oof , le t us fir s t r e cal l the definit ion s of ope r a t or e n t a n gle me n t [ 202 ] a nd e n t a n gl in g
po w e r [ 203 ].
The m ain ide a be hind ope r a t or e n t a n gle me n t i s t o fir s t ex pr e s s the unit a r y e v o lut ion U ( o v e r the b ip a r t it e
H i l be r t sp a c e H
AB
) as a s t a t e in the doub le d sp a c e H
AB
H
A
′
B
′ v i a
j U⟩ = U
I
A
′
B
′j φ
+
⟩ (3.21)
for the m a x im al ly e n t a n gle d s t a t e j φ
+
⟩ =
1
p
d
∑
d
i= 1
j i⟩
AB
j i⟩
A
′
B
′
a nd the n e v alua t e the l ine a r e n tr op y of the
s t a t e σ
U
= T r
BB
′ (j U⟩⟨ Uj) , i . e .,
E
op
( U) := S
lin
( σ
U
) = 1 Tr( σ
2
U
): (3.22)
The e n t a n gl in g po w e r [ 203 ] of a qua n tum e v o lut ion U o v e r a b ip a r t it e qua n tum s ys t e m H =H
A
H
B
i s define d as the a v e r a g e e n t a n gle me n t th a t the e v o lut ion g e ne r a t e s w he n a ct in g on r a ndom s e p a r a b le pur e
s t a t e s . M or e spe c i fical ly ,
e
P
( U) :=
∫
dVdWE
[
U
(
j ψ
V
⟩
A
j ψ
W
⟩
B
)]
; (3.23)
w he r ej ψ
V
⟩
A
= Vj ψ
0
⟩
A
c or r e spond s t o H a a r r a ndom pur e s t a t e s o v e r A (j ψ
0
⟩
A
i s a n ir r e le v a n t r efe r e nc e
s t a t e ), a nd simi l a rly for B , w hi le E(j ψ
AB
⟩) := S
lin
(
T r
B
j ψ
AB
⟩⟨ ψ
AB
j
)
i s the e n t a n gle me n t of the r e s ult in g
s t a t e , as me a s ur e d b y the l ine a r e n tr op y .
100
P r o of . ( i ) The k ey o bs e r v a t ion he r e i s th a t the b ip a r t it e O T O C G
U t
, in t he for m of E q . ( 3.4 ), c o inc ide s
w ith the ope r a t or e n t a n gle me n t E( U
t
) as define d in R ef . [ 202 ] ( s e e E q . (6) the r ein ). E v alua t in g the
ex pr e s sion ( 3.22 ), as in the pr oof of E qua t ion 3.1 , one o bt ain s ex a ctly E q . ( 3.4 ), he nc e E
op
( U
t
) = G
U t
.
( ii ) F or the s y mme tr ic cas e d
A
= d
B
, the r e s ult fo l lo w s b y c omb inin g the fir s t p a r t of the c ur r e n t
The or e m a nd E q . (12) of R ef . [ 202 ].
F in al ly , w e not e th a t b y d ir e ct s ubs t itut ion, one h as G
S AB
= 1 1= d . ■
Pr o positi o n 3.3
The pr oof r e l ie s on me as ur e c onc e n tr a t ion a nd , in p a r t ic ul a r , L e v y ’ s le mm a w hich w e sh al l r e cal l shor tly
( s e e , e . g., [ 246 ]). Be lo w w e a r e al s o g o in g us e v a r ious o pe r a t or ( S ch a tt e n ) k - nor m s [ 214 ]; the l a tt e r a r e
define d as∥ X∥
k
:=
(∑
i
s
k
i
)
1= k
w he r ef s
i
g
i
a r e the sin g ul a r v alue s of X . The cas e ∥ X∥
1
:= max
i
f s
i
g
i
c or r e spond s t o the us ual ope r a t or nor m . F or k l , one alw a ys h as ∥ X∥
k
∥ X∥
l
.
W e al s o r e mind the r e a de r th a t a f unct ion f : U( d)!R i s s aid t o be L ips chitz c on t in uous w ith c on s t a n t
K i f it s a t i sfie s
j f( V) f( W)j K∥ V W∥
2
(3.24)
for al l V; W2 U( d) . F or br e v it y , in thi s s e ct ion w e de not e the H a a r a v e r a g e s as ⟨()⟩
U
a nd al s o oc casion al ly
dr op the ex p l ic it t ime de pe nde nc e .
101
P r opositio n 3.10
L e t U 2 U( d) be d i s tr i but e d a c c or d in g t o the H a a r me as ur e a nd f : U( d) ! R be a L ips chitz
c on t in uous f unct ion . The n for a n y ε > 0
Probfj f( U)⟨ f( U)⟩
U
j εg exp
(
d ε
2
4K
2
)
; (3.25)
w he r e K i s a L ips chitz c on s t a n t.
Dur in g the c our s e of the pr oof of E q ua t ion 3.3 , the fo l lo w in g t w o c on t in uit y r e s ults w i l l c ome in h a ndy .
L e mm a 3.1
( i ) The f unct ion f
W
( V) : U( d
A
) ! R w ith f
W
( V) := C
V A; W B
( t) i s L ips chitz c on t in uous w ith
c on s t a n t K
f
= 2 for al l t2R a nd W2 U( d
B
) .
( i i ) The f unct ion g( W) : U( d
B
) ! R w ith g( W) := ⟨ C
V A; W B
( t)⟩
V
i s L ips chitz c on t in uous w ith
c on s t a n t K
g
= 2= d
A
for al l t2R .
P r o of of lem m a. ( i ) L e t X; Y2 U( d
A
) . W e ne e d t o sho w th a t
j f
W
( X) f
W
( Y)j K
f
∥ X Y∥
2
:
F o l lo w in g the pr oof of E qua t ion 3.1 , w e ca n ex pr e s s
f
W
( V) = 1
1
d
Re T r
[
SU
y
2
t
( V
y
A
V
A
) U
2
t
( W
y
B
W
B
)
]
102
the r efor e
j f
W
( X) f
W
( Y)j
1
d
T r
[
U
2
t
( W
y
B
W
B
) SU
y
2
t
( X
y
A
X
A
Y
y
A
Y
A
)
]
1
d
X
y
A
X
A
Y
y
A
Y
A
1
;
w he r e in the l as t s t e p w e us e d the ine qual it y ∥ Tr( AB)∥∥ A∥
1
∥ B∥
1
a nd the fa ct th a t
U
2
t
( W
y
B
W
B
) SU
y
2
t
1
= 1 sinc e the ope r a t or w ithin the nor m i s unit a r y .
I n or de r t o ex pr e s s the l as t nor m as a f unct ion of the d i ffe r e nc e X
A
Y
A
, w e fir s t a dd a nd s ubtr a ct
Y
y
A
X
A
a nd the n us e the tr i a n gle ine qual it y . Thi s r e s ults in
1
d
X
y
A
X
A
Y
y
A
Y
A
1
1
d
(
( X
y
A
Y
y
A
)
X
A
1
+
Y
y
A
( X
A
Y
A
)
1
)
1
d
(
X
y
A
Y
y
A
1
I
X
A
1
+
X
A
Y
A
1
Y
y
A
I
1
)
w he r e for the l as t s t e p w e ut i l i z e d the ine qual it y ∥ AB∥
1
∥ A∥
1
∥ B∥
1
. N o w not ic e th a t
I
X
A
1
= d
sinc e X
A
i s unit a r y , a nd simi l a rly for
Y
y
A
I
1
. The r efor e w e ca n bound
j f
W
( X) f
W
( Y)j
X
A
Y
A
1
+
X
y
A
Y
y
A
1
2
X
A
Y
A
1
= 2
X Y
1
2
X Y
2
;
f r om w hich cle a rly one ca n t ak e K
f
= 2 .
( ii ) F ir s t not ic e th a t the H a a r a v e r a g e o v e r V
A
= V
I
B
ca n be pe r for me d , as w as done in the pr oof
of E qua t ion 3.1 . The r e s ult i s
g( W) = 1
1
d
Re Tr
[
SU
y
2
t
S
AA
′
d
A
U
2
t
W
y
B
W
B
]
= 1
1
d
Re Tr
[
U
y
2
t
S
BB
′
d
A
U
2
t
W
y
B
W
B
]
:
103
C on side r in g the r e le v a n t d i ffe r e nc e , w e ca n bound
j g( X) g( Y)j
1
d
A
1
d
Tr
[
U
y
2
t
S
BB
′ U
2
t
( X
y
B
X
B
Y
y
B
Y
B
)
]
1
d
A
1
d
X
y
B
X
B
Y
y
B
Y
B
1
:
N o w one ca n fo l lo w the ex a ct s a me s t e ps as in p a r t ( i ); the r e s ult i s ide n t ical exc e pt of the ex tr a fa ct or
1= d
A
th a t ca r r ie s thr ou gh, w hich or i g in a t e s f r om the a v e r a g in g. Thi s r e s ults in
j g( X) g( Y)j
2
d
A
X Y
2
f r om w hich one ca n t ak e K
g
= 2= d
A
. ■
E v e r y thin g i s no w in p l a c e t o g iv e the pr oof of E qua t ion 3.3 .
P r o of . L e t ε > 0 . W e w a n t t o sho w th a t , for V2 U( d
A
) a nd W2 U( d
B
) d i s tr i but e d inde pe nde n tly
a c c or d in g t o the H a a r me as ur e , it ho ld s
Prob( γ ε) exp
(
ε
2
d
max
64
)
w he r e γ :=j C
V A; W B
Gj a nd b y definit ion G =⟨ C
V A; W B
⟩
V; W
.
L e t us c on side r a n y p air V
A
; W
B
th a t s a t i sfie s ε γ . The n, f r om the tr i a n gle ine qual it y al s o
ε α + β;
w he r e w e s e t α :=
C
V A; W B
⟨ C
V A; W B
⟩
V
a nd β :=
⟨ C
V A; W B
⟩
V
G
. H e nc e w e h a v e for the c or r e-
spond in g pr o b a b i l it ie s
Probf γ εg Probf α + β εg:
104
H o w e v e r , i f α + β ε the n ne c e s s a r i ly α ε= 2 or β ε= 2 , the r efor e w e al s o h a v e
Probf α + β εg Prob(f α ε= 2g[f β ε= 2g):
U sin g the s t a nd a r d uni on bound o v e r the l as t ex pr e s sion r e s ults in
Probf γ εg Probf α ε= 2g+ Probf β ε= 2g: (3.26)
The t w o P r o b a b i l it ie s i n E q . ( 3.26 ) c a n be bounde d usin g L e v y ’ s le mm a . F or th a t , le t us fir s t define
the a u x i l i a r y f unct ion s f
W
( V) a nd g( W) as in 3.1 . C omb inin g the L ips chitz c on t in uit y r e s ult f r om the r e
w ith L e v y ’ s le mm a , one g e ts me as ur e c onc e n tr a t ion bound s
Prob
V
f
C
V A; W B
⟨ C
V A; W B
⟩
V
ε= 2g exp
(
d
A
ε
2
64
)
8 W (3.27a)
Probf⟨ C
V A; W B
⟩
V
G ε= 2g exp
(
d
2
A
d
B
ε
2
64
)
(3.27b)
W e a r e almos t done; it s u ffic e s t o not ic e th a t the bound ( 3.27a ) i s uni for m in W , he nc e it i s al s o a pp l i -
ca b le t o Probf α ε= 2g . The r efor e w e a r r iv e a t
Probfj C
V A; W B
( t) G( t)j εg exp
(
d
A
ε
2
64
)
+ exp
(
d
2
A
d
B
ε
2
64
)
2 exp
(
d
A
ε
2
64
)
:
(3.28)
N ot ic e the r e s ult in g bound i s inde pe nde n t of the dy n a mic s, as lon g as the l a tt e r i s unit a r y . F in al ly , one
ca n o bt ain the a n alo g ous bound for A$ B b y in v e r t in g the r o le s of V a nd W in the pr oof . The r efor e
w e o bt ain E q . ( 3.6 ). ■
Pr o positi o n 3.4
H e r e w e g iv e a s tr ai gh tfor w a r d pr oof as s umin g the NR C ho ld s ex a ctly . F or a mor e de t ai le d d i s c us sion, s e e
al s o the s e ct ion of the pr oof of E qua t ion 3.6 .
105
P r o of . Our s t a r t in g po in t i s E q . ( 3.4 ), w hich w e ne e d t o t ime-a v e r a g e . S inc e the H a mi lt oni a n i s b y
as s umpt ion nonde g e ne r a t e , w e ca n spe ctr al ly de c ompos e H =
∑
d
k= 1
E
k
P
k
, w he r e P
k
:= j φ
k
⟩⟨ φ
k
j .
W e the n h a v e
G( t)
NRC
= 1
1
d
2
∑
klmn
exp
[
i( E
k
+ E
l
E
m
E
n
) t
]
T r[ S
AA
′( P
k
P
l
) S
AA
′ ( P
m
P
n
)]:
T ime-a v e r a g in g the ex pone n t i al r e s ults in
exp
[
i( E
k
+ E
l
E
m
E
n
) t
]
= δ
E
k
+ E
l
E m E n; 0
NRC
== δ
k; m
δ
l; n
+ δ
k; n
δ
l; m
δ
k; l
δ
l; m
δ
m; n
w he r e in the l as t s t e p w e us e d the fa ct th a t e ne r g y g a ps a r e nonde g e ne r a t e . Th us
G( t)
NRC
= 1
1
d
2
(
∑
kl
T r[ S
AA
′( P
k
P
l
) S
AA
′ ( P
k
P
l
)]+
∑
kl
T r[ S
AA
′( P
k
P
l
) S
AA
′ ( P
l
P
k
)]
∑
k
T r[ S
AA
′( P
k
P
k
) S
AA
′ ( P
k
P
k
)]
)
= 1
1
d
2
(
∑
kl
T r[( P
k
P
l
) S
AA
′]
2
+
∑
kl
T r[( P
k
P
l
) S
BB
′]
2
∑
k
T r[( P
k
P
k
) S
AA
′]
2
)
;
w he r e for the s e c ond t e r m w e us e d th a t P
l
P
k
= S( P
k
P
l
) S a nd S = S
AA
′ S
BB
′ .
N o w , not ic e th a t p a r t i al tr a c e s ca n be for m al ly pe r for me d , g iv in g
T r
AA
′
BB
′ [( P
k
P
l
) S
AA
′] = T r
AA
′ [ T r
BB
′( P
k
P
l
) S
AA
′] = T r
AA
′
[
( ρ
( A)
k
ρ
( A
′
)
l
) S
AA
′
]
= T r
(
ρ
( A)
k
ρ
( A)
l
)
= R
( A)
kl
;
a nd simi l a rly
T r
AA
′
BB
′ [( P
k
P
l
) S
BB
′] = R
( B)
kl
T r
AA
′
BB
′ [( P
k
P
k
) S
AA
′] = T r
AA
′
BB
′ [( P
k
P
k
) S
BB
′] = R
( A)
kk
= R
( B)
kk
w he r e in the l as t l ine w e us e d the fa ct th a t the spe ctr a of ρ
( A)
k
a nd ρ
( B)
k
a r e e qual , u p t o ( ir r e le v a n t for
106
the tr a c e ) z e r oe s . The r e s ult fo l lo w s b y ex pr e s sin g the m a tr i x 2- nor m as ∥ X∥
2
2
=
∑
ij
X
ij
2
. ■
Pr o positi o n 3.5
Befor e pr oc e e d in g w ith the pr oof , le t us br iefly c omme n t on the ne e d of includ in g the p a r a me t e r α , w hich
c or r e spond s t o the f r a ct ion of the hi ghly e n t a n gle d ei g e n s t a t e s of the H a mi lt oni a n . F or c e r t ain H a mi lt oni a n
mode l s ( e . g., the cl as s of g a ppe d , local H a mi lt oni a n s o v e r one- d ime n sion al l a tt ic e s ys t e m s ) it i s w e l l kno w n
th a t the gr ound s t a t e fo l lo w s a n a r e a l a w for the e n t a n gle me n t e n tr op y [ 247 ]. Th us for l a r g e r s ys t e m si z e s ε
ca nnot be chos e n t o be s m al l for the gr ound s t a t e ( a nd al s o pos si b ly for the lo w ly in g exc it e d s t a t e s ), e v e n
for the s y mme tr ic d
A
= d
B
b ip a r t it ion . N e v e r the le s s, in the bul k of the spe ctr um, t y p ical ei g e n s t a t e s a r e
ex pe ct e d t o o bey in s t e a d a v o lume l a w , w hich i s c omp a t i b le w ith a n ε th a t ca n be chos e n t o be s uit a b ly
s m al l . The r efor e , w e ex pe ct th a t , for c e r t ain p h ysical ly r e le v a n t mod e l s, a l a r g e f r a ct ion α ca n be as s ume d
t o s a t i sf y thi s c ond it ion .
P r o of . T o simp l i f y the not a t ion, w e as s ume d
A
d
B
. L e t us al s o define I =f k : E
max
E(j φ
k
⟩) εg ,
i . e ., I i s the index s e t of thos e H a mi lt oni a n ei g e n s t a t e s th a t de v i a t e a t mos t b y ε f r om E
max
, w hi le w e us e
I t o l a be l the r e s t of the ei g e n s t a t e s . B y as s umpt ion, j Ij α d .
F ir s t of al l , not ic e th a t one ca n ex pr e s s the d i ffe r e nc e E
max
E(j ψ
AB
⟩) as the d i s t a nc e
E
max
E(j ψ
AB
⟩) = Tr( ρ
2
B
) 1= d
B
=
ρ
B
I= d
B
2
2
ρ
A
I= d
A
2
2
= Tr( ρ
2
A
) 1= d
A
:
S e tt in g for br e v it y Δ
( χ)
k
:= ρ
( χ)
k
I= d
χ
( χ = A; B ), w e h a v e for al l k 2 I th a t E
max
E(j φ
k
⟩) =
Δ
( B)
k
2
2
ε a nd he nc e al s o
Δ
( A)
k
2
2
=
ρ
( A)
k
I= d
A
2
2
ε . I t w i l l be c on v e nie n t for l a t e r t o ex pr e s s
⟨ ρ
( χ)
k
; ρ
( χ)
l
⟩
2
=
⟨ I= d
χ
+ Δ
( χ)
k
; I= d
χ
+ Δ
( χ)
l
⟩
2
=
1
d
χ
+⟨ Δ
( χ)
k
; Δ
( χ)
l
⟩
2
(3.29)
=
1
d
2
χ
+
2
d
χ
⟨ Δ
( χ)
k
; Δ
( χ)
l
⟩+⟨ Δ
( χ)
k
; Δ
( χ)
l
⟩
2
: (3.30)
107
M or e o v e r , b y the C a uch y - S ch w a r tz ine qual it y ,
⟨ Δ
( χ)
k
; Δ
( χ)
l
⟩
Δ
( χ)
k
2
Δ
( χ)
l
2
(3.31a)
w hi le
Δ
( χ)
k
2
2
8
>
>
<
>
>
:
ε i f k2 I;
1
1
d χ
othe r w i s e .
(3.31b)
L e t ’ s s t a r t f r om E q . ( 3.7 ). U sin g the fa ct th a t
R
( A)
D
2
2
=
R
( B)
D
2
2
a nd r e cal l in g G
ME
( t)
NRC
=
( 1 1= d)
2
w e g e t b y the tr i a n gle ine qual it y
G
ME
( t)
NRC
G( t)
NRC
1
d
2
R
( A)
2
2
1
d
+
1
d
2
R
( B)
2
2
1
d
+
1
d
2
R
( A)
D
2
2
1
: (3.32)
T o bound the fir s t t e r m w e w r it e
1
d
2
R
( A)
2
2
1
d
=
1
d
2
∑
kl
⟨ ρ
( A)
k
; ρ
( A)
l
⟩
2
1
d
1
d
2
A
1
d
+
1
d
2
∑
kl
(
2
d
A
⟨ Δ
( A)
k
; Δ
( A)
l
⟩
+⟨ Δ
( A)
k
; Δ
( A)
l
⟩
2
)
w he r e w e us e d E q . ( 3.29 ). S p l itt in g both of the s um s as
∑
k
=
∑
k2 I
+
∑
k= 2 I
a nd usin g E qs . ( 3.31 )
w e h a v e
1
d
2
∑
kl
⟨ Δ
( A)
k
; Δ
( A)
l
⟩
ε α
2
+ 2 α( 1 α)
√
ε
(
1
1
d
A
)
+( 1 α)
2
(
1
1
d
A
)
a nd
1
d
2
∑
kl
⟨ Δ
( A)
k
; Δ
( A)
l
⟩
2
ε
2
α
2
+ 2 α( 1 α) ε
(
1
1
d
A
)
+( 1 α)
2
(
1
1
d
A
)
2
:
108
P utt in g the m t o g e the r , a nd r e l a x in g s ome ine qual it ie s for cl a r it y , w e o bt ain for the fir s t t e r m of E q . ( 3.32 )
1
d
2
R
( A)
2
2
1
d
1
d
2
A
1
d
+ α ε
(
2
d
A
+ ε
)
+( 1 α)
2
( 1+
2
d
A
)+ 2( 1 α)( ε +
p
ε):
A n alo g ously for the s e c ond t e r m of E q . ( 3.32 ),
1
d
2
R
( B)
2
2
1
d
1
d
1
d
2
B
+ α ε
(
2
d
A
+ ε
)
+( 1 α)
2
( 1+
2
d
A
)+ 2( 1 α)( ε +
p
ε):
F or the thir d one , w e h a v e
R
( A)
D
2
2
=
∑
k
⟨ ρ
( A)
k
; ρ
( A)
k
⟩
2
=
d
B
d
A
+
2
d
A
∑
k
⟨ Δ
( A)
k
; Δ
( A)
k
⟩+
∑
k
⟨ Δ
( A)
k
; Δ
( A)
k
⟩
2
:
U sin g simi l a r m a nipul a t ion s as a bo v e , a nd unde r the c on v e n t ion d
A
d
B
,
1
d
2
R
( A)
D
2
2
1
1
d
2
(
d
B
d
A
1
)
+
1
d
[
α
(
2 ε
d
A
+ ε
2
)
+( 1 α)
(
2
d
A
+ 1
)]
P utt in g the ine qual it ie s t o g e the r , w e h a v e
G
ME
( t)
NRC
G( t)
NRC
λ 1
d
2
+
λ
2
1
d
2
B
+ α
[
2 ε
(
2
d
A
+
1
d
2
A
d
B
)
+ ε
2
(
2+
1
d
)
]
+( 1 α)
[
2( 1 α)
(
1+
2
d
A
)
+
2
d
+ 4
(
ε +
p
ε
)
]
(3.33)
w hich ca n be r e l a xe d t o g iv e the fin al r e s ult b y usin g
λ
2
1
d
2
B
λ 1
d
2
. ■
T h eo r e m 3.6
Befor e g iv in g the pr oof of the The or e m, w e fir s t br iefly d i s c us s s ome g e ne r al fa cts r e g a r d in g infinit e t ime-
a v e r a g e s, their c onne ct ion w ith the NR C a nd the NR C+, a nd ho w they g iv e r i s e t o the c or r e spond in g
e s t im a t e s .
109
L e t us c on side r unit a r y qua n tum dy n a mic s U
t
() = U
t
() U
y
t
g e ne r a t e d b y a H a mi lt oni a n H =
∑
k
~
E
k
Π
k
,
w he r e Π
k
de not e s the pr oj e ct or on t o the kth ei g e n sp a c e . A s a w a r m- u p , le t us calc ul a t e the t ime-a v e r a g e
of the s u pe r ope r a t or U
t
. The l a tt e r ca n be e asi ly pe r for me d b y not ic in g th a t exp
[
i(
~
E
k
~
E
l
) t
]
= δ
kl
. I t
r e s ults t o
P
H
:=U
t
=
∑
k
Π
k
() Π
k
(3.34)
w hich i s the ( H i l be r t - S chmidt or tho g on al ) pr oj e ct or on t o the c omm ut a n t of the al g e br a g e ne r a t e d b y
f Π
k
g
k
, i . e ., the pr oj e ct or w hos e r a n g e i s the sp a c e of ope r a t or s c omm ut in g w ith H .
The o bj e ct of in t e r e s t for us i s, in fa ct , U
2
t
sinc e
G( t) = 1
1
d
2
⟨ S
AA
′;U
2
t
( S
AA
′)⟩: (3.35)
R e as onin g as a bo v e , it fo l lo w s th a t the r e s ult in g s u pe r ope r a t or i s a g ain a pr oj e ct or , w hos e r a n g e i s the sp a c e
of ope r a t or s o v e r the r e p l ica t e d H i l be r t sp a c e H
2
th a t c omm ut e w ith H
( 2)
:= H
I+ I
H . The pr oj e ct or
ca n be ex p l ic itly ex pr e s s e d as
P
H
( 2) := U
2
t
=
∑
klmn
δ
~
E
k
~
E m;
~
E
l
~
E n
Π
k
Π
l
() Π
m
Π
n
(3.36)
T o e v alua t e the a bo v e s um, le t us for a mome n t ex a mine w h a t h a ppe n s w he n the e ne r g y g a ps f
~
E
k
~
E
l
g
kl
a r e nonde g e ne r a t e . i . e .,
NRC
+
:
~
E
k
+
~
E
l
=
~
E
m
+
~
E
n
() ( k = m ^ l = n) _ ( k = n ^ l = m): (3.37)
W e w i l l r efe r t o thi s c ond it ion o v e r the spe ctr um as NRC
+
, sinc e it c on s t itut e s a r e l a xe d v e r sion of the
NRC . W ithout a n y as s umpt ion o v e r the spe ctr um, one ca n alw a ys s e p a r a t e t w o c on tr i but ion s
P
H
( 2) =P
NRC
+ +P
NRC
+
(3.38)
110
w he r e
P
NRC
+ :=
∑
kl
Π
k
Π
l
() Π
k
Π
l
+
∑
kl
Π
k
Π
l
() Π
l
Π
k
∑
k
Π
k
Π
k
() Π
k
Π
k
(3.39)
a ndP
NRC
+
i s a n y pos si b ly r e m ainin g p ie c e , w hich v a ni she s i f a nd only i f the H a mi lt oni a n doe s inde e d
s a t i sf y NRC
+
.
Di s r e g a r d in g P
NRC
+
, one g e ts the e s t im a t e
G( t)
NRC
+
:= 1
1
d
2
T r[ S
AA
′P
NRC
+ ( S
AA
′)] (3.40)
= 1
1
d
2
(
∑
kl
T r[ S
AA
′( Π
k
Π
l
) S
AA
′ ( Π
k
Π
l
)]+
∑
kl
T r[ S
AA
′( Π
k
Π
l
) S
AA
′ ( Π
l
Π
k
)]
∑
k
T r[ S
AA
′( Π
k
Π
k
) S
AA
′ ( Π
k
Π
k
)]
)
;
(3.41)
w he r e the s e c ond e qua t ion fo l lo w s f r om the pr oof of E qua t ion 3.4 . C le a rly , i f al l pr oj e ct or s f Π
k
g a r e r a nk -
1, the n E q . ( 3.41 ) c o l l a ps e s t o the c or r e spond in g one for NR C , E q . ( 3.7). N ot ic e th a t one ca n e v alua t e
G( t)
NRC
+
r e g a r d le s s of w he the r the H a mi lt oni a n spe ctr um a ctual ly s a t i sfie s NR C+, a nd o bt ain the NR C+
e s t im a t e me n t ione d in the m ain t ex t.
E v ide n tly , one ca n al s o ex pr e s s the NR C t ime-a v e r a g e , E q . ( 3.7), in t e r m s of the c or r e spond in g pr oj e ct or
G( t)
NRC
= 1
1
d
2
T r[ S
AA
′P
NRC
( S
AA
′)]: (3.42)
I f the H a mi lt oni a n doe s not s a t i sf y NR C , pe r for min g a ( pos si b ly non unique ) de c omposit ion H =
∑
k
E
k
j φ
k
⟩⟨ φ
k
j
a nd e v alua t in g E q . ( 3.7 ) g iv e s r i s e t o the c or r e spond in g NR C e s t im a t e .
F in al ly , for the cas e of H a a r r a ndom unit a r ie s, one h as the c or r e spond in g pr oj e ct or U
2
Haar
:= P
Haar
w hos e r a n g e i s g iv e n b y the al g e br a g e ne r a t e d b y f I; Sg [ 248 ]. W e e v alua t e its ex p l ic it ex pr e s sion in the
nex t s e ct ion .
W e a r e no w r e a dy t o g iv e the pr oof of E qua t i on 3.6 .
111
P r o of . The k ey o bs e r v a t i on he r e i s th a t , b y c on s tr uct ion, the r a n g e of e a ch pr oj e ct or s a t i sfie s
Ran(P
H
( 2)) Ran(P
NRC
+) Ran(P
NRC
) Ran(P
Haar
): (3.43)
S inc e al l of the a bo v e a r e H i l be r t - S chmidt or tho g on al pr oj e ct or s, it al s o fo l lo w s th a t
P
H
( 2) P
NRC
+ P
NRC
P
Haar
: (3.44)
A s a r e s ult ,
⟨ S
AA
′;P
H
( 2)( S
AA
′)⟩⟨ S
AA
′;P
NRC
+( S
AA
′)⟩⟨ S
AA
′; P
NRC
( S
AA
′)⟩⟨ S
AA
′;P
Haar
( S
AA
′)⟩; (3.45)
f r om w hich E q . ( 3.11 ) fo l lo w s imme d i a t e ly . ■
Pr o o f o f Eq . ( 3.10 )
The H a a r a v e r a g e
G
Haar
=
( d
2
A
1)( d
2
B
1)
d
2
1
ca n be de r iv e d usin g fa ct th a t U
2
Haar
i s the C P TP or tho g on al pr oj e ct or o v e r the al g e br a g e ne r a t e d b y
f I; Sg [ 248 ], i . e .,
P
Haar
( X) :=U
2
Haar
( X) =
1
2
∑
α= 1
I+ α S
d( d+ α)
⟨ I+ α S; X⟩; (3.46)
w he r e S s w a psH a nd its du p l ica t e H
′
, as us ual . P lu gg in g the a bo v e in t o E q . ( 3.4), o ne g e ts
G
Haar
= 1
1
2d
2
∑
α= 1
j⟨ I+ α S; S
AA
′⟩j
2
d( d+ α)
w hich, a ft e r s ome simp le al g e br a , simp l i fie s t o the a nnounc e d r e s ult.
112
T h eo r e m 3.8
P r o of . L e t us do the χ = A cas e . The r e s ult r e l ie s on the o bs e r v a t ion th a t one ca n ex pr e s s S
AA
′ in
E q . ( 3.12 ) thr ou gh the H a a r a v e r a g e [ 248 ]
∫
dU
(
j ψ
U
⟩⟨ ψ
U
j
)
2
=
1
d
A
( d
A
+ 1)
( I
AA
′ + S
AA
′): (3.47)
P e r for min g the s ubs t itut ion r e s ults in
G( t) = 1+
1
d
2
A
T r( S
AA
′)
d
A
+ 1
d
A
∫
dU T r
(
S
AA
′
[
Λ
( A)
t
(j ψ
U
⟩⟨ ψ
U
j)
]
2
)
=
d
A
+ 1
d
A
(
1
∫
dU T r
[
(
Λ
( A)
t
(j ψ
U
⟩⟨ ψ
U
j)
)
2
]
)
=
d
A
+ 1
d
A
∫
dU S
lin
[
Λ
( A)
t
(j ψ
U
⟩⟨ ψ
U
j)
]
w he r e w e us e d the fa ct th a t Λ
( A)
t
( I) = I a nd the ide n t it y of E q . ( 3.16 ).
The χ = B cas e fo l lo w s simi l a rly . ■
Pr o o f o f Eq . ( 4.17 )
W e ne e d t o pr o v e th a t
Prob
{
S
lin
[
Λ
( χ)
t
(
j ψ⟩⟨ ψj
)]
d
χ
d
χ
+ 1
G( t)
ε
}
exp
(
d
χ
ε
2
64
)
(3.48)
w he r ej ψ⟩ i s a H a a r r a ndom pur e s t a t e . W e w i l l m ak e us e of the c onc e n tr a t ion of me as ur e m a chine r y , br iefly
pr e s e n t e d befor e the pr oof of E qua t ion 3.3 .
The r e s ult fo l lo w s b y the us e of L e v y ’ s le mm a a nd E qua t ion 3.8 , i f one sho w s th a t the f unct ion f :
U( d
χ
) ! R w ith f( V) := S
lin
[
Λ
( χ)
t
(j ψ
V
⟩⟨ ψ
V
j)
]
i s L ips chitz c on t in uous w ith K = 4 . A s befor e , w e
de not ej ψ
V
⟩ := Vj ψ
0
⟩ for s ome ( ir r e le v a n t ) r efe r e nc e s t a t e j ψ
0
⟩ .
113
I nde e d , le t us sho w the L ips chitz c on t in uit y . W e h a v e
f( V) f( W)
=
Λ
( χ)
t
(j ψ
V
⟩⟨ ψ
V
j)
2
2
Λ
( χ)
t
(j ψ
W
⟩⟨ ψ
W
j)
2
2
=
(
Λ
( χ)
t
(j ψ
V
⟩⟨ ψ
V
j)
2
+
Λ
( χ)
t
(j ψ
W
⟩⟨ ψ
W
j)
2
)
Λ
( χ)
t
(j ψ
V
⟩⟨ ψ
V
j)
2
Λ
( χ)
t
(j ψ
W
⟩⟨ ψ
W
j)
2
2
Λ
( χ)
t
(j ψ
V
⟩⟨ ψ
V
j) Λ
( χ)
t
(j ψ
W
⟩⟨ ψ
W
j)
1
2
U
t
(
j ψ
V
⟩⟨ ψ
V
j
I
d
χ
d
χ
)
U
t
(
j ψ
W
⟩⟨ ψ
W
j
I
d
χ
d
χ
)
1
2
(
j ψ
V
⟩⟨ ψ
V
jj ψ
W
⟩⟨ ψ
W
j
)
I
d
χ
d
χ
1
= 2
j ψ
V
⟩⟨ ψ
V
jj ψ
W
⟩⟨ ψ
W
j
1
;
w he r e in the s e c ond t o l as t l ine w e us e d the monot onic it y of the 1- nor m unde r the p a r t i al tr a c e a nd in the
l a s t l ine th a t it i s unit a r i ly in v a r i a n t. U t i l i z in g the ine qual it y
X
1
√
Rank( X)∥ X∥
2
, w e h a v e
f( V) f( W)
2
p
2
j ψ
V
⟩⟨ ψ
V
jj ψ
W
⟩⟨ ψ
W
j
2
= 4
√
1j⟨ ψ
V
j ψ
W
⟩j
2
4
√
2( 1j⟨ ψ
V
j ψ
W
⟩j) 4
√
2( 1 Re⟨ ψ
V
j ψ
W
⟩)
4∥j ψ
V
⟩j ψ
W
⟩∥ 4∥ V W∥
1
4∥ V W∥
2
he nc e one c a n t ak e K = 4 .
Pr o positi o n ??
P r o of . L e t us fir s t ex pr e s s the C ho i s t a t e s ex p l ic itly as
ρ
Λ
( χ)
t
=
(
Λ
( χ)
t
I
)
j φ
+
⟩⟨ φ
+
j =
1
d
χ
∑
ij
Λ
( χ)
t
(
j i⟩⟨ jj
)
j i⟩⟨ jj
ρ
T
( χ)
=
(
T
( χ)
I
)
j φ
+
⟩⟨ φ
+
j =
(
I
χ
d
χ
)
2
:
114
W r it in g S
χ χ
′ =
∑
d χ
i; j= 1
j i⟩⟨ jj
j j⟩⟨ ij one al s o h as f r om E q . ( 3.12 )
G( t) = 1
1
d
2
χ
∑
ij
Λ
( χ)
t
(
j i⟩⟨ jj
)
2
2
:
Th us, ex p a nd in g the C ho i s t a t e d i s t a nc e ,
ρ
Λ
( χ)
t
ρ
T
( χ)
2
2
=⟨ ρ
Λ
( χ)
t
ρ
T
( χ)
; ρ
Λ
( χ)
t
ρ
T
( χ)
⟩ =⟨ ρ
Λ
( χ)
t
; ρ
Λ
( χ)
t
⟩ 2⟨ ρ
Λ
( χ)
t
; ρ
T
( χ)
⟩+⟨ ρ
T
( χ)
; ρ
T
( χ)
⟩
=
ρ
Λ
( χ)
t
2
2
1
d
2
χ
=
1
d
2
χ
∑
ij
Λ
( χ)
t
(
j i⟩⟨ jj
)
2
2
1
d
2
χ
= 1 G( t)
1
d
2
χ
w hich i s w h a t w e w a n t e d . ■
Pr o o f o f
Λ
( χ)
t
T
( χ)
♢
d
3= 2
χ
√
G
( χ)
max
G( t) a nd a n a ppli c a ti o n o n i nf o rma ti o n spr e a d i n g
W e fir s t r e mind the r e a de r th a t the d i a mond nor m ca n be define d as ∥X∥
♢
:= ∥X
I
d
∥
1; 1
w he r eI
d
de n ot e s the ide n t it y qua n tum ch a nne l o v e r H
= C
d
a nd∥X∥
1; 1
:= sup
∥ A∥
1
= 1
∥X( A)∥
1
. One of the
r e as on s for thi s definit ion i s the pr ope r t y th a t ∥X
Y∥
♢
= ∥X∥
♢
∥Y∥
♢
, w hich in g e ne r al fai l s for the
∥()∥
1; 1
nor m ( s e e , e . g., [ 114 ]).
L e t us no w pr o v e th a t
√
G
( χ)
max
G( t)
Λ
( χ)
t
T
( χ)
♢
d
3= 2
χ
√
G
( χ)
max
G( t):
P r o of . The r e s ult fo l lo w s e asi ly b y ut i l i z in g the ine qual it ie s
ρ
E 1
ρ
E 2
1
E
1
E
2
♢
d
ρ
E 1
ρ
E 2
1
(3.49)
th a t ho ld for a n y p air of C P TP m a ps . The ine qual it y w as r e por t e d b y J o hn W a tr ous in [ 249 ]. The r e s ult
fo l lo w s b y us e of the ine qual it y
X
1
p
d
X
2
a nd ?? . ■
115
A s a n a dd it ion al a pp l ica t ion of E q . ( 3.49 ), w e ca n ut i l i z e it t o bound f r om a bo v e the f r a ct ion of t ime
s uch th a t
Λ
( χ)
t
T
( χ)
♢
ε ho ld s tr ue . Thi s ca n be done b y c omb inin g E q . ( 3.49 ) w ith our e a rl ie r
t ime-a v e r a g e s . The r e s ult
Prob
{
t
Λ
( χ)
t
T
( χ)
♢
ε
}
2d
3= 2
χ
ε d
χ
κ; (3.50)
w he r e κ :=
√
1+
d
2
χ
2
(
G
Haar
G( t)
)
, de mon s tr a t e s in y e t a nothe r w a y th a t i f d
χ
≫ d
χ
a nd κ = O( 1) ( i . e .,
the e qui l i br a t ion i s s u ffic ie n tly clos e t o the H a a r e s t im a t e ), the n the r e duc e d e v o lut ion i s ne c e s s a r i ly clos e
t o the m a x im al ly mi x in g one for a l a r g e f r a ct ion of t ime .
P r o of . Our s t a r t in g po in t w i l l be ine qual it y ( 3.49 ),
Λ
( χ)
t
T
( χ)
♢
d
3= 2
χ
√
G
( χ)
max
G( t) . B y t ak in g
the t ime-a v e r a g e of both side s, a nd the n usin g the c onca v it y of the s qua r e r oot , w e o bt ain
Λ
( χ)
t
T
( χ)
♢
d
3= 2
χ
√
G
( χ)
max
G( t) d
3= 2
χ
√
(
G
( χ)
max
G
Haar
)
+
(
G
Haar
G( t)
)
2
d
3= 2
χ
d
χ
κ;
w he r e w e a ppr o x im a t e d the d i ffe r e nc e
G
( χ)
max
G( t)
Haar
=
( d
2
χ
1)
2
d
2
χ
( d
2
1)
2
d
2
χ
:
F in al ly , E q . ( 3.50 ) fo l lo w s b y the us e of M a rk o v ’ s ine qual it y . ■
3.3 Ha ar me a s u re , u nit ar y k - de sig n s a nd the b ip ar tite O T O C
H e r e w e d i s c us s in mor e de t ai l s ho w the H a a r me as ur e in the definit ion of the b ip a r t it e O T O C , E q . ( 3.3 ),
ca n be r e p l a c e d b y othe r pos si b le a v e r a g in g cho ic e s, in a w a y th a t E q . ( 3.4) ( a nd e v e r y thin g th a t s t e m s f r om
it ) r e m ain s v al id .
L e t us fir s t r e cal l the definit ion of a ( unit a r y ) k - de si gn [ 95 – 98 , 190 ]. C on side r a n e n s e mb le of unit a r y
116
ope r a t or s Λ =f( p
i
; U
i
)g
i
a nd define the fa mi ly of C P TP m a ps
E
( k)
Λ
:=
∑
i
p
i
U
k
i
() U
y
k
i
(3.51)
E
( k)
Haar
:=
∫
dU U
k
() U
y
k
(3.52)
for k2N . The e n s e mb le Λ for m s a k - de si gn i f E
( k)
Λ
=E
( k)
Haar
. I n w or d s, a k - de si gn e m ul a t e s H a a r a v e r a g in g
u p t o ( a t le as t ) the kth mome n t.
N o w , le t us in v e s t i g a t e w h a t i s the f r e e dom o v e r the pos si b le pr o b a b i l it y me as ur e s of V
A
a nd W
B
in
E q . ( 3.3 ), s uch th a t E q . ( 3.4) ho ld s tr ue w ithout mod i fica t ion . I t i s e as y t o s e e , b y the pr oof of E qua t ion 3.1 ,
th a t w e a r e in fa ct loo k in g for a unit a r y e n s e mb le Λ r e t ainin g the v al id it y of E q . ( 3.17 ). I n tur n, the l a tt e r i s
jus t a v e ct or i z e d for m of the 1 - de si gn c ond it ion E
( 1)
Λ
=E
( 1)
Haar
. One ca n the r efor e s ubs t itut e the H a a r me a -
s ur e o v e r U( d
A
) a nd U( d
B
) w ith 1 - de si gn s o v e r the c or r e spond in g sp a c e s; the f ul l H a a r r a ndomne s s i s not
pr o be d b y the O T O C [ 190 ].
M or e o v e r , 1 - de si gn s fa ct or i z e , i . e ., i f Λ
1
=f( p
( 1)
i
; U
( 1)
i
)g
i
a nd Λ
2
=f( p
( 2)
j
; U
( 2)
j
)g
j
a r e 1- de si gn s o v e r H
A
a n dH
B
r e spe ct iv e ly , the n Λ
1
Λ
2
:=f( p
( 1)
i
p
( 2)
j
; U
( 1)
i
U
( 2)
j
)g
ij
i s a 1- de si gn o v e r H =H
A
H
B
. Thi s
fo l lo w s jus t b y the 1- de si gn c ond it ion in the for m of E q . ( 3.17 ) a nd the fa ct th a t the s w a p ope r a t or o v e r the
du p l ica t e d sp a c e H
H
′
fa ct o r i z e s S
AB; A
′
B
′ = S
AA
′ S
BB
′ .
Thi s l as t fa ct h as a n impor t a n t imp l ica t ion for the p h ysical ly r e le v a n t cas e of m a n y - body s ys t e m s . C on-
side r the cas e w he r e H
χ
=
⊗
i
H
( i)
χ
for χ = A; B , i . e ., w he n A a nd B a r e m a de u p of ( not ne c e s s a r -
i ly ide n t ical ) ind iv idual s ubs ys t e m s . The n the O T O C of E q . ( 3.3) r e m ain s unch a n g e d i f the a v e r a g e s
∫
dV
A
a nd
∫
dW
b
a r e r e p l a c e d b y the unit a r y e n s e mb le
⊗
i
Λ
( i)
χ
, w he r e e a ch Λ
( i)
χ
i s a 1 - de si gn on H
( i)
χ
. I n
othe r w or d s, it i s alw a ys e nou gh t o a v e r a g e o v e r unit a r y ope r a t or s th a t fa ct or i z e c omp le t e ly . F or in s t a nc e ,
in the cas e of a sp in- 1= 2 m a n y - body s ys t e m H
( i)
χ
= C
2
s uch a n ex a mp le i s g iv e n b y the P a ul i 1 - de si gn
Λ
( i)
χ; Pauli
:=f 1= 4; σ
k
g
3
k= 0
[ 250 ].
117
3.4 Es tima ting the b ip ar tite O T O C vi a l ine ar en trop y me a s u re men ts of r an -
d om pu re s t a te s
H e r e w e pr e s e n t a b asic pr ot oc o l , s t e mmin g d ir e ctly f r om E qua t ion 3.8 , for the e s t im a t ion of the b ip a r t it e
O T O C v i a r e pe a t e d me as ur e me n ts of a sin gle ex pe ct a t ion v alue .
Figure 3.4.1: Proto col to fo r the estimation of the purit y 1 S
lin
[
Λ
( A)
t
(j ψ⟩⟨ ψj)
]
acco rding to
Eq. ( 3.13 ) . The resulting purit y constitutes also an estimate of the bipa rtite OTOC, up to a simple
p r op o rtionalit y facto r. The final measurement of the sw ap op erato r can b e realized, fo r instance, b y
measuring the exp ectation value of A and A
′
over any p referred p ro duct basisfj i⟩
j j⟩g
d A
i; j= 1
, without
the need fo r coherences.
.
A s po in t e d out in the m ain t ex t , the l ine a r e n tr op y of a s t a t e ca n be ex pr e s s e d as a n ex pe ct a t ion v alue , 1
S
lin
( ρ) = T r( S ρ
2
) a t the ex pe n s e of r e quir in g t w o c op ie s of the s t a t e ρ , thou gh unc or r e l a t e d . C omb inin g
E qua t ion 3.8 w ith the a bo v e o bs e r v a t ion, one ca n r e al i z e a simp le pr ot oc o l for e s t im a t in g the b ip a r t it e
O T O C v i a me as ur in g the ex pe ct a t ion v alue of the s w a p ope r a t or o v e r p air s of r a ndomly g e ne r a t e d s t a t e s
j ψ⟩2H
A
. W e s che m a t ical ly dr a w the pr ot oc o l in F i g ur e 3.4.1 .
A v e r a g in g the r e s ult in g ex pe ct a t ion v alue o v e r H a a r r a ndom pur e s t a t e s j ψ⟩ c on v e r g e s t o the ex a ct v alue
of the b ip a r t it e O T O C . I n l i gh t of E q . ( 4.17 ), the ex pe ct e d n umbe r of s a mp le for thi s c on v e r g e nc e t o a
g iv e n a c c ur a c y dr ops fas t as d
A
incr e as e s . C le a rly , the c or r e spond in g pr ot oc o l w ith the r o le s of A a nd B
in t e r ch a n g e d i s for m al ly e quiv ale n t.
A lon g c onc e ptual ly simi l a r l ine s, the r e h a v e be e n a n umbe r of pr opos al s for pr o b in g the l ine a r e n tr op y
of a s t a t e in a n ex pe r ime n t al ly a c c e s si b le w a y . F or ex a mp le , in a r e c e n t ex pe r ime n t [ 231 ] qua n tum pur it y
118
(w hich i s d ir e ctly r e l a t e d t o the s e c ond - or de r R é n y i e n t a n gle me n t e n tr op y ) w as me as ur e d b y in t e r fe r in g
t w o unc or r e l a t e d but ide n t ical c op ie s of a m a n y - body qua n tum s t a t e; simi l a r ide as h a v e al s o be e n c on sid -
e r e d pr e v iously [ 227 – 230 ]. I n p a r t ic ul a r , thi s s che me neithe r r e quir e s f ul l qua n tum s t a t e t omo gr a p h y nor
the us e of e n t a n gle me n t w itne s s e s t o e s t im a t e e n t a n gle me n t of a qua n tum s t a t e .
F ur the r mor e , the r e h a v e be e n r e c e n t pr opos al s b as e d on me as ur e me n ts o v e r r a ndom local b as e s th a t ca n
pr o be e n t a n gle me n t g iv e n jus t a sin gle c op y of the qua n tum s t a t e , a nd , in thi s s e n s e , g o bey ond tr a d it ion al
qua n tum s t a t e t omo gr a p h y . The m ain ide a c on si s ts of d ir e ctly ex pr e s sin g the l ine a r e n tr op y [ 232 , 233 ],
as w e l l as othe r f unct ion s of the s t a t e [ 234 ], as a n e n s e mb le a v e r a g e of me as ur e me n ts o v e r r a ndom b as e s .
R e l a t e d ide as h a v e al s o be e n a d a pt e d t o pr o be O T O C s [ 236 , 237 ] a nd mi xe d s t a t e e n t a n gle me n t [ 235 ].
119
4
I nfor m a tion S cr a mblin g a nd C h a o s in O p e n Qua n tum
S yst e ms
4.1 A bs tr a c t
Out - of -t ime- or de r e d c or r e l a t or s ( O T O C s ) h a v e be e n ex t e n siv e ly us e d o v e r the l as t fe w y e a r s t o s tudy in-
for m a t ion s cr a mb l in g a nd qua n tum ch a os in m a n y - body s ys t e m s . I n thi s ch a pt e r , w e ex t e nd the for m al i s m
of the a v e r a g e d b ip a r t it e O T O C of S t y l i a r i s et a l [ P h ys . R e v . L e tt. 126 , 030601 (2021)] t o the cas e of
ope n qua n tum s ys t e m s . The dy n a mic s i s no lon g e r unit a r y but it i s de s cr i be d b y mor e g e ne r al qua n tum
ch a nne l s ( tr a c e pr e s e r v in g , c omp le t e ly posit iv e m a ps ). Thi s “ ope n b ip a r t it e O T O C” ca n be tr e a t e d in a n
120
ex a ct a n aly t ical fashion a nd i s sho w n t o a moun t t o a d i s t a nc e be t w e e n t w o qua n tum ch a nne l s . M or e o v e r ,
our a n aly t ical for m un v ei l s c ompe t in g e n tr op ic c on tr i but ion s f r om infor m a t ion s cr a mb l in g a nd e n v ir on-
me n t al de c o he r e nc e s uch th a t the l a tt e r ca n o bf us ca t e the for me r . T o e luc id a t e thi s s ubtle in t e r p l a y w e a n-
aly t ical ly s tudy spe c i al cl as s e s of qua n tum ch a nne l s, n a me ly , de p h asin g ch a nne l s, e n t a n gle me n t - br e ak in g
ch a nne l s, a nd othe r s . F in al ly , as a p h ysical a pp l ica t ion w e n ume r ical ly s tudy d i s sip a t iv e m a n y - body sp in-
ch ain s a nd sho w ho w the c ompe t in g e n tr op ic effe cts ca n be us e d t o d i ffe r e n t i a t e be t w e e n in t e gr a b le a nd
ch a ot ic r e g ime s .
T ex t for thi s C h a pt e r i s a d a pt e d f r om [ 251 ].
4.2 In troduc tion
M a n y - body qua n tum ch a os h as w itne s s e d a r e n ai s s a nc e in r e c e n t y e a r s, spe a rhe a de d b y the s tudy of the out -
of -t ime- or de r e d c or r e l a t or ( O T O C ) a nd its in t e r p l a y w ith infor m a t ion s cr a mb l in g [ 17 , 18 , 76 – 79 , 190 ].
The pr e c i s e r o le th a t the O T O C p l a ys in ch a r a ct e r i z in g qua n tum ch a os, v i a its shor t -t ime ex pone n t i al
gr o w th, i s w e l l - unde r s t ood in s ys t e m s w ith eithe r ( i ) a s e micl as sical l imit , or ( i i ) w ith a l a r g e n umbe r of
local de gr e e s of f r e e dom [ 18 , 76 ].
H o w e v e r , its r o le in finit e s ys t e m s, s uch as qua n tum sp in- ch ain s i s s t i l l unde r clos e ex a min a t ion [ 19 –
24 ]; s e e al s o R ef . [ 25 ] de b a t in g s ome of the s e r e s ults . O T O C s h a v e al s o be e n a pp l ie d t o s tudy a v a r ie t y
of m a n y - body p he nome n a , r a n g in g f r om qua n tum p h as e tr a n sit ion s [ 212 ] al l the w a y t o m a n y - body lo -
cal i za t ion [ 193 , 252 – 256 ]. R e c e n tly , a c onne ct ion be t w e e n O T O C s, c o he r e nc e- g e ne r a t in g po w e r , a nd
g e ome tr y w as un v ei le d in R ef . [ 2 ]. Thi s f ur the r qual i fie s the in tuit ion th a t the O T O C me as ur e s inc omp a t -
i b i l it y be t w e e n o bs e r v a b le s [ 257 ]. M or e o v e r , in R ef s . [ 258 , 259 ] v a r ious qua n t i fie r s of ch a os w e r e uni fie d
unde r the f r a me w ork of iso s p e c tr a l t w i r l i n g . The O T O C s ’ the or e t ical in v e s t i g a t ion s h a v e al s o be e n c om-
p le me n t e d w ith s e v e r al s t a t e- of -the-a r t ex pe r ime n ts, w he r e dy n a mical fe a tur e s of the O T O C w e r e s tud ie d
usin g s u pe r c onduct in g qub its [ 260 , 261 ], n ucle a r m a gne t ic r e s on a nc e [ 262 – 265 ], ion-tr a p qua n tum sim-
ul a t or s [ 266 , 267 ], a mon g othe r s [ 268 , 269 ].
I n r e c e n t w ork s it w as not e d th a t , for v a r ious finit e- d ime n sion al m a n y - body s ys t e m s w ith sp a t i al local it y ,
the e q u i l i br a t io n v a l u e of O T O C s ca n d i a gnos e the ch a ot ic - v s -in t e gr a b le n a tur e of dy n a mic s [ 81 , 82 , 168 ].
121
I n p a r t ic ul a r , thi s e mp h asi s on local it y w as e s s e n t i al in e s t a b l i shin g the c onne ct ion [ 197 ] be t w e e n O T O C s
a nd L os chmidt E cho [ 32 – 35 ], a w e l l - e s t a b l i she d si gn a tur e of qua n tum ch a os . M a n y qual it a t iv e fe a tur e s of
the O T O C a r e in s e n sit iv e t o the spe c i fic cho ic e of ope r a t or s, as lon g as their local it y i s fi xe d . The r efor e , it
c on s t itut e s a me a nin g f ul simp l i fica t ion t o foc us on O T O C s a v e r a g e d o v e r ( s uit a b ly d i s tr i but e d ) r a ndom
ope r a t or s .
G iv e n a b ip a r t it ion of the s ys t e m H i l be r t sp a c e , one ca n a n aly t ical ly pe r for m the uni for m a v e r a g e o v e r
p air s of r a ndom unit a r y ope r a t or s, s u ppor t e d o v e r eithe r side of the b ip a r t it ion [ 168 ]. Thi s a v e r a g e d b i -
p a r t it e O T O C h as a t w o - fo ld ope r a t ion al si gni fica nc e: ( i ) it qua n t i fie s the ope r a t or e n t a n gle me n t of the
dy n a mic s [ 202 , 270 ], a nd ( i i ) it qua n t i fie s a v e r a g e e n tr op y pr oduct ion as w e l l the s cr a mb l in g of infor m a -
t ion a t the le v e l of qua n tum ch a nne l s .
M or e o v e r , the e qui l i br a t ion v alue of the O T O C s w as sho w n t o be s e n sit iv e t o the a moun t of s tr uctur e
in the spe ctr um ( for e . g., quasi - f r e e v e r s us nonin t e gr a b le mode l s h a v e de g e ne r a t e v e r s us g e ne r ic spe ctr um,
r e spe ct iv e ly ). Thi s induc e s a hie r a r ch y of c on s tr ain ts th a t ca n be ut i l i z e d t o bound the O T O C’ s e qui l i br a -
t ion v alue . R e m a rk a b ly , the e qui l i br a t ion v alue of the O T O C al s o c on t ain s infor m a t ion a bout the e n t a n-
gle me n t of the f u ll s ys t e m of H a mi lt oni a n ei g e n s t a t e s [ 168 ]. N ot e th a t , a v e r a g in g the O T O C o v e r local ,
r a ndom ope r a t or s, s u ppor t e d on a b ip a r t it ion w as al s o s tud ie d in R ef s . [ 171 , 193 ].
A l l the a bo v e pr o v ide s c ompe l l in g e v ide nc e th a t the a v e r a g e d b ip a r t it e O T O C i s a po w e r f ul t oo l t o in-
v e s t i g a t e infor m a t ion s cr a mb l in g a nd ch a os in m a n y - body qua n tum s ys t e m s . I n thi s ch a pt e r , w e w i l l ex -
t e nd thi s for m al i s m t o ope n qua n tum s ys t e m s, i . e ., s ys t e m s c ou p le d t o a n e n v ir onme n t , w hich unde r g o a
n o n -u n it a r y t ime e v o lut ion . I n fa ct , the s e a r e the s ys t e m s th a t a r e d ir e ctly r e le v a n t t o ex pe r ime n t al situa -
t ion s [ 271 , 272 ] a nd t o c ur r e n t , as w e l l as f utur e-t e chno lo g ie s for qua n tum infor m a t ion pr oc e s sin g [ 263 ,
266 , 271 ].
W e not e th a t ope n- s ys t e m effe cts in infor m a t ion s cr a mb l in g h a v e al s o be e n r e por t e d befor e in R ef s . [ 273 –
281 ]. H o w e v e r , our foc us i s on the ope n- s ys t e m v e r sion of the bip a r t it e a v e r a g e d O T O C , w hich, as me n-
t i one d befor e , h as a cle a r ope r a t ion al c on t e n t [ 168 ].
122
4.3 Gener al re s u l ts
L e tH
=C
d
be the H i l be r t sp a c e c or r e spond in g t o a d - d ime n sion al qua n tum s ys t e m w ith L(H) de not in g
the sp a c e of l ine a r ope r a t or s on H . Qua n tum s t a t e s a r e r e pr e s e n t e d b y ρ 2 L(H) , s uch th a t ρ 0 a nd
Tr ρ = 1 . The sp a c e L(H) ca n be e ndo w e d w ith a H i l be r t - S chmidt inne r pr oduct ⟨ X; Y⟩ := Tr
[
X
y
Y
]
,
tr a n sfor min g it in t o a H i l be r t sp a c e .
1 Pr elimi n a r i e s
The e v o lut ion of qua n tum s t a t e s i s de s cr i be d v i a q u a n t u m c h a n n e ls , l ine a r s u pe r ope r a t or s E : L(H) !
L(K) th a t a r e c omp le t e ly posit iv e a nd tr a c e pr e s e r v in g ( C P TP ). The t ime e v o lut ion of o bs e r v a b le s i s v i a
the a d j o in t ch a nne l , E
y
w hich i s define d as,
⟨ X;E( Y)⟩ =
⟨
E
y
( X); Y
⟩
8 X2L(H); Y2L(K): (4.1)
F or clos e d qua n tum s ys t e m s, the dy n a mic s i s de s cr i be d b y a fa mi ly of unit a r y ch a nne l s, U
t
( X) :=
U
y
t
XU
t
, w he r e U
t
2U(H) (= unit a r y gr ou p o v e r the H i l be r t sp a c e H )8 t:
G iv e n a unit a r y dy n a mic s f U
t
g
t 0
o v e rH , the f und a me n t al qua n t it y th a t w e w i l l us e t o qua n t i f y in-
for m a t ion s cr a mb l in g i s g iv e n b y the “ the s qua r e of the c omm ut a t or ” be t w e e n a n ope r a t or W a nd a t ime-
e v o lv e d one V( t) := U
y
t
VU
t
,
C
V; W
( t) :=
1
2d
∥[ V( t); W]∥
2
2
; (4.2)
w he r e∥ X∥
2
:=
√
⟨ X; X⟩ . I f w e choos e V; W t o be unit a r y , the n the c omm ut a t or C
V; W
( t) i s r e l a t e d t o the
four -po in t c or r e l a t ion f unct ion,
F
V; W
( t) :=
1
d
Tr
(
V
y
( t) W
y
V( t) W
)
; (4.3)
123
as
C
V; W
( t) = 1
1
d
ReF
V; W
( t): (4.4)
The four -po in t f unct ion F
V; W
( t) w ith un us ual t ime- or de r in g i s the s o - cal le d cal le d the “ out - of -t ime- or de r e d
c or r e l a t or ” ( O T O C ). N ot e th a t , w e w i l l be w ork in g w ith the infinit e-t e mpe r a tur e cas e thr ou ghout thi s
ch a pt e r , he nc e the fa ct or of 1= d in the O T O C ( a nd the as s oc i a t e d s qua r e d c omm ut a t or ).
F o l lo w in g [ 168 ], w e w i l l f r om no w on c on side r a b ip a r t it e H i l be r t sp a c e , H
AB
=H
A
H
B
=C
d A
C
d B
a nd define the a v er a ge d bip a r t it e O T O Cs b y
G(U
t
) :=E
V A; W B
[ C
V A; W B
( t)]; (4.5)
w he r e , V
A
= V
I
B
; W
B
= I
A
W , w ith V2U(H
A
); W2U(H
B
); a ndE
V; W
[] :=
∫
Haar
dV dW[]
de not e s H a a r -a v e r a g in g o v e r the s t a nd a r d uni for m me as ur e o v e r U(H
A( B)
) . W e e mp h asi z e th a t , in thi s
w ork ( a nd R ef . [ 168 ]), the H a a r -a v e r a g e s a r e pe r for me d o v e r the ope r a t or s V; W in the O T O C but n o t
o v e r the dy n a mical unit a r y U
t
, w hich i s left as a n input t o thi s c or r e l a t ion f unct ion . e q . ( 5.8) define s the
k ey qua n t it y of thi s ch a pt e r . I n R ef . [ 168 ] w e sho w e d th a t the doub le-a v e r a g e in e q . ( 5.8 ) ca n be pe r for me d
a n aly t ical ly a nd for unit a r y dy n a mic s, the a v e r a g e d b ip a r t it e O T O C t ak e s the fo l lo w in g for m . Thr ou ghout
thi s c h a pt e r , w e w i l l us e pr ime d s ubs ys t e m s A
′
t o r efe r t o a r e p l ica of a s ubs ys t e m A , i . e .,H
A
= H
A
′ ,
H
B
=H
B
′ , a nd s o on .
P r opositio n 4.1: [ 168 ]
L e t S
AA
′ be the ope r a t or o v e r H
AB
H
A
′
B
′ th a t s w a ps A w ith its r e p l ica A
′
, o ne h as
G(U
t
) = 1
1
d
2
Tr
(
S
AA
′ U
2
t
S
AA
′ U
y
2
t
)
: (4.6)
Thi s simp le for m ul a — w hich, quit e s ur pr i sin gly , co i n ci de s w ith the ope r a t or e n t a n gle me n t of U
t
as or i g i -
124
n al ly define d in R ef . [ 202 ] — pr o v ide s the s t a r t in g po in t of the a n alysi s in [ 168 ]. I t al lo w s one t o c onne ct
the a v e r a g e d b ip a r t it e O T O C t o a v a r ie t y of p h ysical a nd infor m a t ion-the or e t ic qua n t it ie s e . g., e n tr op y
pr oduct ion, ch a nne l d i s t in g ui sh a b i l it y , a mon g othe r s . F or c omp le t e ne s s, w e r e v ie w s ome of the s e ide as
in s e ct ion 4.7 .
W e a r e no w r e a dy t o d i s c us s the g e ne r al i za t ion of the b ip a r t it e O T O C for m al i s m t o ope n qua n tum s ys -
t e m s, w he r e , unit a r y tr a n sfor m a t ion s a r e r e p l a c e d b y mor e g e ne r al qua n tum ope r a t ion s .
2 O pen O T O C
A s s umin g th a t s t a nd a r d M a rk o v i a n pr ope r t ie s ho ld , the s ys t e m dy n a mic s in the S chr öd in g e r p ictur e i s the n
de s cr i be d b y a tr a c e-pr e s e r v in g , c omp le t e ly posit iv e ( C P ) m a p , al s o kno w n as a qua n tum ch a nne l E
y
[ 282 ].
I t fo l lo w s th a t in the H ei s e nbe r g p ictur e ( i . e ., the one a dopt e d thr ou ghout thi s ch a pt e r ), the o bs e r v a b le
dy n a mic s i s de s cr i be d b y the u n it a l C P m a pE . R e cal l th a t a qua n tum ch a nne l E i s cal le d unit al i f a nd only
i fE(
I
d
) =
I
d
, w he r e
I
d
i s the m a x im al ly mi xe d s t a t e ( or the G i b bs s t a t e a t infinit e t e mpe r a tur e ). N a me ly , s uch
a m a p h as the m a x im al ly mi xe d s t a t e as a fi xe d po in t. S e v e r al impor t a n t p h ysical ope r a t ion s th a t one ca n
pe r for m on a qua n tum s ys t e m a r e unit al , for ex a mp le , unit a r y e v o lut ion, pr oj e ct iv e me as ur e me n ts w ithout
pos t - s e le ct ion, de p h asin g ch a nne l s, a mon g othe r s . A qua n tum ch a nne l E i s tr a c e pr e s e r v in g i f a nd only i f
E
y
i s its e l f unit al . W hi le m a n y of the r e s ults a nd ide as w hich fo l lo w do not r e ly on thi s as s umpt ion, for the
s a k e o f simp l ic it y , w e w i l l as s ume th a t E
y
i s inde e d unit a l ( )E i s a qua n tum c h a nne l ).
W e define the ope n ( a v e r a g e d ) b ip a r t it e O T O C b y ,
G(E) :=
1
2d
E
V A; W B
∥[E( V
A
); W
B
]∥
2
2
; (4.7)
w he r e V
A
, W
B
a nd the a v e r a g e a r e as define d in e q . ( 5.8). The fir s t s t e p i s t o g e ne r al i z e e q . ( 4.6) t o the ope n
cas e .
125
P r opositio n 4.2
L e t S S
AA
′
BB
′ be the s w a p ope r a t or o v e r H
AB
H
A
′
B
′ , the n for a qua n tum ch a nne l E :L(H
AB
)!
L(H
AB
) , the ope n b ip a r t it e O T O C t ak e s the fo l lo w in g for m,
G(E) =
1
d
2
Tr
(
( d
B
S S
AA
′)E
2
( S
AA
′)
)
: (4.8)
A fe w r e m a rk s a r e in or de r :
( a ) I f L
S
( X) := SX ¹ one h as th a t [E
2
; L
S
] = 0 , i f a nd only i f E i s unit a r y ( s e e the s e ct ion 5.6 for a
pr oof ). I n thi s cas e the fir s t t e r m in e q . ( 4.8 ) be c ome s e qual t o one , g iv in g b a ck e q . ( 4.6).
( b ) F r om[E
2
; L
S
] = 0 a nd S
BB
′ = S
AA
′ S = SS
AA
′ , one s e e s th a t the s e c ond t e r m in e q . ( 4.8) ca n be
w r itt e n S
BB
′E
2
( S
BB
′): Thi s me a n s th a t in the unit a r y cas e the r e a s y mme tr y be t w e e n the s ubs ys t e m s A a nd
B w hich i s los t in the g e ne r al ope n cas e .
( c ) S inc e , for unit a r y dy n a mic s, e q . ( 4.6) c o inc ide s w ith ope r a t or e n t a n gle me n t [ 202 ] of U , one h as th a t
G(U) = 0 () U = U
A
U
B
: (4.9)
H o w e v e r , for non- unit a r y dy n a mic s, E =E
A
E
B
=) G(E) = 0 , but the c on v e r s e i s not tr ue . N a me ly ,
one ca n h a v e z e r o G(E) e v e n for E ̸=E
A
E
B
. L a t e r , w e w i l l i l lus tr a t e thi s p he nome non b y a n ex a mp le of
a d e p h asin g ch a nne l .
( d ) L e t us r e mind th a t g iv e n the qua n tum ch a nne l E:L(H)!L(K) , one define s the C h o i s t a t e as s o -
c i a t e d t o it b y ,
ρ
E
:= (E
I)(j Φ
+
⟩⟨ Φ
+
j)2L(K)
L(H); (4.10)
w he r ej Φ
+
⟩ = d
1= 2
∑
d
i= 1
j i⟩
2
2L(H)
2
; ( d = dimH):
¹ H e r e , L
S
i s a s u pe r ope r a t or w hos e a ct ion i s t o left m ult ip ly w ith the s w a p ope r a t or S , th a t i s, L
S
( X) := SX . The c omm u-
t a t or i s a t the le v e l of s u pe r ope r a t or s, n a me ly , [E
2
; L
S
] = E
2
◦ L
S
L
S
◦E
2
, w he r e w e h a v e e mp h asi z e d the s u pe r op -
e r a t or c omposit ion v i a the ◦ s y mbo l . Thi s c omm ut a t or ca n be unde r s t ood b y its a ct ion on a n ope r a t or X as[E
2
; L
S
]( X) =
E
2
L
S
( X) L
S
E
2
( X) .
126
N ot ic e th a t in the unit a r y cas e , E =U = U U
y
; e q . ( 4.6 ) ca n be w r itt e n as [ 202 ]
G(U
t
) = 1∥ tr
BB
′ ρ
U t
∥
2
2
= S
L
( tr
BB
′ ρ
U t
); (4.11)
w he r e S
L
i s the s o - cal le d l ine a r e n tr op y i . e ., S
L
( ρ) := 1 Tr( ρ
2
): Thi s sho w s w h y the a v e r a g e d b ip a r t it e
O T O C c or r e spond s t o a me as ur e of ope r a t or e n t a n gle me n t for U
t
a cr os s the A : B b ip a r t it ion [ 202 ].
The fo l lo w in g r e s ult ca n be s e e n as a n ex t e n sion of e q . ( 4.11 ) t o g e ne r al qua n tum ch a nne l s .
P r opositio n 4.3
( i) G(E) = d
B
Tr
B
′ ρ
E
2
2
Tr
BB
′ ρ
E
2
2
: (4.12)
( ii) G(E) = d
B
ρ
e
E
ρ
T◦
e
E
2
2
= d
B
(
ρ
e
E
2
2
ρ
T◦
e
E
2
2
)
; (4.13)
w he r e
e
E :L(H
A
)!L(H
AB
) : X!E( X
I
d B
) a ndT :L(H
AB
)!L(H
AB
) : X7! Tr
B
( X)
I
d B
.
I n w or d s: the a v e r a g e d b ip a r t it e O T O C ( 4.8 ) for a ch a nne l E ca n be ex pr e s s e d as a d i ffe r e nc e of pur it i e s
of ( r e duc e d ) C ho i m a tr ic e s of E or as a ( s qua r e d ) d i s t a nc e be t w e e n the C ho i m a tr ic e s of ch a nne l s
e
E a nd
T ◦
e
E . M or e pr e c i s e ly , sinc e the m a p be t w e e n ch a nne l s a nd the c or r e spond in g C ho i s t a t e i s in j e ct iv e , the
R HS of e q . ( 4.13 ) me as ur e s the d i s t a nc e be t w e e n the ch a nne l s
e
E a ndT ◦
e
E: H e nc e , w e s e e th a t G(E) = 0
i f a nd only i f
~
E =T ◦
~
E: N a me ly ,8 X2L(H
A
);
E( X
I
d
B
) = Tr
B
E( X
I
d
B
)
I
d
B
: (4.14)
I n p as sin g , w e o bs e r v e th a t the m a p T i s a ( s u pe r ) pr oj e ct ion th a t ca n be r e al i z e d as a gr ou p a v e r a g e T ( X) =
E
U
[
(I
A
U) X(I
A
U
y
)
]
; w ith U2U(H
B
) .
127
F r om the p h ysical po in t of v ie w one of the m ain find in gs in [ 168 ] w as t o sho w th a t the b ip a r t it e O T O C
G(U
t
) i s nothin g but a me as ur e of the a v e r a g e e n tr op y pr oduct i on b y Tr
B
[
e
E] o v e r pur e s t a t e s . O pe r -
a t ion al ly , one pr e p a r e s pur e s t a t e s in the A - s ubs ys t e m t e n s or i z e d w ith the t ot al ly mi xe d one in the B -
s ubs ys t e m a nd le ts the j o in t s ys t e m e v o lv e a c c or d in g t o the ch a nne l E: The e n tr op y th a t i s the n o bs e r v e d
in the A - s ubs ys t e m alone i s the r e s ult , in the unit a r y cas e , of the infor m a t ion los s due t o le ak in g in t o the
B - s ubs ys t e m induc e d b y the e v o lut ion i . e ., qua n tum infor m a t ion s cr a mb l in g.
One ca n ex t e nd th a t k ey r e s ult t o the ope n s ys t e m cas e .
P r opositio n 4.4
W e de not e b y ψ :=j ψ⟩⟨ ψj w ithj ψ⟩2H
A
. The n,
G(E) = N
A
E
ψ
[
S
L
( Tr
B
e
E ( ψ)) d
B
( S
L
(
e
E( ψ)) S
min
L
)
]
; (4.15)
w he r eE
ψ
i s the the H a a r a v e r a g e o v e r H
A
; N
A
:=
d A+ 1
d A
, a nd S
min
L
:= 1
1
d B
:
W e not e th a t for E =U , th a t i s, clos e d s ys t e m dy n a mic s, the s e c ond t e r m in e q . ( 4.15 ) i s z e r o . I n g e ne r al ,
sinc eE i s unit al , one h as th a t
e
E( ψ)) S
m
L
(8 ψ): H e nc e ,
G(E) G
scra
(E) G
max
:= 1
1
d
2
A
; (4.16)
w he r e , the “ s cr a mb l in g e n tr op y ” pr oduct ion G
scra
i s g iv e n b y the fir s t t e r m in e q . ( 4.15 ).
C r uc i al ly , e q . ( 4.15 ) sho w s th a t in the ope n s ys t e m cas e in G(E) , the r e i s a c ompe t it ion be t w e e n the
e n tr op y pr oduct ion, qua n t i fie d b y the fir s t t e r m G
scra
due t o s cr a mb l in g , a nd the s e c ond one due t o de c o -
he r e nc e . F or ex a mp le , i f
e
E ( ψ) =
I
d
;8 ψ , the n, the s cr a mb l in g t e r m a tt ain s its m a x im um v alue G
max
, but
thi s i s e xa c t l y c a n ce le d b y the de c o he r e nc e c on tr i but ion . Thi s situa t ion, as sho w n in the nex t s e ct ion, ca n
be p h ysical ly r e al i z e d b y a de p h asin g ch a nne l in the m a x im al ly e n t a n gle d b asi s .
W e s tr e s s th a t t o o bt ain a s a t i sfa ct or y e s t im a t e of the a v e r a g e in the R HS of E q . ( 4.15 ), one doe s not , in
128
pr a ct ic e , ne e d t o s a mp le o v e r the f ul l H a a r e n s e mb le . A n a de qua t e e s t im a t e ca n be o bt aine d w ith a r a p id ly
de cr e asin g n umbe r of ne c e s s a r y s a mp le s, as the d ime n sion d
χ
gr o w s . F or ex a mp le , in the unit a r y cas e ,
i f
~
P( ε) i s the pr o b a b i l it y of the e n tr op y S
lin
[
E
t
(
j ψ⟩⟨ ψj
)]
de v i a t in g f r om
d A
d A+ 1
G(U
t
) mor e th a n ε for a n
in s t a nc e of a r a ndom s t a t e , one h as th a t [ 168 ]:
~
P( ε) exp
(
d
A
ε
2
64
)
: (4.17)
I t i s al s o impor t a n t t o not ic e th a t the t w o t e r m s in e q . ( 4.15 ) ca n be , in pr inc ip le , me as ur e d inde pe nde n tly
a nd the r efor e h a v e a w e l l - define d ope r a t ion al me a nin g in their o w n r i gh t s e e s e ct ion 4.8 for a de t ai le d d i s -
c us sion .
4.4 Some spec i al c h annel s
F or c oncr e t e ne s s, le t us no w c on side r a fa mi ly of m a ps w hich include s s e v e r al one s of p h ysical in t e r e s t a nd
for w hich e q . ( 4.8) t ak e s a p a r t ic ul a rly in t e r e s t in g for m . L e t us s t a r t w ith de p h asin g ch a nne l s s tr ic t o sen s u .
1 D eph a si n g c h a nnel s
P r opositio n 4.5
C on side r the de p h asin g ch a nne l , E = D
B
, w he r e ,D
B
( ρ) =
d
∑
α= 1
Π
α
ρ Π
α
a ndB = f Π
α
g
d
α= 1
w ith
Π
α
=j ψ
α
⟩⟨ ψ
α
j , a n or thonor m al b asi s . The n,
G(D
B
) =
1
d
2
A
e
X
B
e
X
2
B
1
; (4.18)
w he r e (
e
X
B
)
α; β
:= d
1
B
⟨
ρ
α
; ρ
β
⟩
i s the r e nor m al i z e d Gr a m m a tr i x of the s ys t e m a nd ρ
α
= Tr
B
( Π
α
) 8 α .
e q . ( 4.18 ) de s cr i be s a n “ ide mpot e nc y defic it , ” n a me ly ho w fa r a w a y
e
X
B
i s f r om bein g e qual t o its o w n s qua r e
129
Figure 4.4.1: Non-unita ry OTOC G( e
L t
) withL = i ad S+ λ(D
B
I) where S is the sw ap op erato r, B
is the Bell basis ( d
A
= d
B
= 2 ). The different curves co rresp ond to different choices of the dephasing
pa rameter λ: Over the time scale λ
1
on which dephasing b ecomes relevant the “scrambling entrop y”
(first term in eq. ( 4.15 )) is balanced, and eventually overwhelmed, b y the decoherence-induced en-
trop y p ro duction (second term in eq. ( 4.15 )). F o r any fixed time t the OTOC supp ression is exp onen-
tial in the dephasing strength λ: Mo reover, in sha rp contrast with the unita ry case, fo r any λ ̸= 0; the
infinite time limit of the OTOC is vanishing.
e
X
2
B
. H e nc e , G(D
B
) = 0 i f a nd only i f
e
X
B
i s a pr oj e ct or .
D efine j φ
s
⟩
X
:=
1
p
d X
∑
d X
j= 1
j j⟩; ( X = A; B) a nd c on side r the fo l lo w in g t w o ex a mp le s of v a ni shin g G .
( i ) A pr oduct de p h asin g ch a nne l , i . e ., D
B
=D
B A
D
B A
. L e tB
A
=f P
j
g
d A
j= 1
a ndB
B
=f Q
j
g
d B
j= 1
, the n
the pr oj e ct or s c or r e spond in g t o D
B
a r ef P
j
Q
k
g
d A; d B
j; k= 1
. I t i s e as y t o sho w th a t the Gr a m m a tr i x
c or r e spond in g t o B t ak e s the for m ,
e
X =I
A
j φ
s
⟩
B
⟨ φ
s
j .
( i i ) A m a x im al ly e n t a n gle d de p h asin g b asi s, i . e ., e a ch of the ρ
α
=I
A
= d
A
a nd the r efor e , a simp le calc ul a -
t ion sh o w s th a t the Gr a m m a tr i x t ak e s the for m,
e
X =j φ
s
⟩
A
⟨ φ
s
j
j φ
s
⟩
B
⟨ φ
s
j .
Quit e in t e r e s t in gly , e q . ( 4.18 ) al lo w s one t o c onne ct G(D
B
) t o the e n t a n gle me n t of the s t a t e s c ompr i sin g
130
B .
P r opositio n 4.6
L e tB =f Π
α
g , ρ
α
= Tr
B
[ Π
α
] , a nd , Δ
α
:= ρ
α
I
d A
. The n i f ∥ Δ
α
∥
2
2
"(8 α) , one h as the fo l lo w in g
u ppe r bound on the ope n O T O C for de p h asin g ch a nne l s, G(D
B
)
"
d A
.
S inc e Π
α
a r e pur e s t a t e s, i f Δ
α
i s s m al l , the n the s t a t e s Π
α
a r e ne a rly m a x im al ly e n t a n gle d a cr os s the A : B
p a r t it ion . The r efor e , the bound the n t e l l s us th a t , t h e m o r e en t a n g le d t h e dep h asi n g b asis s t a t e s, t h e s m a ller t h e
O T O C . N ot e th a t the as s umpt ion a bo v e , in or de r t o m ak e its c onne ct ion t o e n t a n gle me n t cle a r e r , ca n al s o
be r e cas t as
S
L
( ρ
α
) S
max
L
" ( α = 1;:::; d);
w he r e S
max
L
:= 1 1= d
A
:
A nothe r us ef ul w a y of r e w r it in g e q . ( 4.18 ) i s o bt aine d b y in tr oduc in g the fo l lo w in g B - de pe nde n t s t a t e ,
R
B
2H
2
=H
A
H
B
H
A
′
H
B
′ s uch th a t
R
B
:=
1
d
d
∑
α= 1
Π
α
Π
α
= (D
B
I)(j Φ
+
AB
⟩⟨ Φ
+
AB
j); (4.19)
w he r ej Φ
+
AB
⟩ := d
1= 2
∑
d
α= 1
j ψ
α
⟩
2
: The s e c ond e qual it y a bo v e sho w s th a t R
B
i s nothin g but the C ho i
s t a t e as s oc i a t e d t o D
B
: U sin g E qs . ( 4.12 ) ( or ( 4.18 )) a nd ( 4.19 ) one ca n w r it e
G(D
B
) =
1
d
A
⟨ S
AA
′; R
B
⟩∥ R
AA
′
B
∥
2
2
; (4.20)
w he r e R
AA
′
B
:= Tr
BB
′ R
B
: S inc e the fir s t t e r m in e q . ( 4.20 ) i s u ppe r bounde d b y 1 a nd the s e c ond t e r m i s
lo w e r bounde d b y d
2
A
, one imme d i a t e ly o bt ain s the B -inde pe nde n t u ppe r bound
G(D
B
)
1
d
A
(
1
1
d
A
)
= O(
1
d
A
): (4.21)
131
Thi s ine qual it y sho w s th a t the m a x im al v alue of the O T O C th a t i s a chie v a b le b y de p h asin g ch a nne l s i s w e l l
be lo w the u ppe r bound e q . ( 4.16 ), G
max
= 1 1= d
2
A
.
T o ex p lor e thi s p he nome non w e no w mo v e t o c on side r r a n do m de p h asin g ch a nne l s . The s e t of B ’ s i s
n a tur al ly a ct e d u pon b y the unit a r y gr ou p U(H) ² :
B
0
:=f Π
( 0)
α
g
d
α= 1
7! UB
0
:=f U Π
( 0)
α
U
y
g
d
α= 1
:
I n t e r m s of the R
B
m a tr ic e s: R
B 0
7! U
2
R
B 0
U
y
2
: B y c on side r in g the U ’ s H a a r d i s tr i but e d one o bt ain s
the de sir e d e n s e mb le of r a ndom de p h asin g ch a nne l s . The nex t pr oposit ion sho w s the a v e r a g e a nd me as ur e
c onc e n tr a t ion for G(D
B
) for s uch a n e n s e mb le w ith d
A
d
B
.
P r opositio n 4.7
i )E
U
[
G(D
UB 0)
]
7
4 d
2
A
= O(
1
d
2
A
):
i i ) Probf G(D
B
)
7
4d
2
A
+ εg exp[ d ε
2
= K
2
]; w he r e K i s the L ips chitz c on s t a n t of the f unct ion
F( U) := G(D
UB 0
) a nd c a n be chos e n K 100:
I n w or d s: in l a r g e d ime n sion the o v e r w he lmin g m a j or it y of r a ndom de p h asin g ch a nne l s h a v e a G(D
B
)
w hich i s ( 1= d
2
A
): Thi s i s the r e s ult of de c o he r e nc e w hich m ak e s the fir s t t e r m in e q . ( 4.20 ) ( or e q . ( 4.12 ))
bein g O( 1= d
2
A
) for t y p ical de p h asin g ch a nne l s . On the othe r h a nd , s uch a t e r m in the clos e d cas e i s ide n t i -
cal ly one a nd t y p ical unit a r ie s h a v e a G(U) w hich i s clos e t o G
max
[ 168 ].
² The k ey ide a i s th a t a n y t w o b as e s in the H i l be r t sp a c e ca n be c onne ct e d v i a a unit a r y . The r efor e , s t a r t in g f r om a fi xe d b asi s
B
0
, the a ct ion of the unit a r y gr ou p g e ne r a t e s a ll b ase s in the H i l be r t sp a c e . The n, ut i l i z in g the uni for m ( H a a r ) me as ur e on U(H)
al lo w s us t o define a not ion of ( uni for mly d i s tr i but e d ) r a ndom b as e s .
132
2 Ent a n gle ment - b r e a ki n g c h a nnel s
H e r e w e d i s c us s the cl as s of ch a nne l s cal le d e n t a n gle me n t - br e ak in g or me as ur e-a nd -pr e p a r e , define d ( in
the H ei s e nbe r g p ictur e ) as,
Φ
EB
( X) =
∑
k
M
k
Tr[ δ
k
X]; w he r e
∑
k
M
k
=I: (4.22)
H e r e , Φ
EB
:L(H
AB
)!L(H
AB
) w he r ef M
k
g;f δ
k
g a r e l ine a r ope r a t or s on L(H
AB
) w ith the a dd it ion al
c on s tr ain t th a t f M
k
g
k
for m a PO VM a nd f δ
k
g
k
i s a s e t of qua n tum s t a t e s .
F or g e ne r al EB ch a nne l s, w e h a v e the fo l lo w in g for m .
P r opositio n 4.8
C on side r a g e ne r al e n t a n gle me n t - br e ak in g ( EB ) ch a nne l as in e q . ( 4.22 ) the n,
G( Φ
EB
) =
1
d
2
∑
k; k
′
⟨
δ
A
k
; δ
A
k
′
⟩[
d
B
⟨ M
k
; M
k
′⟩
⟨
M
A
k
; M
A
k
′
⟩]
; (4.23)
w he r e M
A
k
Tr
B
M
k
a nd δ
A
k
Tr
B
δ
k
.
N ot e th a t de p h asin g ch a nne l s a r e a spe c i al cas e of EB ch a nne l s w he n the me as ur e me n ts a r e r a nk - 1 pr oj e c -
t or s a nd the pr e p a r e d s t a t e s a r e ( the s a me ) pur e s t a t e s; th a t i s, le t B =fj ψ
k
⟩⟨ ψ
k
jg
d
k= 1
a nd δ
k
=j ψ
k
⟩⟨ ψ
k
j =
M
k
8 k =f 1; 2; ; dg , the n, Φ
EB
=D
B
. The r efor e , G( Φ
EB
) t ak e s the a n aly t ical for m in e q . ( 4.18 ).
A s a n ex a mp le , one ca n c on side r the fo l lo w in g for m of EB ch a nne l . L e t B =f Π
α
g; Π
α
=j ψ
α
⟩⟨ ψ
α
j a nd
e
B =f
f
Π
α
g;
f
Π
α
=j φ
α
⟩⟨ φ
α
j be t w o b as e s for H
AB
. The n,
Φ
(B!
e
B)
EB
( X) :=
d
∑
k= 1
e
Π
α
⟨ ψ
α
j Xj ψ
α
⟩: (4.24)
F or thi s cl as s of ch a nne l s, w e h a v e the fo l lo w in g for m of the ope n O T O C . L e t ρ
α
:= Tr
B
Π
α
;e ρ
α
:= Tr
B
e
Π
α
.
133
The n,
G( Φ
(B!
e
B)
EB
) =
1
d
2
(
d
B
d
∑
k= 1
ρ
k
2
2
d
∑
k; k
′
= 1
⟨
ρ
k
; ρ
k
′
⟩⟨
e ρ
k
;e ρ
k
′
⟩
)
: (4.25)
F orB =
e
B (w ith the ide n t ical or de r in g of s t a t e s ), thi s t ak e s the for m of th e de p h asin g ch a nne l .
I t i s e as y t o s e e th a t al s o for e q . ( 4.25 ) the bound ( 4.21 ) ho ld s . I nde e d , the fir s t t e r m in e q . ( 4.25 ) i s cle a rly
u ppe r bounde d b y 1= d
A
i . e ., w he n al l the ρ
k
’ s a r e pur e ) a nd the s e c ond ca n be w r itt e n ∥ d
1
∑
k
ρ
k
e ρ
k
∥
2
2
a nd the r efor e i s lo w e r bounde d b y 1= d
2
A
: Thi s i s a chie v e d forB bein g a pr oduct b asi s a nd
e
B m a x im al ly
e n t a n gle d i . e ., e ρ
k
=I= d
A
;(8 k):
3 B - d i a go n a l c h a nnel s
L e t us no w mo v e t o a n aly z e a g e ne r al i za t ion of the a bo v e w hich w e r efe r t o as B - d i a g on al ch a nne l s . C on-
side r a b asi s B :=fj α⟩g
d
α= 1
ofH
AB
a nd m a pE
^
Φ
s uch th a t
E
^
Φ
(j α⟩⟨ α
′
j) = φ
α; α
′
j α⟩⟨ α
′
j (8 α; α
′
); (4.26)
w ith φ
α; α
′
2 C 8 α; α
′
. Thi s fa mi ly , for ex a mp le , c ompr i s e s unit a r y ch a nne l s, de p h asin g ch a nne l s a nd
qua n tum me as ur e me n ts . W e ca n the n pr o v e the fo l lo w in g.
P r opositio n 4.9
i ) I f
^
Φ :=
(
φ
α; α
′
)
α; α
′
0; a nd φ
α; α
= 1(8 α); the n e q . ( 4.26 ) define s a ( unit al ) qua n tum ch a nne l
w hos e ei g e n v alue s a r e e nc ode d in the m a tr i x
^
Φ:
134
i i ) ρ
α; α
′
= Tr
B
j α⟩⟨ α
′
j , the n
G(E
^
Φ
) =
d
B
d
2
∑
α; α
′
φ
α; α
′
2
ρ
α; α
′
2
2
1
d
2
∑
α; α
′
; β; β
′
φ
α; α
′
φ
β; β
′
⟨
ρ
α; α
′
; ρ
β; β
′
⟩
2
: (4.27)
W e not e the fo l lo w in g fa cts:
( a ) F or
^
Φ = 1 , w e r e c o v e r the de p h asin g ch a nne l D
B
a nd e q . ( 4.27 ) be c ome s e q . ( 4.18 ) .
( b ) F or φ
α; α
′
= e
i( θ α θ
α
′)
w ithf θ
α
g
α
2 [ 0; 2 π) , one r e c o v e r s unit a r y ch a nne l s a nd e q . ( 4.27 ) be-
c ome s e q . ( 4.6 ). I n p a r t ic ul a r i f φ
α; α
′
= 1(8 α; α
′
) w e h a v e ,E
^
Φ
=I a nd the r efor e G v a ni she s .
( c ) S u ppos e th a t the dy n a mic s i s g e ne r a t e d b y a L ind b l a d i a n
L( X) =
∑
μ
(
L
μ
XL
y
μ
1
2
f L
μ
L
y
μ
; Xg
)
;
w he r e the L ind b l a d ope r a t or s L
μ
for m a n a be l i a n al g e br a a nd f X; Yg := XY+ YX . The n one one h as th a t
E
t
= e
tL
i s of the for m e q . ( 4.26 ) w ith
φ
α; α
′
= exp
2
4
1
2
∑
μ
(j α
μ
α
′
μ
j
2
2iIm( α
′
μ
α
μ
))
3
5
;
bein g the j α⟩ a j o in t ei g e nb asi s of the L
μ
i . e ., L
μ
j α⟩ = α
μ
j α⟩; L
y
μ
j α⟩ = α
μ
j α⟩;(8 μ; α) .
T o i l lus tr a t e the p h ysical r e le v a nc e of the fa mi ly of ch a nne l s in e q . ( 4.27 ) w e no w pr o v ide a c ou p le of
simp le a n aly t ical ex a mp le s a r i sin g f r om a dy n a mical s e mi gr ou p . They a r e aime d a t m ak in g m a ni fe s t non-
unit a r y effe cts a nd their in t e r p l a y w ith unit a r y one s . F or both ex a mp le s be lo w , H
A
=H
B
a nd the r e le v a n t
b asi sfj α⟩g i s a n ei g e nb asi s of the s w a p o pe r a t or S , fo r e . g., the Be l l b asi s for d
A
= d
B
= 2:
Exa m p le 1.– L e t us c on side r the L ind b l a d i a n
L = AdS I;
w he r e AdS( X) := SXS: The n, b y a s tr ai gh tfor w a r d ex pone n t i a t ion one find s a c on v ex c omb in a t ion of
135
unit a r ie s
E
t
= e
L t
= a( t) I+ b( t) AdS
w ith a( t) =
1
2
( 1 + e
t
); b( t) =
1
2
( 1 e
t
) . The L ind b l a d i a n he r e i s de si gne d t o g e ne r a t e a n e v o lut ion
w hi ch i s a mi x tur e of the I de n t it y a nd the S W A P unit a r ie s . The ide a i s th a t the s w a p unit a r y m a x imi z e s the
( u nit a r y ) b ip a r t it e O T O C , w hi le the I de n t it y ch a nne l c or r e spond s t o z e r o b ip a r t it e O T O C . The pr o b a b i l i -
t i e s for the s e t w o e v o lut ion s a r e t ime- de pe nde n t a nd , as t ime e v o lv e s, the w ei gh t c or r e spond in g t o the s w a p
unit a r y incr e as e s ex pone n t i al ly ( f r om z e r o ) w hi le th a t of the I de n t it y de ca ys ( f r om one ) t o z e r o . N a me ly ,
it g e ne r a t e s a m a x im al ly s cr a mb l in g e v o lut ion w ith incr e asin g t ime . W e h a v e ,
^
Φ
α α
′ = a( t)+ b( t) λ
α
λ
α
′ w ith
λ
α= α
′ = 1: The ope n a v e r a g e d b ip a r t it e O T O C for thi s ch a nne l i s
G(E
t
) = b
2
( t) G
max
:
N ot e th a t the ide n t it y c ompone n t of E
t
doe s not c on tr i but e t o the a v e r a g e d b ip a r t it e O T O C a nd G(E
1
) =
1
4
G
max
:
Exa m p le 2.– L e t us c on side r the L ind b l a d i a n
L = i adH+ λ(D
B
I);
w he r e adH( X) := [ H; X] a ndD
B
i s the de p h asin g s u pe r ope r a t or . W e as s ume th a t the de p h asin g b asi s i s
the s a me as the H a mi lt oni a n ei g e nb asi s, i . e ., B =f Π
j
g w ith Π
j
the H a mi lt oni a n ei g e n s t a t e s . I n thi s cas e
[ adH;D
B
] = 0 a nd the r efor e the dy n a mic s i s g iv e n b y a c on v ex c omb in a t ion of a unit a r y a nd a de p h asin g
ch a nne l
E
t
= e
tL
=~ a( t) e
it ad H
+
~
b( t)D
B
w he r e ~ a( t) := e
λ t
a nd
~
b( t) := 1~ a( t): Thi s c or r e spond s t o
^
Φ
α α
′ = ~ a( t) e
it( λ α λ
α
′)
e +
~
b( t) δ
α α
′; w ith
λ
α= α
′ = 1: M or e o v e r , i f w e as s ume th a t D
B
(
X
I
d B
)
= Tr( X)
I
d
. The n, the b ip a r t it e O T O C be c ome s,
G(E
t
) = ~ a
2
( t) G( e
it adH
):
136
0 10 20 30 40 50
0.0
0.2
0.4
0.6
(a) integrable
0 10 20 30 40 50
0.0
0.2
0.4
0.6
(b) chaotic
Figure 4.5.1: T emp o ral va riation of the op en OTOC G(E
t
) fo r the TFIM e q. ( 4.30 ) with L = 6 spins.
The three curves co rresp ond to va rying choices of the dissipation strength α in the Lindblad op era-
to rs eq. ( 4.28 ). The chaotic ( g = 1: 05; h = 0: 5 ) and integrable ( g = 1; h = 0 ) phases a re clea rly
distinguishable fo r the α = 0 case (closed system), ho w ever, increasing the dissipation strength to
α = 0: 05 mak es them fairly indiscernible and destro ys the revivals (o r fluctuations) cha racteristic of
integrable systems.
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
(a) integrable
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
(b) chaotic
Figure 4.5.2: T emp o ral va riation of the op en OTOC G(E
t
) fo r the XX Z-NNN mo del eq. ( 4.31 ) with
L = 6 spins. The thre e curves co rresp ond to va rying choices of the dissipation strength α in the
Lindblad op erato rs eq. ( 4.28 ) . The nonintegrable ( J = 1; Δ = 0: 5; J = 1; Δ
′
= 0: 5 ) and integrable
( J = 1; Δ = 0 = J = Δ
′
) phases a re clea rly distinguishable fo r the α = 0 = γ case (closed system) . The
integrable mo del here can b e mapp ed onto free fermions and hence unlik e the TFIM case, even after
increasing the dissipation strength ( α = 0: 1 = γ ), the system demonstrates revivals (o r fluctuations)
cha racteristic of integrable systems.
I f the H a mi lt oni a n i s the s w a p ope r a t or , S one g e ts G( e
it adH
) = ( 1 cos
4
( t)) G
max
: F i g. 4.4.1 sho w s the
c or r e spond in g p a tt e r n of ex pone n t i al ly d a mpe d os c i l l a t ion s .
137
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
0.8
1.0
(a) TFIM
0 5 10 15 20 25 30
0.0
0.2
0.4
0.6
0.8
1.0
(b) XXZ-NNN mo del
Figure 4.5.3: T emp o ral va riation of the individual terms of the op en OTOC G
( 1)
(E
t
) =
d B
d
2
Tr[ SE
2
( S
AA
′)] and G
( 2)
(E
t
) =
1
d
2
Tr[ S
AA
′E
2
( S
AA
′)] with G(E
t
) = G
( 1)
(E
t
) G
( 2)
(E
t
) . The t w o
figures co rresp ond to the integrable and chaotic limits as considered ab ove fo r the (a) TFIM and (b)
XXZ-NNN mo del with L = 6 spins, resp ectively . The dissipation pa rameters a re α = 0: 01; γ = 0: 01.
The first term G
( 1)
(E
t
) o riginates fr om environmental decoherence and is simila r fo r b oth the inte-
grable and the chaotic case. Ho w ever, the second term, G
( 2)
(E
t
) is clea rly distinct fo r the t w o phases
and can diagnose quantum chaos even in the p resence of dissipation.
4.5 Qu an tum Spin Ch ain s
A s a p h ysical a pp l ica t ion of the ope n O T O C , w e s tudy p a r a d i gm a t ic qua n tum sp in- ch ain mode l s of qua n-
tum ch a os in the pr e s e nc e of ope n- s ys t e m dy n a mic s . F or s ys t e m s in t e r a ct in g w ith a M a rk o v i a n e n v ir on-
me n t , the dy n a mic s ca n be de s cr i be d b y a L ind b l a d m as t e r e qua t ion ( s ome t ime s al s o cal le d the GK S L
for m ) [ 282 ],
d ρ( t)
dt
=L
y
( ρ( t)) i[ H; ρ( t)]
+
∑
j
(
L
j
ρ( t) L
y
j
1
2
f L
y
j
L
j
; ρ( t)g
)
; (4.28)
w he r eL
y
i s the L ind b l a d i a n, H i s the H a mi lt oni a n, ρ( t) i s the qua n tum s t a t e a t t ime t , a ndf L
j
g a r e cal le d
the L ind b l a d ( or jump ) ope r a t or s, w hich c on s t itut e the s ys t e m- e n v ir onme n t in t e r a ct ion . The m as t e r e qua -
t i on a bo v e g iv e s r i s e t o a one-p a r a me t e r fa mi ly of t ime- e v o lut ion s u pe r ope r a t or s ( in the S chr öd in g e r p ic -
138
tur e ),
E
y
t
= e
tL
y
; t 0: (4.29)
W e c on side r t w o qua n tum sp in- 1= 2 ch ain s on L sit e s, ( i ) the tr a n s v e r s e- fie ld I sin g mode l ( TFI M ) w ith
a n on sit e m a gne t i za t ion a nd ( i i ) the nex t -t o - ne a r e s t nei ghbor H ei s e nbe r g X X Z mode l (X X Z - NNN ).
H
TFIM
=
0
@
∑
j
σ
z
j
σ
z
j+ 1
+ g σ
x
j
+ h σ
z
j
1
A
: (4.30)
H
XXZ
= J
L 1
∑
j= 1
(
σ
x
j
σ
x
j+ 1
+ σ
y
j
σ
y
j+ 1
+ Δ σ
z
j
σ
z
j+ 1
)
+ J
′
L 2
∑
j= 1
(
σ
x
j
σ
x
j+ 2
+ σ
y
j
σ
y
j+ 2
+ Δ
′
σ
z
j
σ
z
j+ 2
)
: (4.31)
H e r e , the σ
α
j
; α2f x; y; zg a r e the P a ul i m a tr ic e s . F or the TFI M , g; h de not e s the s tr e n g th of the tr a n s v e r s e
fie ld a nd the local fie ld , r e spe ct iv e ly . The TFI M H a mi lt oni a n i s in t e gr a b le for h = 0 a nd nonin t e gr a b le
w he n both g; h a r e nonz e r o . W e c on side r as the in t e gr a b le po in t , g = 1; h = 0 a nd the nonin t e gr a b le
po in t g = 1: 05; h = 0: 5 . F or the X X Z - NNN mode l , J( J
′
) de not e s the s tr e n g th of the ne a r e s t - ( nex t -t o -
ne a r e s t -) nei ghbor c ou p l in g , a nd Δ( Δ
′
) de not e s the a ni s otr op y alon g the z -a x i s . The X X Z - NNN mode l
H a mi lt oni a n i s in t e gr a b le b y Be the A n s a tz for J
′
= 0 = Δ
′
. W e c on side r as the in t e gr a b le po in t , J = 1; Δ =
0 = J = Δ
′
w hich ca n be m a ppe d on t o f r e e fe r mion s a nd as the nonin t e gr a b le po in t , J = 1; Δ = 0: 5; J =
1; Δ
′
= 0: 5 [ 283 ].
W e c on side r t w o t y pe s of jump pr oc e s s e s a t the bound a r y: ( i ) a mp l itude d a mp in g , w ith L ind b l a d op -
e r a t or s
p
α σ
1
a n d
p
α σ
L
; a nd ( i i ) bound a r y de p h asin g , w ith L ind b l a d ope r a t or s
p
γ σ
z
1
;
p
γ σ
z
L
. N ot e th a t
simi l a r mode l s h a v e be e n c on side r e d befor e t o s tudy non- e qui l i br ium sp in tr a n spor t [ 284 – 286 ] a nd d i s -
sip a t iv e qua n tum ch a os [ 287 ]. T o n ume r ical ly sim ul a t e the e v o lut ion, w e “ v e ct or i z e ” the L ind b l a d i a n s u-
139
pe r ope r a t or L in t o a 4
L
4
L
d ime n sion al m a tr i x r e pr e s e n t a t ion,
jL⟩⟩ = i
(
I
H
y
H
I
)
+
∑
j
( L
T
j
L
y
j
1
2
L
j
L
T
j
I
1
2
I
L
y
j
L
j
); (4.32)
w he r e X
T
; X
de not e s the m a tr i x tr a n spos e a nd c omp lex c on ju g a t ion, r e spe ct iv e ly ³ .
W e sim ul a t e ex a ct dy n a mic s for the ope n s ys t e m a nd c omput e G(E
t
) for L = 6 sp in s a cr os s the b i -
p a r t it ion 1 : L 1 . I n fi gs . 4.5.1 a nd 4.5.2 w e c on side r the t w o mode l s in e qs . ( 4.30 ) a nd ( 4.31 ) w ith
their in t e gr a b le a nd ch a ot ic l imits . A s w e incr e as e the s tr e n g th of s ys t e m- e n v ir onme n t c ou p l in g , n a me ly ,
in fi g. 4.5.1 the p a r a me t e r α a nd in fi g. 4.5.2 the p a r a me t e r s α; γ , the ope n O T O C G(E
t
) s t a r ts de ca y in g
f r om its clos e d s ys t e m v alue , G(U
t
) . I n fi g. 4.5.1 the in t e gr a b le a nd ch a ot ic p h as e s a r e cle a rly d i s t in g ui sh-
a b le for the clos e d s ys t e m cas e ( α = 0 ), ho w e v e r , for α = 0: 05 , the p h as e s be c ome ind i s c e r ni b le due t o
ope n- s ys t e m effe cts . S imi l a rly , in fi g. 4.5.2 , the r e v iv al s in the f r e e fe r mion s r e g ime i s cle a rly d i s t in g ui sh a b le
f r om the nonin t e gr a b le r e g ime for the clos e d s ys t e m ( α = 0 = γ ). H o w e v e r , a t α = 0: 1 = γ , the t w o a r e
le s s d i s c e r ni b le . N ot e , ho w e v e r , in thi s “ s tr on gly in t e gr a b le ” r e g ime ( sinc e the s ys t e m ca n be m a ppe d on t o
f r e e fe r mion s ), e v e n b y incr e asin g the d i s sip a t ion s tr e n g th, one ca n s e e r e v iv al s ( or fluctua t ion s ).
F ur the r mor e , fo l lo w in g the in tuit ion de v e lope d in 4.2 w e ca n s e p a r a t e the c on tr i but ion s due t o e n v ir on-
me n t al de c o he r e nc e a nd the dy n a mical e n t a n gle me n t g e ne r a t ion . The ope n O T O C G(E
t
) = G
( 1)
(E
t
)
G
( 2)
(E
t
) i s the d i ffe r e nc e of t w o t e r m s, G
( 1)
(E
t
)
d B
d
2
Tr[ SE
2
( S
AA
′)] a nd G
( 2)
(E
t
)
1
d
2
Tr[ S
AA
′E
2
( S
AA
′)] .
A s i l lus tr a t e d in fi g. 4.5.3 , the fir s t t e r m G
( 1)
(E
t
) d i sp l a ys a simi l a r be h a v ior in the in t e gr a b le a nd ch a ot ic
r e g ime s for both the TFI M a nd the X X Z - NNN mode l , ho w e v e r , the s e c ond t e r m, G
( 2)
(E
t
) ca n s t i l l d i a g -
nos e qua n tum ch a os, e v e n in the pr e s e nc e of d i s sip a t ion . I n fa ct , as w e kno w f r om 4.3, thi s i s the ope n-
s ys t e m v a r i a n t of the ope r a t or e n t a n gle me n t - O T O C c onne ct ion for the unit a r y cas e a nd i s ex pe ct e d t o be
the d i a gnos t ic of the s e t w o p h as e s . M or e o v e r , not ic e th a t a ft e r s e p a r a t in g the s e t w o c on tr i but ion s, one i s
a b le t o d i s t in g ui sh the ch a ot ic a nd in t e gr a b le p h as e s for the TFI M w hich w e r e le s s d i s c e r ni b le pr e v iously .
³ Thi s m a tr i x r e pr e s e n t a t ion i s al s o s ome t ime s kno w n as the L io u v i lle r epr e sen t a t io n . I t i s clos e ly r e l a t e d t o the C ho i -
J a mio l k o w sk i for m v i a , jE⟩⟩
R
= ρ
E
, w he r ej jk⟩⟨ lmj
R
=j jl⟩⟨ kmj for al l b asi s s t a t e s fj j⟩g i s kno w n as the “ r e sh u ffl in g ” ope r a t ion .
140
4.6 Conc l u sion s
I n thi s w ork w e g e ne r al i z e the b ip a r t it e O T O C t o the cas e of ope n qua n tum dy n a mic s de s cr i be d b y qua n-
tum ch a nne l s . W e pr o v ide a n ex a ct a n aly t ical ex pr e s sion for thi s o p en b ip a r t it e O T O C w hich al lo w s us
t o unde r s t a nd the c ompe t in g e n tr op ic c on tr i but ion s f r om e n v i r onme n t al de c o he r e nc e a nd infor m a t ion
s cr a mb l in g. The s e p a r a t e c on tr i but ion s t o e n tr op y pr oduct ion ca n be unde r s t ood v i a ( a ) 4.3, as the d i ffe r -
e nc e of pur it ie s of the C ho i s t a t e ( c or r e spond in g t o the dy n a mical m a p ) a cr os s d i ffe r e n t p a r t it ion s a nd ( b )
in 4.4 as the a v e r a g e e n tr op y pr oduct ion unde r the r e duc e d dy n a mic s a nd th a t due t o the (glo b al ) mi xe d -
ne s s of the e v o lut ion .
A s a c oncr e t e ex a mp le , w e s tudy spe c i al cl as s e s of ch a nne l s, n a me ly de p h asin g ch a nne l s, e n t a n gle me n t -
br e ak in g ch a nne l s, a nd B - d i a g on al ch a nne l s . F or de p h asin g ch a nne l s, the ope n O T O C ca n be ex pr e s s e d in
t e r m s of the “ ide mpot e nc y defic it ” of the Gr a m m a tr i x of r e duc e d s t a t e s of the s t a t e s in the de p h asin g b asi s .
M or e o v e r , i f the ( de p h asin g ) b asi s s t a t e s a r e hi ghly e n t a n gle d the n a n u ppe r bound on the ope n O T O C ca n
be o bt aine d in t e r m s of their de v i a t ion f r om m a x im al e n t a n gle me n t. F ur the r mor e , w e pr o v ide a n a n aly t i -
cal e s t im a t e of the ope n O T O C for r a ndom de p h asin g ch a nne l s, de v i a t ion s f r om w hich a r e ex pone n t i al ly
s u ppr e s s e d due t o me as ur e c onc e n tr a t ion .
F in al ly , as a p h ysical a pp l ica t ion of our a n aly t ical r e s ults, w e c on side r p a r a d i gm a t ic qua n tum sp in- ch ain
mode l s of qua n tum ch a os in the pr e s e nc e of ope n s ys t e m dy n a mic s . A s ex pe ct e d , the d i s sip a t ion effe cts
o bf us ca t e the dy n a mical s cr a mb l in g of infor m a t ion a nd the in t e gr a b le a nd ch a ot ic p h as e s be c ome le s s d i s -
c e r ni b le as the s tr e n g th of d i s sip a t ion i s incr e as e d . H o w e v e r , our a n aly t ical r e s ults al lo w us t o s e p a r a t e the
e n tr op ic c on tr i but ion s m ak in g d i s c e r ni b le the “ s cr a mb l in g e n tr op y ” e v e n in the pr e s e nc e of d i s sip a t ion .
I n closin g , w e l i s t t w o pr omi sin g d ir e ct ion s for f utur e in v e s t i g a t ion s . F ir s t , ex p lor in g f ur the r the in t e r -
p l a y of the t w o d i s t inct c on tr i but ion s t o e n tr op y pr oduct ion — th a t ca n o bf us ca t e the effe ct of infor m a t ion
s cr a mb l in g — a nd ho w t o bui ld r o bus t t e chnique s t o de l ine a t e the m in a n ex pe r ime n t al ly a c c e s si b le w a y .
A nd s e c ond , the a v e r a g e d ( b ip a r t it e ) ope n O T O C d i s c us s e d in thi s ch a pt e r h as a w e l l define d qua n tum-
infor m a t ion the or e t ic me a nin g in t e r m s of ope r a t ion al pr ot oc o l s ( s e e s e ct ion 4.8) w hich m ak e no d ir e ct
r efe r e nc e t o the s ys t e m ’ s t e mpe r a tur e . H o w e v e r , it i s a c ompe l l in g t op ic for f utur e r e s e a r ch t o s tudy ex t e n-
141
sion s w he r e the for m al infinit e-t e mpe r a tur e a v e r a g e i s r e p l a c e d b y ex pe ct a t ion s o v e r othe r s t e a dy s t a t e s of
the qua n tum ch a nne l unde r ex a min a t ion, for ex a mp le , finit e-t e mpe r a tur e G i b bs s t a t e s for D a v ie s g e ne r a -
t or s [ 288 ] ⁴ .
4.7 R evie w of oper a t or en t a ng l e men t and en t ang l ing po wer
L e t us br iefly r e cal l the ide as as s oc i a t e d t o ope r a t or e n t a n gle me n t a nd e n t a n gl in g po w e r ; s e e R ef s . [ 202 ,
203 , 270 ] for a de t ai le d d i s c us sion . G iv e n a d - d ime n sion al H i l be r t sp a c e H the al g e br a of l ine a r ope r a t or s
o v e rH ,L(H) i s e ndo w e d w ith a H i l be r t sp a c e s tr uctur e its e l f de not e d as H
HS
, induc e d v i a the H i l be r t -
S chmidt inne r pr oduct. M or e o v e r , H
HS
i s i s omor p hic ( both al g e br aical ly a nd as a H i l be r t sp a c e ) t o H
2
,
the r efor e , one ca n as s oc i a t e b ip a r t it e s t a t e s t o l ine a r ope r a t or s . Thi s i s a n alo g ous t o the C ho i - J a mio l k o w sk i
i s omor p hi s m .
F or m al ly , g iv e n U 2 H
HS
, one ca n definej U⟩ := ( U
I)j Φ
+
⟩ , w he r ej Φ
+
⟩ :=
1
p
d
d
∑
j= 1
j j⟩j j⟩ i s the
m a x im al ly e n t a n gle d s t a t e a cr os s H
2
. N o w , i f the H i l be r t sp a c e H its e l f h as a b ip a r t it e s tr uctur e , th a t i s,
H
= H
A
H
B
the n the c or r e spond in g s t a t e- r e pr e s e n t a t ion of U U
AB
( sinc e it g e ne r ical ly a cts on
the t ot al sp a c e ) i s a four -p a r t y s t a t e . N a me ly , j U⟩
ABA
′
B
′ = ( U
AB
I
A
′
B
′)j Φ
+
⟩
ABA
′
B
′ w ithj Φ
+
⟩
ABA
′
B
′ =
1
p
d
d
∑
j= 1
j j⟩
AB
j j⟩
A
′
B
′ . M or e o v e r , not ic e th a t for the s t a t e j U⟩
ABA
′
B
′ , the e n t a n gle me n t a cr os s the ABj A
′
B
′
p a r -
t it ion i s m a x im al ( sinc e it i s local unit a r i ly e quiv ale n t t o the m a x im al ly e n t a n gle d s t a t e ). H o w e v e r , the
e n t a n gle me n t a cr os s the AA
′
j BB
′
p a r t it ion i s non tr iv i al a nd one w a y t o qua n t i f y thi s w ould be t o c omput e
the l ine a r e n tr op y a cr os s thi s b ip a r t it ion . Thi s i s pr e c i s e ly the ope r a t or e n t a n gle me n t. Th a t i s, tr a c in g out
o v e r BB
′
w e o bt ain σ
U
:= Tr
BB
′ [j U⟩⟨ Uj] a nd c omput in g its l ine a r e n tr op y define d as S
lin
( ρ) := 1 Tr[ ρ
2
] ,
w e h a v e ,
E
op
( U) := S
lin
( σ
U
) = 1 Tr[( Tr
BB
′j U⟩⟨ Uj)
2
]: (4.33)
A nothe r k ey qua n t it y th a t i s r e l a t e d t o the ope r a t or e n t a n gle me n t i s the en t a n g l i n g p o w er of a unit a r y U
a ct in g on a ( s y mme tr ic ) b ip a r t it e sp a c e H
AB
= H
A
H
B
w ith d
A
= d
B
=
p
d , define d as the a v e r a g e
⁴ A finit e-t e mpe r a tur e g e ne r al i za t ion of P r oposit ion 1 for the cas e of unit a r y e v o lut ion s w as alr e a dy r e por t e d in the S u pp le-
me n t al M a t e r i al of R ef . [ 168 ], fo l lo w in g the pr oof of The or e m 1.
142
a moun t of e n t a n gle me n t g e ne r a t e d b y U v i a its a ct ion on pur e pr oduct s t a t e s . F or m al ly ,
e
p
( U) :=E
V2U( H A); W2U(H B)
[
E
op
(
Uj ψ
V A
⟩j ψ
W B
⟩
)]
; (4.34)
w he r ej ψ
V A
⟩ = Vj ψ
0
⟩ ( a nd simi l a rly for j ψ
W B
⟩ ). Quit e r e m a rk a b ly , the e n t a n gl in g po w e r a nd the ope r a t or
e n t a n gle me n t a r e r e l a t e d as,
e
p
( U) =
d
2
( d+ 1)
2
[
E
op
( U)+ E
op
( US) E
op
( S)
]
; (4.35)
w he r e S i s the s w a p ope r a t or be t w e e n s ubs ys t e m s A; B ( as s ume d t o be s y mme tr ic for the c onne ct ion t o
e n t a n gl i n g po w e r ).
4.8 A prot o c ol for e s tima ting the O pen O T O C
4.4 e s t a b l i she s the ope n O T O C , G(E) as the d i ffe r e nc e of t w o t e r m s, e a ch of w hich qua n t i f y the a v e r a g e
e n tr op y pr oduct ion of ch a nne l s
e
E a nd Tr
B
[
e
E
]
, r e spe ct iv e ly . L e t us br iefly r e v ie w the cas e for unit a r y
ch a nne l s fir s t , w hich w as fir s t d i s c us s e d in R ef . [ 168 ], s e e S e ct ion I I I of the S u pp le me n t al M a t e r i al for
mor e de t ai l s .
F or a unit a r y t ime e v o lut ion fU
t
g
t 0
, the b ip a r t it e O T O C ca n be ex pr e s s e d as,
G(U
t
) =
d
A
+ 1
d
A
E
ψ2H A
[
S
lin
(
Λ
( A)
t
( ψ)
)]
; (4.36)
w he r e Λ
( A)
t
(
ρ
A
)
:= Tr
B
[
U
t
(
ρ
A
I
B
= d
B
)
U
y
t
]
,E
ψ2H A
de not e s r a ndom pur e s t a t e s uni for mly d i s tr i but e d
inH
A
, a nd S
lin
() i s the l ine a r e n tr op y . The b asic pr ot oc o l i s t o ( i ) init i al i z e a r a ndom s t a t e in s ubs ys t e m A
a nd a m a x im al ly mi xe d s t a t e in s ubs ys t e m B , ( i i ) a pp ly the ch a nne l U
t
t o the e n t ir e s ys t e m AB , ( i i i ) tr a c e out
s ubs ys t e m B , ( iv ) me as ur e the l ine a r e n tr op y of the r e s ult in g s t a t e , a nd ( v ) r e pe a t for m a n y r a ndom init i al
s t a t e s uni fo r mly d i s tr i but e d in H
A
.
The k ey ide a i s th a t ( i ) due t o me as ur e c onc e n tr a t ion, as d
A
gr o w s, fe w e r r a ndom s t a t e s a r e ne e de d t o
e s t im a t e G(U
t
) ex pone n t i al ly w e l l , a nd ( i i ) l ine a r e n tr op y of a qua n tum s t a t e ca n be me as ur e d in a n ex pe r -
143
ime n t al ly a c c e s si b le w a y , s e e , for ex a mp le , the s e min al ex pe r ime n t in R ef . [ 231 ] w he r e the pur it y (w hich i s
e qual t o one min us the l ine a r e n tr op y ) w as me as ur e d b y in t e r fe r in g t w o unc or r e l a t e d but ide n t ical c op ie s
of a m a n y - body qua n tum s t a t e; simi l a r ide as h a v e al s o be e n c on side r e d pr e v iously [ 227 – 230 ].
F ur the r mor e , the r e h a v e al s o be e n r e c e n t pr opos al s b as e d on me as ur e me n ts o v e r r a ndom local b as e s th a t
ca n pr o be e n t a n gle me n t g iv e n jus t a sin gle c op y of the qua n tum s t a t e , a nd , in thi s s e n s e , g o bey ond tr a d i -
t ion al qua n tum s t a t e t omo gr a p h y . The m ain ide a c on si s ts of d ir e ctly ex pr e s sin g the l ine a r e n tr op y [ 232 ,
233 ], as w e l l as othe r f unct ion s of the s t a t e [ 234 ], as a n e n s e mb le a v e r a g e of me as ur e me n ts o v e r r a ndom
b as e s .
N o w , for the ope n- s ys t e m cas e , t o e s t im a t e E
ψ
[
S
lin
(
Tr
B
e
E ( ψ)
)]
, w e r e p l a c e in the pr ot oc o l a bo v e , U
t
w i th the ch a nne l E . T o unde r s t a nd thi s, r e cal l th a t
e
E i s define d s uch th a t its a ct ion i s
e
E( ρ
A
)7!E
(
ρ
A
I
d B
)
,
a n a lo g o us t o the ch a nne l Λ
( A)
t
for the unit a r y cas e . F or the s e c ond t e r m th a t i s pr opor t ion al t o E
ψ
[
S
lin
(
e
E ( ψ)
)]
,
w e s imp ly dr op the p a r t i al tr a c in g o v e r s ubs ys t e m B a bo v e a nd e v e r y thin g e l s e in the pr ot o c o l i s the s a me .
4.9 Pro of s
Pr o o f o f 4.2
P r o of . L e t us s t a r t b y simp l i f y i n g G(E) fir s t ,
G(E) :=
1
2( d
A
d
B
)
E
A2U A; B2U B
∥[E( A); B]∥
2
2
; (4.37)
w he r eE
X2G
()
∫
X Haar
dX() de not e s the H a a r a v e r a g e a nd the fa ct or of
1
2
or i g in a t e s f r om
the s qua r e d c omm ut a t or , w hi le the fa ct or of
1
d A d B
i s for the infinit e-t e mpe r a tur e s t a t e .
N o w , ex p a nd in g the c omm ut a t or g iv e s us,
G(E) =
1
d
[
Tr
(
E( A
y
)E( A)
)
Re Tr
(
E( A) B
y
E( A
y
) B
)]
; (4.38)
144
w he r e d d
A
d
B
.
U sin g the ide n t it y ,
Tr( XY) = Tr( SX
Y); (4.39)
w e h a v e ,
Tr
(
E( A
y
)E( A)
)
= Tr
(
SE
2
( A
y
A)
)
: (4.40)
A nd ,
Tr
(
E( A) B
y
E( A
y
) B
)
= Tr
(
SE
2
((
A
A
y
))(
B
y
B
))
(4.41)
W e no w us e a nothe r k ey ide n t it y ,
E
A2U A
(
A
y
A
)
=
S
AA
′
d
A
; (4.42)
w he r e S
AA
′ i s the ope r a t or th a t s w a ps the r e p l icas A w ith A
′
. The a n alo g ous ex pr e s sion for BB
′
al s o
ho ld s .
The n, w e h a v e ,
E
A2U A
Tr
(
SE
2
(
A
y
A
))
= Tr
(
SE
2
(
S
AA
′
d
A
))
; (4.43)
a nd ,
E
A2U A; B2U B
Tr
(
E( A) B
y
E( A
y
) B
)
= Tr
(
SE
2
(
S
AA
′
d
A
)(
S
BB
′
d
B
))
=
1
d
Tr
(
S
AA
′E
2
( S
AA
′)
)
;
(4.44)
w he r e in the l as t e qual it y w e h a v e us e d the fa ct th a t S = S
AA
′ S
BB
′ .
145
P utt in g e v e r y thin g t o g e the r , w e h a v e ,
G(E) =
1
( d
A
d
B
)
2
Tr
(
( d
B
S S
AA
′)E
2
( S
AA
′)
)
: (4.45)
N ot e th a t i f E = U , the n, SU
2
S
AA
′ U
y
2
= U
2
S
BB
′ U
y
2
usin g[ S; U
2
] = 0 a nd S = S
AA
′ S
BB
′ ,
the n, the fir s t t e r m of G(E) be c ome s one .
W e no w sho w th a t [ L
S
;E
2
] = 0 () E i s unit a r y .
L e t L
S
( X) := SX be the s u pe r ope r a t or th a t de not e s the left a ct ion of the s w a p ope r a t or . N ot e th a t ,
L
S
E
2
( S) =E
2
L
S
( S) =E
2
( I): (4.46)
A nd , le t E( X) =
∑
j
A
j
XA
y
j
w ith
∑
j
A
y
j
A
j
= I be its Kr a us r e pr e s e n t a t i on . The n,
LHS = S
0
@
∑
i; j
(
A
i
A
j
)
S
(
A
y
i
A
y
j
)
1
A
=
∑
i; j
A
j
A
y
i
A
i
A
y
j
: (4.47)
N o w , the R HS i s
∑
i; j
A
i
A
y
i
A
j
A
y
j
. T ak in g the tr a c e of both side s, w e h a v e ,
∑
i; j
Tr
(
A
j
A
y
i
)
2
=
∑
i; j
Tr
(
A
i
A
y
i
)
| {z }
∥ A i∥
2
2
Tr
(
A
j
A
y
j
)
| {z }
∥ A j∥
2
2
: (4.48)
U sin g C a uch y - S ch w a r z ine qual it y , w e h a v e ,
⟨
A
j
; A
i
⟩
2
∥ A
i
∥
2
2
A
j
2
2
; (4.49)
w he r e the e qual it y ho ld s i f a nd only i f 8 i; jA
i
= λ
ij
A
j
.
146
S a y A
i
= λ
i
A
0
8 i . The n,
E( X) =
∑
i
j λ
i
j
2
A
0
XA
y
0
=
e
A
0
X
e
A
y
0
; w he r e
e
A
0
√
∑
i
j λ
i
j
2
A
0
: (4.50)
N a me ly ,E i s a C P m a p w ith a sin gle Kr a us ope r a t or , the r efor e , E i s unit a r y .
■
Pr o o f o f 4.3
P r o of . G iv e n a b ip a r t it e c h a nne l , E
AB
, c on side r its C ho i s t a t e ,
ρ
E
= (E
AB
I
A
′
B
′)
(
j φ
+
⟩⟨ φ
+
j
)
; (4.51)
w he r ej φ
+
⟩j φ
+
ABA
′
B
′
⟩ =
1
p
d
d A d B
∑
i; j= 1
j i
A
j
B
⟩
j i
A
′ j
B
′⟩ .
The n,
ρ
E
=
1
d
d A
∑
i; l= 1
d B
∑
j; m= 1
E (j i⟩⟨ lj
j j⟩⟨ mj)
j i⟩⟨ lj
j j⟩⟨ mj: (4.52)
N ot ic e ,
Tr
BB
′
[
ρ
E
]
=
1
d
d A
∑
i; l= 1
Tr
B
[E (j i⟩⟨ lj
I
B
)
j i⟩⟨ lj] ρ
AA
′
E
: (4.53)
A nd ,
Tr
B
′
[
ρ
E
]
=
1
d
d A
∑
i; l= 1
E (j i⟩⟨ lj
I
B
)
j i⟩⟨ lj ρ
ABA
′
E
: (4.54)
The n,
ρ
AA
′
E
2
2
=
1
d
2
d A
∑
i; l= 1
∥ Tr
B
[E (j i⟩⟨ lj
I
B
)]∥
2
2
=
1
d
2
Tr
[
S
AA
′E
2
( S
AA
′)
]
: (4.55)
147
A nd ,
ρ
ABA
′
E
2
2
=
1
d
2
d A
∑
i; l= 1
∥E (j i⟩⟨ lj
I
B
)∥
2
2
=
1
d
2
Tr
[
SE
2
( S
AA
′)
]
: (4.56)
The r efor e , the ope n O T O C ca n be r e ex pr e s s e d as the d i ffe r e nc e of pur it ie s of the C ho i s t a t e ρ
E
a cr os s d i ffe r e n t p a r t it ion s,
G(E) = d
B
Tr
B
′
[
ρ
E
]
2
2
Tr
BB
′
[
ρ
E
]
2
2
: (4.57)
N ot ic e th a t for a de p h asin g ch a nne l , D
B
, one find s,
ρ
D
B
=
1
d
∑
α
Π
α
Π
α
R
B
(4.58)
a nd the G(D
B
) be c ome s the kno w n ex pr e s sion for w ith the “R- m a tr i x ” .
M or e o v e r , for unit a r y ch a nne l s, ρ
ABA
′
U
i s i s ospe ctr al t o ρ
B
′
U
sinc e the s t a t e ρ
ABA
′
B
′
U
i s pur e . A nd , it i s
e as y t o sho w th a t ρ
B
′
U
= I
B
′= d
B
′ , the r efor e , its pur it y i s 1= d
B
. Th a t i s, for unit a r y ch a nne l s the fir s t t e r m
of G(U) i s e qual t o one , as ex pe ct e d . A s a r e s ult , w e h a v e , G(U) = 1
ρ
AA
′
U
2
2
, w hich i s the ope r a t or
e n t a n gle me n t o f the unit a r y ch a nne l U .
T o pr o v e p a r t ( i i ), not ic e th a t ,
Tr
[
SE
2
( S
AA
′)
]
=
d A
∑
i; j= 1
Tr
[
SE
2
(j i⟩
A
⟨ jj
I
B
j j⟩
A
′⟨ ij
I
B
′)
]
(4.59)
=
d A
∑
i; j= 1
Tr[ SE (j i⟩⟨ jj
I
B
)
E (j j⟩⟨ ij
I
B
′)] (4.60)
= d
2
B
d A
∑
i; j= 1
E
(
j i⟩⟨ jj
I
B
d
B
)
2
2
: (4.61)
148
N o w , not ic e th a t , for
e
E( X) =E
(
X
I B
d B
)
, w e h a v e ,
ρ
e
E
=
1
d
A
d A
∑
i; j
(
e
E
I
)
j i⟩
A
⟨ jj
j i⟩
A
′⟨ jj =
1
d
A
d A
∑
i; j
E
(
j i⟩⟨ jj
I
d
B
)
j i⟩
A
′⟨ jj: (4.62)
The n,
ρ
e
E
2
2
=
1
d
2
A
d A
∑
i; j
E
(
j i⟩⟨ jj
I
d B
)
j i⟩
A
′⟨ jj
2
2
.
The r efor e ,
d
B
d
2
Tr
[
SE
2
( S
AA
′)
]
=
d
B
d
2
A
d A
∑
ij
E
(
j i⟩⟨ jj
I
B
d
B
)
2
2
= d
B
ρ
e
E
2
2
: (4.63)
S imi l a rly , w e h a v e ,
Tr
[
S
AA
′E
2
( S
AA
′)
]
=
1
d
2
A
d A
∑
i; j
Tr
[
S
AA
′E
(
j i⟩⟨ jj
I
B
d
B
)
E
(
j j⟩⟨ ij
I
B
d
B
)]
(4.64)
=
1
d
2
A
d A
∑
i; j
Tr
B
[
E
(
j i⟩⟨ jj
I
B
d
B
)]
2
2
= d
B
ρ
T◦
e
E
2
2
: (4.65)
P utt in g e v e r y thin g t o g e the r , w e h a v e the de sir e d pr oof . ■
Pr o o f o f 4.4
P r o of . L e tj φ
A
⟩ be a n a r b itr a r y s t a t e a nd j ψ
A
⟩ := Uj φ
A
⟩ c or r e spond t o H a a r r a ndom pur e s t a t e s o v e r
H
A
. The n, the k ey ide a of the pr oof i s the o bs e r v a t ion th a t S
AA
′ ca n be ex pr e s s e d v i a the ide n t it y ,
E
ψ
A
Haar
(
j ψ
A
⟩⟨ ψ
A
j
)
2
=
1
d
A
( d
A
+ 1)
( I
AA
′ + S
AA
′): (4.66)
P lu gg in g thi s in t o E q (2), w e h a v e ,
1
d
A
+ 1
d
A
{
d
B
Tr
(
SE
2
(
ψ
2
A
I
BB
′
d
2
B
ψ
A
))
Tr
(
S
AA
′E
2
(
ψ
2
A
I
BB
′
d
2
B
ψ
A
))}
: (4.67)
149
The n, usin g ,
S
L
( X) = 1 Tr( X
2
) = 1 Tr( SX
X); (4.68)
w e h a v e ,
1
d
A
+ 1
d
A
{
S
L
(
Tr
B
(
e
E( ψ
A
)
)) ψ
A
d
B
[
S
L
(
e
E( ψ
A
)
) ψ
A
(
1
1
d
B
)]}
; (4.69)
w he r e
e
E( ψ
A
) =E
(
ψ
A
I B
d B
)
.
N o w , not ic e th a t ,
S
L
(
e
E( ψ
A
)
)
1
1
d
B
S
min
L
; (4.70)
a nd sinc e E i s unit al , S
L
(
E( ψ
A
I B
d B
)
)
m us t incr e as e w ith t ime , sinc e e n tr op y ca nnot de cr e as e unde r
a u nit al m a p . ■
Pr o o f o f 4.5
P r o of . C on side r the de p h asin g ch a nne l , E =D
B
, w he r eD
B
( X) =
d
∑
α= 1
Π
α
X Π
α
=
d
∑
α= 1
j ψ
α
⟩⟨ ψ
α
j
⟨
ψ
α
j Xj ψ
α
⟩
,
w he r ef Π
α
g
α
i s a b asi s ( of r a nk - 1 pr oj e ct or s ).
F ir s t , not e th a t ,
Tr
(
SE
2
( S
AA
′)
)
=
d A
∑
i; j= 1
Tr( SE (j i⟩⟨ jj
I
B
)
E (j j⟩⟨ ij
I
B
)) =
d A
∑
i; j= 1
∥E (j i⟩⟨ jj
I
B
)∥
2
2
; (4.71)
w he r e w e h a v e us e d S
AA
′ =
d A
∑
i; j= 1
j ij⟩
AA
′⟨ jij
I
BB
′ .
150
N o w ,
E (j i⟩⟨ jj
I
B
) =
∑
α
j ψ
α
⟩⟨ ψ
α
j
⟨
ψ
α
j(j i⟩⟨ jj
I
B
)j ψ
α
⟩
(4.72)
=
∑
α
Π
α
Tr
(
ρ
α α
j i⟩⟨ jj
)
=
∑
α
⟨
jj ρ
α α
j i
⟩
Π
α
; (4.73)
w he r e ρ
α α
:= Tr
B
(
j ψ
α
⟩⟨ ψ
α
j
)
.
The r efor e ,
d A
∑
i; j= 1
∥E (j i⟩⟨ jj
I
B
)∥
2
2
=
d A
∑
i; j= 1
∑
α
⟨
ij ρ
α α
j j
⟩
2
=
∑
α
ρ
α α
2
2
: (4.74)
S imi l a rly ,
Tr
(
S
AA
′E
2
( S
AA
′)
)
=
d A
∑
i; j= 1
∥ Tr
B
(E (j i⟩⟨ jj
I
B
))∥
2
2
=
d A
∑
i; j= 1
∑
α
ρ
α α
⟨
jj ρ
α α
j i
⟩
2
2
(4.75)
=
d A
∑
i; j= 1
∑
α; β
⟨
jj ρ
α α
j i
⟩
⟨
ij ρ
β β
j j
⟩⟨
ρ
α α
; ρ
β β
⟩
=
∑
α; β
⟨
ρ
α α
; ρ
β β
⟩
2
: (4.76)
P utt in g e v e r y thin g t o g e the r , w e h a v e the de sir e d r e s ult ,
G(D
B
) =
1
d
2
2
4
d
B
∑
α
ρ
α α
2
2
∑
α; β
⟨
ρ
α α
; ρ
β β
⟩
2
3
5
: (4.77)
D efine the r e nor m al i z e d Gr a m m a tr i x as, X
α β
=
⟨ ρ
α α
; ρ
β β
⟩
d B
, the n,
G(D
B
) =
1
d
2
A
2
4
∑
α
X
α α
∑
α β
X
2
α β
3
5
=
1
d
2
A
( Tr( X) Tr( X
2
)) =
1
d
2
A
∥ X X
2
∥
1
: (4.78)
F or the bound , not e th a t X X
2
sinc e X i s b i s t och as t ic . The r efor e , one h as th a t spe c ( X) [ 0; 1] .
151
The n,
∥ X X
2
∥
1
=
∑
α
x
α
( 1 x
α
) r a nk ( X)= 4: (4.79)
A nd , r a nk ( X) min( d
2
A
; d) sinc e it i s a Gr a m m a tr i x of v e ct or s in a d
2
A
- d ime n sion al sp a c e . The r efor e ,
w e h a v e the bound ,
G(D
B
)
1
4
min
(
1;
d
d
2
A
)
=
1
4
min
(
1;
d
B
d
A
)
: (4.80)
■
Pr o o f o f 4.6
P r o of . F ir s t not ic e ,
X
α β
=
⟨
I
A
d
A
+ Δ
α
;
I
B
d
B
+ Δ
β
⟩
=
1
d
A
d
B
+
⟨
Δ
α
; Δ
β
⟩
d
B
( 8 α; β): (4.81)
N a me ly ,
^
X
B
=j φ
s
AB
⟩⟨ φ
s
AB
j+
^
δ
B
w he r ej φ
s
AB
⟩ =
1
p
d A d B
d A
∑
i= 1
d B
∑
j= 1
j i⟩
j j⟩ a nd[
^
δ]
α β
=
⟨ Δ α; Δ
β⟩
d B
.
The n, usin g ,
∑
α
Δ
α
=
∑
α
ρ
α
d
B
I
A
= Tr
B
[
∑
α
j ψ
α
⟩⟨ ψ
α
j
]
= d
B
I
A
d
B
I
A
= 0; (4.82)
w e find th a t
1
d
2
A
Tr
[
^
X
B
^
X
2
B
]
=
1
d
2
A
Tr
[
^
δ
B
δ
2
B
]
= G(D
B
): (4.83)
I gnor in g the s qua r e d t e r m, it fo l lo w s th a t
G(D
B
)
1
d
2
A
Tr
[
^
δ
B
]
=
1
d
2
A
∑
α
⟨ Δ
α
; Δ
α
⟩
d
B
1
d
2
A
∑
α
ε
d
B
= ε
d
d
2
A
d
B
=
ε
d
A
: (4.84)
152
■
Pr o o f o f 4.7
P r o of . W e h a v e
G(D
B
) =
1
d
A
⟨ S
AA
′; R
B
⟩∥ R
AA
′
B
∥
2
2
; (4.85)
L e t us c on side r the t w o t e r m s s e p a r a t e ly .
1
d
A
Tr[ S
AA
′ U
2
(
1
d
∑
α
j α⟩⟨ αj
2
)
| {z }
Ω 0
U
y
2
]: (4.86)
The n, not ic e th a tE
U
[
U
2
Ω
0
U
y
2
]
=
I+ S
d( d+ 1)
, he nc e ,
1
d
A
d( d+ 1)
Tr[ S
AA
′ ( I+ S)] =
1+
d B
d A
d+ 1
2
d
2
A
: (4.87)
S e c ond t e r m . U sin g c on v ex it y , w e h a v e ,
∥ Tr
BB
′ R
B
∥
2
2
B
Tr
BB
′ R
B
B
2
2
: (4.88)
R e cal l th a t ,
Tr
BB
′ R
B
B
= Tr
BB
′
[
I+ S
d( d+ 1)
]
=
d
2
B
I
AA
′ + d
B
S
AA
′
d( d+ 1)
: (4.89)
The r efor e ,
R
B
B
2
2
=
d
2
B
d
2
( d+ 1)
2
∥ d
B
I
AA
′ + S
AA
′∥
2
2
=
d
2
B
d
2
( d+ 1)
2
[ d
2
+ d
2
A
+ 2d]
d
2
B
( d+ 1)
2
: (4.90)
153
A nd fin al ly , putt in g E qs . ( 4.87 ) a nd ( 4.90 ) t o g e the r
G(D
B
)
B
2
d
2
A
d
2
B
( d+ 1)
2
2
d
2
A
d
2
B
( 2d)
2
=
7
4
d
2
A
= O(
1
d
2
A
): (4.91)
i i )
f(B)
1
d
A
⟨ S
AA
′; R
B
⟩
R
AA
′
B
2
2
α(B)+ β(B): (4.92)
W e fir s t c o l le ct a fe w r e s ults . F ir s t ,
α(B) α(
e
B)
=
1
d
A
⟨
S
AA
′; R
B
R
e
B
⟩
1
d
A
∥ S
AA
′∥
1
R
B
R
e
B
1
1
d
A
R
B
R
e
B
1
;
(4.93)
w he r e in the fir s t ine qual it y w e h a v e us e d the H o lde r -t y pe ine qual it y ( for m a tr ic e s ),
Tr
[
A
y
B
]
∥ A∥
1
∥ B∥
1
. A nd in the s e c ond ine qual it y , ∥ U∥
1
= 1 for a n y unit a r y U .
S e c ond ,
β(B) β(
e
B)
=
R
AA
′
B
2
2
R
AA
′
e
B
2
2
=
(
R
AA
′
B
2
+
R
AA
′
e
B
2
)(
R
AA
′
B
2
R
AA
′
e
B
2
)
(4.94)
= 2
R
AA
′
B
R
AA
′
e
B
2
2
R
AA
′
B
R
AA
′
e
B
1
2
R
B
R
e
B
1
; (4.95)
w he r e in the fir s t ine qual it y w e h a v e bounde d the 2 - nor m w ith the 1 - nor m d i s t a nc e a nd in the s e c ond
ine qual it y w e h a v e us e d the fa ct th a t p a r t i al tr a c e i s a C P m a p a nd the 1 - nor m i s c on tr a ct iv e unde r C P
m a ps .
N o w , w e h a v e t o bound ,
R
B
R
e
B
1
=
R
B 0
(
V
y
U
)
2
R
B 0
(
V
y
U
)
2
1
; (4.96)
154
w he r e w e h a v e us e the unit a r y in v a r i a nc e of the 1 - nor m .
D efine , U V Δ =) V
y
U = I+ Δ . The n,
( I+ Δ)
2
= I
I+ Δ
I+ I
Δ+ Δ
Δ I+ X: (4.97)
U sin g thi s, w e h a v e ,
∥ R
B
R
e
B
∥
1
=∥ XR
B 0
+ R
B 0
X+ XR
B 0
X∥
1
2∥ X∥
1
+∥ X∥
2
1
=∥ X∥
1
( 2+∥ X∥
1
); (4.98)
w he r e w e h a v e r e pe a t e d ly us e d ∥ AB∥
1
∥ A∥
1
∥ B∥
1
, s ubm ult ip l ica t iv it y of nor m s a nd the fa ct th a t
R
B 0
i s a qua n tum s t a t e , ∥ R
B 0
∥
1
= 1 .
N o w ,
∥ X∥
1
=∥ Δ
I+ I
Δ+ Δ
Δ∥
1
2∥ Δ∥
1
+∥ Δ∥
2
1
(4.99)
=∥ Δ∥
1
( 2+∥ Δ∥
1
) 4∥ Δ∥
1
= 4∥ U V∥
1
: (4.100)
The r efor e ,
R
B
R
e
B
1
4∥ Δ∥
1
( 2+ 4∥ Δ∥
1
) 4∥ Δ∥
1
( 2+ 4 2) = 40∥ Δ∥
1
; (4.101)
w he r e w e h a v e us e d ∥ Δ∥
1
2 .
B r in g in g e v e r y thin g t o g e the r , w e h a v e ,
F(B) F(
e
B)
α(B) α(
e
B)
+
β(B) β(
e
B)
(
1
d
A
+ 2
)
R
B
R
e
B
1
40
(
1
d
A
+ 2
)
∥ Δ∥
1
(4.102)
40
5
2
∥ Δ∥
1
= 100∥ U V∥
1
100∥ U V∥
2
: (4.103)
■
155
Pr o o f o f 4.8
P r o of . T o c omput e the ope n O T O C for the g e ne r al cas e , G( Φ
EB
) , w e ne e d t o c omput e , Tr
(
SΦ
2
EB
( S
AA
′)
)
=
d A
∑
i; j= 1
∥ Φ
EB
(j i⟩⟨ jj
I
B
)∥
2
2
a nd Tr
(
S
AA
′ Φ
2
EB
( S
AA
′)
)
=
d A
∑
i; j= 1
∥ Tr
B
( Φ
EB
(j i⟩⟨ jj
I
B
))∥
2
2
.
( i) Φ
EB
(j i⟩⟨ jj
I
B
) =
∑
k
M
k
Tr[ δ
k
j i⟩⟨ jj
I
B
] =
∑
k
M
k
⟨ jj δ
A
k
j i⟩; (4.104)
w he r e δ
A
k
Tr
B
[ δ
k
] .
The r efor e ,
d A
∑
i; j= 1
∥ Φ
EB
(j i⟩⟨ jj
I
B
)∥
2
2
=
d A
∑
i; j= 1
Tr
[
∑
k; k
′
M
k
M
k
′⟨ ij δ
A
k
j j⟩⟨ jj δ
A
k
′j i⟩
]
(4.105)
=
∑
k; k
′
⟨
δ
A
k
; δ
A
k
′
⟩
⟨ M
k
; M
k
′⟩: (4.106)
S imi l a rly ,
( ii) Tr
B
[ Φ
EB
(j i⟩⟨ jj
I
B
)] = Tr
B
[
∑
k
M
k
Tr[ δ
k
j i⟩⟨ jj
I
B
]
]
=
∑
k
M
A
k
⟨ jj δ
A
k
j i⟩; (4.107)
w he r e M
A
k
Tr
B
[ M
k
] .
The r efor e ,
d A
∑
i; j= 1
∥ Tr
B
[ Φ
EB
(j i⟩⟨ jj
I
B
)]∥
2
2
=
d A
∑
i; j= 1
Tr
[
∑
k; k
′
M
A
k
M
A
k
′⟨ ij δ
A
k
j j⟩⟨ jj δ
A
k
′j i⟩
]
(4.108)
=
∑
k; k
′
⟨
δ
A
k
; δ
A
k
′
⟩⟨
M
A
k
; M
A
k
′
⟩
: (4.109)
156
P utt in g e v e r y thin g t o g e the r , w e h a v e ,
G( Φ
EB
) =
1
d
2
∑
k; k
′
⟨
δ
A
k
; δ
A
k
′
⟩[
d
B
⟨ M
k
; M
k
′⟩
⟨
M
A
k
; M
A
k
′
⟩]
; (4.110)
■
Pr o o f o f 4.9
P r o of . T o pr o v e ( i ), w e ne e d t o sho w th a t g iv e n, B = fj α⟩g a b asi s of H
AB
, d = dim(H
AB
) w ith
^
Φ =
[
φ
α; α
′
]
d
α; α
′
= 1
s uch th a t
^
Φ 0 a nd φ
α; α
= 1 8 α , the m a pE
^
Φ
( X) =
d
∑
α; α
′
φ
α; α
′
X
α; α
′j α⟩⟨ α
′
j i s a
qua n tum ch a nne l .
F ir s t , not ic e th a t E
^
Φ
define s a l ine a r m a p on L(H
AB
) s uch th a t
Tr[E
^
Φ
( X)] =
∑
α; α
′
φ
α; α
′
X
α; α
′ δ
α; α
′ =
∑
α
φ
α; α
X
α; α
=
∑
α
X
α
= Tr[ X]: (4.111)
H e nc e ,E
^
Φ
i s a tr a c e-pr e s e r v in g m a p .
The n, sinc e
^
Φ 0 , one ca n w r it e ,
^
Φ = S
^
Φ
D
S
y
w he r e
^
Φ
D
diag( φ
μ
); φ
μ
0 a nd S i s a unit a r y .
The n, E
^
Φ
ca n be ex pr e s s e d as,
E
^
Φ
( X) =
∑
μ; α; α
′
λ
μ
S
α; μ
S
α
′
; μ
X
α; α
′j α⟩⟨ α
′
j: (4.112)
W e no w define A
μ
j α⟩ :=
√
λ
μ
S
α; μ
j α⟩ 8 α; α
′
. The r efor e ,
E
^
Φ
( X) =
∑
μ; α; α
′
X
α; α
′ A
μ
j α⟩⟨ α
′
j A
y
μ
=
∑
μ
A
μ
XA
y
μ
: (4.113)
157
M or e o v e r ,
⟨ αj
∑
μ
A
y
μ
A
μ
j α
′
⟩ =
∑
μ
λ
μ
S
α; μ
S
α
′
; μ
⟨ αj α
′
⟩ = δ
α; α
′
∑
μ
λ
μ
S
α; μ
2
= φ
α; α
δ
α; α
= δ
α; α
(8 α; α
′
):
(4.114)
The r efor e ,
∑
μ
A
y
μ
A
μ
=I a nd sinc e E
^
Φ
ca n be ex pr e s s e d in a Kr a us for m, it i s C P .
R e m a rk : L e tF =f
^
Φ2M
C
d
j
^
Φ 0 a nd φ
α; α
= 1(8 α)g . The n, F i s a c on v ex s ubs e t of M
C
d
, the
s e t of d d m a tr ic e s o v e r C . M a ps of the for m E
^
Φ
a r e p a r a me tr i z e d b y e le me n ts in F a nd b as e sB .
F or a fi xe dB , the m a p ,
^
Φ2F !E
^
Φ
i s a n a ffine m a p of c on v ex bod ie s .
T o pr o v e ( i i ), the pr oof s tr a t e g y i s simi l a r t o P r oposit ion 4. The k ey o bs e r v a t ion i s th a t the a ct ion
of the m a p , E
^
Φ
ca n be ex pr e s s e d as,
E
^
Φ
( X) =
d
∑
α; β= 1
φ
α; β
j α⟩⟨ αj Xj β⟩⟨ βj =
d
∑
α; β
φ
α; β
x
α; β
j α⟩⟨ βj; (4.115)
w he r e x
α; β
⟨ αj Xj β⟩ . Thi s fo l lo w s f r om the a ct ion E
^
Φ
(j α⟩⟨ α
′
j) = φ
α; α
′
j α⟩⟨ α
′
j .
W e ne e d t o e v alua t e
Tr
(
SE
2
( S
AA
′)
)
=
d A
∑
i; j= 1
∥E (j i⟩⟨ jj
I
B
)∥
2
2
a nd
Tr
(
S
AA
′E
2
( S
AA
′)
)
=
d A
∑
i; j= 1
∥ Tr
B
(E (j i⟩⟨ jj
I
B
))∥
2
2
:
158
N o w ,
E (j i⟩⟨ jj
I
B
) =
∑
α; β
φ
α; β
j α⟩⟨ αj(j i⟩⟨ jj
I
B
)j β⟩⟨ βj =
∑
α; β
φ
α; β
Π
α; β
Tr
[
ρ
α; β
j i⟩⟨ jj
]
; (4.116)
w he r e Π
α; β
j α⟩⟨ βj a nd ρ
α; β
Tr
B
[j α⟩⟨ βj] .
A nd ,
d A
∑
i; j= 1
∥E (j i⟩⟨ jj
I
B
)∥
2
2
=
d A
∑
i; j
Tr
2
4
∑
α; β; γ; δ
φ
α; β
φ
γ; δ
⟨ jj ρ
α; β
j i⟩ Π
β; α
⟨ jj ρ
γ; δ
j i⟩ Π
γ; δ
3
5
(4.117)
=
d A
∑
i; j
∑
α; β; γ; δ
φ
α; β
φ
γ; δ
⟨ jj ρ
α; β
j i⟩ Tr
[
Π
β; α
⟨ jj ρ
γ; δ
j i⟩ Π
γ; δ
]
| {z }
= δ α; γ δ
β; δ
(4.118)
=
d A
∑
i; j
∑
α; β
φ
α; β
2
⟨ jj ρ
α; β
j i⟩
2
=
∑
α; β
φ
α; β
2
ρ
α; β
2
2
: (4.119)
S imi l a rly , w e h a v e ,
Tr
B
[E (j i⟩⟨ jj
I
B
)] =
∑
α; β
φ
α; β
⟨ jj ρ
α; β
j i⟩ Tr
B
[
Π
α; β
]
| {z }
= ρ
α; β
=
∑
α; β
φ
α; β
⟨ jj ρ
α; β
j i⟩ ρ
α; β
: (4.120)
The n,
∥ Tr
B
[E (j i⟩⟨ jj
I
B
)]∥
2
2
=
∑
α; β; γ; δ
φ
α; β
φ
γ; δ
⟨ ij ρ
β; α
j j⟩⟨ jj ρ
γ; δ
j i⟩ Tr
[
ρ
β; α
ρ
γ; δ
]
: (4.121)
A nd ,
d A
∑
i; j= 1
∥ Tr
B
[E (j i⟩⟨ jj
I
B
)]∥
2
2
=
d A
∑
i; j= 1
∑
α; β; γ; δ
φ
α; β
φ
γ; δ
⟨ ij ρ
β; α
j j⟩⟨ jj ρ
γ; δ
j i⟩ Tr
[
ρ
β; α
ρ
γ; δ
]
(4.122)
=
∑
α; β; γ; δ
φ
α; β
φ
γ; δ
⟨
ρ
α; β
; ρ
γ; δ
⟩
2
: (4.123)
P utt in g e v e r y thin g t o g e the r , w e h a v e the de sir e d pr oof . ■
159
5
BR O T O C s a nd Qua n tum I nfor m a tion S cr a mblin g a t
F init e T e mp e r a tur e
5.1 A bs tr a c t
Out - of -t ime- or de r e d c or r e l a t or s ( O T O C s ) h a v e be e n ex t e n siv e ly s tud ie d in r e c e n t y e a r s as a d i a gnos t ic of
qua n tum infor m a t ion s cr a mb l in g. I n thi s ch a pt e r , w e s tudy qua n tum infor m a t ion-the or e t ic aspe cts of the
r e g u l a r i ze d finit e-t e mpe r a tur e O T O C . W e in tr oduc e a n aly t ical r e s ults for the bip a r t it e r e g u l a r i ze d O T O C
( BR O T O C ): the r e g ul a r i z e d O T O C a v e r a g e d o v e r r a ndom unit a r ie s s u ppor t e d o v e r a b ip a r t it ion . W e
sho w th a t the BR O T O C h as s e v e r al in t e r e s t in g pr ope r t ie s, for ex a mp le , it qua n t i fie s the pur it y of the as s o -
160
c i a t e d the r mofie ld doub le s t a t e a nd the “ ope r a t or pur it y ” of the a n aly t ical ly c on t in ue d t ime- e v o lut ion op -
e r a t or . A t infinit e-t e mpe r a tur e , it r e duc e s t o one min us the ope r a t or e n t a n gle me n t of the t ime- e v o lut ion
ope r a t or . I n the z e r o -t e mpe r a tur e l imit a nd for nonde g e ne r a t e H a mi lt oni a n s, the BR O T O C pr o be s the
gr ound s t a t e e n t a n gle me n t. B y c omput in g lon g -t ime a v e r a g e s, w e sho w th a t the e qui l i br a t ion v alue of the
BR O T O C i s in t im a t e ly r e l a t e d t o ei g e n s t a t e e n t a n gle me n t. F in al ly , w e n ume r ical ly s tudy the e qui l i br a t ion
v alue of the BR O T O C for v a r ious p h ysical ly r e le v a n t H a mi lt oni a n mode l s a nd c omme n t on its a b i l it y t o
d i s t in g ui sh in t e gr a b le a nd ch a ot ic dy n a mic s .
T ex t for thi s C h a pt e r i s a d a pt e d f r om [ 289 ].
5.2 In troduc tion
The the r m al i za t ion of clos e d qua n tum s ys t e m s h as be e n a lon g s t a nd in g puz z le in the or e t ical p h ysic s [ 13 ,
46 , 290 , 291 ]. R e c e n tly , the not ion of “ infor m a t ion s cr a mb l in g ” as the unde rly in g me ch a ni s m for the r m al -
i za t ion h as g aine d pr omine nc e . The ide a i s th a t c omp lex qua n tum s ys t e m s quick ly d i s s e min a t e local i z e d
infor m a t ion thr ou gh the ( nonlocal ) de gr e e s of f r e e dom, m ak in g it in a c c e s si b le t o a n y lo c a l pr o be s t o the
s ys t e m . The infor m a t ion i s not los t , sinc e the glo b al e v o lut ion i s s t i l l unit a r y , r a the r , it i s e nc ode d in non-
local c or r e l a t ion s a cr os s the s ys t e m . A qua n t i fica t ion of thi s dy n a mical p he nome n a h as init a t e d a r ich d i s -
c us sion s ur r ound in g ope r a t or gr o w th [ 109 – 112 , 292 – 294 ], ei g e n s t a t e the r m al i za t ion h y pothe si s ( E TH )
[ 295 ], qua n tum ch a os [ 296 , 297 ], a mon g othe r s; s e e al s o R ef s . [ 298 , 299 ] for a r e c e n t r e v ie w . One of
the c e n tr al o bj e cts in thi s qua n t i fica t ion a r e the s o - cal le d out - of -t ime- or de r e d c or r e l a t or s ( O T O C s ). The
O T O C i s us ual ly define d as a four po in t f unct ion w ith un us ual t ime- or de r in g [ 17 , 18 ],
F
β
( t) := Tr
[
W
y
t
V
y
W
t
V ρ
β
]
; (5.1)
w he r e W
t
:= U
y
t
WU
t
i s the H ei s e nbe r g - e v o lv e d ope r a t or a nd ρ
β
= exp[ β H]=Z( β) i s the G i b bs s t a t e a t
in v e r s e t e mpe r a tur e β w ithZ( β) := Tr[ exp[ β H]] . A n in t im a t e ly r e l a t e d qua n t it y t o the a bo v e O T O C
161
i s the fo l lo w in g nor m of the c omm ut a t or ,
C
β
( t) :=
1
2
Tr
[
[ W
t
; V]
y
[ W
t
; V] ρ
β
]
=
1
2
[ W
t
; V]
√
ρ
β
2
2
: (5.2)
H e r e w e h a v e us e d the H i l be r t - S chmidt nor m ∥∥
2
, w hich or i g in a t e s f r om the ( H i l be r t - S chmidt ) inne r
pr oduct ⟨ A; B⟩ := Tr
[
A
y
B
]
. The t w o qua n t it ie s a r e r e l a t e d v i a the simp le for m ul a ,
C
β
( t) = 1 ReF
β
( t): (5.3)
The r efor e , the gr o w th of the nor m of the c omm ut a t or i s as s oc i a t e d t o the de ca y of the O T O C s .
The ide a be hind usin g the nor m of the c omm ut a t or t o qua n t i f y s cr a mb l in g i s the fo l lo w in g : le t V a nd
W be t w o local ope r a t or s th a t init i al ly c omm ut e ( for ex a mp le , c on side r local ope r a t or s on t w o d i ffe r e n t
sit e s of a qua n tum sp in- ch ain ). U nde r H ei s e nbe r g t ime- e v o lut ion, the s u ppor t of W
t
gr o w s a nd a ft e r a
tr a n sie n t pe r iod , it w i l l s t a r t nonc omm ut in g w ith the ope r a t or V a nd one ca n ut i l i z e the c omm ut a t or C
β
( t)
t o qua n t i f y thi s gr o w th . I n tuit iv e ly , i f the H a mi lt oni a n of thi s s ys t e m i s local , the n L ei b - R o b in s on t y pe
bound s ca n pr o v ide a n e s t im a t e for the t ime it t ak e s for the gr o w th of thi s c omm ut a t or [ 86 , 300 , 301 ].
U nde r s t a nd in g qua n t it a t iv e ly , the s cr a mb l in g of qua n tum infor m a t ion h as le a d t o a p le thor a of the or e t i -
cal in si gh ts [ 104 , 109 – 112 , 292 – 296 , 302 ]. Thi s w as s w i ftly fo l lo w e d b y s e v e r al s t a t e- of -the-a r t ex pe r ime n-
t al in v e s t i g a t ion s [ 260 – 262 , 264 , 265 , 267 – 269 , 303 , 304 ]. F ur the r mor e , s e v e r al w ork s h a v e no w e luc id a t e d
qua n tum infor m a t ion the or e t ic aspe cts unde rly in g the O T O C , for ex a mp le , b y c onne ct in g it t o L os chmidt
E cho [ 83 ], ope r a t or e n t a n gle me n t a nd e n tr op y pr oduct ion [ 305 , 306 ], qua n tum c o he r e nc e [ 307 ], e n tr op ic
unc e r t ain t y r e l a t ion s [ 308 ], a mon g othe r s .
I n R ef s . [ 83 , 305 ], the a uthor s define d a “ b ip a r t it e O T O C , ” o bt aine d b y a v e r a g in g the infinit e-t e mpe r a tur e
O T O C uni for mly o v e r local r a ndom unit a r ie s s u ppor t e d on a b ip a r t it ion . I n R ef . [ 305 ], thi s b ip a r t it e
O T O C w as sho w n t o h a v e the fo l lo w in g ope r a t ion al in t e r pr e t a t ion s: ( i ) it i s ex a ctly the o p er a t o r en t a n g le-
m en t [ 309 , 310 ] of the dy n a mical unit a r y U
t
, ( i i ) it c onne cts in a simp le w a y t o the e n t a n gl in g po w e r [ 36 ]
of the dy n a mical unit a r y U
t
, ( i i i ) it i s ex a ctly e qual t o the a v e r a g e l ine a r e n tr op y pr oduct ion po w e r of the r e-
162
duc e d dy n a mic s, a mon g othe r s . F ur the r mor e , s e v e r al of the s e c onne ct ion s w e r e g e ne r al i z e d t o the cas e of
ope n- s ys t e m dy n a mic s in R ef . [ 306 ], w he r e , in p a r t ic ul a r , a c ompe t it ion be t w e e n infor m a t ion s cr a mb l in g
a nd e n v ir onme n t al de c o he r e nc e w as unc o v e r e d [ 311 ].
U nfor tun a t e ly , as w e mo v e a w a y f r om the infinit e-t e mpe r a tur e as s umpt ion, the c onne ct ion s un v ei le d in
R ef . [ 305 ] do not ca r r y o v e r their ope r a t ion al aspe cts a n y mor e . F or ex a mp le , a s tr ai gh tfor w a r d g e ne r a t ion
t o the finit e t e mpe r a tur e cas e , s a y , b y usin g the O T O C as define d in e q . ( 5.1) fai l s t o r e t ain the ope r a t or
e n t a n g le m e n t or e n tr op y pr oduct ion c onne ct ion . N ot al l i s los t , ho w e v e r , as it i s the r e g u l a r i ze d O T O C
[ 296 ] th a t n a tur al ly le nd s its e l f t o the s e ope r a t ion al c onne ct ion s . Eluc id a t in g thi s c onne ct ion i s the k ey
t e chnical c on tr i but ion of thi s w ork . F or e as e of r e a d a b i l it y , the pr oof s of k ey P r oposit ion s a ppe a r in the
A ppe nd i x .
5.3 Prel imin arie s
The O T O C in tr oduc e d in e q . ( 5.1 ) w i l l he r e a ft e r be r efe r r e d t o as the u n r e g u l a r i ze d O T O C . I n c on tr as t , the
r e g u l a r i ze d ( or s y mme tr ic ) O T O C i s define d as [ 296 ],
F
( r)
β
( t) := Tr
[
W
y
t
yV
y
yW
t
yVy
]
w ith y
4
= ρ
β
: (5.4)
E quiv ale n tly ,
F
( r)
β
( t) =
1
Z( β)
Tr
[
W
y
t
xV
y
xW
t
xVx
]
; (5.5)
w ith x = exp[ β H= 4] . W e al s o define the as s oc i a t e d d i s c onne ct e d c or r e l a t or [ 296 ],
F
( d)
β
( t) := Tr
[
√
ρ
β
W
y
t
√
ρ
β
W
t
]
Tr
[
√
ρ
β
V
y
√
ρ
β
V
]
: (5.6)
I n R ef . [ 296 ], a bound on the gr o w th of the c or r e l a t or F
( d)
β
( t) F
( r)
β
( t) w as o bt aine d unde r c e r t ain
163
as s umpt ion s as
@
@ t
log
(
F
( d)
β
( t) F
( r)
β
( t)
)
2 π
β
: (5.7)
W e al s o r efe r the r e a de r t o R ef . [ 295 ] the s a me bound w as de r iv e d for s ys t e m s s a t i sf y in g E TH, alon g w ith
s ome ex tr a as s umpt ion s . I n thi s w ork w e foc us on the qua n t it y F
( d)
β
( t) F
( r)
β
( t) a r i sin g f r om thi s bound a nd
c onne ct it t o ope r a t ion al , qua n tum infor m a t ion-the or e t ic qua n t it ie s . N ot ic e th a t , for a t ime-inde pe nde n t
H a mi lt oni a n, the d i s c onne ct e d c or r e l a t or F
( d)
β
( t) i s t ime inde pe nde n t ( b y usin g the c omm ut a t ion of [ y
2
; U
t
]
a n d t he c y cl ic it y of tr a c e ). The r efor e , w e ca n define , F
d
β
F
d
β
( t) = Tr
[
y
2
W
y
y
2
W
]
Tr
[
y
2
V
y
y
2
V
]
.
F o l lo w in g R ef . [ 305 ], w e w i l l c on side r the fo l lo w in g s e tu p: le t H
AB
= H
A
H
B
= C
d A
C
d B
be
a b ip a r t it ion of the H i l be r t sp a c e . D efine as U(H
A( B)
) , the unit a r y gr ou p o v e r H
A( B)
. W e w a n t t o unde r -
s t a nd the qual it a t iv e a nd qua n t it a t iv e fe a tur e s of the O T O C for a gen er ic cho ic e of local ope r a t or s V a nd
W . The r efor e , w e a v e r a g e o v e r unit a r y ope r a t or s s u ppor t e d on the b ip a r t it ion Aj B . W e define the b ip a r t it e
a v e r a g e d , u n r e g u l a r i ze d O T O C ( he r e a ft e r , b ip a r t it e unr e g ul a r i z e d O T O C ) as [ 305 ]
G
β
( t) :=E
V A; W B
[
C
β
( t)
]
; (5.8)
w he r e , V
A
= V
I
B
; W
B
= I
A
W , w ith V2U(H
A
); W2U(H
B
); a ndE
V; W
[] :=
∫
Haar
dV dW[]
de not e s H a a r -a v e r a g in g o v e r the s t a nd a r d uni for m me as ur e o v e r U(H
A( B)
) [ 312 ]. I n R ef . [ 305 ] it w as
sho w n th a t one ca n a n aly t ical ly pe r for m the H a a r a v e r a g e s t o o bt ain the fo l lo w in g ex pr e s sion,
G
β
( t) = 1
1
d
Re Tr
((
ρ
β
I
A
′
B
′
)
U
y
2
t
S
AA
′ U
2
t
S
AA
′
)
; (5.9)
w he r eS
AA
′ i s the ope r a t or th a t s w a ps the A$ A
′
sp a c e s inH
A
H
B
H
A
′
H
B
′ . Thi s e qua t ion r e pr e-
s e n ts the finit e t e mpe r a tur e v e r sion of the unr e g ul a r i z e d b ip a r t it e O T O C . F or β = 0 , thi s i s the ope r a t or
e n t a n gle me n t of the t ime e v o lut ion ope r a t or U
t
as w i l l be d i s c us s e d shor tly . H o w e v e r , for β ̸= 0 , it doe s
not h a v e a cle a r qua n tum infor m a t ion-the or e t ic c or r e sponde nc e .
R ef . [ 305 ] s tud ie d G
β
( t) in ex t e n siv e de t ai l a t β = 0 . H e r e , w e w i l l c on tr as t the dy n a mical be h a v ior of
164
the b ip a r t it e unr e g ul a r i z e d O T O C w ith th a t of the r e g ul a r i z e d cas e , w hich w e a r e no w r e a dy t o in tr oduc e .
P e r for min g b ip a r t it e a v e r a g e s in a simi l a r w a y for the r e g ul a r i z e d cas e , w e h a v e ,
N
β
( t) := G
( d)
β
G
( r)
β
( t); (5.10)
w ith G
( d)
β
:=E
V A; W B
[
F
( d)
β
]
; (5.11)
a nd G
( r)
β
( t) :=E
V A; W B
[
F
( r)
β
( t)
]
: (5.12)
I n the nex t s e ct ion, w e w i l l d i s c us s infor m a t ion-the or e t ic aspe cts of the s e qua n t it ie s . W e al s o r efe r the
r e a de r t o R ef s . [ 294 , 313 – 317 ] for a d i s c us sion of v a r ious infor m a t ion s cr a mb l in g/ope r a t or gr o w th aspe cts
of the r e g ul a r i z e d v e r s us unr e g ul a r i z e d O T O C s .
O p er a t o r Sc h m i d t de co m p o sit io n.— W e t ak e a s m al l de t our t o r e mind the r e a de r a fe w k ey fa cts a bout
ope r a t or e n t a n gle me n t befor e de lv in g in t o out m ain r e s ults . G iv e n a pur e qua n tum s t a t e in a b ip a r t it e
H i l be r t sp a c e , j ψ⟩2H
=H
A
H
B
, the r e ex i s ts a S chmidt de c omposit ion o f thi s s t a t e [ 3 ],
j ψ⟩ =
r
∑
j= 1
√
λ
j
j j
A
⟩
j j
B
⟩: (5.13)
H e r e ,f λ
j
g
j
a r e nonne g a t iv e c oeffic ie n ts w ith r = min( d
A
; d
B
) the S chmidt r a nk a nd fj j
A
⟩g
d A
j= 1
;fj j
B
⟩g
d B
j= 1
b as e s for the s ubs ys t e m s A; B , r e spe ct iv e ly . The S chmidt c oeffic ie n ts ca n be us e d t o c omput e v a r ious e n-
t a n gle me n t me as ur e s for the b ip a r t it e s t a t e j ψ⟩ [ 318 ]. The k ey ide a be hind S chmidt de c omposit ion i s t o
us e the sin g ul a r v alue de c omposit ion for the m a tr i x of c oeffic ie n ts o bt aine d f r om ex pr e s sin g the s t a t e j ψ⟩
w i th r e spe ct t o local or thonor m al b as e s . I n fa ct , one ca n g e ne r al i z e thi s ide a t o the ope r a t or sp a c e . N a me ly ,
c on side r b ip a r t it e ope r a t or s, i . e ., e le me n ts of L(H
A
H
B
) , the n w e ca n define a n o p er a t o r Sc h m i d t de co m p o-
sit io n [ 309 , 319 , 320 ]. F or m al ly , g iv e n a b ip a r t it e ope r a t or X2L(H
A
H
B
) , the r e ex i s t or tho g on al b as e s
f U
j
g
d
2
A
j= 1
a ndf W
j
g
d
2
B
j= 1
forL(H
A
);L(H
B
) , r e spe ct iv e ly , s uch th a t
⟨
U
j
; U
k
⟩
= d
A
δ
jk
a nd
⟨
W
j
; W
k
⟩
= d
B
δ
jk
.
M or e o v e r ,
X =
~ r
∑
j= 1
√
λ
j
U
j
W
j
: (5.14)
165
The c oeffic ie n ts f λ
j
g
j
a r e nonne g a t iv e a nd a r e cal le d the ope r a t or S chmidt c oeffic ie n ts a nd ~ r = minf d
2
A
; d
2
B
g
i s the ope r a t or S chmidt r a nk . I n fa ct , the ope r a t or e n t a n gle me n t of a unit a r y in tr oduc e d in R ef . [ 309 ] i s
ex a ctly the l ine a r e n tr op y of the pr o b a b i l it y v e ct or ⃗ p = ( λ
1
; λ
2
; ; λ
~ r
) a r i sin g f r om the ope r a t or S chmidt
c oeffic ie n ts . A k ey r e s ult o bt aine d in R ef . [ 309 ] w as th a t the ope r a t or e n t a n gle me n t of a unit a r y ope r a t or
ca n be e quiv ale n tly ex pr e s s e d as,
E
op
( U) = 1
1
d
2
Tr
[
S
AA
′ U
2
S
AA
′ U
y
2
]
: (5.15)
I n a simi l a r sp ir it , w e define the o p er a t o r pu r it y of a l ine a r ope r a t or as the pur it y of the pr o b a b i l it y v e ct or ⃗ p
o bt aine d fo l lo w in g the ope r a t or S chmidt de c omposit ion . N a me ly ,
P
op
( X) :=
1
∥ X∥
4
2
Tr
[
S
AA
′ X
2
S
AA
′ X
y
2
]
; (5.16)
w he r e w e h a v e ex p l ic itly in tr oduc e d the nor m al i za t ion ∥ X∥
4
2
for a r b itr a r y ope r a t or s ( it i s e qual t o d
2
for
unit a r ie s w hich r e c o v e r s the pr e v ious for m ul a a bo v e ).
L as tly , w e r e mind the r e a de r th a t , for unit a r y dy n a mic s, infor m a t ion s cr a mb l in g i s us ual ly qua n t i fie d v i a
the O T O C s, the ope r a t or e n t a n gle me n t of the t ime- e v o lut ion ope r a t or U
t
, a nd the qua n tum m utual infor -
m a t ion [ 104 ]. Our w ork , in p a r t ic ul a r , foc us e s ex t e n siv e ly on the in t e r p l a y be t w e e n O T O C s a nd ope r a t or
e n t a n g le m e n t.
5.4 Main re s u l ts
1 O per a t o r ent a n gle ment
Our fir s t r e s ult i s t o br in g N
β
( t) in t o a n ex a ct a n aly t ical for m . W e in tr oduc e s ome not a t ion fir s t. L e t
P
χ
( ρ) :=
ρ
χ
2
2
be the s qua r e d 2 - nor m of the ope r a t or ρ
χ
w ith ρ
χ
:= Tr
χ
[ ρ] , χ = f A; Bg , a nd χ the
c omp le me n t of χ . I f ρ i s a qua n tum s t a t e the n P
χ
( ρ) i s the pur it y a cr os s the Aj B p a r t it ion .
166
P r opositio n 5.1
The r e g ul a r i z e d b ip a r t it e O T O C a t finit e t e mpe r a tur e i s
N
β
( t) =
1
d
P
A
(
√
ρ
β
)P
B
(
√
ρ
β
) (5.17)
1
dZ
β
Tr
[
S
AA
′U
2
β; t
(S
AA
′)
]
;
w he r e ,U
β; t
:=V
β
◦U
t
w ithV
β
( X) := exp[ β H= 4] X exp[ β H= 4] the im a g in a r y t ime- e v o lut ion,
U
t
( X) := U
y
t
XU
t
the r e al t ime- e v o lut ion, a nd U
t
= exp[ iHt] the us ual t ime- e v o lut ion ope r a t or .
L e t us not e a fe w simp le thin gs a bout thi s r e s ult : ( i ) a t infinit e t e mpe r a tur e ( β = 0 ), thi s r e duc e s t o the
ope r a t or e n t a n gle me n t of the t ime e v o lut ion ope r a t or [ 305 ] G
β= 0
( t) . The e qui l i br a t ion v alue of thi s qua n-
t i t y w as us e d t o d i s t in g ui sh v a r ious in t e gr a b le a nd ch a ot ic mode l s, s e e R ef s . [ 305 , 321 ] for mor e d e t ai l s . ( i i )
I n qua n tum infor m a t ion the or y [ 3 ], the mos t g e ne r al de s cr ipt ion of the dy n a mic s of a qua n tum s ys t e m i s
g iv e n b y a c omp le t e ly posit iv e ( C P ) a nd tr a c e- nonincr e asin g m a p , al s o cal le d a q u a n t u m o p er a t io n . F ur the r -
mor e , i f the e v o lut ion i s not only tr a c e non-incr e asin g , r a the r , tr a c e-pr e s e r v in g ( TP ), the n s uch dy n a mical
m a ps a r e cal le d q u a n t u m c h a n n e ls . I n the A ppe nd i x , w e sho w th a t U
β; t
i s a qua n tum ope r a t ion . M or e o v e r ,
the s e c ond t e r m, G
( r)
β
( t) i s r e al a nd pr opor t ion al t o the ope r a t or pur it y of U
β
( t) := exp[( β it) H= 4] ,
the a n aly t ical ly c on t in ue d t ime- e v o lut ion ope r a t or , w ith Z( β= 2)
2
=( dZ( β)) as the pr opor t ion al it y fa ct or .
( i i i ) The fo l lo w in g simp le u ppe r bound ho ld s for the BR O T O C: N
β
( t) = G
( d)
β
G
( r)
β
( t) G
( d)
β
Z( β= 2)
4
=( dZ( β)
2
) . ( iv ) F or a non- e n t a n gl in g H a mi lt oni a n, w e h a v e , N
β
= 0 8 β . N a me ly , i f H =
H
A
+ H
B
, the n a simp le calc ul a t ion r e v e al s th a t , G
( d)
β
=
Z( β= 2)
2
dZ( β)
= G
( r)
β
a nd the r efor e , N
β
( t) i s ide n t ical ly
v a ni shin g a t al l β . O f c our s e , the fa ct th a t a t β = 0 , N
β
= 0 al s o fo l lo w s f r om the c onne ct ion t o ope r a t or
e n t a n gle me n t [ 305 ].
W e e mp h asi z e th a t , althou gh s e v e r al pr e v ious w ork s h a v e foc us s e d on unde r s t a nd in g the gr o w th of local
O T O C s in t e r m s of L ie b - R o b in s on bound s [ 253 , 322 – 326 ], thi s a n alysi s do e s n o t a pp ly t o our b ip a r t it e
O T O C s ( r e g ul a r i z e d or unr e g ul a r i z e d ). The k ey d i s t inct ion he r e i s th a t , our a v e r a g in g i s o v e r o bs e r v a b le s
s u p por t e d on a b ip a r t it ion Aj B of the en t i r e s ys t e m ( a nd not s ome s ubs e t of the t ot al H i l be r t sp a c e ). A s a
167
r e s ult , e v e n i f one of the s ubs ys t e m s i s local its c omp le me n t i s ( hi ghly ) nonlocal . A s a r e s ult , L ie b - R o b in s on
t y pe bound s a r e not ne c e s s a r i ly us ef ul in unde r s t a nd in g the gr o w th of thi s qua n t it y .
2 BR O T O C , th ermo fi eld d o u b le , a nd th e spec tr a l f o rm f a c t o r
I n thi s s e ct ion, w e foc us on the qua n tum ope r a t ion U
β; t
, the ope r a t or pur it y of w hich i s qua n t i fie d b y the
c onne ct e d BR O T O C . W e w i l l sho w th a t the m a p U
β; t
c on t ain s infor m a t ion a bout both spe ctr al a nd ei g e n-
s t a t e si gn a tur e s of qua n tum ch a os [ 29 , 30 , 46 , 327 ]. I n p a r t ic ul a r , w e w i l l e s t a b l i sh its r e l a t ion t o the spe ctr al
for m fa ct or ( S FF ) [ 101 ] a nd the the r mofie ld doub le s t a t e ( TD S ) [ 328 ]. R e c e n tly , s e v e r al w ork s h a v e e lu-
c id a t e d the a b i l it y of the TD S t o pr o be s cr a mb l in g a nd qua n tum ch a os [ 302 , 329 – 331 ]. I n its simp le s t
for m, the TD S c or r e spond s t o a “ ca nonical ” pur i fica t ion of the G i b bs s t a t e ρ
β
= exp[ β H]=Z( β) . G iv e n
the c onne ct ion s t o s cr a mb l in g a nd ch a os, the a b i l it y t o ex pe r ime n t al ly pr e p a r e TD S al lo w s us t o d ir e ctly
pr o be the s e pr ope r t ie s; s e e for e . g., R ef s . [ 332 – 336 ] for a d i s c us sion a bout ho w t o pr e p a r e s uch s t a t e s on a
qua n tum c omput e r .
M or e for m al ly , le t j Γ⟩ :=
d
∑
j= 1
j j⟩j j⟩ be the u n n o r m a l i ze d m a x im al ly e n t a n gle d v e ct or in H
2
, the n, the
TD S i s define d as,
j ψ( β)⟩ :=
(
√
ρ
β
I
)
j Γ⟩: (5.18)
B y c on s tr uct ion, j ψ( β)⟩2H
2
a nd tr a c in g out eithe r s ubs ys t e m g iv e s us b a ck the or i g in al G i b bs s t a t e . F or
simp l ic it y , c on side r a nonde g e ne r a t e H a mi lt oni a n w ith a spe ctr al de c omposit ion H =
d
∑
j= 1
E
j
j j⟩⟨ jj , the n, b y
c on side r in g the j Γ⟩ m a tr i x ex pr e s s e d w ith r e spe ct t o the H a mi l t oni a n ei g e nb asi s, w e h a v e ,
j ψ( β)⟩ =
1
√
Z( β)
d
∑
j= 1
exp
[
β E
j
= 2
]
j j⟩j j⟩: (5.19)
W r itt e n in thi s for m, it i s e as y t o s e e th a t p a r t i al tr a c in g eithe r s ubs ys t e m of j ψ( β)⟩ g e ne r a t e s the G i b bs s t a t e
ρ
β
. I n R ef . [ 330 ], the s ur v iv al pr o b a b i l it y ( or L os chmidt E cho ) o f the t ime- e v o lv in g TD S w as r e l a t e d t o the
a n aly t ical ly c on t in ue d p a r t it ion f unct ion [ 296 , 302 , 329 ]. N a me ly , le t the t ime- e v o lv e d TD S be define d as
168
[ 328 , 330 ],
j ψ( β; t)⟩ := ( U
t
I)j ψ( β)⟩
=
1
√
Z( β)
d
∑
j= 1
exp
[
( β= 2+ it) E
j
]
j j⟩j j⟩; (5.20)
the n its s ur v iv al pr o b a b i l it y i s
j⟨ ψ( β; 0)j ψ( β; t)⟩j
2
=
jZ( β+ it)j
2
Z( β)
2
: (5.21)
Thi s i s cle a rly r e l a t e d t o the t w o -po in t , a n aly t ical ly c on t in ue d S FF , w hich i s define d as [ 99 , 302 ],
R
2
( β; t) :=⟨jZ( β+ it)j
2
⟩
RMT
; (5.22)
w he r e⟨⟩
RMT
de not e s a n e n s e mb le a v e r a g e , us ual ly o v e r a r a ndom m a tr i x e n s e mb le of H a mi lt oni a n s
[ 30 ].
W e w i l l no w sho w th a t a n a n alo g ous, thou gh, not ide n t ical , r e s ult ho ld s for the qua n tum ope r a t ion U
β; t
.
N a me ly , w e w i l l c on side r the fide l it y be t w e e n the C ho i - J a mio l k o w sk i ( C J ) m a tr i x [ 337 ] c or r e spond in g t o
U
β; t
a ndU
β; 0
a nd sho w th a t it i s r e l a t e d t o the t w o -po in t S FF . R e cal l th a t the C ho i - J a mio l k o w sk i i s omor -
p hi s m i s a n i s omor p hi s m be t w e e n l ine a r m a ps E : L(H) ! L(K) t o m a tr ic e s ρ
E
2 L(H)
L(K)
[ 337 ]. L e tj φ
+
⟩ :=
1
p
d
j j⟩j j⟩ be the n o r m a l i ze d m a x im al ly e n t a n gle d s t a t e in H
2
, the n,
ρ
E
:=E
I
(
j φ
+
⟩⟨ φ
+
j
)
: (5.23)
A l ine a r m a p E i s C P() ρ
E
0 . N o w , a simp le calc ul a t ion sho w s th a t the C J m a tr i x c or r e spond in g t o
the qua n tum ope r a t ion U
β; t
i s,
ρ
U
β; t
=
Z( β= 2)
d
j ψ( β= 2; t)⟩⟨ ψ( β= 2; t)j: (5.24)
169
T o qua n t i f y ho w clos e t w o pur e qua n tum s t a t e s a r e , w e ca n c omput e the fide l it y [ 3 ] be t w e e n the m .
R e cal l th a t the fide l it y be t w e e n t w o pur e qua n tum s t a t e s j ψ⟩;j φ⟩ i s g iv e n as,
F(j ψ⟩;j φ⟩) =j⟨ ψj φ⟩j
2
; (5.25)
w ith F(j ψ⟩;j φ⟩) = 1 () j ψ⟩ = j φ⟩ . S inc e the C ho i m a tr i x ρ
U
β; t
i s pr opor t ion al t o a pur e- s t a t e
pr oj e ct or , the fide l it y be t w e e n the m a tr ic e s ρ
U
β; t
a nd ρ
U
β; 0
ca n be define d as,
F( ρ
U
β; t
; ρ
U
β; 0
)
(
Z( β= 2)
d
)
2
F(j ψ( β= 2; t)⟩;j ψ( β= 2; 0)⟩)
=
(
Z( β= 2)
d
)
2
j⟨ ψ( β= 2; t)j ψ( β= 2; 0)⟩j
2
=
R
H
2
( β= 2; t)
d
2
; (5.26)
w he r eR
H
2
i s the t w o -po in t S FF b ef o r e e n s e mb le a v e r a g in g [ 99 ], a n alo g ous t o the r e s ult o bt aine d in R ef .
[ 330 ].
The a bo v e r e s ult c onne ct in g the qua n tum ope r a t ion U
β; t
t o the t w o -po in t S FF m ak e s one w onde r i f a
d ir e ct r e l a t ion be t w e e n the S FF a nd the r e g ul a r i z e d O T O C ca n be o bt aine d , sinc e the U
β; t
or i g in a t e s in the
cho ic e of the r e g ul a r i za t ion for the the r m al O T O C [ 296 ]. W e w i l l no w sho w th a t the r e g ul a r i z e d O T O C ,
a v e r a g e d o v e r g lo b a l r a ndom unit a r ie s i s r e l a t e d t o the f o u r-p o i n t S FF . N ot ic e th a t , unl i k e the b ip a r t it e a v -
e r a g in g th a t w e w i l l foc us on thr ou ghout thi s ch a pt e r , thi s r e l ie s on g lo b a l a v e r a g e s o v e r the unit a r y gr ou p .
The ne c e s sit y of pe r for min g glo b al a v e r a g e s t o c onne ct w ith S FF s ubtly hin ts a t the nonlocal it y ( in both
sp a c e a nd t ime ) of the S FF , s e e R ef s . [ 99 , 302 ] for a de t ai le d d i s c us sion . L e t
F
( A; B; C; D)
β
( t) := Tr[ yA
t
yByC
t
yD] (5.27)
w ith y = ρ
1= 4
β
, the n w e h a v e the fo l lo w in g r e s ult.
170
P r opositio n 5.2
The r e g ul a r i z e d four -po in t O T O C a v e r a g e d glo b al ly o v e r H a a r - r a ndom unit a r ie s i s r e l a t e d t o the four -
po in t spe ctr al for m fa ct or as, E
A 1; B 1; A 22U(H)
[
F
( A 1; B 1; A 2; B 2)
β
( t)
]
=R
( H)
4
( β= 4; t)=( d
3
Z( β)) , w he r e B
2
=
A
y
2
B
y
1
A
y
1
a ndR
( H)
4
( β; t) :=
(
Z
β
( t)Z
β
( t)
)
2
w ithZ
β
( t) = Tr[ exp[( β+ it) H]] , the a n aly t ical ly
c on t in ue d p a r t it ion f unct ion .
M or e o v e r , not ic e th a t thi s for m ul a ca n be e asi ly g e ne r al i z e d t o the cas e of d i ffe r e n t r e g ul a r i za t ion s of the
O T O C , for ex a mp le , i f w e h a v e 2 -po in t f unct ion s w ith
p
ρ
β
in s e r t e d be t w e e n the m, the n w e ca n g e t the
R
2
( β= 2; t) . I n the mos t g e ne r al cas e , i f w e h a v e , 2k -po in t the r m al ly r e g ul a t e d O T O C s (w hich w i l l h a v e
ρ
1= 2k
β
in s e r t e d be t w e e n the m ), the n, thi s w i l l c onne ct w ith R
2k
( β= 2k; t) .
P u r it y of t h e t h er m ofie l d do u b le .— W e a r e no w r e a dy t o foc us a g ain on the local pr ope r t ie s th a t a r e qua n-
t i fie d b y the BR O T O C . L e t us c on side r a b ip a r t it ion of the or i g in al H i l be r t sp a c e , H
=H
A
H
B
. The n
the C ho i m a tr i x c or r e spond in g t o the C P m a p U
β; t
i s a four -p a r t it e s t a t e , sinc e ρ
U
β; t
2 L(H
A
H
B
)
L(H
A
′
H
B
′) , w he r e the pr ime d H i l be r t sp a c e s r e pr e s e n t a r e p l ica of the or i g in al H i l be r t sp a c e . W e ca n
the n c omput e the 2 - nor m s qua r e d of the r e duc e d C ho i m a tr i x ρ
AA
′
U
β; t
Tr
BB
′
[
ρ
U
β; t
]
( or the pur it y i f the
m a tr i x w as nor m al i z e d ; it i s alr e a dy posit iv e s e midefinit e ). The n, a k ey le mm a f r om R ef . [ 309 ] sho w s th a t
ρ
AA
′
U
β; t
2
2
=
1
d
2
Tr
[
S
AA
′U
2
β; t
(S
AA
′)
]
. The r efor e , the ( c onne ct e d c ompone n t of the ) r e g ul a r i z e d O T O C ,
G
( r)
β
( t) =
d
Z( β)
ρ
AA
′
U
β; t
2
2
: (5.28)
Th a t i s, it i s pr opor t ion al t o the 2 - nor m s qua r e d of the r e duc e d C ho i m a tr i x for the qua n tum ope r a t ion U
β; t
.
N o w , le t P
AA
′(j ψ⟩
ABA
′
B
′) :=∥ Tr
BB
′ [j ψ⟩⟨ ψj
ABA
′
B
′]∥
2
2
be the pur it y of the the r mofie ld doub le a cr os s the
AA
′
j BB
′
p a r t it ion . The n, usin g the fa ct th a t the C ho i m a tr i x of U
β; t
i s pr opor t ion al t o the t ime- e v o lv e d
the r mofie ld doub le s t a t e j ψ( β= 2; t)⟩⟨ ψ( β= 2; t)j , w e h a v e ,
G
( r)
β
( t) =
Z( β= 2)
2
dZ( β)
P
AA
′ (j ψ( β= 2; t)⟩
ABA
′
B
′): (5.29)
171
F in al ly , not ic e th a t P a g e s cr a mb l i n g of a qua n tum s t a t e [ 26 , 104 , 338 ] i s define d as al l s ubs ys t e m s c on t ain-
in g le s s th a n h al f the de gr e e s of f r e e dom bein g ne a rly m a x im al ly mi xe d . S inc e the pur it y i s minim al for
m a x im al ly mi xe d s t a t e s, the clos e r the v alue of G
( r)
β
( t) t o the lo w e r bound
Z( β= 2)
2
dd
2
A
Z( β)
, the mor e infor m a t ion
s cr a mb l in g w e h a v e in the s ys t e m ’ s dy n a mic s . Th a t i s, the c onne ct e d c ompone n t of the BR O T O C qua n-
t i fie s the de gr e e of P a g e s cr a mb l in g in the t ime- e v o lv in g TD S . F ur the r mor e , the c onne ct ion t o the pur it y
of the the r mofie ld doub le imme d i a t e ly al lo w s us t o infe r the fo l lo w in g bound s (w hich al s o fo l lo w f r om the
c onne ct ion t o ope r a t or pur it y a bo v e ),
Z( β= 2)
2
dd
2
A
Z( β)
G
( r)
β
( t)
Z( β= 2)
2
dZ( β)
; (5.30)
w he r e w e h a v e us e d the fa ct th a t the pur it y of a qua n tum s t a t e in H
AA
′ i s bounde d b e t w e e n
1
d
2
A
a nd 1 .
N o n - H er m it i a n ev o l u t io n.— The c onne ct e d BR O T O C h as the for m
G
( r)
β
( t) =
1
dZ( β)
⟨
S
AA
′;U
2
β; t
(S
AA
′)
⟩
; (5.31)
w hich qua n t i fie s the a ut oc or r e l a t ion f unct ion be t w e e n the o bs e r v a b le S
AA
′ a nd its e v o lv e d v e r sion U
2
β; t
(S
AA
′) .
N o w , r e cal l th a t a non- H e r mit i a n H a mi lt oni a n i s us ual ly define d t o be of the for m, H = H
0
i Γ , w he r e
H
0
; Γ a r e H e r mit i a n ope r a t or s a nd w e h a v e s e p a r a t e d the H e r mit i a n a nd non- H e r mit i a n p a r ts ex p l ic itly .
A s s ume th a t w e a r e in the simp le s c e n a r io w he r e the H e r mit i a n a nd a n t i - H e r mit i a n p a r ts c omm ut e , n a me ly ,
[ H
0
; Γ] = 0 . The r efor e , the t ime- e v o lut ion of a n o bs e r v a b le X unde r s uch dy n a mic s i s g iv e n as X
t
=
e
Γ t
e
iH 0 t
Xe
iH 0 t
e
Γ t
. F or the c onne ct e d BR O T O C , i f w e ide n t i f y Γ = β H=( 4t) a t t > 0 , the n w e ca n think
ofU
β; t
as a simp le n o n - H er m it i a n e v o lut ion ( a nd the c omm ut a t ion as s umpt ion a bo v e i s tr iv i al ly s a t i sfie d ).
I n thi s cas e , the BR O T O C qua n t i fie s the s cr a mb l in g po w e r of non- H e r mit i a n dy n a mic s . Thi s ide n t i fica -
t ion ope n s u p the pos si b i l it y of ut i l i z in g t oo l s f r om the the or y of d i s sip a t iv e qua n tum ch a os [ 339 – 345 ] s uch
as c omp lex sp a c in g r a t ios[ 345 ], t o a n aly z e d ir e ctly the spe ctr al c or r e l a t ion s e nc ode d in the non- H e r mit i a n
dy n a mic sU
β; t
as a me a n s of d i s t in g ui shin g in t e gr a b le a nd ch a ot ic dy n a mic s . F ur the r mor e , the a b i l it y t o d i s -
t in g ui sh qua n tum ch a os f r om de c o he r e nc e i s a fas c in a t in g que s t ion w ith a lon g hi s t or y [ 29 , 346 ]. R e w r it -
in g the qua n tum ope r a t ion U
β; t
= V
β
◦U
t
as a c omposit ion of a qua n tum ope r a t ion V
β
(w hich si gni fie s
172
de c o he r e nc e ) a nd the unit a r y dy n a mic s U
t
m a y al lo w for d i s e n t a n gl in g the de c o he r e nc e effe cts f r om the
u n it a r y scr a m b l i n g .
3 Z er o - te mper a tu r e limit
A s w e d i s c us s e d a bo v e , the infinit e-t e mpe r a tur e l imit of N
β
( t) i s the ope r a t or e n t a n gle me n t of the unit a r y
U
t
a nd e n j o ys s e v e r al infor m a t ion-the or e t ic c onne ct ion s [ 305 ]. W h a t a bout the othe r l imit , n a me ly , β!
1 ? H e r e , w e sho w th a t in the z e r o -t e mpe r a tur e l imit , the r e g ul a r i z e d O T O C pr o be s the ope r a t or pur it y
of the gr ound s t a t e pr oj e ct or , de pe nd in g on the de g e ne r a c y of the gr ound s t a t e m a ni fo ld . L e t Π
0
be the
gr ound s t a t e pr oj e ct or , the n, r e cal l th a t , lim
β!1
ρ
β
! Π
0
= g
0
, w he r e g
0
i s the gr ound s t a t e de g e ne r a c y . Th a t
i s, a t z e r o t e mpe r a tur e , the G i b bs s t a t e i s pr opor t ion al t o the pr oj e ct or on t o the gr ound s t a t e m a ni fo ld .
M or e o v e r , sinc e Π
0
i s a pr oj e ct or , w e h a v e , Π
2
0
= Π
0
. The r efor e , the d i s c onne ct e d c or r e l a t or simp l i fie s t o ,
F
( d)
β!1
=
1
g
2
0
Tr
[
Π
0
V
y
Π
0
V
]
Tr
[
Π
0
W
y
Π
0
W
]
: (5.32)
S imi l a rly , for the r e g ul a r i z e d p a r t w e h a v e ,
F
( r)
β!1
( t) =
1
g
0
Tr
[
Π
0
W
y
t
Π
0
V
y
Π
0
W
t
Π
0
V
]
: (5.33)
N o w , le t H =
∑
j
E
j
Π
j
be the spe ctr al de c omposit ion of the H a mi lt oni a n, the n, the pr oj e ct or s f Π
j
g
j
a r e
or thonor m al ( but not ne c e s s a r i ly r a nk - 1 ). P lu gg in g in U
t
=
∑
j
exp
[
iE
j
t
]
Π
j
, w e g e t ,
F
( r)
β!1
( t) =
1
g
0
Tr
[
Π
0
W
y
Π
0
V
y
Π
0
W Π
0
V
]
: (5.34)
N o w , i f the gr ound s t a t e i s nonde g e ne r a t e , the n, w e h a v e , Π
0
=j ψ
gs
⟩⟨ ψ
gs
j , w he r ej ψ
gs
⟩ i s the gr ound s t a t e
w a v ef unct ion a nd g
0
= 1 . The n, a simp le calc ul a t ion sho w s th a t , for thi s cas e ,
F
( d)
β
=
⟨ ψ
gs
j Vj ψ
gs
⟩
2
⟨ ψ
gs
j Wj ψ
gs
⟩
2
= F
( r)
β
(5.35)
173
a nd the r efor e , their d i ffe r e nc e v a ni she s . I n fa ct , not ic e th a t , the four -po in t c or r e l a t or h as no w r e duc e d t o
a pr oduct of 1 -po in t c or r e l a t or s . I n s umm a r y , a t z e r o t e mpe r a tur e , for nonde g e ne r a t e H a mi lt oni a n s, the
c or r e l a t or F
( d)
β
F
( r)
β
( t) v a ni she s, a nd s o doe s the r e g ul a r i z e d b ip a r t it e O T O C N
β!1
( t) .
W e no w pe r for m the b ip a r t it e a v e r a g in g for the z e r o -t e mpe r a tur e cas e , w ithout the as s umpt ion of nonde-
g e ne r a c y . The fo l lo w in g r e s ult e s t a b l i she s th a t i f the gr ound s t a t e i s de g e ne r a t e , the n, both the d i s c onne ct e d
a nd c onne ct e d c ompone n ts of the r e g ul a r i z e d b ip a r t it e O T O C pr o be the e n t a n gle me n t in the gr ound s t a t e
pr oj e ct or . M or e o v e r , for the nonde g e ne r a t e cas e , both t e r m s a r e pr opor t ion al t o the s qua r e of the pur it y of
the gr ound s t a t e a nd ca n be ut i l i z e d t o de t e ct qua n tum p h as e tr a n sit ion s [ 347 ]. The a b i l it y of gr ound s t a t e
O T O C s t o de t e ct qua n tum p h as e tr a n sit ion s w as ex p lor e d in R ef . [ 348 ]. Es t a b l i shin g a pos si b le c onne c -
t ion t o finit e-t e mpe r a tur e p h as e tr a n sit ion s i s a n in t e r e s t in g que s t ion for f utur e in v e s t i g a t ion s .
P r opositio n 5.3
The d i s c onne ct e d a nd c onne ct e d c ompone n ts of the b ip a r t it e a v e r a g e d O T O C a t z e r o t e mpe r a tur e
a r e ,
G
( d)
β!1
=
1
dg
2
0
P
A
( Π
0
)P
B
( Π
0
); a nd
G
( r)
β!1
=
1
dg
0
Tr
[
S
AA
′ Π
2
0
S
AA
′ Π
2
0
]
: (5.36)
N ot e th a t both qua n t it ie s be c ome s t i m e-i n dep en den t a nd the c on v e r g e nc e t o the gr ound s t a t e i s e x p o n en -
t i a l in β , g iv e n the G i b bs w ei gh ts . F in al ly , w e not e th a t for a pur e qua n tum s t a t e Π = j ψ⟩⟨ ψj , w e h a v e ,
Tr[S
AA
′ Π
2
S
AA
′ Π
2
] =
ρ
A
4
2
, w he r e ρ
A
Tr
B
[j ψ⟩⟨ ψj] [ 309 ]. Th a t i s, the o p er a t o r pur it y t e r m r e duc e s
t o the s t a t e pur it y s qua r e d . The r efor e , the c onne ct e d ( a nd d i s c onne ct e d ) BR O T O C a t z e r o t e mpe r a tur e ,
for a nonde g e ne r a t e H a mi lt oni a n pr o be s its gr ound s t a t e pur it y .
174
4 L o n g - time limit a nd ei gen s t a te ent a n gle ment
The e qui l i br a t ion v alue of c or r e l a t ion f unct ion s h as lon g be e n s tud ie d as a pr o be t o the r m al i za t ion a nd
ch a os [ 46 , 291 ]. A lthou gh, for finit e- d ime n sion al qua n tum s ys t e m s, c or r e l a t ion f unct ion s t y p ical ly do not
c on v e r g e t o a l imit for t!1 . I n s t e a d , a ft e r a tr a n sie n t init i al pe r iod , they os c i l l a t e a r ound s ome e qui l i b -
r ium v alue [ 14 , 15 , 349 , 350 ], w hich ca n be ex tr a ct e d v i a l on g -t ime a v e r a g in g ( al s o kno w n as infinit e-t ime
a v e r a g in g ), define d as, A( t) := lim
T!1
1
T
T
∫
0
A( τ) d τ . I n R ef s . [ 305 , 307 , 351 – 353 ], the e qui l i br a t ion v alue of
the O T O C ( or the a v e r a g e d O T O C ) w as us e d t o d i s t in g ui sh in t e gr a b le v e r s us ch a ot ic qua n tum s ys t e m s .
H e r e , w e d i s c us s ho w the lon g -t ime a v e r a g e of the BR O T O C ca n al s o r e v e al the de gr e e of in t e gr a b i l it y for
H a mi lt oni a n qua n tum s ys t e m s, a nd d i s c us s the β - de pe nde nc e .
A k ey as s umpt ion on the e ne r g y spe ctr um th a t w e w i l l us e in thi s s e ct ion i s the s o - cal le d no - r e s on a nc e
c ond it ion ( NR C ) or nonde g e ne r a t e e ne r g y g a ps c ond it ion [ 14 , 16 ]. S imp ly put , both th e e ne r g y le v e l s a nd
the e ne r g y g a ps be t w e e n the s e le v e l s i s nonde g e ne r a t e . M or e for m al ly , c on side r the spe ctr al de c omposit ion
of the H a mi lt oni a n, H =
∑
d
j= 1
E
j
j φ
j
⟩⟨ φ
j
j . The n, H o beys NR C i f E
l
+ E
k
= E
n
+ E
m
() l = n; k =
m or l = m; k = n8 j; k; l; m . The NR C c ond it ion i s s a t i sfie d b y gen er ic qua n tum s ys t e m s a nd in p a r t ic ul a r ,
ch a ot ic qua n tum s ys t e m s s a t i sf y s uch a c ond it ion eithe r ex a ctly or t o a clos e a ppr o x im a t ion . L e t us de not e
b y ρ
χ
j
:= Tr
χ
[
j φ
j
⟩⟨ φ
j
j
]
; χ = f A; Bg the r e duc e d de n sit y m a tr i x c or r e spond in g t o the j -th H a mi lt oni a n
ei g e n s t a t e . M or e o v e r , w e in tr oduc e a Gr a m m a tr i x c or r e spond in g t o the inne r pr oduct be t w e e n the r e duc e d
s t a t e s, R
( χ)
jk
:=
⟨
ρ
j
; ρ
k
⟩
w ith χ =f A; Bg a nd⟨;⟩ the H i l be r t - S chmidt inne r pr oduct. The n, w e h a v e the
fo l lo w i n g r e s ult.
175
P r opositio n 5.4
G
( r)
β
( t) =
1
dZ( β)
2
4
d
∑
j; k= 1
exp
[
β( E
j
+ E
k
)= 2
]
(
R
A
jk
2
+
R
B
jk
2
δ
jk
R
A
jk
2
)]
: (5.37)
Thi s r e s ult g e ne r al i z e s t o finit e-t e mpe r a tur e the P r oposit ion 4 o bt aine d in [ 305 ] a nd the r efor e a t β = 0 ,
r e duc e s t o the for m de s cr i be d the r e . W e ca n r e s cale the r e duc e d s t a t e s as σ
χ
j
:= exp
[
β E
j
= 4
]
ρ
χ
j
, w hich
g e ne r a t e s a r e s cale d Gr a m m a tr i x
e
R
( χ)
jk
:=
⟨
σ
χ
j
; σ
χ
k
⟩
= exp
[
β
(
E
j
+ E
k
)
= 4
]⟨
ρ
χ
j
; ρ
χ
k
⟩
. The r efor e , w e ca n
r e w r it e the t ime-a v e r a g e as,
G
( r)
β
( t) =
1
dZ( β)
∑
χ2f A; Bg
(
e
R
( χ)
2
2
1
2
e
R
( χ)
D
2
2
)
; (5.38)
w ith [
e
R
( χ)
D
]
jk
= [
e
R
( χ)
D
]
jk
δ
jk
.
S imi l a rly , for the d i s c onne ct e d c or r e l a t or , w e h a v e ,
G
( d)
β
=
1
d
Tr
A
[
√
ρ
β
]
2
2
Tr
B
[
√
ρ
β
]
2
2
=
1
d
d
∑
j= 1
exp
[
β E
j
= 2
]
√
Z( β)
ρ
B
j
2
2
d
∑
k= 1
exp[ β E
k
= 2]
√
Z( β)
ρ
A
k
2
2
=
Z( β= 2)
4
dZ( β)
2
d
∑
j= 1
p
j
( β= 2) ρ
B
j
2
2
d
∑
k= 1
p
k
( β= 2) ρ
A
k
2
2
; (5.39)
w he r e p
j
( β) := exp
[
β E
j
]
=Z( β) i s the G i b bs pr o b a b i l it y as s oc i a t e d t o the e ne r g y le v e l j a t in v e r s e t e m-
pe r a t ur e β .
M ax i m a ll y- en t a n g le d m o de ls .— 5.4 al lo w s us t o c onne ct the e qui l i br a t ion v alue of the r e g ul a r i z e d O T O C
w ith the e n t a n gle me n t in the H a mi lt oni a n ei g e n s t a t e s . A s a c oncr e t e ex a mp le , w e e v alua t e thi s e qui l i br a -
t ion v alue for a s y mme tr ic b ip a r t it ion, th a t i s, d
A
= d
B
=
p
d a nd a H a mi lt oni a n w hos e ei g e n s t a t e s
176
a r e m a x im al ly e n t a n gle d , th a t i s, fj φ
k
⟩g
d
k= 1
a r e m a x im al ly e n t a n gle d a cr os s the Aj B p a r t it ion . W e t e r m
thi s H a mi lt oni a n a “ m a x im al ly e n t a n gle d H a mi lt oni a n ” for br e v it y . F or s uch a H a mi lt oni a n, w e h a v e ,
ρ
A
k
= I=
p
d = ρ
B
k
8 k a nd the r efor e , R
A
k; l
= Tr
[
ρ
A
k
ρ
A
l
]
=
1
d
Tr
[
I
p
d
]
=
1
p
d
= R
B
kl
8 k; l . The n, w e
h a v e ,
G
( r)
β
( t)j
ME
=
1
d
2
(
2Z( β= 2)
2
Z( β)
1
)
: (5.40)
N ot ic e th a t thi s e qui l i br a t ion v alue i s clos e t o the lo w e r bound ( for a s y mme tr ic b ip a r t it ion ): Z( β= 2)
2
= d
2
Z( β)
G
( r)
β
( t) .
S imi l a rly , for the d i s c onne ct e d c or r e l a t or , one ca n sho w th a t ,
G
( d)
β
j
ME
=
Z( β= 2)
4
d
2
Z( β)
2
: (5.41)
P utt in g e v e r y thin g t o g e the r , w e h a v e , the e qui l i br a t ion v alue of the BR O T O C for a m a x im al ly - e n t a n gle d
H a mi lt oni a n i s,
N
β
( t)j
ME
=
1
d
2
(
Z( β= 2)
2
Z( β)
1
)
2
: (5.42)
N ot ic e th a t a t β = 0 ,Z( β) = Tr[ I] = d = Z( β= 2) . The r efor e , the a bo v e e v alua t e s t o N
β= 0
( t)j
ME
=
(
1
1
d
)
2
= G
NRC
ME
for β = 0 as in R ef . [ 305 ], w hich sho w s th a t the e qui l i br a t ion v alue i s ne a rly m a x i -
m al ; w hich for a s y mme tr ic b ip a r t it ion i s e qual t o 1 1= d . F or qua n tum ch a ot ic s ys t e m s, r a ndom m a tr i x
the or y pr e d icts th a t the spe ctr al a nd ei g e n s t a t e pr ope r t ie s of the H a mi lt oni a n r e s e mb le thos e of the G a us -
si a n r a ndom m a tr i x e n s e mb le s ( de pe nd in g on the univ e r s al it y cl as s ) [ 30 , 327 ], w hich t y p ical ly h a v e ne a rly
m a x im al ly e n t a n gle d ei g e n s t a t e s . The r efor e , one ca n ex pe ct the e qui l i br a t ion v alue t o be clos e t o e q . ( 5.42 ).
W e outl ine he r e a qual it a t iv e a r g ume n t t o unde r s t a nd the de cr e as e of G
( r)
β
( t) w ith β as w i l l be c ome e v -
ide n t in the s e ct ion on N ume r ical S im ul a t ion s . I n fa ct , thi s monot onic it y i s a n en tr o pic effe ct due t o the
fa ct th a t , b y incr e asin g β , le s s a nd le s s s t a t e s c on tr i but e t o the s um in 5.4, the g e ne r al for m ul a for al l H a mi l -
t oni a n s th a t s a t i f y NR C . W e no w m ak e a qua n t it a t iv e a r g ume n t : le t p( β= 2) be a pr o b a b i l it y v e ct or w hos e
177
c ompone n ts a r e p
i
=
e
β E
i
= 2
Z( β= 2)
. The n, ∥ p( β= 2)∥
2
= Z( β) Z( β= 2)
2
a nd w e ca n r e ex pr e s s the BR O T O C as,
G
( r)
β
( t) =
⟨ p( β= 2);
^
C p( β= 2)⟩
d∥ p( β= 2)∥
2
; (5.43)
w he r e C
ij
:=j R
A
ij
j
2
+j R
B
ij
j
2
δ
ij
j R
A
ij
j
2
.
The de nomin a t or of e q . ( 5.43 ) i s pr opor t ion al t o the pur it y of p( β= 2) a nd it i s the r efor e monot onical ly
incr e asin g w ith β ( d
1
a t β = 0 , a nd 1 a t β = 1 .) On the othe r h a nd , the n ume r a t or of e q . ( 5.43 )
ca n ch a n g e f r om O( d
2
A
) a t β = 0 t o O( 1) for local mode l s ( O( d
2
A
) for non-local one s ). Thi s ch a n g e i s
alw a ys domin a t e d b y the pur it y incr e as e in the de nomin a t or . F or ex a mp le in the m a x im al ly e n t a n gle d cas e
( d
A
= d
B
=
p
d) ) one h as C
ij
= d
1
( 2 δ
ij
) a nd the r efor e one ca n r e w r it e e q . ( 5.40 ) as
G
( r)
β
( t)j
ME
=
2∥ p( β= 2)∥
2
d
2
∥ p( β= 2)∥
2
=
1
d
2
1+ S
lin
( β= 2)
1 S
lin
( β= 2)
; (5.44)
w he r e S
lin
( β) := 1∥ p( β)∥
2
i s the l i n e a r en tr o p y of ρ( β) . Thi s f unct ion i s cle a rly monot onical ly de cr e asin g
w ith β a nd sho w s, onc e a g ain, th a t the incr e as e of of the t ime-a v e r a g e d c onne ct e d O T O C w ith t e mpe r a tur e
i s a n en tr o pic effe ct.
N e a r l y m ax i m a ll y- en t a n g le d m o de ls .— W e w i l l no w sho w th a t , i f the H a mi lt oni a n ei g e n s t a t e s a r e hi ghly
e n t a n gle d the n it imp l ie s a bound on the e qui l i br a t ion v alue th a t i s clos e t o the m a x im al ly e n t a n gle d cas e .
R e cal l th a t a qua n tum s t a t e i s cal le d “P a g e s cr a mb le d ” [ 26 , 104 , 338 ] i f a n y a r b itr a r y s ubs ys t e m th a t c on-
si s ts of u p t o h al f of the s t a t e ’ s de gr e e s of f r e e dom a r e ne a rly m a x im al ly mi xe d . I n the fo l lo w in g pr oposit ion,
w e as s ume P a g e s cr a mb l in g of al l H a mi lt oni a n ei g e n s t a t e s a cr os s a s y mme tr ic b ip a r t it ion d
A
= d
B
=
p
d
a nd sho w th a t the e qui l i br a t ion v alue i s clos e t o th a t of hi ghly e n t a n gle d mode l s . L e t P(j ψ
AB
⟩) de not e the
pur it y of the r e duc e d s t a t e of j ψ
AB
⟩ a cr os s the b ip a r t it ion Aj B a ndP
min
= minf
1
d A
;
1
d B
g be the minim um
pur it y of a qua n tum s t a t e a cr os s the Aj B b ip a r t it ion . R e cal l th a t a pur e s t a t e j ψ
AB
⟩ i s m a x im al ly e n t a n-
gle d a cr os s Aj B () P(j ψ
AB
⟩) = P
min
. The n, the de v i a t ion s f r om the m a x im al ly e n t a n gle d v alue a r e
bounde d as fo l lo w s .
178
P r opositio n 5.5
F or a s y mme tr ic b ip a r t it ion of the H i l be r t sp a c e , d
A
= d
B
=
p
d i fP(j ψ
AB
⟩)P
min
ε ho ld s
for al l ei g e n s t a t e s, the n for s ys t e m s s a t i sf y in g NR C , the e qui l i br a t ion v alue i s bounde d a w a y f r om the
m a x im al ly e n t a n gle d cas e as fo l lo w s,
G
( r)
β
( t)j
ME
G
( r)
β
( t)j
NRC
Z( β= 2)
2
dZ( β)
(
6 ε
p
d
+ 3 ε
2
)
.
U n r e g u l a r i ze d v s r e g u l a r i ze d O T O C .— W e hi ghl i gh t a k ey d i ffe r e nc e be t w e e n the b ip a r t it e r e g ul a r i z e d
v e r s us unr e g ul a r i z e d O T O C s . A s w e w i l l not e , fo r ne a rly m a x im al ly e n t a n gle d mode l s, the G
β
( t) i s ne a rly
β -inde pe nde n t , w hi le the G
( r)
β
( t) sho w s a cle a r β - de pe nde nc e as w e h a v e s e e n a bo v e . The pr oof r e l ie s on
usin g a n ope r a t or S chmidt de c omposit ion for the unit a r y U
t
, s e e the A ppe nd i x for mor e de t ai l s . W e o bt ain
th a t ,
G
β
( U) = 1 F
β
( U) = 1
1
d
2
A
; (5.45)
a nd i s inde pe nde n t of β . C on tr as t thi s, w ith the e qui l i br a t ion v alue for ne a rly m a x im al ly e n t a n gle d H a mi l -
t oni a n s e q . ( 5.42 ), as c omput e d a bo v e . L e t us c on side r H a mi lt oni a n s f r om the G a us si a n U nit a r y E n s e mb le
( G UE ) as a n ex a mp le . The ei g e n s t a t e s of the s e a r e kno w n t o h a v e ne a r m a x im al e n t a n gle me n t a nd the r e-
for e , w e ca n a ppr o x im a t e the N
β
( t)
1
d
2
(
Z( β= 2)
2
Z( β)
1
)
2
. M or e o v e r , for l a r g e- d , the p a r t it ion f unct ion
a ft e r e n s e mb le a v e r a g in g ca n be ex pr e s s e d as [ 315 , 327 ],⟨Z( β)⟩
GUE
=
dI 1( 2 β)
β
, w he r e I
1
( β) i s the mod i fie d
Be s s e l f unct ion of the fir s t k ind . The r efor e , the e n s e mb le a v e r a g e d e qui l i br a t ion v alue of N
β
( t) for the G UE
i s ¹
⟨
N
β
( t)
⟩
GUE
[
4I
1
( β)
2
β I
1
( 2 β)
1
d
]
2
: (5.46)
N ot ic e th a t , I
1
( 2 β)= β =
1
∑
n= 0
β
2n
( n!)
2
( n+ 1)
. The r efor e , w e ca n ex tr a ct f r om thi s, both the lo w - a nd hi gh-t e mpe r a tur e
¹ W e h a v e imp l ic itly as s ume d he r e th a t the e n s e mb le a v e r a g in g a nd the l a r g e- d l imits c omm ut e , s e e [ 302 , 354 ] for a d i s c us sion .
179
e s t im a t e s . I n fi g. 5.4.1 w e p lot the Be s s e l f unct ion for m alon g w ith the n ume r ical e s t im a t e of the lon g -t ime
a v e r a g e of the c onne ct e d BR O T O C for the G UE, o bt aine d b y a v e r a g in g ( n ume r ical ly g e ne r a t e d ) G UE
H a mi lt oni a n s for d = 100 .
Figure 5.4.1: A log-log plot of the equilib ration value (long-time average) of the G
( r)
β
( t) fo r the GUE
Hamiltonian at d = 100 fo r 10
10
β 10
3
compa ring the numerical estimate to the Bessel function
fo rm ab ove.
N R C -pr o d u c t s t a t e s ( N R C - PS).— W e in tr oduc e a H a mi lt oni a n mode l th a t h as a g e ne r ic spe ctr um, n a me ly ,
one th a t s a t i sfie s NR C but w ith al l ei g e n s t a t e s as pr oduct s t a t e s ( for ex a mp le , the c omput a t ion al b asi s
s t a t e s ), th a t w e c al l “NR C - PS” . The H a mi lt oni a n ca n be ex pr e s s e d as,
H
NRC - PS
:=
d A; d B
∑
j; k= 1
E
j; k
j φ
( A)
j
⟩⟨ φ
( A)
j
j
j φ
( B)
k
⟩⟨ φ
( B)
k
j; (5.47)
w he r e the spe ctr um f E
j; k
g
j; k
s a t i sfie s NR C ² ; for ex a mp le , c on side r the spe ctr um of a H a mi lt oni a n f r om
a G a us si a n U nit a r y E n s e mb le ( G UE ). The r e as on t o in tr oduc e s uch a mode l i s t w ofo ld : fir s t , it al lo w s us
t o d isen t a n g le the spe ctr al a nd ei g e n s t a t e c on tr i but ion s t o the e qui l i br a t ion v alue G
( r)
β
( t) sinc e , it h as the
spe ctr um of a “ ch a ot ic ” mode l a nd the ei g e n s t a t e pr ope r t ie s of a “ f r e e ” mode l . S e c ond , as w e sho w no w ,
thi s mode l i s a n aly t ical ly tr a ct a b le . The k ey r e as on for thi s i s th a t the NR C - PS mode l h as a n ex t e n siv e
n umbe r of c on s e r v e d qua n t it ie s, d = 2
L
of the m in fa ct. A local ope r a t or of the for m, A
j
=j φ
A
j
⟩⟨ φ
A
j
j
I
² N ot e th a t a n y H a mi lt oni a n th a t s a t i sfie s NR C ca nnot be nonin t e r a ct in g , i . e ., ca nnot be of the for m H = H
A
I
B
+I
A
H
B
sinc e s uch a H a mi lt oni a n w ould , b y c on s tr uct ion, v io l a t e NR C . A s a n ex a mp le , c on side r pr oduct ei g e n s t a t e s of the for m,
fj φ
( A)
j
⟩
j χ
( B)
k
⟩g
j; k
the n it i s e as y t o find p air s of ei g e n s t a t e s for w hich the e ne r g y g a ps a r e e qual [ 16 ]. H o w e v e r , the c on v e r s e i s
not tr ue , n a me ly , the r e ex i s t in t e r a ct in g H a mi lt oni a n s, i . e ., of the for m H̸= H
A
I
B
+I
A
H
B
th a t h a v e pr oduct ei g e n s t a t e s .
180
c omm ut e s w ith the H a mi lt oni a n a bo v e ,
[
H; A
j
]
= 08 j2f 1; 2; ; d
A
g . S imi l a rly , ope r a t or s of the for m,
B
k
=I
j φ
B
k
⟩⟨ φ
B
k
j al s o c omm ut e w ith the H a mi lt oni a n, [ H; B
k
] = 08 k2f 1; 2; ; d
B
g . The r efor e , the
H a mi l t oni a n h as d = d
A
d
B
n umbe r of local c on s e r v e d qua n t it ie s . I n thi s s e n s e , thi s i s a n in t e gr a b le mode l ,
not ic e ho w e v e r , th a t its spe ctr um i s in t e n t ion al ly chos e n t o s a t i sf y NR C .
The pr e s e nc e of c on s e r v e d qua n t it ie s e n a b le a n ex a ct calc ul a t ion for the e qui l i br a t ion v alue of the BR O -
T O C in thi s mode l . A de t ai le d pr oof of thi s ca n be found in the A ppe nd i x .
G
( r)
β
( t)j
NRC PS
=
1
d
(
∥ p
A
( β= 2)∥
2
+∥ p
B
( β= 2)∥
2
∥ p( β= 2)∥
2
1
)
(5.48)
w he r e the pr o b a b i l it y v e ct or p( β) i s as in the a bo v e a nd p
A= B
( β) a r e its m a r g in al s e . g., p
A
j
( β) =
∑
d B
k= 1
p
jk
( β) =
1
Z( β)
∑
d B
k= 1
e
β E
jk
:
F r om thi s E qua t ion one find s imme d i a t e ly G
( r)
β= 0
( t)j
NRC PS
= 1= d
A
+ 1= d
B
1= d > 1= d a t infinit e
t e mpe r a tur e a nd , G
( r)
β=1
( t)j
NRC PS
= 1= d a t z e r o t e mpe r a tur e . F or a s y mme tr ic b ip a r t it ion d
A
= d
B
=
p
d , the for me r simp l i fie s t o 2=
p
d 1= d = O( 1=
p
d) . A nd , as w e w i l l s e e in the nex t s e ct ion, the n ume r ical
d a t a o bt aine d f r om finit e- si z e s cal in g in t a b le s 5.5.1 a nd 5.6.1 i s c on si s t e n t w ith thi s as G
( r)
β
( t)
2
p
d
.
5.5 N umeric al sim u l a tion s
I n thi s s e ct ion w e s tudy n ume r ical ly v a r ious dy n a mical fe a tur e s of the BR O T O C . I n p a r t ic ul a r , w e v a r y
the de g r e e of in t e gr a b i l it y of H a mi lt oni a n s ys t e m s a nd qua n t i f y it ’ s effe ct on the e qui l i br a t ion v alue G
( r)
β
( t) .
A t β = 0 , thi s i s e qual t o one min us the ope r a t or e n t a n gle me n t of the dy n a mical unit a r y , w hos e e qui l i -
br a t ion v alue w as us e d t o d i s t in g ui sh v a r ious in t e gr a b le a nd ch a ot ic mode l s in R ef . [ 305 ], s e e al s o R ef s .
[ 321 , 351 , 352 , 355 ] for d i s t in g ui shin g in t e gr a b le a nd ch a ot ic mode l s v i a t ime-a v e r a g e s of the O T O C or
ope r a t or e n t a n gle me n t. W e al s o r efe r the r e a de r t o R ef . [ 356 ], w he r e bound s on de ca y of O T O C s in t ime
w e r e o bt aine d usin g the s cal in g of the t ime-a v e r a g e d O T O C . H e r e , w e pe r for m mor e ex t e n siv e n ume r ical
s tud ie s, c on side r mor e g e ne r al ly the β - de pe nde nc e of thi s qua n t it y , a nd foc us on the fo l lo w in g H a mi lt o -
ni a n mode l s of in t e r e s t :
181
1. I n t e gr a b le mode l : The tr a n s v e r s e- fie ld I sin g mode l ( TFI M ) w ith the H a mi lt oni a n, H
TFIM
=
L 1
∑
j= 1
σ
z
j
σ
z
j+ 1
g
L
∑
j= 1
σ
x
j
h
L
∑
j= 1
σ
z
j
; as a p a r a d i gm a t ic qua n tum sp in- ch ain mode l . H e r e , the σ
α
j
; α2f x; y; zg a r e the
P a ul i m a tr ic e s . F or the TFI M , g; h de not e s the s tr e n g th of the tr a n s v e r s e fie ld a nd the local fie ld , r e-
spe ct iv e ly . The TFI M H a mi lt oni a n i s in t e gr a b le for eithe r h = 0 or g = 0 a nd nonin t e gr a b le w he n
both g; h a r e nonz e r o . W e c on side r as the in t e gr a b le po in t , g = 1; h = 0 a nd the nonin t e gr a b le
po in t g = 1: 05; h = 0: 5 . A t the in t e gr a b le po in t , thi s mode l ca n be m a ppe d on t o f r e e fe r mion s
v i a the J or d a n- W i gne r tr a n sfor m a t ion a nd i s “ hi ghly in t e gr a b le ” in thi s s e n s e . A t the nonin t e gr a b le
po in t , the mode l i s qua n tum ch a ot ic, in the s e n s e of r a ndom m a tr i x spe ctr al s t a t i s t ic s [ 357 , 358 ] a nd
v o lume-l a w e n t a n gle me n t o f ei g e n s t a t e s [ 359 ].
2. L ocal i z e d mode l s: W e s tudy A nde r s on a nd m a n y - body local i za t ion ( MBL ) w ith the H a mi lt oni a n,
H
MBL
=
L 1
∑
j= 1
σ
z
j
σ
z
j+ 1
L
∑
j= 1
g
j
σ
x
j
h
L
∑
j= 1
σ
z
j
; w he r e w e dr a w f r om the uni for m d i s tr i but ion, e a ch
g
j
2 [ η; η] . I n the a bs e nc e of the lon g itud in al fie ld , i . e ., h = 0 , a nd for nonz e r o d i s or de r , thi s
( d i s or de r e d ) f r e e fe r mion mode l i s A nde r s on local i z e d . I n the pr e s e nc e of the lon g itud in al fie ld , the
fe r mio n s a r e in t e r a ct in g a nd a t s u ffic ie n tly s tr on g d i s ode r , the mode l i s m a n y - body local i z e d ( MBL ).
A s i s w e l l -kno w n, MBL e s ca pe s the r m al i za t ion b y e me r g e n t in t e gr a b i l it y [ 46 , 360 ]. W e r efe r the
r e a de r t o R ef . [ 361 ] for a d i s c us sion of the lon g -t ime v alue s of the u n r e g u l a r i ze d O T O C in local i z e d
p h as e s . I n our n ume r ical sim ul a t ion s, w e foc us on η = 10 for the d i s or de r s tr e n g th a nd h = 0: 1
for the MBL cas e . W e a v e r a g e e a ch in s t a nc e of the d i s or de r e d mode l o v e r ⌊ 200= L⌋ inde pe nde n t
r e al i za t i on s . I n e a ch cas e , the e r r or b a r s a r e t oo s m al l t o p lot alon gside the d a t a po in ts .
3. NR C -pr o duct s t a t e s ( NR C - PS ): A s in tr oduc e d befor e thi s mode l al lo w s us t o s e p a r a t e the s p e c tr a l
a nd ei gen s t a t e c on tr i b ut ion s t o the BR O T O C’ s e qui l i br a t ion v alue . W e choos e a “ ch a ot ic ” spe ctr um
( in the s e n s e th a t it c or r e spond s t o a G UE H a mi lt oni a n a nd he nc e i s a n in s t a nc e of a mode l th a t
o beys W i gne r - D ys on s t a t i s t ic s ), w hi le h a v in g the ei g e n s t a t e s of a n o n i n t er a c t i n g mode l , th a t i s, sim-
p le pr oduct s t a t e s . T o s tudy the NR C - PS mode l n ume r ical ly , w e g e ne r a t e a r a ndom m a tr i x f r om
182
the G a us si a n U nit a r y E n s e mb le ( G UE ) a nd us e its spe ctr um, w hi le k e e p in g pr oduct ei g e n s t a t e s .
W e a v e r a g e thi s n ume r ical ly o v e r ⌊ 200= L⌋ inde pe nde n t r e al i za t ion s . Thi s y ie ld s n ume r ical r e s ults
c on si s t e n t w ith th e a n aly t ical ex pr e s sion o bt aine d f r om e q . ( 5.48 ).
4. R a ndom m a tr ic e s: A s a be nchm a rk for a “ m a x im al ly ch a ot ic ” mode l , w e c on side r H a mi lt oni a n s
dr a w n f r om the G UE . F or H a mi lt oni a n s ys t e m s, the s e min al w ork s of Be r r y a nd T a bor [ 362 ] a nd
th a t of Bo hi g as, G i a nnoni , a nd S chmit [ 28 ] e s t a b l i she s th a t P o i s s on le v e l - s t a t i s t ic s i s a ch a r a ct e r i s t ic
fe a tur e of in t e gr a b le , w hi le for the r m al i z in g s ys t e m s, W i gne r - D ys on s t a t i s t ic s a r e the nor m [ 13 , 360 ].
F ur the r mo r e , the ei g e n s t a t e s of G UE a r e ne a rly m a x im al ly e n t a n gle d a nd w i l l pr o v ide a n a n aly t ical ly
tr a ct a b le ex a mp le of a hi ghly ch a ot ic mode l . F or our n ume r ical sim ul a t ion s, w e g e ne r a t e r a ndom
m a tr ic e s a nd a v e r a g e o v e r ⌊ 200= L⌋ inde pe nde n t r e al i za t ion s . The e r r or b a r s a r e t oo s m al l t o p lot
alon gside the d a t a po in ts in thi s cas e as w e l l . The “ME ” in th e p lots c or r e spond s t o a m a x im al ly e n-
t a n gle d mode l for w hich w e us e the a n aly t ical ex pr e s sion s f r om e q . ( 5.40 ). F or thi s w e g e ne r a t e the
spe ctr um f r om the G UE a nd a v e r a g e o v e r ⌊ 200= L⌋ inde pe nde n t r e al i za t ion s .
Thr ou ghout thi s s e ct ion, t o e v alua t e the t ime-a v e r a g e s G
( r)
β
( t) n ume r ical ly , w e us e t w o d i ffe r e n t me th-
od s . F ir s t , for the A nde r s on a nd TFI M in t e gr a b le mode l , sinc e it doe s not s a t i sf y NR C ( the H a mi lt oni a n
h as s y mme tr ie s ), w e pe r for m ex a ct t ime e v o lut ion . W e do thi s for a t ime in t e r v al of t2 [ 10; 10
3
] w ith a 10
6
t ime s t e ps in be t w e e n . Thi s i s fi xe d for al l the s ys t e m si z e s L a nd in v e r s e t e mpe r a tur e s β . A l l mode l s exc e pt
the s e t w o s a t i sf y NR C ( al s o v e r i fie d n ume r ical ly ) a nd s o w e c omput e G
( r)
β
( t) usin g the a n aly t ical ex pr e s -
sion in 5.4 . T o do thi s, w e pe r for m ex a ct d i a g on al i za t ion of the f ul l H a mi lt oni a n for thi s a nd c omput e the
r e duc e d s t a t e s . A t l a r g e β , it i s e as y t o sho w th a t one only ne e d s the gr ound s t a t e alon g w ith a fe w exc it e d
s t a t e s t o e s t im a t e the t ime-a v e r a g e in 5.4 . The r efor e , for L = 13; 14 a nd β = 1 , w e only ex tr a ct the lo w e s t
20 H a mi lt oni a n ei g e n s t a t e s .
The fir s t n ume r ical r e s ult foc us s e s on L = 6 qub its a nd w e s tudy the v a r i a t ion of G
( r)
β
( t) as a f unct ion of
β . I n fi g. 5.5.1 , w e not ic e the fo l lo w in g u n i v er sa l f e a t u r e s of G
( r)
β
( t) as a f unct ion of β : the e qui l i br a t ion v alue
i s v e r y slo w ly de ca y in g as β v a r ie s f r om z e r o t o O( 1) . A r ound β = O( 1) , the e qui l i br a t ion v alue quick ly
de ca ys t o the as y mpt ot ic v alue . U sin g , 5.3 , w e not e th a t the as y mpt ot ic v alue β ! 1 i s pr opor t ion al t o
183
the ope r a t or pur it y for the gr ound s t a t e pr oj e ct or . W ith the s e univ e r s al fe a tur e s a t h a nd , w e s ys t e m a t ical ly
s tudy the e qui l i br a t ion v alue G
( r)
β
( t) for thr e e r e pr e s e n t a t iv e cho ic e s of β : β = 0; β = 1 a nd β!1 . W e
n ume r ical ly s tudy their s cal in g as a f unct ion of the s ys t e m si z e for a s y mme tr ic b ip a r t it ion of the l a tt ic e ,
⌊ L= 2⌋ :⌈ L= 2⌉ . F or the cas e w he r e L i s not e v e n, the n ume r ical r e s ults a r e v e r y simi l a r for ei the r cho ic e of
b ip a r t it ion, ⌊ L= 2⌋ : ⌈ L= 2⌉ or⌈ L= 2⌉ : ⌊ L= 2⌋ , a nd the r efor e , w e choos e the for me r thr ou ghout. W e al s o
l a be l as “ lo g p lot ” a p lot w ith lo g a r ithmic s cale on the y -a x i s a nd “ lo glo g p lot ” thos e w ith lo g a r ithmic s cale
on both x - a nd y -a xe s .
The r e s ults for β = 0 a r e d i s c us s e d in fi g. 5.5.2 . W e not ic e th a t the s cal in g w . r . t. the s ys t e m si z e i s ef -
fe ct iv e ly d iv ide d in t o t w o cl as s e s: the qua n tum ch a ot ic mode l s, n a me ly , the nonin t e gr a b le TFI M a nd the
G UE . A nd , the s e c ond cl as s i s al l the othe r s, n a me ly , the f r e e fe r mion s, the A nde r s on a nd MBL , a nd the
NR C - PS . The s e t w o cl as s e s a r e pr im a r i ly d i s t in g ui she d b y their ei g e n s t a t e e n t a n gle me n t , n a me ly , the s cal -
in g of the e n t a n gle me n t a cr os s the e n t ir e spe ctr um . Thi s, pe rh a ps, c ome s as no s ur pr i s e sinc e the infinit e-
t e mpe r a tur e O T O C b y c on s tr uct ion pr o be s the e n t a n gle me n t a cr os s al l ei g e n s t a t e s .
A t β = 1 , f r om fi g. 5.5.3 , w e not ic e th a t the MBL a nd qua n tum ch a ot ic mode l s h a v e me r g e d , w hi le
h a v in g a d i s t inct s cal in g f r om the othe r in t e gr a b le mode l s a nd the G UE . R e cal l th a t a t β = 0 , The or e m 6 of
R ef . [ 305 ] e s t a b l i she s a hie r a r ch y be t w e e n the e qui l i br a t ion v alue s of v a r ious e s t im a t e s for the e qul i br a t ion
v alue . H o w e v e r , for the r e g ul a r i z e d O T O C , thi s r e s ult doe s not ne c e s s a r i ly ho ld a w a y f r om the β = 0 cas e
sinc e no w w e h a v e ex tr a H - de pe nde n t t e r m s in the NR C e s t im a t e; s e e 5.3.
A nd fin al ly , the s cal in g for β =1 ca n be unde r s t ood usin g 5.3 . F or the nonde g e ne r a t e H a mi lt oni a n s,
thi s simp ly pr o be s the gr ound s t a t e e n t a n g le m e n t. W e not ic e th a t al l c ur v e s c o ale s c e in t o t w o gr ou ps, one
for the in t e gr a b le/local i z e d mode l s a nd the s e c ond for the G UE a nd ME mode l s, r e spe ct iv e ly . W hi le the s e
mode l s v a r y in their de gr e e of in t e gr a b i l it y , their gr ound s t a t e s ( a p a r t f r om the G UE/ME ) al l fo l lo w a n a r e a
l a w [ 359 ] a nd he nc e o bey a d i ffe r e n t de ca y r a t e w ith L f r om the G UE . N ot e th a t the G UE gr ound s t a t e i s
a H a a r r a ndom s t a t e a nd the r efor e , should s cale as G
( r)
β!1
( t)
1
d
2
, w hich i s c on si s t e n t w ith the finit e- si z e
s cal in g r e s ults .
F i n it e-si ze sc a l i n g .— T o qua n t it a t iv e ly unde r s t a nd the n ume r ical r e s ults, w e pe r for m finit e- si z e s cal in g
a n alysi s for e a ch cho ic e of β . L e t us s t a r t w ith the infinit e-t e mpe r a tur e cas e ( β = 0 ). W e c on side r a n
184
M ode l β = 0 β = 1 β =1
TFI M in t e gr a b le 0.507672 0.687979 1.01858
NR C - PS 0.495827 0.7218 1.00
A nde r s on 0.557617 0.655576 1.00
MBL 0.491745 0.883465 1.00075
TFI M ch a ot ic 1.00781 0.884371 0.999999
G UE 0.999992 1.76251 2.00016
T able 5.5.1: The deca y rate γ fo r va rious Ha miltonian mo dels at β = 0; 1;1 , with resp ect to the
Ansatz G
( r)
β
( t) = α d
γ
. The p refacto r α is nonuniversal , the details of which can b e found in the
App endix.
A n s a tz of the for m,
G
( r)
β
( t) = α d
γ
+ G
( r)
β
( t)
1
; (5.49)
w he r e d i s the H i l be r t sp a c e d ime n sion a nd G
( r)
β
( t)
1
i s the as y mpt ot ic v alue , i . e ., as d!1 . F r om s e v e r al
a n aly t ical a nd n ume r ical r e s ults, w e kno w th a t G
( r)
β
( t)
1
= 0 . Th a t i s, the BR O T O C de ca ys for al l mode l s,
f r e e , in t e gr a b le , or ch a ot ic . The r efor e , w e r e duc e the A n s a tz t o
G
( r)
β
( t) = α d
γ
: (5.50)
A s a r e s ult , w e h a v e , log
2
( G
( r)
β
( t)) = log
2
( α) + ( γ) L w he r e 2
L
= d . The n ume r ical r e s ults n a tur al ly
m a ni fe s t thi s A n s a tz as i s e v ide n t f r om the ne a rly l ine a r fi g ur e s . The r efor e , pe r for min g a l ine a r fit t o the
log
2
( G
( r)
β
( t)) v e r s us L p lots y ie ld s the de ca y r a t e s c or r e spond in g t o v a r ious mode l s . W e foc us on the l as t 5
d a t a po in ts t o o bt ain the fit p a r a me t e r s, s e e the A ppe nd i x for mor e de t ai l s .
The finit e- si z e s cal in g r e s ults a r e s umm a r i z e d in t a b le 5.5.1 . The de ca y r a t e s a r e u n i v er sa l a t β = 0 w ith
γ 0: 5 for the in t e gr a b le mode l s a nd γ 1 for the ch a ot ic mode l s . A r ound β = O( 1) , thi s univ e r s al it y
be g i n s t o br e ak do w n a nd a t l a r g e β , the e qui l i br a t ion v alue G
( r)
β
( t) only d i ffe r e n t i a t e s local mode l s f r om the
nonlocal G UE mode l .
F r om the a n aly t ical r e s ults a bout NR C - PS a nd G UE, w e kno w th a t a t β = 0 , the γ
NRC - PS
=
1
2
γ
GUE
.
A nd , f r om the finit e- si z e s cal in g r e s ults, w e o bt ain, γ
NRC - PS
= γ
GUE
0: 495831 . A t β ! 1 , usin g 5.3,
185
for both NR C - PS a nd G UE, the e qui l i br a t ion v alue i s de t e r mine d b y the gr ound s t a t e pur it y . The r efor e ,
for NR C - PS , it s cale s as
1
d
a nd for G UE it s cale s as
1
d
2
sinc e G UE gr ound s t a t e s a r e H a a r - r a ndom s t a t e s,
their pur it y i s ne a r minim um, w ith O(
1
d
) c or r e ct ion s, w hich i s al s o c on si s t e n t w ith the finit e- si z e s cal in g
r e s ults . The r efor e , the r a t io of the r a t e s in thi s cas e i s al s o
1
2
. A nd fin al ly , f r om the n ume r ical v alue s l i s t e d in
t a b le 5.5.1 , w e s e e th a t the r a t io i s
1
2
a t β = 1 as w e l l .
10
-11
10
-9
10
-7
10
-5
0.001 0.100 10
5.×10
-4
0.001
0.005
0.010
0.050
0.100
Figure 5.5.1: A log-log plot of the equilib ration value (long-time average) of the G
( r)
β
( t) fo r va rious
Hamiltonian mo dels at L = 6 as a function of the inverse temp erature β across a symmetric bipa rti-
tion L= 2 : L= 2 . W e use exact time evolution fo r the integrable TFIM and Anderson since they do not
satisfy NRC. F o r Anderson, MBL, and GUE, w e p erfo rm exact diagonalization of the full Hamiltonian
and use the analytical exp ression in 5.4 . F o r NRC-PS and ME (maxim ally entangled mo del), w e use
the analytical exp ressions in eq. ( 5.40 ) and eq. ( 5.48 ).
5.6 Conc l u sion s
I n thi s w ork w e in tr oduc e the b ip a r t it e r e g ul a r i z e d O T O C th a t al lo w s us t o o bt ain a w e alth of a n aly t ical a nd
n ume r ical r e s ults t o aid our unde r s t a nd in g of r e g ul a r i z e d O T O C s for local qua n tum s ys t e m s . The infinit e-
t e mpe r a tur e O T O C h as s e v e r al ope r a t ion al in t e r pr e t a t ion s in t e r m s of ope r a t or e n t a n gle me n t , e n tr op y
pr oduct ion, a nd othe r s . P r oposit ion 1 e s t a b l i she d the c onne ct e d c ompone n t of the BR O T O C as pr o b in g
the ope r a t or pur it y of a qua n tum ope r a t ion . W e the n sho w e d th a t the qua n tum ope r a t ion U
β; t
i s in t im a t e ly
r e l a t e d t o the t w o -po in t spe ctr al for m fa ct or a nd glo b al ly a v e r a g e d r e g ul a r i z e d O T O C s a r e c onne ct e d t o
186
2 4 6 8 10 12 14
10
-4
0.001
0.010
0.100
1
Figure 5.5.2: A logplot of the equilib ration value (long-time average) of the G
( r)
β
( t) fo r va rious
Hamiltonian mo dels as a function of the system size L at β = 0 across a symmetric bipa rtition
⌊ L= 2⌋ : ⌈ L= 2⌉ . W e use exact time evolution fo r the integrable TFIM and Anderson since they do not
satisfy NRC. F o r Anderson, MBL, and GUE, w e p erfo rm exact diagonalization of the full Hamiltonian
and use the analytical exp ression in 5.4 . F o r NRC-PS and ME (maxim ally entangled mo del), w e use
the analytical exp ressions in eq. ( 5.40 ) and eq. ( 5.48 ).
2 4 6 8 10 12 14
10
-7
10
-5
0.001
0.100
Figure 5.5.3: A logplot of the equilib ration value (long-time average) of the G
( r)
β
( t) fo r va rious
Hamiltonian mo dels as a function of the system size L at β = 1 across a symmetric bipa rtition
⌊ L= 2⌋ : ⌈ L= 2⌉ . W e use exact time evolution fo r the integrable TFIM and Anderson since they do not
satisfy NRC. F o r Anderson, MBL, and GUE, w e p erfo rm exact diagonalization of the full Hamiltonian
and u se the analytical exp ression in 5.4 . F o r NRC-PS and ME (maxim ally entangled mo del), w e use
the analytical exp ressions in eq. ( 5.40 ) and eq. ( 5.48 ).
the four -po in t spe ctr al for m fa ct or , r e spe ct iv e ly . M or e o v e r , the c onne ct e d BR O T O C pr o be s the pur it y of
the as s oc i a t e d the r mofie ld doub le s t a t e .
M o v in g a w a y f r om the infinit e-t e mpe r a tur e as s umpt ion, w e in v e s t i g a t e the z e r o -t e mpe r a tur e cas e , w he r e ,
in P r oposit ion 3, w e sho w e d th a t , in thi s l imit , both the d i s c onne ct e d a nd c onne ct e d c ompone n ts of the
BR O T O C pr o be the gr ound s t a t e e n t a n gle me n t for nonde g e ne r a t e H a mi lt oni a n s . Thi s al lo w s us t o think
187
2 4 6 8 10 12 14 16
10
-8
10
-5
0.01
Figure 5.5.4: A logplot of the equilib ration value (long-time average) of the G
( r)
β
( t) fo r va rious
Hamiltonian mo dels as a function of the system size L at β = 1 across a symmetric bipa rtition
⌊ L= 2⌋ : ⌈ L= 2⌉ . The va rious data p oints have coalesced into t w o curves, first consisting of all the i n-
tegrable mo dels, whose ground state follo w a rea-la w entanglement. And second, fo r the GUE and ME
(maximally entangled mo del), whose ground states follo w volume-la w entanglement. Using 5.3 , w e
simply compute the ground state p rojecto r fo r va rious mo dels to compute this numerically .
of the m as pr o be s t o qua n tum p h as e tr a n sit ion s in the s ys t e m .
I n P r oposit ion s 4 a nd 5, w e s tudy the e qui l i br a t ion v alue of the BR O T O C a nd ho w it c onne cts t o ei gen -
s t a t e en t a n g lem en t . I n fa ct , w e sho w th a t i f the r e i s s u ffic ie n t e n t a n gle me n t in al l ei g e n s t a t e s a cr os s the
spe ctr um, the n the e qui l i br a t ion v alue m us t be ne a rly m a x im al . W e al s o o bt ain a n aly t ical clos e d - for m
ex pr e s sion s for the e qui l i br a t ion v alue of ne a rly m a x im al ly - e n t a n gle d H a mi lt oni a n s a nd c on tr as t the β -
de pe nde nc e in the unr e g ul a r i z e d v e r s us the r e g ul a r i z e d cas e .
F in al ly , w e pe r for m n ume r ical sim ul a t ion s on v a r ious in t e gr a b le a nd ch a ot ic H a mi lt oni a n mode l s t o
s tudy the e qui l i br a t ion v alue of the c onne ct e d c ompone n t of the BR O T O C . U sin g a mi x of finit e- si z e s cal -
in g a nd a n aly t ical e s t im a t e s, w e c on tr as t the de ca y r a t e s of the BR O T O C for v a r ious mode l s . W hi le a t
β = 0 , the de ca y r a t e i s univ e r s al a nd d i s t in g ui she s in t e gr a b le , ch a ot ic, a nd r a ndom m a tr i x e v o lut ion s; as
w e r e a ch β = O( 1) , thi s univ e r s al it y be g in s t o br e ak do w n . A nd , in fa ct , a t β!1 , the e qui l i br a t ion v alue
only d i s t in g ui she s local mode l s f r om the G UE, a nd i s the r efor e no lon g e r a r e l i a b le si gn a tur e of ch a ot ic -
v s -in t e gr a b le dy n a mic s . A n in t e r e s t in g f utur e w ork w ould be t o c on tr as t v a r ious cho ic e s of r e g ul a r i za t ion s
a nd their a b i l it y t o d i s t in g ui sh ch a ot ic a nd in t e gr a b le dy n a mic s .
188
A ppendic e s
Pr o o f o f 5.1
C on side r a b ip a r t it e H i l be r t sp a c e , H
AB
=H
A
H
B
. L e t V2U(H
A
); W2U(H
B
) a ndE
U2U(H)
[ f( U)]
de not e th e H a a r a v e r a g e of f( U) o v e rU(H) . The n, usin g the le mm a [ 312 ],
E
U2U(H)
[
U
U
y
]
=
S
d
; (5.51)
w he r eS i s the s w a p ope r a t or on H
H
′
w ithH
′
r e pr e s e n t in g a r e p l ica of the or i g in al H i l be r t sp a c e H .
G iv e n a n or thonor m al b asi s of H ,B =fj j⟩g
d
j= 1
( a nd a r e p l ica of the b asi s for H
′
), the s w a p ope r a t or ca n
be r e pr e s e n t e d as S =
d
∑
j; k= 1
j j⟩⟨ kj
j k⟩⟨ jj . N o w , i f w e c on side r H a a r a v e r a g e s o v e r a s ubs ys t e m in s t e a d ,
the n the le mm a a bo v e i s mod i fie d as,
E
V2U(H A)
V
y
V =
S
AA
′
d
A
a ndE
W2U(H B)
W
y
W =
S
BB
′
d
B
: (5.52)
F ir s t , w e c omput e the d i s c onne ct e d BR O T O C ,
G
( d)
β
:=E
V2U(H A); W2U(H B)
F
( d)
β
(5.53)
=E
V2U(H A); W2U(H B)
Tr
[(
√
ρ
β
√
ρ
β
)
(
W
y
W
)
S
]
Tr
[(
√
ρ
β
√
ρ
β
)
(
V
y
V
)
S
]
(5.54)
=
1
d
Tr
[
S
(
√
ρ
β
)
2
S
BB
′
]
Tr
[
S
(
√
ρ
β
)
2
S
AA
′
]
=
1
d
Tr
[
(
√
ρ
β
)
2
S
AA
′
]
Tr
[
(
√
ρ
β
)
2
S
BB
′
]
;
(5.55)
w he r e in the s e c ond l ine , w e h a v e us e d the le mm a , Tr[( A
B)S] = Tr[ AB] a nd in the thir d l ine w e h a v e
us e dS =S
AA
′S
BB
′ ,S
2
=I ,S
2
AA
′ =I
AA
′ , a ndS
2
BB
′ =I
BB
′ .
189
W e ca n for m al ly pe r for m p a r t i al tr a c e s a bo v e , for ex a mp le ,
Tr
[(
√
ρ
β
√
ρ
β
)
S
AA
′
]
= Tr
AA
′
[
Tr
BB
′
[(
√
ρ
β
√
ρ
β
)]
S
AA
′
]
= Tr
AA
′ [ σ
A
σ
A
′S
AA
′] = Tr[ σ
2
A
];
(5.56)
w he r e σ
A
Tr
B
[
p
ρ
β
]
a bo v e for br e v it y . L e t P
χ
( ρ) :=
ρ
χ
2
2
be the s qua r e d 2 - nor m of the ope r a t or ρ
χ
w ith ρ
χ
:= Tr
χ
[ ρ] a nd χ =f A; Bg w ith chi the c omp le me n t of χ . The n,
G
d
( β) =
1
d
P
A
(
√
ρ
β
)P
B
(
√
ρ
β
): (5.57)
S imi l a rly , for the c onne ct e d BR O T O C w e h a v e ,
G
( r)
β
( t) :=E
V2U(H A); W2U(H B)
F
( r)
β
( t) (5.58)
=E
V2U(H A); W2U(H B)
Tr
[
S
(
W
y
t
W
t
)
( y
y)
(
V
y
V
)
( y
y)
]
(5.59)
=
1
d
Tr
[
S
AA
′ y
2
U
y
2
t
S
AA
′ U
2
t
y
2
]
(5.60)
=
1
dZ( β)
Tr
[
S
AA
′U
2
β; t
(S
AA
′)
]
; (5.61)
w he r eU
β; t
:=V
β
◦U
t
w ithV
β
( X) := exp[ β H= 4] X exp[ β H= 4] the im a g in a r y t ime- e v o lut ion, U
t
( X) :=
U
y
t
XU
t
the r e al t ime- e v o lut ion, a nd U
t
= exp[ iHt] the us ual t ime- e v o lut ion ope r a t or . Thi s c omp le t e s
the pr oof . s
Pr o o f o f 5.4
R e cal l th a t NR C imp l ie s nonde g e ne r a c y of the spe ctr um, th e r efor e , usin g the spe ctr al de c omposit ion of a
H a mi lt oni a n, H =
d
∑
j= 1
E
j
Π
j
w ith Π
j
=j φ
j
⟩⟨ φ
j
j , w e h a v e ,
U
2
β; t
( A) =
d
∑
j; k; l; m
exp
[
β= 4
(
E
j
+ E
k
+ E
l
+ E
m
)
it
(
E
j
+ E
k
E
l
E
m
)](
Π
j
Π
k
)
A( Π
l
Π
m
):
(5.62)
190
U sin g the NR C as s umpt ion, w e h a v e the fo l lo w in g , exp
[
it
(
E
j
+ E
k
E
l
E
m
)] t
= δ
jl
δ
km
+ δ
jm
δ
kl
δ
jk
δ
kl
δ
lm
. The r efor e ,
G
( r)
β
( t) =
1
dZ( β)
0
@
d
∑
j; k
exp
[
β= 2
(
E
j
+ E
k
)]
Tr
[(
Π
j
Π
k
)
S
AA
′
(
Π
j
Π
k
)
S
AA
′
]
(5.63)
+
d
∑
j; k
exp
[
β= 2
(
E
j
+ E
k
)]
Tr
[(
Π
j
Π
k
)
S
BB
′
(
Π
j
Π
k
)
S
BB
′
]
(5.64)
d
∑
j
exp
[
β E
j
]
Tr
[
Π
2
j
S
AA
′ Π
2
j
S
AA
′
]
1
A
: (5.65)
N o w , for pur e s t a t e s Π
j
; Π
k
one ca n sho w th a t Tr
[(
Π
j
Π
k
)
S
AA
′
(
Π
j
Π
k
)
S
AA
′
]
=
Tr
[(
Π
j
Π
k
)
S
AA
′
]
2
.
W e no w define the R - m a tr i x in tr oduc e d in R ef . [ 305 ] w hi le c omput in g the infinit e-t e mpe r a tur e v a r i a n t
of thi s P r oposit ion . L e t ρ
A( B)
j
:= Tr
A( B)
[
Π
j
]
be the r e duc e d s t a t e , the n, for m al ly pe r for min g the p a r t i al
tr a c e w e h a v e ,
Tr
AA
′
BB
′ [ Π
k
Π
l
S
AA
′] = Tr
AA
′ [ Tr
BB
′ [ Π
k
Π
l
] S
AA
′] = Tr
AA
′
[
ρ
A
k
ρ
A
′
l
S
AA
′
]
= Tr
[
ρ
A
k
ρ
A
l
]
=: R
A
kl
:
(5.66)
The r efor e , the t ime-a v e r a g e ca n be simp l i fie d as,
G
( r)
β
( t) =
1
dZ( β)
0
@
d
∑
j; k
exp
[
β= 2( E
j
+ E
k
)
]
(
R
A
jk
2
+
R
B
jk
2
δ
jk
R
A
jk
2
)
1
A
: (5.67)
N o w w e in tr oduc e a mor e c omp a ct not a t ion for the e qui l i br a t ion v alue . L e t
~
R
χ
jk
:= exp
[
β
(
E
j
+ E
k
)
= 4
]⟨
ρ
χ
j
; ρ
χ
k
⟩
,
the n,
G
( r)
β
( t) =
1
dZ( β)
∑
χ2f A; Bg
(
e
R
( χ)
2
2
1
2
e
R
( χ)
D
2
2
)
; (5.68)
w ith [
e
R
( χ)
D
]
jk
= [
e
R
( χ)
D
]
jk
δ
jk
; w he r e w e h a v e us e d the fa ct th a t R
A
kk
= R
B
kk
, th a t i s, the the r e duc e d s t a t e s ρ
A
j
a nd ρ
B
j
a r e i s ospe ctr al ( u p t o ir r e le v a n t z e r os ).
191
Pr o o f o f 5.5
W e pr o v e thi s for the g e ne r al cas e d
A
̸= d
B
, a nd the or i g in al P r oposit ion ca n be r e c o v e r e d b y s e tt in g d
A
=
d
B
a t the e nd . L e t us as s ume w ithout los s of g e ne r al it y th a t d
A
d
B
. The spe ctr al de c omposit ion of the
H a mi lt oni a n i s H =
d
∑
j= 1
E
j
j φ
j
⟩⟨ φ
j
j a nd its r e duc e d s t a t e s a r e l a be l le d as ρ
A
j
:= Tr
B
[
j φ
j
⟩⟨ φ
j
j
]
.
N ot ic e th a t sinc e P(j ψ
AB
⟩)P
min
= Tr
[
ρ
2
A
]
1
d A
, the as s umpt ion th a t the pur it ie s a r e ne a rly minim um
ca n be e quiv ale n tly ex pr e s s e d as: P( ρ
A
k
)P
min
ε () ∥ Δ
A
k
∥
2
2
ε w ith Δ
A
k
:= ρ
A
k
I= d
A
8 k.
D efine Δ
B
k
a n alo g ously , i . e ., Δ
B
k
:= ρ
B
k
I= d
B
. The n, usin g the fa ct th a t the r e duc e d s t a t e s of a pur e
s t a t e a r e i s ospe ctr al ( the r efor e , P( ρ
A
k
) = P( ρ
B
k
) 8 k ) a nd d
A
d
B
, w e h a v e ,∥ Δ
B
k
∥
2
2
=
ρ
B
k
2
2
1
d B
=
ρ
A
k
2
2
1
d B
ρ
A
k
2
2
1
d A
ε 8 k .
N o w , r e cal l th a t , G
( r)
β
( t)j
ME
=
1
d
2
(
2Z( β= 2)
2
Z( β)
1
)
a nd G
( r)
β
( t) =
1
dZ( β)
(
~
R
A
2
2
+
~
R
B
2
2
~
R
A
D
2
2
)
,
w he r e w e h a v e us e d the fa ct th a t
~
R
( A)
D
2
2
=
~
R
( B)
D
2
2
. The n,
G
( r)
β
( t)j
ME
G
( r)
β
( t)j
NRC
=
1
d
2
(
2Z
2
( β= 2)
Z( β)
1
)
1
dZ( β)
(
e
R
( A)
2
2
+
e
R
( B)
2
2
e
R
( A)
D
2
2
)
(5.69)
1
d
2
Z( β)
Z
2
( β= 2)
1
dZ( β)
e
R
( A)
2
2
| {z }
( i)
+
1
d
2
Z( β)
Z
2
( β= 2)
1
dZ( β)
e
R
( B)
2
2
| {z }
( ii)
+
1
dZ( β)
e
R
( A)
D
2
2
1
d
2
| {z }
( iii)
;
(5.70)
w he r e w e h a v e sp l it the t e r m s in G
( r)
β
( t)j
ME
a nd the n us e d the tr i a n gle ine qual it y . W e no w w a n t t o bound
e a ch of the t e r m s ( i);( ii); a nd( iii) .
S inc e
e
R
( χ)
2
2
=
d
∑
j; k= 1
exp
[
β
(
E
j
+ E
k
)
= 2
]
(⟨
ρ
( χ)
j
; ρ
( χ)
k
⟩)
2
( not e th a t w e do not ne e d a n a bs o lut e
v alue he r e sinc e the ex pone n t i al t e r m i s nonne g a t iv e a nd A; B 0 =) ⟨ A; B⟩ 0 s o the inne r
pr oducts be t w e e n the r e duc e d s t a t e s i s nonne g a t iv e as w e l l ); w e ne e d t o bound
(⟨
ρ
( χ)
j
; ρ
( χ)
k
⟩)
2
. N ot e
192
th a t ,
(⟨
ρ
( χ)
k
; ρ
( χ)
l
⟩)
2
=
(⟨
I= d
χ
+ Δ
( χ)
k
; I= d
χ
+ Δ
( χ)
l
⟩)
2
=
(
1
d
χ
+
⟨
Δ
( χ)
k
; Δ
( χ)
l
⟩
)
2
=
1
d
2
χ
+
2
d
χ
⟨
Δ
( χ)
k
; Δ
( χ)
l
⟩
+
⟨
Δ
( χ)
k
; Δ
( χ)
l
⟩
2
:
(5.71)
A nd , usin g C a uch y - S ch w a r z ine qual it y alon g w ith the fa ct th a t
⟨
Δ
χ
k
; Δ
χ
k
⟩
=
ρ
( χ)
k
2
2
1
d χ
ε 8 k ( as
sho w n a bo v e ), w e h a v e ,
⟨
Δ
χ
k
; Δ
χ
l
⟩
√
⟨
Δ
χ
k
; Δ
χ
k
⟩⟨
Δ
χ
l
; Δ
χ
l
⟩
ε 8 k; l . The r efor e ,
(⟨
ρ
( χ)
k
; ρ
( χ)
l
⟩)
2
1
d
2
χ
+
2 ε
d χ
+ ε
2
f( d
χ
; ε) 8 k; l .
P lu gg in g thi s b a ck in
~
R
( χ)
2
2
, w e h a v e ,
e
R
( χ)
2
2
=
d
∑
j; k= 1
exp
[
β
(
E
j
+ E
k
)
= 2
]
f( d
χ
; ε) =Z( β= 2)
2
f( d
χ
; ε) .
N o w , for the d i a g on al p a r t ,
~
R
A
D
2
2
=
d
∑
j= 1
exp
[
β E
j
]
(⟨
ρ
A
j
; ρ
A
j
⟩)
2
d
∑
j= 1
exp
[
β E
j
]
f( d
A
; ε) =Z( β) f( d
A
; ε) .
The r efor e , t e r m( i) a bo v e be c ome s
Z
2
( β= 2)
dZ( β)
1
d
f( d
A
; ε)
a nd t e r m ( ii) be c ome s,
Z
2
( β= 2)
dZ( β)
1
d
f( d
B
; ε)
.
A nd , t e r m ( iii) be c ome s,
1
d
1
d
f( d
A
; ε)
. N o w , not ic e th a t
Z( β= 2)
2
=
d
∑
j; k= 1
exp
[
β E
j
= 2
]
exp[ β E
k
= 2] =
d
∑
j= k
exp
[
β E
j
= 2
]
exp[ β E
k
= 2] (5.72)
+
d
∑
j̸= k
exp
[
β E
j
= 2
]
exp[ β E
k
= 2]
d
∑
j= 1
exp
[
β E
j
]
=Z( β); (5.73)
w he r e w e h a v e dr oppe d the j ̸= k t e r m s in the s umm a t ion a nd us e d their nonne g a t iv it y . The r efor e ,
Z( β= 2)
2
=Z( β) 1 a nd s o , t e r m ( i i i ) ca n be u ppe r bounde d as
1
d
1
d
f( d
A
; ε)
Z( β= 2)
2
dZ( β)
1
d
f( d
A
; ε)
.
P utt in g e v e r y thin g t o g e the r , w e h a v e
G
( r)
β
( t)j
ME
G
( r)
β
( t)j
NRC
Z( β= 2)
2
dZ( β)
(
2
1
d
f( d
A
; ε)
+
1
d
f( d
B
; ε)
)
: (5.74)
N o w , i f w e s e t d
A
= d
B
=
p
d , w e h a v e ,
1
d
f( d
χ; ε
)
=
1
d
(
1
d
+
2 ε
p
d
+ ε
2
)
=
(
2 ε
p
d
+ ε
2
)
. The r e-
for e ,
G
( r)
β
( t)j
ME
G
( r)
β
( t)j
NRC
Z( β= 2)
2
dZ( β)
(
6 ε
p
d
+ 3 ε
2
)
: (5.75)
193
Pr o o f o f 5.2
W e h a v e ,
F
( A 1; B 1; A 2; B 2)
β
( t) =
1
Z( β)
Tr
[
xU
t
( A
1
) xB
1
xU
t
( A
2
) x
(
A
y
2
B
y
1
A
y
1
)]
; (5.76)
w he r e w e h a v e us e d B
2
= A
y
2
B
y
1
A
y
1
. N o w , usin g the c y cl ic it y of tr a c e a nd the le mm a E
A2U(H)
AXA
y
=
Tr[ X]
d
,
w e h a v e ,
E
A 12U(H)
F
( A 1; B 1; A 2; B 2)
β
( t) =
1
dZ( β)
Tr
[
xU
y
t
]
Tr
[
U
t
xB
1
xU
t
( A
2
) x
(
A
y
2
B
y
1
)]
(5.77)
=
1
dZ( β)
Tr
[
xU
y
t
]
Tr
[
B
y
1
U
t
xB
1
xU
t
( A
2
) x
(
A
y
2
)]
; (5.78)
w he r e in the s e c ond l ine w e h a v e us e d the c y cl ic it y of tr a c e t o mo v e B
y
1
t o the f r on t. N o w , pe r for min g E
B 1
usin g the le mm a a bo v e , w e h a v e ,
E
B 12U(H)
Tr
[
B
y
1
U
t
xB
1
xU
t
( A
2
) x
(
A
y
2
)]
=
1
d
Tr[ U
t
x] Tr
[
xU
y
t
A
2
U
t
xA
y
2
]
: (5.79)
S imi l a rly , pe r for min g the a v e r a g e o v e r A
2
w e o bt ain the fin al t e r m s th a t a r e pr opor t ion al t o Tr[ U
t
x] Tr
[
U
y
t
x
]
.
Th us, w e h a v e the de sir e d r e s ult.
Unr eg u l a r ized O T O Cs f o r maxima ll y ent a n gled mo d el s
C on side r a n ope r a t or S chmidt de c omposit ion of U
t
U =
∑
j
√
λ
j
U
j
W
j
w ith U
j
2 L(H
A
) a nd
W
j
2 L(H
B
) s uch th a t
⟨
U
j
; U
k
⟩
= d
A
δ
jk
a nd
⟨
W
j
; W
k
⟩
= d
B
δ
jk
. F or r a unit a r y ope r a t or , ∥ U∥
2
2
= d ,
the r efor e , w e h a v e ,
∥ U∥
2
2
= Tr
2
4
∑
j; k
√
λ
j
λ
k
U
j
U
y
k
W
j
W
y
k
3
5
=
∑
j; l
√
λ
j
λ
k
⟨
U
j
; U
k
⟩⟨
W
j
; W
k
⟩
= d
A
d
B
∑
j
λ
j
=)
∑
j
λ
j
= 1:
(5.80)
194
N o w , c on side r the b ip a r t it e unr e g ul a r i z e d O T O C , F
β
( U) =
1
d
Tr
[(
ρ
β
I
)
U
2
S
AA
′ U
y
2
S
AA
′
]
. P lu g -
g in g in the ope r a t or S chmidt de c omposit ion of U , w e h a v e ,
F
β
( U) =
1
d
Tr
2
4
(
ρ
β
I
)
0
@
∑
j
√
λ
j
U
j
W
j
1
A
2
S
AA
′
(
∑
k
√
λ
k
U
y
k
W
y
k
)
2
S
AA
′
3
5
=
1
d
∑
jklm
√
λ
j
λ
k
λ
l
λ
m
Tr
[(
ρ
β
I
)
(
U
j
W
j
U
k
W
k
)
S
AA
′
(
U
y
l
W
y
l
U
y
m
W
y
m
)
S
AA
′
]
=
1
d
∑
jklm
√
λ
j
λ
k
λ
l
λ
m
Tr
[(
ρ
β
I
)
(
U
j
W
j
U
k
W
k
)
(
U
y
m
W
y
l
U
y
l
W
y
m
)]
;
w he r e w e h a v e us e d the a d j o in t a ct ion S
AA
′ ( X
A
Y
A
′)S
AA
′ = Y
A
X
A
′ . The n,
F
β
( U) =
∑
jk
√
λ
j
( λ
k
)
3= 2
Tr
[
ρ
β
(
U
j
W
j
)
(
U
y
k
W
y
k
)]
; (5.81)
w he r e w e h a v e us e d the fa ct th a t
⟨
U
j
; U
k
⟩
= d
A
δ
jk
a nd
⟨
W
j
; W
k
⟩
= d
B
δ
jk
a nd s umme d o v e r t w o of the
ind ic e s of the s umm a t ion a bo v e . N o w , i f U i s m a x im al ly e n t a n gle d ( n a me ly , its ei g e n s t a t e s a r e m a x im al ly
e n t a n gle d ) a cr os s d
A
= d
B
=
p
d a nd one h as
√
λ
j
=
√
1
d
2
A
, the n
F
β
( U) =
1
d
2
A
Tr
2
4
ρ
β
0
@
∑
j
√
λ
j
U
j
W
j
1
A
(
∑
k
√
λ
j
U
k
W
k
)
3
5
=
1
d
2
A
Tr
[
ρ
β
UU
y
]
=
1
d
2
A
: (5.82)
The r efor e , G
β
( U) = 1 F
β
( U) = 1
1
d
2
A
a nd i s inde pe nde n t of β .
T h e d y n a mi c a l ma pU
β; t
is a q u a ntum o per a ti o n
R e cal l th a t the C ho i - J a mio l k o w sk i ( C J ) i s omor p hi s m i s a n i s omor p hi s m be t w e e n l ine a r m a ps E :L(H)!
L(K) t o m a tr ic e s ρ
E
2L(H)
L(K) . L e tj φ
+
⟩ :=
1
p
d
j j⟩j j⟩ be the n o r m a l i ze d m a x im al ly e n t a n gle d s t a t e
inH
2
, the n,
ρ
E
:=E
I
(
j φ
+
⟩⟨ φ
+
j
)
: (5.83)
195
A l ine a r m a p E i s C P() ρ
E
0 . C omput in g the C J m a tr i x c or r e spond in g t o the l ine a r m a p V
β
( X) :=
exp[ β H= 4] X exp[ β H= 4] , w e h a v e ,
ρ
V
β
=
1
d
d
∑
j; k= 1
V
β
I (j j⟩⟨ kj
j j⟩⟨ kj) (5.84)
=
1
d
d
∑
j; k= 1
exp
[
β
(
E
j
+ E
k
)
= 4
]
(j j⟩⟨ kj
j j⟩⟨ kj) =
Z( β= 2)
d
j ψ( β= 2)⟩⟨ ψ( β= 2)j; (5.85)
w he r e in the s e c ond e qual it y w e h a v e us e d the ex p a n sion of the TD S e q . ( 5.19 ). N o w , sinc e j ψ( β= 2)⟩⟨ ψ( β= 2)j
i s a pur e s t a t e ( or a r a nk - 1 pr oj e ct or ), it i s posit iv e s e midefinit e . M or e o v e r ,
Z( β= 2)
d
0 , the r efor e , ρ
V
β
0 .
A s a r e s ult , V
β
i s a C P m a p .
T o sho w th a t the m a p V
β
i s tr a c e- nonincr e asin g , w e h a v e t o sho w th a t Tr
[
V
β
( ρ)
]
Tr[ ρ] 8 ρ2B(H) .
N a me ly ,
() Tr
[
V
β
( ρ) ρ
]
0 8 ρ; (5.86)
() Tr[ exp[ β H= 4] ρ exp[ β H= 4] ρ] 0 8 ρ (5.87)
() Tr[( exp[ β H= 2]I) ρ] 0 8 ρ; (5.88)
() exp[ β H= 2]I; (5.89)
w he r e in the l as t l ine , w e h a v e us e d the definit ion of posit iv e s e midefinit e ne s s . N o w , as s umin g H 0 , w e
h a v e , exp[ β H= 2]I sinc e exp
[
β E
j
= 2
]
1 i f β; E
j
0 8 j . N ot ic e th a t , the m a tr i x ρ
V
β
i s s u bn o r m a l-
i ze d w ith Tr
[
ρ
V
β
]
=
Z( β= 2)
d
1 . The r efor e , the dy n a mical m a p , V
β
i s a C P a nd tr a c e- nonincr e asin g l ine a r
m a p , th a t i s, it i s a p h ysical qua n tum ope r a t ion [ 3 ] a nd w e ca n think of ρ
V
β
as a s ubnor m al i z e d de n sit y
m a tr i x c or r e spond in g t o a qua n tum pr oc e s s [ 363 , 364 ].
Eq u i li b r a ti o n v a l u e o f NR C - PS
T o c omput e the e qui l i br a t ion v alue , w e ne e d t o e v alua t e the R - m a tr i x . R e cal l th a t H
NRC - PS
:=
d A; d B
∑
j; k= 1
E
j; k
j φ
( A)
j
⟩⟨ φ
( A)
j
j
j φ
( B)
k
⟩⟨ φ
( B)
k
j , w ith the a dd it ion al as s umpt ion th a t the spe ctr um f E
j; k
g
j; k
s a t i sfie s NR C . D efine the index
196
α ( j; k) w ith α 2 f 1; 2; ; dg a nd j 2 f 1; 2;; d
A
g; k 2 f 1; 2; ; d
B
g . The n, ρ
A
α
= Tr
B
[
ρ
α
]
=
Tr
B
[
ρ
jk
]
= j φ
A
j
⟩⟨ φ
A
j
j , th a t i s for al l p air e d ind ic e s ( j; k) a nd( j; k
′
) the r e duc e d s t a t e s a r e the s a me . F or
the R - m a tr i x , c on side r α = ( j; k) a nd β = ( l; m) , w e h a v e , R
A
α; β
=
⟨
ρ
A
α
; ρ
A
β
⟩
= δ
j; l
. S imi l a rly , w e h a v e ,
ρ
B
α
= Tr
A
[
ρ
α
]
= Tr
A
[
ρ
jk
]
=j φ
B
k
⟩⟨ φ
B
k
j a nd R
B
α; β
= δ
k; m
.
W e a r e no w r e a dy t o e v alua t e G
( r)
β
( t) . N ot ic e th a t
d
∑
α; β= 1
exp
[
β= 2
(
E
α
+ E
β
)]
R
A
α; β
2
=
d A; d B
∑
j; k; l; m= 1
exp
[
β= 2
(
E
j; k
+ E
l; m
)]
δ
j; l
2
(5.90)
=
d A; d B; d B
∑
j; k; m= 1
exp
[
β= 2
(
E
j; k
+ E
j; m
)]
=
d A
∑
j= 1
(
d B
∑
k= 1
exp
[
β E
j; k
= 2
]
)
2
: (5.91)
D efinin g , Θ
A
j
:=
(
d B
∑
k= 1
exp
[
β E
j; k
= 2
]
)
2
, w e h a v e ,
d
∑
α; β= 1
exp
[
β= 2
(
E
α
+ E
β
)]
R
A
α; β
2
=
d A
∑
j= 1
Θ
A
j
. S imi l a r
al g e b r aic m a nipul a t ion s pr o v e the fin al r e s ult ,
G
( r)
β
( t) =
1
dZ( β)
0
@
d A
∑
j= 1
Θ
A
j
+
d B
∑
k= 1
Θ
B
k
Z( β)
1
A
: (5.92)
Thi s ca n the n be r e w r itt e n in the for m in the m ain t ex t , n a me ly ,
G
( r)
β
( t)j
NRC PS
=
1
d
(
∥ p
A
( β= 2)∥
2
+∥ p
B
( β= 2)∥
2
∥ p( β= 2)∥
2
1
)
; (5.93)
w he r e the pr o b a b i l it y v e ct or p( β) i s define d b y the c ompone n ts p
j
=
e
β E
j
Z( β)
a nd p
A= B
( β) a r e its m a r g in al s .
M or e o v e r , for the d i s c onne ct e d c or r e l a t or , a simi l a r calc ul a t ion sho w s th a t ,
G
( d)
β
=
1
dZ( β)
2
0
@
d A
∑
j= 1
Θ
A
j
1
A
(
d B
∑
k= 1
Θ
B
k
)
: (5.94)
N umer i c a l d e t a i l s
The R
2
v alue for al l l ine a r fits w as ⪆ 0: 99 for al l d a t a a nd he nc e w e do not r e por t the n umbe r s he r e .
197
M ode l β = 0 β = 1 β =1
TFI M in t e gr a b le 0.523785 0.0341617 -0.525957
NR C - PS 0.971448 0.83213 1.00
A n de r s on 1.13615 0.742852 6: 79567 10
7
MBL 0.700658 0.213039 -0.012799
TFI M ch a ot ic 1.60648 0.327334 -0.00502613
G UE 1.12868 2.56073 2.10215
T able 5.6.1: The log
2
( α) fo r va rious Hamiltonian mo dels at β = 0; 1;1 , given the Ansatz G
( r)
β
( t) =
α d
γ
.
198
Outlo ok
The c e n tr al aim of thi s the si s h as be e n t o ut i l i z e po w e r f ul t oo l s or i g in a t in g f r om qua n tum infor m a t ion
the or y , a nd the unique le n s they pr o v ide , t o unde r s t a nd qual it a t iv e ly a nd qua n t it a t iv e ly v a r ious qua n tum
m a n y - body p he nome n a . A r g ua b ly , the t w o ca r d in al fe a tur e s of qua n tum the or y a r e the s u pe r posit ion pr in-
c ip le ( qua n t i fie d v i a the not ion of qua n tum c o he r e nc e ) a nd qua n tum e n t a n gle me n t (w hich m a ni fe s ts v i a
the t e n s or pr oduct s tr uctur e of qua n tum s ys t e m s ). U nde r s t a nd in g their r o le in the k ine m a t ical a nd dy -
n a mical be h a v ior of qua n tum s ys t e m s - a nd th us unr a v e l in g their de p a r tur e f r om cl as sical s ys t e m s - h as le d
t o m a n y k ey p h ysical in si gh ts in m a n y - body p h ysic s in the l as t fe w de ca de s . Thi s the si s asp ir e s t o be a n-
othe r infinit e sim al s t e p in thi s d ir e ct ion . F or e a ch of the p he nome n a w e h a v e s tud ie d in thi s the si s, n a me ly ,
ch a os, local i za t ion, a nd s cr a mb l in g , w e sho w e d th a t both qua n tum c o he r e nc e a nd qua n tum e n t a n gle me n t
a r e po w e r f ul d i a gnos t ic s a nd he nc e pr o v ide n a tur al qua n t i fie r s of the inhe r e n t q u a n t u m n e ss for the s e m a n y -
body p he nome n a .
I t i s b y no w w e l l e s t a b l i she d th a t qua n tum infor m a t ion-the or e t ic qua n t it ie s s uch as e n t a n gle me n t , tr i -
p a r t it e qua n tum m utual infor m a t ion a nd s o on h a v e be c ome a n e s s e n t i al t oo l in the s tudy of m a n y - body
p h ysic s . F r om de t e ct in g qua n tum p h as e tr a n sit ion s al l the w a y t o the qua n t i fica t ion a nd , in fa ct , defi n it io n
of t opo lo g ical p h as e s h as be e n a chie v e d usin g the s e ide as . Qua n tum infor m a t ion the or y s t a nd s he r e as a
unique s ubj e ct sinc e it not only pr o v ide s thi s infor m a t ion the or y b as e d le n s for unde r s t a nd in g qua n tum
s ys t e m s - fo l lo w in g in the foots t e ps of J o hn A . W he e le r ’ s t ime le s s r e m a rk , “ I t fr o m bit ” - mor e o v e r , it pr o -
v ide s a unique c omput a t ion al c omp lex it y le n s . A s a n ex a mp le , the not ion s of in t e gr a b i l it y a nd ch a ot ic it y
199
ca n the m s e lv e s be r e e x pr e sse d in t e r m s of their ( cl as sical or qua n tum ) c omput a t ion al c omp lex it y . Thi s i s
in t im a t e ly r e l a t e d t o the e n t a n gle me n t gr o w th of ope r a t or s a nd s t a t e s w he n e v o lv e d v i a a ch a ot ic ( as op -
pos e d t o in t e gr a b le ) H a mi lt oni a n . M or e o v e r , thi s ide a of usin g c omp lex it y as a pr o be t o ide n t i f y , in a br o a d
s e n s e , q ua n tum p h as e s of m a tt e r , h as al s o be e n a pp l ie d t o t opo lo g ical p h as e s .
W hi le the r e a r e m a n y ope n que s t ion s on the hor i z on of qua n tum m a n y - body the or y , the r e i s one in
p a r t ic ul a r th a t i s clos e ly r e l a t e d t o thi s the si s . A s it tur n s out , find in g a n un a mb i g uous a nd s a t i sfa ct or y
defi n it io n of qua n tum ch a os its e l f h as tur ne d out t o be quit e a ch al le n g in g pr o b le m . I t i s w or th spe c ul a t in g
th a t , pe rh a ps, a r e s o lut i on mi gh t be o bt aine d v i a eithe r qua n tum i nfor m a t ion- or qua n tum c omp lex it y -
the or e t ic ide as . Thi s w ould a moun t t o f und a me n t al pr o gr e s s in thi s fie ld a nd w e hope th a t thi s the si s ca n
pr o v ide s ome non v a ni shin g in sp ir a t ion ( i f not c on tr i but ion ) in thi s d ir e ct ion .
200
R e fe r e nces
[1] A v iv a G ub in a nd L e a F . S a n t os . Qua n tum ch a os: A n in tr oduct ion v i a ch ain s of in t e r a ct in g sp in s
1/2. A m er ic a n J o u r n a l of Ph y sic s , 80(3):246–251, M a r ch 2012. doi: 10 . 1119 / 1 . 3671068 .
[2] N a mit A n a nd , Ge or g ios S t y l i a r i s, M e e n u K um a r i , a nd P a o lo Za n a r d i . Qua n tum c o he r e nc e as a
si gn a tur e of ch a os . Ph y s . R ev . R e se a r c h , 3:023214, J un 2021. UR L : https://link.aps.org/
doi/ 10 . 1103 /PhysRevResearch. 3 . 023214 , doi: 10 . 1103 /PhysRevResearch. 3 . 023214 .
[3] M ich a e l A . N ie l s e n a nd I s a a c L . C h ua n g. Q u a n t u m Co m pu t a t io n a n d Q u a n t u m I nf o r m a t io n . C a m-
br id g e U niv e r sit y P r e s s, C a mbr id g e ; N e w Y ork , 10th a nniv e r s a r y e d e d it ion, 2010.
[4] A lex a nde r S tr e lts o v , Ge r a r do Ade s s o , a nd M a r t in B . P le nio . Co llo q u i u m : Qua n tum c o he r e nc e as a
r e s our c e . R ev iew s of M o der n Ph y sic s , 89(4), O ct o be r 2017. doi: 10 . 1103 /RevModPhys. 89 . 041003 .
[5] R ys za r d H or ode ck i , P a w e ł H or ode ck i , M ich ał H or ode ck i , a nd K a r o l H or ode ck i . Qua n tum e n t a n-
gle me n t. R ev iew s of M o der n Ph y sic s , 81(2):865–942, J une 2009. doi: 10 . 1103 /RevModPhys. 81 .
865 .
[6] X i a o g ua n g W a n g , S ho hini Ghos e , B a r r y C . S a nde r s, a nd B a mb i H u . E n t a n gle me n t as a si gn a tur e of
qua n tum ch a os . Ph y sic a l R ev iew E , 70(1), J uly 2004. doi: 10 . 1103 /PhysRevE. 70. 016217 .
[7] L e v V idm a r a nd M a r c os R i g o l . E n t a n gle me n t e n tr op y of ei g e n s t a t e s of qua n tum ch a ot ic h a mi lt o -
ni a n s . Ph y s . R ev . Lett . , 119:220603, N o v 2017. UR L : https://link.aps.org/doi/ 10 . 1103 /
PhysRevLett. 119 . 220603 , doi: 10 . 1103 /PhysRevLett. 119 . 220603 .
[8] M e e n u K um a r i a nd S ho hini Ghos e . U n t a n gl in g e n t a n gle me n t a nd ch a os . Ph y sic a l R ev iew A ,
99(4):042311, A pr i l 2019. arXiv: 1806 . 10545 , doi: 10 . 1103 /PhysRevA. 99 . 042311 .
[9] S C h a ud h ur y , A S mith, BE A nde r s on, S Ghos e , a nd PS J e s s e n . Qua n tum si gn a tur e s of ch a os in a
k ick e d t op . N a t u r e , 461(726 5):768, 2009.
[10] C . N ei l l , P . R oush a n, M . F a n g , Y . C he n, M . Ko lodr ube tz , Z. C he n, A . M e gr a n t , R . B a r e nd s, B . C a mp -
be l l , B . C hi a r o , A . Dun s w or th, E . J eff r ey , J . Ke l ly , J . M utus, P . J . J . O ’ M al ley , C . Quin t a n a , D . S a nk ,
A . V ain s e nche r , J . W e nne r , T . C . W hit e , A . P o l k o v ni k o v , a nd J . M . M a r t ini s . E r g od ic dy n a mic s a nd
the r m al i za t ion in a n i s o l a t e d qua n tum s ys t e m . N a t u r e Ph y sic s , 12(11):1037–1041, N o v e mbe r 2016.
doi: 10 . 1038 /nphys 3830 .
[11] M a rk S r e dnick i . C h a os a nd qua n tum the r m al i za t ion . Ph y sic a l R ev iew E , 50(2):888–901, Au g us t
1994. doi: 10 . 1103 /PhysRevE. 50 . 888 .
201
[12] J . M . D e uts ch . Qua n tum s t a t i s t ical me ch a nic s in a clos e d s ys t e m . Ph y sic a l R ev iew A , 43(4):2046–
2049, F e br ua r y 1991. doi: 10 . 1103 /PhysRevA. 43 . 2046 .
[13] M a r c os R i g o l , V a n j a Dun jk o , a nd M a x im Ol sh a ni i . The r m al i za t ion a nd its me ch a ni s m for g e ne r ic
i s o l a t e d qua n tum s ys t e m s . N a t u r e , 452(7189):854–858, A pr i l 2008. doi: 10 . 1038 /nature 06838 .
[14] P e t e r R eim a nn . F ound a t ion of S t a t i s t ical M e ch a nic s unde r E x pe r ime n t al ly R e al i s t ic C ond it ion s .
Ph y sic a l R ev iew Lett er s , 101(19), N o v e mbe r 2008. doi: 10 . 1103 /PhysRevLett. 101 . 190403 .
[15] N o ah L inde n, S a ndu P ope s c u , A n thon y J . S hor t , a nd A ndr e as W in t e r . Qua n tum me ch a nical e v o lu-
t ion t o w a r d s the r m al e qui l i br ium . Ph y sic a l R ev iew E , 79(6), J une 2009. doi: 10 . 1103 /PhysRevE.
79 . 061103 .
[16] A n thon y J S hor t. E qui l i br a t ion of qua n tum s ys t e m s a nd s ubs ys t e m s . N ew J o u r n a l of Ph y sic s ,
13(5):053009, M a y 2011. doi: 10 . 1088 / 1367 - 2630/ 13 / 5 / 053009 .
[17] I A L a rk in a nd Y u N O v chinni k o v . Quasicl as sical M e thod in the The or y of S u pe r c onduct iv it y . J o u r-
n a l of Ex p er i m en t a l a n d Th e o r et ic a l Ph y sic s , 28(6):2262, J une 1969.
[18] A lexei Kit a e v . A simp le mode l of qua n tum ho lo gr a p h y ( p a r t 1).
h tt p://onl ine .k it p . uc s b . e du/onl ine/e n t a n gle d15/k it a e v/, 2015.
[19] S i lv i a P a pp al a r d i , A n g e lo R us s om a nno , Boj a n Ž unk o v ič, F e r n a ndo I e mini , A le s s a ndr o S i lv a , a nd
R os a r io F a z io . S cr a mb l in g a nd e n t a n gle me n t spr e a d in g in lon g - r a n g e sp in ch ain s . Ph y sic a l R ev iew
B , 98:134303, O ct 2018. doi: 10 . 1103 /PhysRevB. 98 . 134303 .
[20] Quir in H umme l , Be n j a min Gei g e r , J ua n Die g o U r b in a , a nd K l a us R ich t e r . R e v e r si b le qua n tum
infor m a t ion spr e a d in g in m a n y - body s ys t e m s ne a r cr it ical it y . Ph y sic a l R ev iew Lett er s , 123:160401,
2019. doi: 10 . 1103 /PhysRevLett. 123 . 160401 .
[21] D a v id J L uitz a nd Y e v g e n y B a r L e v . I nfor m a t ion pr op a g a t ion in i s o l a t e d qua n tum s ys t e m s . Ph y sic a l
R ev iew B, 96(2):020406, 2017. doi: 10 . 1103 /PhysRevB. 96 . 020406 .
[22] S a úl P i l a t o w sk y - C a me o , J or g e C h á v ez - C a rlos, M i g ue l A . B as t a r r a che a - M a gn a ni , P a v e l S tr á n sk ý ,
S e r g io L e r m a - H e r n á ndez , L e a F . S a n t os, a nd J or g e G . H ir s ch . P osit iv e qua n tum L ya puno v ex po -
ne n ts in ex pe r ime n t al s ys t e m s w ith a r e g ul a r c l as sical l imit. Ph y sic a l R ev iew E , 101:010202, J a n 2020.
doi: 10 . 1103 /PhysRevE. 101 . 010202 .
[23] T i a nr ui X u , Thom as S ca ffid i , a nd X i a n g y u C a o . D oe s s cr a mb l in g e qual ch a os? Ph y sic a l R ev iew
Lett er s , 124:140602, 2020. doi: 10 . 1103 /PhysRevLett. 124 . 140602 .
[24] Koji H ashimot o , K y oun g - B um H uh, Ke un- Y oun g Kim, a nd R y ot a W a t a n a be . E x pone n t i al gr o w th of
out - of -t ime- or de r c or r e l a t or w ithout ch a os: in v e r t e d h a r monic os c i l l a t or . a r X i v:2007.04746 , 2020.
UR L : https://arxiv.org/abs/ 2007 . 04746 .
[25] J i a o z i W a n g , G iul i a no Be ne n t i , G iul io C as a t i , a nd W e n g e W a n g. Qua n tum ch a os a nd the c or r e-
sponde nc e pr inc ip le , 2020. arXiv: 2010 . 10360 .
202
[26] N im a L ashk a r i , D ou gl as S t a nfor d , M a tthe w H as t in gs, T o b i as O s bor ne , a nd P a tr ick H a y de n . T o -
w a r d s the fas t s cr a mb l in g c on j e ctur e . J o u r n a l of H i g h E n er g y Ph y sic s , 2013(4):22, A pr i l 2013.
doi: 10 . 1007 /JHEP 04( 2013 ) 022 .
[27] Ge or g ios S t y l i a r i s a nd P a o lo Za n a r d i . Qua n t i f y in g the I nc omp a t i b i l it y of Qua n tum M e as ur e me n ts
R e l a t iv e t o a B asi s . Ph y sic a l R ev iew Lett er s , 123(7), Au g us t 2019. doi: 10 . 1103 /PhysRevLett.
123 . 070401 .
[28] O . Bo hi g as, M . J . G i a nnoni , a nd C . S chmit. C h a r a ct e r i za t ion of C h a ot ic Qua n tum S pe ctr a a nd
U niv e r s al it y of L e v e l Fluctua t ion L a w s . Ph y sic a l R ev iew Lett er s , 52(1):1–4, J a n ua r y 1984. doi:
10 . 1103 /PhysRevLett. 52 . 1 .
[29] F r itz H a ak e . Q u a n t u m Si g n a t u r e s of C h a o s . N umbe r 54 in S pr in g e r S e r ie s in S y ne r g e t ic s . S pr in g e r ,
Be rl in ; N e w Y ork , 3r d r e v . a nd e nl . e d e d it ion, 2010.
[30] Thom as G uhr , A xe l M ül le r – Gr oe l in g , a nd H a n s A . W eide nm ül le r . R a ndom- m a tr i x the or ie s in
qua n tum p h ysic s: C ommon c onc e pts . Ph y sic s R ep o r ts , 299(4-6):189–425, J une 1998. doi:
10 . 1016 /S 0370 - 1573 ( 97 ) 00088 - 4 .
[31] J . E i s e r t , M . C r a me r , a nd M . B . P le nio . Co llo q u i u m : A r e a l a w s for the e n t a n gle me n t e n tr op y . R ev iew s
of M o der n Ph y sic s , 82(1):277–306, F e br ua r y 2 010. doi: 10 . 1103 /RevModPhys. 82 . 277 .
[32] A she r P e r e s . S t a b i l it y of qua n tum mot ion in ch a ot ic a nd r e g ul a r s ys t e m s . Ph y sic a l R ev iew A ,
30(4):1610, 1984. doi: 10 . 1103 /PhysRevA. 30 . 1610 .
[33] R odo l fo A . J al a be r t a nd H or a c io M . P as t a w sk i . E n v ir onme n t -inde pe nde n t de c o he r e nc e r a t e in cl as -
sical ly ch a ot ic s ys t e m s . Ph y s . R ev . Lett . , 86(12):2490–2493, M a r 2001. UR L : https://link.
aps.org/doi/ 10 . 1103 /PhysRevLett. 86 . 2490 , doi: 10 . 1103 /PhysRevLett. 86 . 2490 .
[34] A . Gous s e v , R . A . J al a be r t , H. M . P as t a w sk i , a nd D . A r ie l W i s ni a ck i . L os chmidt e cho . Sc h o l a r p e d i a ,
7(8):11687, 2012. r e v i sion #127578. doi: 10 . 4249 /scholarpedia.11687 .
[35] Thom as Gor in, T om a ž P r os e n, Thom as H S e l i gm a n, a nd M a rk o Žnid a r ič . D y n a mic s of los chmidt
e choe s a nd fide l it y de ca y . Ph y sic s R ep o r ts , 435(2-5):33–156, 2006.
[36] P a o lo Za n a r d i , C hr i s t of Zal k a , a nd L a r a F a or o . E n t a n gl in g po w e r of qua n tum e v o lut ion s . Ph y sic a l
R ev iew A , 62(3), Au g us t 2000. doi: 10 . 1103 /PhysRevA. 62 . 030301 .
[37] P a o lo Za n a r d i . E n t a n gle me n t of qua n tum e v o lut ion s . Ph y sic a l R ev iew A , 63(4):040304, M a r ch
2001. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevA. 63 . 040304 , doi: 10 . 1103 /
PhysRevA. 63 . 040304 .
[38] A J S c ott a nd C a rlt on M C a v e s . E n t a n gl in g po w e r of the qua n tum b ak e r s m a p . J o u r n a l of Ph y sic s
A : M a t h em a t ic a l a n d Gen er a l , 36(36):9553–9576, S e pt e mbe r 2003. doi: 10 . 1088 / 0305 - 4470 / 36 /
36 / 308 .
[39] A r ul L ak shmin a r a ya n . E n t a n gl in g po w e r of qua n t i z e d ch a ot ic s ys t e m s . Ph y s . R ev . E , 64(3):036207,
Au g 2001. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevE. 64. 036207 , doi: 10 .
1103 /PhysRevE. 64. 036207 .
203
[40] V ai b h a v M a d ho k , V i b h u G u pt a , D e ni s - A lex a ndr e T r ott ie r , a nd S ho hini Ghos e . S i gn a tur e s of ch a os
in the dy n a mic s of qua n tum d i s c or d . Ph y s ic a l R ev iew E , 91(3), M a r ch 2015. doi: 10 . 1103 /
PhysRevE. 91 . 032906 .
[41] T . B a um gr a tz , M . C r a me r , a nd M . B . P le nio . Qua n t i f y in g C o he r e nc e . Ph y sic a l R ev iew Lett er s ,
113(14), S e pt e mbe r 2014. doi: 10 . 1103 /PhysRevLett. 113 . 140401 .
[42] J o h a n A be r g. Qua n t i f y in g S u pe r posit ion . a r X i v: q u a n t-p h/0612146 , D e c e mbe r 2006. arXiv:
quant- ph/ 0612146 .
[43] Ge or g ios S t y l i a r i s, N a mit A n a nd , L or e nz o C a mpos V e n ut i , a nd P a o lo Za n a r d i . Qua n tum c o he r e nc e
a nd the local i za t ion tr a n sit ion . Ph y sic a l R ev iew B , 100(22):224204, D e c e mbe r 2019. arXiv: 1906 .
09242 , doi: 10 . 1103 /PhysRevB. 100 . 224204 .
[44] A ndr e as W in t e r a nd D on g Y a n g. O pe r a t ion al R e s our c e The or y of C o he r e nc e . Ph y sic a l R ev iew Let-
t er s , 116(12), M a r ch 2016. doi: 10 . 1103 /PhysRevLett. 116 . 120404 .
[45] C é s a r A . R odr í g uez - R os a r io , Thom as F r a ue nheim, a nd A l á n A spur u- G uz i k . The r mody n a mic s of
qua n tum c o he r e nc e . a r X i v e-pr i n ts , p a g e a r X iv :1308.1245, Au g us t 2013. arXiv: 1308 . 1245 .
[46] L uca D ’ A le s sio , Y a r iv K a f r i , A n a t o l i P o l k o v ni k o v , a nd M a r c os R i g o l . F r om qua n tum ch a os a nd
ei g e n s t a t e the r m al i za t ion t o s t a t i s t ical me ch a nic s a nd the r mody n a mic s . A d v a n ce s i n Ph y sic s ,
65(3):239–362, M a y 2016. doi: 10 . 1080 / 00018732 . 2016 . 1198134 .
[47] F . Bor g ono v i , F . M . I zr ai le v , L .F . S a n t os, a nd V . G . Z e le v in sk y . Qua n tum ch a os a nd the r m al -
i za t ion in i s o l a t e d s ys t e m s of in t e r a ct in g p a r t icle s . Ph y sic s R ep o r ts , 626:1–58, 2016. Qua n-
tum ch a os a nd the r m al i za t ion in i s o l a t e d s ys t e m s of in t e r a ct in g p a r t icle s . UR L : https:
//www.sciencedirect.com/science/article/pii/S 0370157316000831 , doi:https://
doi.org/ 10 . 1016 /j.physrep. 2016 . 02 . 005 .
[48] V .K .B . Kot a . E m b e d de d R a n do m M a tr i x E n sem b le s i n Q u a n t u m Ph y sic s , v o lume 884 of Le c t u r e N o t e s
i n Ph y sic s . S pr in g e r I n t e r n a t ion al P ub l i shin g , C h a m, 2014. doi: 10 . 1007 / 978 - 3 - 319 - 04567 - 2 .
[49] A l f r é d R é n y i e t al . On me as ur e s of e n tr op y a nd infor m a t ion . I n P r o ce e d i n g s of t h e F o u r t h Ber k e-
le y S y m p o si u m o n M a t h em a t ic a l S t a t is t ic s a n d P r o b a bi l it y , V o l u m e 1: Co n tr i bu t io n s t o t h e Th e o r y of
S t a t is t ic s . The R e g e n ts of the U niv e r sit y of C al i for ni a , 1961.
[50] E r ic C hit a mb a r a nd G i l a d Gour . Qua n tum r e s our c e the or ie s . R ev iew s of M o der n Ph y sic s , 91(2),
A pr i l 2019. doi: 10 . 1103 /RevModPhys. 91 . 025001 .
[51] A l be r t W . M a r sh al l , I n gr a m Ol k in, a nd B a r r y C . A r no ld . I n e q u a l it ie s: Th e o r y of M aj o r i za t io n a n d I ts
App l ic a t io n s . S pr in g e r S e r ie s in S t a t i s t ic s . S pr in g e r S c ie nc e+B usine s s M e d i a , LL C , N e w Y ork , 2nd
e d e d it ion, 2011.
[52] S h ua np in g Du , Z h a ofa n g B ai , a nd Y u G uo . C ond it ion s for c o he r e nc e tr a n sfor m a t ion s unde r inc o -
he r e n t ope r a t ion s . Ph y sic a l R ev iew A , 91(5), M a y 2015. doi: 10 . 1103 /PhysRevA. 91 . 052120 .
[53] A lex a nde r S tr e lts o v , U tt a m S in gh, H im a dr i S he k h a r Dh a r , M a n a be ndr a N a th Be r a , a nd Ge r a r do
Ade s s o . M e as ur in g Qua n tum C o he r e nc e w ith E n t a n gle me n t. Ph y sic a l R ev iew Lett er s , 115(2), J uly
2015. doi: 10 . 1103 /PhysRevLett. 115 . 020403 .
204
[54] E r ic C hit a mb a r a nd M in- H siu H sie h . R e l a t in g the R e s our c e The or ie s of E n t a n gle me n t a nd Qua n-
tum C o he r e nc e . Ph y sic a l R ev iew Lett er s , 117(2):020402, J uly 2016. arXiv: 1509 . 07458 , doi:
10 . 1103 /PhysRevLett. 117 . 020402 .
[55] B a r t os z R e g ul a . C on v ex g e ome tr y of qua n tum r e s our c e qua n t i fica t ion . J o u r n a l of Ph y sic s A : M a t h -
em a t ic a l a n d Th e o r et ic a l , 51(4):045303, J a n ua r y 2018. doi: 10 . 1088 / 1751 - 8121 /aa 9100 .
[56] A lex a nde r S tr e lts o v , H e r m a nn K a mpe r m a nn, S a b ine W ö l k , M a n ue l Ge s s ne r , a nd D a gm a r B r u ß .
M a x im al c o he r e nc e a nd the r e s our c e the or y of pur it y . N ew J o u r n a l of Ph y sic s , 20(5):053058, M a y
2018. doi: 10 . 1088 / 1367 - 2630/aac 484 .
[57] J a me s R . G a r r i s on a nd T a r un Gr o v e r . D oe s a sin gle ei g e n s t a t e e nc ode the f ul l h a mi lt oni a n?
Ph y s . R ev . X , 8:021026, A pr 2018. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevX.
8 . 021026 , doi: 10 . 1103 /PhysRevX. 8 . 021026 .
[58] J . M . D e uts ch . The r mody n a mic e n tr op y of a m a n y - body e ne r g y ei g e n s t a t e . N ew J o u r n a l of Ph y sic s ,
12(7):075021, J uly 2010. arXiv: 0911 . 0056 , doi: 10 . 1088 / 1367 - 2630/ 12 / 7 / 075021 .
[59] Y iche n H ua n g. U niv e r s al ei g e n s t a t e e n t a n gle me n t of ch a ot ic local h a mi lt oni a n s . N u c le a r Ph y sic s B ,
938:594–604, 2019. doi: 10 . 1016 /j.nuclphysb. 2018 . 09 . 013 .
[60] L e a F . S a n t os, F r a nc i s c o P é r ez - Be r n al , a nd E . J on a th a n T or r e s - H e r r e r a . S pe ck of C h a os . a r X i v e-
pr i n ts , p a g e a r X iv :2006.10779, J une 2020. arXiv: 2006 . 10779 .
[61] LF S a n t os . I n t e gr a b i l it y of a d i s or de r e d hei s e nbe r g sp in-1/2 ch ain . J o u r n a l of Ph y sic s A : M a t h em a t-
ic a l a n d Gen er a l , 37(17 ):4723, 2004.
[62] E . J . T or r e s - H e r r e r a a nd L e a F . S a n t os . L ocal que nche s w ith glo b al effe cts in in t e r a ct in g qua n-
tum s ys t e m s . Ph y s . R ev . E , 89:062110, J un 2014. UR L : https://link.aps.org/doi/ 10 . 1103 /
PhysRevE. 89 . 062110 , doi: 10 . 1103 /PhysRevE. 89 . 062110 .
[63] E J T or r e s - H e r r e r a , D a v id a Ko l lm a r , a nd L e a F S a n t os . R e l a x a t ion a nd the r m al i za t ion of i s o l a t e d
m a n y - body qua n tum s ys t e m s . Ph y sic a Scr ip t a , T165:014018, oct 2015. UR L : https://doi.
org/ 10 . 1088 / 0031 - 8949 / 2015 /t 165 / 014018 , doi: 10 . 1088 / 0031 - 8949 / 2015 /t 165 / 014018 .
[64] E J T or r e s - H e r r e r a a nd L e a F S a n t os . D y n a mical m a ni fe s t a t ion s of qua n tum ch a os: c or r e l a t ion ho le
a nd bul g e . Ph i lo so p h ic a l T r a n sa c t io n s of t h e R oy a l So ciet y A : M a t h em a t ic a l , Ph y sic a l a n d E n g i n e er i n g
Scien ce s, 375(2108):20160434, 2017.
[65] M o hit P a ndey , P ie t e r W . C l a eys, D a v id K . C a mp be l l , A n a t o l i P o l k o v ni k o v , a nd Dr ie s S e l s . Ad i -
a b a t ic ei g e n s t a t e defor m a t ion s as a s e n sit iv e pr o be for qua n tum ch a os . Ph y s . R ev . X , 10:041017,
O ct 2020. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevX. 10 . 041017 , doi: 10 . 1103 /
PhysRevX. 10 . 041017 .
[66] M a rlon B r e ne s, T y le r L eB lond , J o hn Goo ld , a nd M a r c os R i g o l . E i g e n s t a t e the r m al i za t ion in a local ly
pe r tur be d in t e gr a b le s ys t e m . Ph y s . R ev . Lett . , 125:070605, Au g 2020. UR L : https://link.aps.
org/doi/ 10 . 1103 /PhysRevLett. 125 . 070605 , doi: 10 . 1103 /PhysRevLett. 125 . 070605 .
205
[67] J on as R ich t e r , A n a t o ly D y m a r sk y , R o b in S t eini g e w e g , a nd J oche n Ge mme r . E i g e n s t a t e the r m al -
i za t ion h y pothe si s bey ond s t a nd a r d ind ica t or s: E me r g e nc e of r a ndom- m a tr i x be h a v ior a t s m al l f r e-
que nc ie s . Ph y s . R ev . E , 102:042127, O ct 2020. UR L : https://link.aps.org/doi/ 10 . 1103 /
PhysRevE. 102 . 042127 , doi: 10 . 1103 /PhysRevE. 102 . 042127 .
[68] M a rk o Žnid a r ič . W e ak in t e gr a b i l it y br e ak in g : C h a os w ith in t e gr a b i l it y si gn a tur e in c o he r e n t d i f -
f usion . Ph y s . R ev . Lett . , 125:180605, O ct 2020. UR L : https://link.aps.org/doi/ 10 . 1103 /
PhysRevLett. 125 . 180605 , doi: 10 . 1103 /PhysRevLett. 125 . 180605 .
[69] L e a F . S a n t os a nd M a r c os R i g o l . On s e t of qua n tum ch a os in one- d ime n sion al bos onic a nd fe r mionic
s ys t e m s a nd its r e l a t ion t o the r m al i za t ion . Ph y sic a l R ev iew E , 81(3), M a r ch 2010. doi: 10 . 1103 /
PhysRevE. 81 . 036206 .
[70] L . F . S a n t os, F . Bor g ono v i , a nd F . M . I zr ai le v . On s e t of ch a os a nd r e l a x a t ion in i s o l a t e d s ys t e m s of
in t e r a ct in g sp in s: E ne r g y she l l a ppr o a ch . Ph y sic a l R ev iew E , 85(3), M a r ch 2012. doi: 10 . 1103 /
PhysRevE. 85 . 036209 .
[71] V l a d imir Z e le v in sk y , B A lex B r o w n, N j e m a F r a z ie r , a nd M i h ai H or o i . The n ucle a r she l l mode l as a
t e s t in g gr ound for m a n y - body qua n tum ch a os . Ph y sic s r ep o r ts , 276(2-3):85–176, 1996.
[72] E u g e ne P . W i gne r . C h a r a ct e r i s t ic V e ct or s of Bor de r e d M a tr ic e s W ith I nfinit e Dime n sion s . A n n a ls
of M a t h em a t ic s , 62(3):548–564, 1955. doi: 10 . 2307 / 1970079 .
[73] E u g e ne P . W i gne r . C h a r a ct e r i s t ic s V e ct or s of Bor de r e d M a tr ic e s w ith I nfinit e Dime n sion s I I. A n n a ls
of M a t h em a t ic s , 65(2):203–207, 1957. doi: 10 . 2307 / 1969956 .
[74] E u g e ne P . W i gne r . On the Di s tr i but ion of the R oots of C e r t ain S y mme tr ic M a tr ic e s . A n n a ls of
M a t h em a t ic s , 67(2):325–327, 1958. doi: 10 . 2307 / 1970008 .
[75] F r e e m a n J . D ys on . S t a t i s t ical The or y of the E ne r g y L e v e l s of C omp lex S ys t e m s . I. J o u r n a l of M a t h -
em a t ic a l Ph y sic s , 3(1):140–156, J a n ua r y 1962. doi: 10 . 1063 / 1 . 1703773 .
[76] J ua n M ald a c e n a , S t e p he n H. S he nk e r , a nd D ou gl as S t a nfor d . A bound on ch a os . J o u r n a l of H i g h
E n er g y Ph y sic s , 2016(8):106, Au g us t 2016. arXiv: 1503 . 01409 , doi: 10 . 1007 /JHEP 08 ( 2016 ) 106 .
[77] D a nie l A . R o be r ts a nd D ou gl as S t a nfor d . Di a gnosin g ch a os usin g four -po in t f unc -
t ion s in t w o - d ime n sion al c onfor m al fie ld the or y . Ph y s . R ev . Lett . , 115(13):131603,
S e p 2015. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevLett. 115 . 131603 ,
doi: 10 . 1103 /PhysRevLett. 115 . 131603 .
[78] J os e p h P o lchin sk i a nd V l a d imir R os e nh a us . The spe ctr um in the S a chde v - Y e- Kit a e v mode l . J o u r n a l
of H i g h E n er g y Ph y sic s , 2016(4):1, A pr i l 2016. arXiv: 1601. 06768 , doi: 10 . 1007 /JHEP 04 ( 2016 )
001 .
[79] M á rk M ez ei a nd D ou gl as S t a nfor d . On e n t a n gle me n t spr e a d in g in ch a ot ic s ys t e m s . J o u r n a l of H i g h
E n er g y Ph y sic s , 2017(5):65, M a y 2017. arXiv: 1608 . 05101 , doi: 10 . 1007 /JHEP 05 ( 2017 ) 065 .
[80] D a nie l A . R o be r ts a nd Be ni Y oshid a . C h a os a nd c omp lex it y b y de si gn . J o u r n a l of H i g h E n er g y Ph y sic s ,
2017(4):121, A pr i l 2017. arXiv: 1610 . 04903 , doi: 10 . 1007 /JHEP 04 ( 2017 ) 121 .
206
[81] I gn a c io G a r c í a - M a t a , M a r c os S a r a c e no , R odo l fo A . J al a be r t , Au g us t o J . R onca gl i a , a nd Die g o A .
W i s ni a ck i . C h a os si gn a tur e s in the shor t a nd lon g t ime be h a v ior of the out - of -t ime or de r e d c or r e-
l a t or . Ph y sic a l R ev iew Lett er s , 121:210601, 2018. UR L : https://link.aps.org/doi/ 10 . 1103 /
PhysRevLett. 121 . 210601 , doi: 10 . 1103 /PhysRevLett. 121 . 210601 .
[82] E mi l i a no M . F or t e s, I gn a c io G a r c í a - M a t a , R odo l fo A . J al a be r t , a nd Die g o A . W i s ni a ck i . G a u g -
in g cl as sical a nd qua n tum in t e gr a b i l it y thr ou gh out - of -t ime- or de r e d c or r e l a t or s . Ph y sic a l R ev iew
E , 100:042201, 2019. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevE. 100 . 042201 ,
doi: 10 . 1103 /PhysRevE. 100 . 042201 .
[83] B in Y a n, L uk as z C inc io , a nd W oj c ie ch H. Z ur e k . I nfor m a t ion S cr a mb l in g a nd L os chmidt E cho .
Ph y sic a l R ev iew Lett er s , 124(16):160603, A pr i l 2020. doi: 10 . 1103 /PhysRevLett. 124 . 160603 .
[84] Ge or g ios S t y l i a r i s, N a mit A n a nd , a nd P a o lo Za n a r d i . I nfor m a t ion s cr a mb l in g o v e r b ip a r t i -
t ion s: E qui l i br a t ion, e n tr op y pr oduct ion, a nd t y p ical it y . Ph y s . R ev . Lett . , 126:030601, J a n
2021. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevLett. 126. 030601 , doi: 10 .
1103 /PhysRevLett. 126. 030601 .
[85] El l iott H L ie b a nd D e r e k W R o b in s on . The finit e gr ou p v e loc it y of qua n tum sp in s ys t e m s . I n
S t a t is t ic a l m e c h a n ic s , p a g e s 425–431. S pr in g e r , 1972.
[86] M a tthe w B . H as t in gs a nd T o hr u Kom a . S pe ctr al G a p a nd E x pone n t i al D e ca y of C or r e l a -
t ion s . Co m m u n ic a t io n s i n M a t h em a t ic a l Ph y sic s , 265(3):781–804, Au g us t 2006. doi: 10 . 1007 /
s 00220 - 006 - 0030 - 4 .
[87] D a nie l A . R o be r ts a nd B r i a n S w in gle . L ie b - r o b in s on bound a nd the butt e r fly effe ct in qua n tum fie ld
the or ie s . Ph y s . R ev . Lett . , 117(9):091602, Au g 2016. UR L : https://link.aps.org/doi/ 10 .
1103 /PhysRevLett. 117 . 091602 , doi: 10 . 1103 /PhysRevLett. 117 . 091602 .
[88] P a o lo Za n a r d i , Ge or g ios S t y l i a r i s, a nd L or e nz o C a mpos V e n ut i . C o he r e nc e- g e ne r a t in g po w e r of
qua n tum unit a r y m a ps a nd bey ond . Ph y s ic a l R ev iew A , 95(5), M a y 2017. doi: 10 . 1103 /PhysRevA.
95 . 052306 .
[89] P a o lo Za n a r d i , Ge or g ios S t y l i a r i s, a nd L or e nz o C a mpos V e n ut i . M e as ur e s of c o he r e nc e- g e ne r a t in g
po w e r for qua n tum unit al ope r a t ion s . Ph y sic a l R ev iew A , 95(5), M a y 2017. doi: 10 . 1103 /
PhysRevA. 95 . 052307 .
[90] P a o lo Za n a r d i a nd L or e nz o C a mpos V e n ut i . Qua n tum c o he r e nc e g e ne r a t in g po w e r , m a x im al ly
a be l i a n s ub al g e br as, a nd Gr as s m a nni a n g e ome tr y . J o u r n a l of M a t h em a t ic a l Ph y sic s , 59(1):012203,
J a n ua r y 2018. doi: 10 . 1063 / 1 . 4997146 .
[91] F a us t o Bor g ono v i , F e l i x M . I zr ai le v , a nd L e a F . S a n t os . T ime s cale s in the que nch dy n a mic s of m a n y -
body qua n tum s ys t e m s: P a r t ic ip a t ion r a t io v e r s us out - of -t ime or de r e d c or r e l a t or . Ph y sic a l R ev iew
E , 99(5), M a y 2019. doi: 10 . 1103 /PhysRevE. 99 . 052143 .
[92] R . J . L e w i s - S w a n, A . S a fa v i - N aini , J . J . Bo l l in g e r , a nd A . M . R ey . U ni f y in g fas t s cr a mb l in g ,
the r m al i za t ion a nd e n t a n gle me n t thr ou gh the me as ur e me n t of F O T O C s in the Dick e mode l .
N a t u r e Co m m u n ic a t io n s , 10(1):1581, D e c e mbe r 2019. arXiv: 1808 . 07134 , doi: 10 . 1038 /
s 41467 - 019 - 09436 - y .
207
[93] P hi l l ip Gr i ffith s a nd J os e p h H a r r i s . P r i n cip le s of a l ge br a ic ge o m etr y , v o lume 19. W i ley Onl ine L i br a r y ,
1978.
[94] ’ A n g e l R iv as a nd S us a n a F H ue l g a . O p en Q u a n t u m S y s t em s a n I n tr o d u c t io n . S pr in g e r Be rl in H eide l -
be r g , Be rl in, H eide l be r g , 2012.
[95] D a v id P DiV inc e nz o , D e b b ie W L e un g , a nd B a r b a r a M T e rh al . Qua n tum d a t a hid in g. I E E E T r a n s-
a c t io n s o n I nf o r m a t io n Th e o r y , 48(3):580–598, 2002. doi: 10 . 1109 / 18 . 985948 .
[96] J os e p h M R e ne s, R o b in B lume- Ko hout , A ndr e w J S c ott , a nd C a rlt on M C a v e s . S y mme tr ic infor -
m a t ion al ly c omp le t e qua n tum me as ur e me n ts . J o u r n a l of M a t h em a t ic a l Ph y sic s , 45(6):2171–2180,
2004. doi: 10 . 1063 / 1 . 1737053 .
[97] A ndr e w J S c ott. T i gh t infor m a t ion al ly c omp le t e qua n tum me as ur e me n ts . J o u r n a l of Ph y sic s A :
M a t h em a t ic a l a n d Gen er a l , 39(43):13507, 2006. doi: 10 . 1088 / 0305 - 4470 / 39 / 43 / 009 .
[98] D a v id Gr os s, Koe nr a a d Aude n a e r t , a nd J e n s E i s e r t. E v e nly d i s tr i but e d unit a r ie s: On the s tr uctur e
of unit a r y de si gn s . J o u r n a l of M a t h em a t ic a l Ph y sic s , 48(5):052104, 2007. doi: 10 . 1063 / 1 . 2716992 .
[99] J or d a n C otle r , N icho l as H un t e r - J one s, J un y u L iu , a nd Be ni Y oshid a . C h a os, c omp lex it y , a nd r a n-
dom m a tr ic e s . J o u r n a l of H i g h E n er g y Ph y sic s , 2017(11):48, N o v e mbe r 2017. doi: 10 . 1007 /
JHEP 11 ( 2017 ) 048 .
[100] S . A . G a r d ine r , J . I. C ir a c, a nd P . Z o l le r . Qua n tum ch a os in a n ion tr a p: The de lt a -k ick e d h a r monic
os c i l l a t or . Ph y s . R ev . Lett . , 79:4790–4793, D e c 1997. UR L : https://link.aps.org/doi/ 10 .
1103 /PhysRevLett. 79 . 4790 , doi: 10 . 1103 /PhysRevLett. 79 . 4790 .
[101] M ich a e l V ict or Be r r y . S e micl as sical the or y of spe ctr al r i g id it y . P r o ce e d i n g s of t h e R oy a l So ciet y of
Lo n do n. A . M a t h em a t ic a l a n d Ph y sic a l Scien ce s , 400(1819):229–251, 1985.
[102] J un y u L iu . S pe ctr al for m fa ct or s a nd l a t e t ime qua n tum ch a os . Ph y s . R ev . D , 98:086026, O ct
2018. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevD. 98 . 086026 , doi: 10 . 1103 /
PhysRevD. 98 . 086026 .
[103] J or d a n S . C otle r , G u y G ur - A r i , M as a nor i H a n a d a , J os e p h P o lchin sk i , P hi l S a a d , S t e p he n H. S he nk e r ,
D ou gl as S t a nfor d , A lex a ndr e S tr eiche r , a nd M as ak i T ez uk a . B l a ck ho le s a nd r a ndom m a tr i -
c e s . J o u r n a l of H i g h E n er g y Ph y sic s , 2017(5):118, M a y 2017. arXiv: 1611 . 04650 , doi: 10 . 1007 /
JHEP 05 ( 2017 ) 118 .
[104] P a v a n H os ur , X i a o - L i a n g Qi , D a nie l A . R o be r ts, a nd Be ni Y oshid a . C h a os in qua n tum ch a nne l s .
J o u r n a l of H i g h E n er g y Ph y sic s , 2016(2):4, F e br ua r y 2016. doi: 10 . 1007 /JHEP 02 ( 2016 ) 004 .
[105] M iche l L e dou x . Th e Co n cen tr a t io n of M e as u r e Ph en o m en o n , v o lume 89 of M a t h em a t ic a l Su r v e y s a n d
M o n o g r a p h s . A me r ica n M a the m a t ical S oc ie t y , P r o v ide nc e , R hode I sl a nd , F e br ua r y 2005. doi:
10 . 1090 /surv/ 089 .
[106] El l iott H L ie b a nd D e r e k W R o b in s on . The finit e gr ou p v e loc it y of qua n tum sp in s ys t e m s . I n
S t a t is t ic a l m e c h a n ic s , p a g e s 425–431. S pr in g e r , 1972.
208
[107] A . V e r sh y nin a a nd E . H. L ie b . L ie b - R o b in s on bound s . Sc h o l a r p e d i a , 8(9):31267, 2013. r e v i sion
#150021. doi: 10 . 4249 /scholarpedia. 31267 .
[108] I v a n K uk ul j a n, S a š o Gr o z d a no v , a nd T om a ž P r os e n . W e ak qua n tum ch a os . Ph y s . R ev . B ,
96:060301, Au g 2017. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevB. 96 . 060301 ,
doi: 10 . 1103 /PhysRevB. 96 . 060301 .
[109] C . W . v on Keys e rl in gk , T i bor R ak o v s z k y , F r a nk P o l lm a nn, a nd S . L . S ond hi . O pe r a t or h y dr o -
dy n a mic s, ot oc s, a nd e n t a n gle me n t gr o w th in s ys t e m s w ithout c on s e r v a t ion l a w s . Ph y s . R ev . X ,
8(2):021013, A pr 2018. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevX. 8 . 021013 ,
doi: 10 . 1103 /PhysRevX. 8 . 021013 .
[110] Ad a m N ah um, S a g a r V i j a y , a nd J e on g w a n H a ah . O pe r a t or spr e a d in g in r a ndom unit a r y c ir -
c uits . Ph y s . R ev . X , 8(2):021014, A pr 2018. UR L : https://link.aps.org/doi/ 10 . 1103 /
PhysRevX. 8 . 021014 , doi: 10 . 1103 /PhysRevX. 8 . 021014 .
[111] V e d i k a K he m a ni , A sh v in V i sh w a n a th, a nd D a v id A . H us e . O pe r a t or spr e a d in g a nd the e me r g e nc e of
d i s sip a t iv e h y dr ody n a mic s unde r unit a r y e v o lut ion w ith c on s e r v a t ion l a w s . Ph y s . R ev . X , 8:031057,
S e p 2018. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevX. 8 . 031057 , doi: 10 . 1103 /
PhysRevX. 8 . 031057 .
[112] T i bor R ak o v s z k y , F r a nk P o l lm a nn, a nd C . W . v on Keys e rl in gk . Di ff usiv e h y dr ody n a mic s of out - of -
t ime- or de r e d c or r e l a t or s w ith ch a r g e c on s e r v a t ion . Ph y s . R ev . X , 8:031058, S e p 2018. UR L : https:
//link.aps.org/doi/ 10 . 1103 /PhysRevX. 8 . 031058 , doi: 10 . 1103 /PhysRevX. 8 . 031058.
[113] P a tr ick J . C o le s, M a r io Be r t a , M a r c o T om a miche l , a nd S t e p h a nie W e hne r . E n tr op ic unc e r t ain t y
r e l a t ion s a nd their a pp l ica t ion s . R ev iew s of M o der n Ph y sic s , 89(1), F e br ua r y 2017. doi: 10 . 1103 /
RevModPhys. 89 . 015002 .
[114] A lexei Y u Kit a e v , A lex a nde r S he n, a nd M i k h ai l N V yaly i . C l assic a l a n d q u a n t u m co m pu t a t io n , v o l -
ume 47. A me r ica n M a the m a t ical S oc ie t y , 2002. doi: 10 . 1090 /gsm/ 047 .
[115] D a nie l M . Gr e e nbe r g e r , M ich a e l A . H or ne , a nd A n t on Z ei l in g e r . Go in g bey ond be l l ’ s the or e m,
2007. arXiv: 0712 . 0921 .
[116] I m a n M a r v i a n, R o be r t W . S pe k k e n s, a nd P a o lo Za n a r d i . Qua n tum spe e d l imits, c o he r e nc e , a nd
as y mme tr y . Ph y sic a l R ev iew A , 93(5):052331, M a y 2016. UR L : https://link.aps.org/doi/
10 . 1103 /PhysRevA. 93 . 052331 , doi: 10 . 1103 /PhysRevA. 93 . 052331 .
[117] I m a n M a r v i a n a nd R o be r t W . S pe k k e n s . H o w t o qua n t i f y c o he r e nc e: Di s t in g ui shin g spe ak a b le a nd
un spe ak a b le not ion s . Ph y sic a l R ev iew A , 94(5), N o v e mbe r 2016. doi: 10 . 1103 /PhysRevA. 94 .
052324 .
[118] Ge or g ios S t y l i a r i s, N a mit A n a nd , a nd P a o lo Za n a r d i . I nfor m a t ion S cr a mb l in g o v e r B ip a r t it ion s:
E qui l i br a t ion, E n tr op y P r oduct ion, a nd T y p ical it y . a r X i v e-pr i n ts , p a g e a r X iv :2007.08570, J uly 2020.
arXiv: 2007 . 08570 .
[119] L or e nz o C a mpos V e n ut i . The r e c ur r e nc e t ime in qua n tum me ch a nic s . a r X i v e-pr i n ts , p a g e
a r X iv :1509.04352, S e pt e mbe r 2015. arXiv: 1509 . 04352 .
209
[120] S hm ue l F i shm a n, D . R . Gr e mpe l , a nd R . E . P r a n g e . C h a os, qua n tum r e c ur r e nc e s, a nd a nde r s on
local i za t ion . Ph y s . R ev . Lett . , 49:509–512, Au g 1982. UR L : https://link.aps.org/doi/ 10 .
1103 /PhysRevLett. 49 . 509 , doi: 10 . 1103 /PhysRevLett. 49 . 509 .
[121] Z b i gnie w P uch ał a a nd J a r osł a w Ad a m M i s z czak . S y mbo l ic in t e gr a t ion w ith r e spe ct t o the H a a r
me as ur e on the unit a r y gr ou p . a r X i v e-pr i n ts , p a g e a r X iv :1109.4244, S e pt e mbe r 2011. arXiv:
1109 . 4244 .
[122] Ge or g ios S t y l i a r i s, N a mit A n a nd , L or e nz o C a mpos V e n ut i , a nd P a o lo Za n a r d i . Qua n tum c o he r e nc e
a nd the local i za t ion tr a n sit ion . Ph y s . R ev . B , 100(22):224204, D e c e mbe r 2019. arXiv: 1906 . 09242 ,
doi: 10 . 1103 /PhysRevB. 100 . 224204 .
[123] D a v id Bo hm . Q u a n t u m t h e o r y . C our ie r C or por a t ion, 1951.
[124] J o h a n Å be r g. Qua n t i f y in g s u pe r posit ion . a r X i v: q u a n t-p h/0612146 , 2 006.
[125] T . B a um gr a tz , M . C r a me r , a nd M . B . P le nio . Qua n t i f y in g c o he r e nc e . Ph y s . R ev . Lett . ,
113(14):140401, S e pt e mbe r 2014. doi: 10 . 1103 /PhysRevLett. 113 . 140401 .
[126] A lex a nde r S tr e lts o v , Ge r a r do Ade s s o , a nd M a r t in B . P le nio . C o l loquium : Qua n tum c o he r e nc e as a
r e s our c e . R ev . M o d. Ph y s . , 89(4):041003, O ct o be r 2017. doi: 10 . 1103 /RevModPhys. 89 . 041003 .
[127] P . W . A nde r s on . A bs e nc e of d i ff usion in c e r t ain r a ndom l a tt ic e s . Ph y s . R ev . , 109:1492–1505, 1958.
doi: 10 . 1103 /PhysRev. 109 . 1492 .
[128] A a r t L a g e nd i jk , B a r t V a n T i gg e le n, a nd Die de r i k S W ie r s m a . F i ft y y e a r s of a nde r s on local i za t ion .
Ph y s . T o d a y , 62(8):24–29, 2009. doi: 10 . 1063 / 1 . 3206091 .
[129] D . M . B ask o , I.L . A leine r , a nd B .L . A ltsh ule r . M e t al –in s ul a t or tr a n sit ion in a w e ak ly in t e r a ct i n g m a n y -
e le ctr on s ys t e m w ith local i z e d sin gle-p a r t icle s t a t e s . A n n a ls of Ph y sic s , 321(5):1126–1205, 2006.
doi:https://doi.org/ 10 . 1016 /j.aop. 2005 . 11 . 014 .
[130] A r i j e e t P al a nd D a v id A . H us e . M a n y - body local i za t ion p h as e tr a n sit ion . Ph y s . R ev . B , 82:174411,
2010. doi: 10 . 1103 /PhysRevB. 82 . 174411 .
[131] R ah ul N a nd k i shor e a nd D a v id A . H us e . M a n y - Body L ocal i za t ion a nd The r m al i za t ion in Qua n tum
S t a t i s t ical M e ch a nic s . A n n u a l R ev iew of Co n den se d M a tt er Ph y sic s , 6(1):15–38, M a r ch 2015. doi:
10 . 1146 /annurev- conmatphys- 031214 - 014726 .
[132] P a o lo Za n a r d i a nd L or e nz o C a mpos V e n ut i . Qua n tum c o he r e nc e g e ne r a t in g po w e r , m a x im al ly
a be l i a n s ub al g e br as, a nd gr as s m a nni a n g e ome tr y . J o u r n a l of M a t h em a t ic a l Ph y sic s , 59(1):012203,
J a n ua r y 2018. doi: 10 . 1063 / 1 . 4997146 .
[133] P e t e r P r e lo v š e k , M a r c in M ie r z ej e w sk i , O B a r i šić, a nd J a c e k H e r br y ch . D e n sit y c or r e l a t ion s a nd
tr a n spor t in mo de l s of m a n y - body local i za t ion . A n n a len der Ph y si k , 529(7):1600362, 2017. doi:
10 . 1002 /andp. 201600362 .
[134] E r ic C hit a mb a r a nd G i l a d Gour . C omp a r i s on of inc o he r e n t ope r a t ion s a nd me as ur e s of c o he r e nc e .
Ph y s . R ev . A , 94(5):052336, N o v e mbe r 2016. arXiv: 1602 . 06969 , doi: 10 . 1103 /PhysRevA. 94 .
052336 .
210
[135] A ndr e as W in t e r a nd D on g Y a n g. O pe r a t ion al r e s our c e the or y of c o he r e nc e . Ph y s . R ev . Lett . ,
116(12):120404, M a r ch 2016. doi: 10 . 1103 /PhysRevLett. 116 . 120404 .
[136] Be n j a min Y a d in, J i a jun M a , D a v ide G ir o l a mi , M i le G u , a nd V l a tk o V e dr al . Qua n tum pr oc e s s e s
w hich do not us e c o he r e nc e . Ph y s . R ev . X , 6(4):041028, N o v e mbe r 2016. doi: 10 . 1103 /
PhysRevX. 6 . 041028 .
[137] Qi Z h a o , Y unch a o L iu , X i a o Y ua n, E r ic C hit a mb a r , a nd X ion g fe n g M a . One- shot c o he r e nc e d i lu-
t ion . Ph y s . R ev . Lett . , 120:070403, 2018. doi: 10 . 1103 /PhysRevLett. 120. 070403 .
[138] I n g e m a r Be n g ts s on a nd K a r o l Ż y cz k o w sk i . Ge o m etr y of q u a n t u m s t a t e s: A n i n tr o d u c t io n t o q u a n t u m
en t a n g lem en t . C a mbr id g e U niv e r sit y P r e s s, 2017. doi: 10 . 1017 /CBO 9780511535048 .
[139] S h ua np in g Du , Z h a ofa n g B ai , a nd Y u G uo . C ond it ion s for c o he r e nc e tr a n sfor m a t ion s unde r inc o -
he r e n t ope r a t ion s . Ph y s . R ev . A , 9 1(5):052120, M a y 2015. doi: 10 . 1103 /PhysRevA. 91 . 052120 .
[140] P a o lo Za n a r d i , Ge or g ios S t y l i a r i s, a nd L or e nz o C a mpos V e n ut i . M e as ur e s of c o he r e nc e- g e ne r a t in g
po w e r for qua n tum unit al ope r a t ion s . Ph y s . R ev . A , 95(5):052307, M a y 2017. doi: 10 . 1103 /
PhysRevA. 95 . 052307 .
[141] A l be r t W M a r sh al l , I n gr a m Ol k in, a nd B a r r y C A r no ld . I n e q u a l it ie s: Th e o r y of M aj o r i za t io n a n d I ts
App l ic a t io n s , v o lume 143. S pr in g e r , 1979.
[142] P e t e r R eim a nn . F ound a t ion of s t a t i s t ical me ch a nic s unde r ex pe r ime n t al ly r e al i s t ic c ond it ion s . Ph y s .
R ev . Lett . , 101(19):190403, N o v e mbe r 2008. doi: 10 . 1103 /PhysRevLett. 101 . 190403 .
[143] N o ah L inde n, S a ndu P ope s c u , A n thon y J . S hor t , a nd A ndr e as W in t e r . Qua n tum me ch a nical
e v o lut ion t o w a r d s the r m al e qui l i br ium . Ph y s . R ev . E , 79(6):061103, J une 2009. doi: 10 . 1103 /
PhysRevE. 79 . 061103 .
[144] L or e nz o C a mpos V e n ut i a nd P a o lo Za n a r d i . U nit a r y e qui l i br a t ion s: P r o b a b i l it y d i s tr i but ion of the
los chmidt e cho . Ph y s . R ev . A , 81(2):022113, F e br ua r y 2010. doi: 10 . 1103 /PhysRevA. 81 . 022113 .
[145] L or e nz o C a mpos V e n ut i , N . T o b i as J a c o bs on, S idd h a r th a S a n tr a , a nd P a o lo Za n a r d i . E x a ct infinit e-
t ime s t a t i s t ic s of the los chmidt e cho for a qua n tum que nch . Ph y s . R ev . Lett . , 107(1):010403, J uly
2011. doi: 10 . 1103 /PhysRevLett. 107 . 010403 .
[146] D a v id J . L uitz a nd Y e v g e n y B a r L e v . The e r g od ic side of the m a n y - body local i za t ion tr a n sit ion .
A n n a len der Ph y si k , 529(7):1600350, 2017. doi: 10 . 1002 /andp. 201600350 .
[147] Dirk H unde r tm a rk . A shor t in tr oduct ion t o a nde r s on local i za t ion . I n A n a l y sis a n d s t o c h as t ic s of
g r o w t h pr o ce sse s a n d i n t er f a ce m o de ls , p a g e s 194–219. Ox for d U niv e r sit y P r e s s, N e w Y ork , 2008.
[148] P L lo y d . E x a ctly s o lv a b le mode l of e le ctr onic s t a t e s in a thr e e- d ime n sion al d i s or de r e d h a mi lt oni a n :
non- ex i s t e nc e of local i z e d s t a t e s . J o u r n a l of Ph y sic s C : So l i d S t a t e Ph y sic s , 2(10):1717–1725, 1969.
doi: 10 . 1088 / 0022 - 3719 / 2 / 10 / 303 .
[149] D J Thoule s s . A r e l a t ion be t w e e n the de n sit y of s t a t e s a nd r a n g e of local i za t ion for one d ime n-
sion al r a ndom s ys t e m s . J o u r n a l of Ph y sic s C : So l i d S t a t e Ph y sic s , 5(1):77–81, 1972. doi: 10 . 1088 /
0022 - 3719 / 5 / 1 / 010 .
211
[150] Dmitr y A . A b a nin, Eh ud A ltm a n, I mm a n ue l B loch, a nd M ak s y m S e r b y n . C o l loquium : M a n y - body
local i za t ion, the r m al i za t ion, a nd e n t a n gle me n t. R ev . M o d. Ph y s . , 91(2):021001, M a y 2019. doi:
10 . 1103 /RevModPhys. 91 . 021001 .
[151] M ak s y m S e r b y n, Z. P a p ić, a nd Dmitr y A . A b a nin . U niv e r s al slo w gr o w th of e n t a n gle me n t in
in t e r a ct in g s tr on gly d i s or de r e d s ys t e m s . Ph y s . R ev . Lett . , 110:260601, 2013. doi: 10 . 1103 /
PhysRevLett. 110 . 260601 .
[152] A . D e L uca a nd A . S ca r d ic chio . E r g od ic it y br e ak in g in a mode l sho w in g m a n y - body local i za t ion .
E PL ( E u r o p h y sic s Lett er s ) , 101(3):37003, 2013. doi: 10 . 1209/ 0295 - 5075 / 101 / 37003 .
[153] J . Goo ld , C . Go g o l in, S . R . C l a rk , J . E i s e r t , A . S ca r d ic chio , a nd A . S i lv a . T ot al c or r e l a t ion s of the
d i a g on al e n s e mb le he r ald the m a n y - body local i za t ion tr a n sit ion . Ph y s . R ev . B , 92:180202, 2015.
doi: 10 . 1103 /PhysRevB. 92 . 180202 .
[154] E . J . T or r e s - H e r r e r a a nd L e a F . S a n t os . D y n a mic s a t the m a n y - body local i za t ion tr a n sit ion . Ph y s .
R ev . B , 92:014208, 2015. doi: 10 . 1103 /PhysRevB. 92 . 014208.
[155] E . J . T or r e s - H e r r e r a a nd L e a F . S a n t os . E x t e nde d none r g od ic s t a t e s in d i s or de r e d m a n y - body qua n-
tum s ys t e m s . A n n a len der Ph y si k , 529(7):1600284, 2017. doi: 10 . 1002 /andp. 201600284 .
[156] M ak s y m S e r b y n a nd Dmitr y A . A b a nin . L os chmidt e cho in m a n y - body local i z e d p h as e s . Ph y s . R ev .
B , 96:014202, 2017. doi: 10 . 1103 /PhysRevB. 96 . 014202 .
[157] P a o lo Za n a r d i a nd N i k o l a P a unk o v ić . Gr ound s t a t e o v e rl a p a nd qua n tum p h as e tr a n sit ion s . Ph y s .
R ev . E , 74:031123, 2006. doi: 10 . 1103 /PhysRevE. 74 . 031123 .
[158] L or e nz o C a mpos V e n ut i a nd P a o lo Za n a r d i . Qua n tum cr it ical s cal in g of the g e ome tr ic t e n s or s .
Ph y s . R ev . Lett . , 99:095701, 2007. doi: 10 . 1103 /PhysRevLett. 99 . 095701 .
[159] W e n- L on g Y ou , Y in g - W ai L i , a nd S hi - J i a n G u . F ide l it y , dy n a mic s tr uctur e fa ct or , a nd s us c e pt i b i l it y
in cr it ical p he nome n a . Ph y s . R ev . E , 76:022101, 2 007. doi: 10 . 1103 /PhysRevE. 76 . 022101 .
[160] P a o lo Za n a r d i , P a o lo G ior d a , a nd M a r c o C o z z ini . I nfor m a t ion-the or e t ic d i ffe r e n t i al g e ome tr y
of qua n tum p h as e tr a n sit ion s . Ph y s . R ev . Lett . , 99(10):100603, S e pt e mbe r 2007. doi: 10 . 1103 /
PhysRevLett. 99 . 100603 .
[161] M ich a e l Ko lodr ube tz , Dr ie s S e l s, P a nk a j M e h t a , a nd A n a t o l i P o l k o v ni k o v . Ge ome tr y a nd non-
a d i a b a t ic r e spon s e in qua n tum a nd cl as sical s ys t e m s . Ph y s ic s R ep o r ts , 697:1–87, 2017. doi:https:
//doi.org/ 10 . 1016 /j.physrep. 2017 . 07 . 001 .
[162] J . S . H e lt on, K . M a t a n, M . P . S hor e s, E . A . N y tk o , B . M . B a r tle tt , Y . Qiu , D . G . N oc e r a , a nd Y . S . L e e .
D y n a mic s cal in g in the s us c e pt i b i l it y of the sp in-
1
2
k a g ome l a tt ic e a n t i fe r r om a gne t he r be r ts mithit e .
Ph y s . R ev . Lett . , 104:147201, 2010. doi: 10 . 1103 /PhysRevLett. 104 . 147201 .
[163] P e n g che n g D ai . A n t i fe r r om a gne t ic or de r a nd sp in dy n a mic s in ir on- b as e d s u pe r c onduct or s . R ev .
M o d. Ph y s . , 87:855–896, 2015. doi: 10 . 1103 /RevModPhys. 87 . 855 .
212
[164] P hi l ipp H a uk e , M a rk us H ey l , L uca T a gl i a c o z z o , a nd P e t e r Z o l le r . M e as ur in g m ult ip a r t it e e n-
t a n gle me n t thr ou gh dy n a mic s us c e pt i b i l it ie s . N a t u r e Ph y sic s , 12(8):778, 2016. doi: 10 . 1038 /
nphys 3700 .
[165] O s or S . B a r i šić, J ur e Ko k al j , I v a n B alo g , a nd P e t e r P r e lo v š e k . D y n a mical c onduct iv it y a nd its fluc -
tua t ion s alon g the cr os s o v e r t o m a n y - body local i za t ion . Ph y s . R ev . B , 94:045126, 2016. doi:
10 . 1103 /PhysRevB. 94 . 045126 .
[166] A le s s a ndr o S i lv a . S t a t i s t ic s of the w ork done on a qua n tum cr it ical s ys t e m b y que nchin g a c on tr o l
p a r a me t e r . Ph y s . R ev . Lett . , 101:120603, S e p 2008. UR L : https://link.aps.org/doi/ 10 .
1103 /PhysRevLett. 101 . 120603 , doi: 10 . 1103 /PhysRevLett. 101 . 120603 .
[167] Thom as M C o v e r a nd J o y A Thom as . E lem en ts of i nf o r m a t io n t h e o r y . J o hn W i ley & S on s, H o bo k e n,
N e w J e r s ey , 2012.
[168] Ge or g ios S t y l i a r i s, N a mit A n a nd , a nd P a o lo Za n a r d i . I nfor m a t ion s cr a mb l in g o v e r b ip a r t i -
t ion s: E qui l i br a t ion, e n tr op y pr oduct ion, a nd t y p ical it y . Ph y s . R ev . Lett . , 126:030601, J a n
2021. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevLett. 126. 030601 , doi: 10 .
1103 /PhysRevLett. 126. 030601 .
[169] D on N P a g e . A v e r a g e e n tr op y of a s ubs ys t e m . Ph y sic a l R ev iew Lett er s , 71(9):1291, 1993. doi:
10 . 1103 /PhysRevLett. 71 . 1291 .
[170] P a tr ick H a y de n a nd J o hn P r e sk i l l . B l a ck ho le s as mir r or s: qua n tum infor m a t ion in r a ndom s ubs ys -
t e m s . J o u r n a l of H i g h E n er g y Ph y sic s , 2007(09):120, 2007. doi: 10 . 1088 / 1126 - 6708 / 2007 / 09 / 120 .
[171] P a v a n H os ur , X i a o - L i a n g Qi , D a nie l A R o be r ts, a nd Be ni Y oshid a . C h a os in qua n tum ch a nne l s .
J o u r n a l of H i g h E n er g y Ph y sic s , 2016(2):4, 2016. doi: 10 . 1007 /JHEP 02 ( 2016 ) 004 .
[172] C W V on Keys e rl in gk , T i bor R ak o v s z k y , F r a nk P o l lm a nn, a nd S hiv a ji L al S ond hi . O pe r a t or h y -
dr ody n a mic s, O T O C s, a nd e n t a n gle me n t gr o w th in s ys t e m s w ithout c on s e r v a t ion l a w s . Ph y sic a l
R ev iew X, 8(2):021013, 2018. doi: 10 . 1103 /PhysRevX. 8 . 021013 .
[173] S a n j a y M oud g alya , T r ithe p D e v ak ul , C W V on Keys e rl in gk , a nd S L S ond hi . O pe r a t or spr e a d in g in
qua n tum m a ps . Ph y sic a l R ev iew B , 99(9):094312, 2019. doi: 10 . 1103 /PhysRevB. 99 . 094312 .
[174] El l iott H L ie b a nd D e r e k W R o b in s on . The finit e gr ou p v e loc it y of qua n tum sp in s ys t e m s . I n
S t a t is t ic a l m e c h a n ic s , p a g e s 425–431. S pr in g e r , 1972. doi: 10 . 1007 /BF 01645779 .
[175] M a tthe w B H as t in gs . L ie b - S ch ultz - M a tt i s in hi ghe r d ime n sion s . Ph y sic a l R ev iew B , 69(10):104431,
2004. doi: 10 . 1103 /PhysRevB. 69 . 104431 .
[176] D a nie l A R o be r ts a nd B r i a n S w in gle . L ie b - R o b in s on bound a nd the butt e r fly effe ct in qua n tum fie ld
the or ie s . Ph y sic a l R ev iew Lett er s , 117(9):091602, 2016. doi: 10 . 1103 /PhysRevLett. 117 . 091602 .
[177] N im a L ashk a r i , D ou gl as S t a nfor d , M a tthe w H as t in gs, T o b i as O s bor ne , a nd P a tr ick H a y de n . T o -
w a r d s the fas t s cr a mb l in g c on j e ctur e . J o u r n a l of H i g h E n er g y Ph y sic s , 2013(4):22, 2013. doi:
10 . 1007 /JHEP 04( 2013 ) 022 .
213
[178] A I L a rk in a nd Y u N O v chinni k o v . Quasicl as sical me thod in the the or y of s u pe r c onduct iv it y . So v
Ph y s JE TP , 28(6):1200–1205, 1969.
[179] A lexei Kit a e v . A simp le mode l of qua n tum ho lo gr a p h y . I n P r o ce e d i n g s of t h e K ITP P r o g r a m : E n t a n -
g lem en t i n S tr o n g l y- Co r r e l a t e d Q u a n t u m M a tt er , v o lume 7, 2015.
[180] S hm ue l F i shm a n, D . R . Gr e mpe l , a nd R . E . P r a n g e . C h a os, qua n tum r e c ur r e nc e s, a nd a nde r s on
local i za t ion . Ph y s . R ev . Lett . , 49(8):509–512, Au g 1982. UR L : https://link.aps.org/doi/
10 . 1103 /PhysRevLett. 49 . 509 , doi: 10 . 1103 /PhysRevLett. 49 . 509 .
[181] S Ad a chi , M T od a , a nd K Ik e d a . Qua n tum- cl as sical c or r e sponde nc e in m a n y - d ime n sion al qua n tum
ch a os . Ph y sic a l R ev iew Lett er s , 61(6):659, 1988. doi: 10 . 1103 /PhysRevLett. 61 . 659 .
[182] M a r t in C G utzw i l le r . C h a o s i n c l assic a l a n d q u a n t u m m e c h a n ic s . S pr in g e r , 1990. doi: 10 . 1007 /
978 - 1 - 4612 - 0983 - 6 .
[183] F r itz H a ak e . Q u a n t u m Si g n a t u r e s of C h a o s . S pr in g e r , 2013. doi: 10 . 1007 / 978 - 3 - 642 - 05428 - 0 .
[184] J ua n M ald a c e n a , S t e p he n H S he nk e r , a nd D ou gl as S t a nfor d . A bound on ch a os . J o u r n a l of H i g h
E n er g y Ph y sic s , 2016(8):106, 2016. doi: 10 . 1007 /JHEP 08 ( 2016 ) 106 .
[185] D a nie l A R o be r ts a nd D ou gl as S t a nfor d . Di a gnosin g ch a os usin g four -po in t f unct ion s in t w o -
d ime n sion al c onfor m al fie ld the or y . Ph y sic a l R ev iew Lett er s , 115(13):131603, 2015. doi: 10 . 1103 /
PhysRevLett. 115 . 131603 .
[186] J os e p h P o lchin sk i a nd V l a d imir R os e nh a us . The spe ctr um in the S a chde v - Y e- Kit a e v mode l . J o u r n a l
of H i g h E n er g y Ph y sic s , 2016(4):1, 2016. doi: 10 . 1007 /JHEP 04 ( 2016 ) 001 .
[187] M á rk M ez ei a nd D ou gl as S t a nfor d . On e n t a n gle me n t spr e a d in g in ch a ot ic s ys t e m s . J o u r n a l of H i g h
E n er g y Ph y sic s , 2017(5):65, 2017. doi: 10 . 1007 /JHEP 05 ( 2017 ) 065 .
[188] Y iche n H ua n g , Y on g - L i a n g Z h a n g , a nd X ie C he n . Out - of -t ime- or de r e d c or r e l a t or s in m a n y - body
local i z e d s ys t e m s . A n n a l en der Ph y si k , 529(7):1600318, 2017. doi: 10 . 1002 /andp. 201600318.
[189] Y on g - L i a n g Z h a n g , Y iche n H ua n g , X ie C he n, e t al . I nfor m a t ion s cr a mb l in g in ch a ot ic s ys t e m s w ith
d i s sip a t ion . Ph y sic a l R ev iew B , 9 9(1):014303, 2019. doi: 10 . 1103 /PhysRevB. 99 . 014303 .
[190] D a nie l A . R o be r ts a nd Be ni Y oshid a . C h a os a nd c omp lex it y b y de si gn . J o u r n a l of H i g h E n er g y Ph y sic s ,
2017(4):121, A pr i l 2017. arXiv: 1610 . 04903 , doi: 10 . 1007 /JHEP 04 ( 2017 ) 121 .
[191] R a v i P r ak ash a nd A r ul L ak shmin a r a ya n . S cr a mb l in g in s tr on gly ch a ot ic w e ak ly c ou p le d b ip a r t it e
s ys t e m s: U niv e r s al it y bey ond the Ehr e nfe s t t ime s cale . Ph y sic a l R ev iew B , 101(12):1211 08, 2020.
doi: 10 . 1103 /PhysRevB. 101 . 121108 .
[192] J or d a n C otle r , N icho l as H un t e r - J one s, J un y u L iu , a nd Be ni Y oshid a . C h a os, c omp lex it y , a nd r a n-
dom m a tr ic e s . J o u r n a l of H i g h E n er g y Ph y sic s , 2017(11):48, 2017. doi: 10 . 1007 /JHEP 11 ( 2017 ) 048.
[193] R ui h ua F a n, P e n g fei Z h a n g , H uit a o S he n, a nd H ui Z h ai . Out - of -t ime- or de r c or r e l a t ion for m a n y -
body local i za t ion . Scien ce bu llet i n , 62(10):707–711, 2017. doi: 10 . 1016/j.scib. 2017 . 04 . 011 .
214
[194] R o be r t de M e l lo Koch, J i a - H ui H ua n g , C he n- T e M a , a nd H e ndr i k JR V a n Z y l . S pe ctr al for m fa ct or
as a n O T O C a v e r a g e d o v e r the H ei s e nbe r g gr ou p . Ph y sic s Lett er s B , 795:183–187, 2019. doi:
10 . 1016 /j.physletb. 2019 . 06 . 025 .
[195] C he n- T e M a . Ea rly -t ime a nd l a t e-t ime qua n tum ch a os . I n t er n a t io n a l J o u r n a l of M o der n Ph y sic s A ,
p a g e 2050082, 2020. doi: 10 . 1142 /S 0217751 X 20500827 .
[196] A kr a m T oui l a nd S e b as t i a n D eff ne r . Qua n tum s cr a mb l in g a nd the gr o w th of m utual infor m a t ion .
Q u a n t u m Scien ce a n d T e c h n o lo g y , 5(3):03 5005, 2020. doi: 10 . 1088 / 2058 - 9565 /ab 8 ebb .
[197] B in Y a n, L uk as z C inc io , a nd W oj c ie ch H Z ur e k . I nfor m a t ion s cr a mb l in g a nd L os chmidt e cho . Ph y s-
ic a l R ev iew Lett er s , 124(16):160603, 2020. doi: 10 . 1103 /PhysRevLett. 124 . 160603 .
[198] J o hn W a tr ous . Th e t h e o r y of q u a n t u m i nf o r m a t io n . C a mbr id g e U niv e r sit y P r e s s, 2018. doi: 10 .
1017 / 9781316848142 .
[199] R odo l fo A J al a be r t a nd H or a c io M P as t a w sk i . E n v ir onme n t -inde pe nde n t de c o he r e nc e r a t e in cl as -
sical ly ch a ot ic s ys t e m s . Ph y sic a l R ev iew Lett er s , 86(12):2490, 2001. doi: 10 . 1103 /PhysRevLett.
86 . 2490 .
[200] Thom as Gor in, T om a z P r os e n, Thom as H S e l i gm a n, a nd M a rk o Žnid a r ič . D y n a mic s of L os chmidt
e choe s a nd fide l it y de ca y . Ph y sic s R ep o r ts , 435(2-5):33–156, 2006. doi: 10 . 1016/j.physrep.
2006 . 09 . 003 .
[201] A . Gous s e v , R . A . J al a be r t , H. M . P as t a w sk i , a nd D . A r ie l W i s ni a ck i . L os chmidt e cho . Sc h o l a r p e d i a ,
7(8):11687, 2012. doi: 10 . 4249 /scholarpedia.11687 .
[202] P a o lo Za n a r d i . E n t a n gle me n t of qua n tum e v o lut ion s . Ph y sic a l R ev iew A , 63(4):040304, 2001. doi:
10 . 1103 /PhysRevA. 63 . 040304 .
[203] P a o lo Za n a r d i , C hr i s t of Zal k a , a nd L a r a F a or o . E n t a n gl in g po w e r of qua n tum e v o lut ion s . Ph y s . R ev .
A , 62(3):030301, Au g us t 2000. doi: 10 . 1103 /PhysRevA. 62 . 030301 .
[204] X i a o g ua n g W a n g , S ho hini Ghos e , B a r r y C . S a nde r s, a nd B a mb i H u . E n t a n gle me n t as a si gn a tur e of
qua n tum ch a os . Ph y s . R ev . E , 70(1):016217, J uly 2004. doi: 10 . 1103 /PhysRevE. 70. 016217 .
[205] A r ul L ak shmin a r a ya n . E n t a n gl in g po w e r of qua n t i z e d ch a ot ic s ys t e m s . Ph y sic a l R ev iew E ,
64(3):036207, 2001. doi: 10 . 1103 /PhysRevE. 64. 036207 .
[206] A ndr e w J S c ott a nd C a rlt on M C a v e s . E n t a n gl in g po w e r of the qua n tum b ak e r ’ s m a p . J o u r n a l of
Ph y sic s A : M a t h em a t ic a l a n d Gen er a l , 36(36):9553, 2003. doi: 10 . 1088 / 0305 - 4470 / 36 / 36 / 308 .
[207] R a j a r shi P al a nd A r ul L ak shmin a r a ya n . E n t a n gl in g po w e r of t ime- e v o lut ion ope r a t or s in in t e gr a b le
a nd nonin t e gr a b le m a n y - body s ys t e m s . Ph y sic a l R ev iew B , 98(17):174304, 2018. doi: 10 . 1103 /
PhysRevB. 98 . 174304 .
[208] M iche l L e dou x . Th e co n cen tr a t io n of m e as u r e p h en o m en o n , v o lume 89 of M a t h em a t ic a l Su r v e y s a n d
M o n o g r a p h s . A me r ica n M a the m a t ic al S oc ie t y , 2001. doi: 10 . 1090 /surv/ 089 .
215
[209] J os e p h E me r s on, Y a ak o v S W ein s t ein, M a r c os S a r a c e no , S e th L lo y d , a nd D a v id G C or y . P s e udo -
r a ndom unit a r y ope r a t or s for qua n tum infor m a t ion pr oc e s sin g. Scien ce , 302(5653):2098–2100,
2003. doi: 10 . 1126 /science. 1090790 .
[210] L or e nz o C a mpos V e n ut i , N T o b i as J a c o bs on, S idd h a r th a S a n tr a , a nd P a o lo Za n a r d i . E x a ct infinit e-
t ime s t a t i s t ic s of the L os chmidt e cho for a qua n tum que nch . Ph y sic a l R ev iew Lett er s , 107(1):010403,
2011. doi:PhysRevLett. 107 . 010403 .
[211] Ad a m N ah um, S a g a r V i j a y , a nd J e on g w a n H a ah . O pe r a t or spr e a d in g in r a ndom unit a r y c ir c uits .
Ph y sic a l R ev iew X , 8(2):021014, 2018. doi: 10 . 1103 /PhysRevX. 8 . 021014 .
[212] C e r e n B D a ğ , K ai S un, a nd L - M Dua n . D e t e ct ion of qua n tum p h as e s v i a out - of -t ime- or de r c or r e-
l a t or s . Ph y sic a l R ev iew Lett er s , 123(14):140602, 2019. doi: 10 . 1103 /PhysRevLett. 123 . 140602 .
[213] Á lv a r o M A l h a mbr a , J on a thon R idde l l , a nd L ui s P e dr o G a r c í a - P in t os . T ime e v o lut ion of c or r e l a t ion
f unct ion s in qua n tum m a n y - body s ys t e m s . Ph y sic a l R ev iew Lett er s , 124(11):110605, 2020. doi:
PhysRevLett. 124 . 110605 .
[214] R a j e ndr a B h a t i a . M a tr i x a n a l y sis , v o lume 169. S pr in g e r - V e rl a g , 2013. doi: 10 . 1007 /
978 - 1 - 4612 - 0653 - 8 .
[215] Y iche n H ua n g , F e r n a ndo GS L B r a nd a o , Y on g - L i a n g Z h a n g , e t al . F init e- si z e s cal in g of out - of -t ime-
or de r e d c or r e l a t or s a t l a t e t ime s . Ph y sic a l R ev iew Lett er s , 123(1):010601, 2019. doi: 10 . 1103 /
PhysRevLett. 123 . 010601 .
[216] A r a m W H a r r o w , L in gh a n g Kon g , Z i - W e n L iu , S a e e d M e hr a b a n, a nd P e t e r W S hor . A s e p a r a t ion
of out - of -t ime- or de r e d c or r e l a t or a nd e n t a n gle me n t. a r X i v:1906.02219 , 2019. UR L : https://
arxiv.org/abs/1906 . 02219 .
[217] J osh M D e uts ch . Qua n tum s t a t i s t ical me ch a nic s in a clos e d s ys t e m . Ph y sic a l R ev iew A , 43(4):2046,
1991. doi: 10 . 1103 /PhysRevA. 43 . 2046 .
[218] M a rk S r e dnick i . C h a os a nd qua n tum the r m al i za t ion . Ph y sic a l R ev iew E , 50(2):888, 1994. doi:
10 . 1103 /PhysRevE. 50 . 888 .
[219] M a r c os R i g o l , V a n j a Dun jk o , a nd M a x im Ol sh a ni i . The r m al i za t ion a nd its me ch a ni s m for g e ne r ic
i s o l a t e d qua n tum s ys t e m s . N a t u r e , 452(7189):854–858, 2008. doi: 10 . 1038 /nature 06838 .
[220] L uca D ’ A le s sio , Y a r iv K a f r i , A n a t o l i P o l k o v ni k o v , a nd M a r c os R i g o l . F r om qua n tum ch a os a nd
ei g e n s t a t e the r m al i za t ion t o s t a t i s t ical me ch a nic s a nd the r mody n a mic s . A d v a n ce s i n Ph y sic s ,
65(3):239–362, 2016. doi: 10 . 1080 / 00018732 . 2016 . 1198134 .
[221] T s un g - C he n g L u a nd T a r un Gr o v e r . R e n y i e n tr op y of ch a ot ic ei g e n s t a t e s . Ph y sic a l R ev iew E ,
99(3):032111, 2019. doi: 10 . 1103 /PhysRevE. 99 . 032111 .
[222] R ys za r d H or ode ck i , P a w e łH or ode ck i , M ich ałH or ode ck i , a nd K a r o l H or ode ck i . Qua n tum e n t a n-
gle me n t. R ev . M o d. Ph y s . , 81(2):865 –942, J une 2009. doi: 10 . 1103 /RevModPhys. 81 . 865 .
[223] S u g a t o Bos e a nd V l a tk o V e dr al . M i xe dne s s a nd t e le por t a t ion . Ph y sic a l R ev iew A , 61(4):040101,
2000. doi: 10 . 1103 /PhysRevA. 61 . 040101 .
216
[224] El i h u L ub k in . E n tr op y of a n n - s ys t e m f r om its c or r e l a t ion w ith a k - r e s e r v o ir . J o u r n a l of M a t h em a t ic a l
Ph y sic s , 19(5):1028–1031, 1 978. doi: 10 . 1063 / 1 . 523763 .
[225] A l ios c i a H a mm a , S idd h a r th a S a n tr a , a nd P a o lo Za n a r d i . Qua n tum e n t a n gle me n t in r a ndom p h ysi -
cal s t a t e s . Ph y sic a l r ev iew lett er s , 109(4):040502, 2012. doi: 10 . 1103 /PhysRevLett. 109 . 040502 .
[226] J os ef R a mme n s e e , J ua n Die g o U r b in a , a nd K l a us R ich t e r . M a n y - body qua n tum in t e r fe r e nc e a nd
the s a tur a t ion of out - of -t ime- or de r c or r e l a t or s . Ph y sisc a l R ev iew Lett er s , 121:124101, 2018. doi:
10 . 1103 /PhysRevLett. 121 . 124101 .
[227] A r tur K Ek e r t , C a r o l in a M our a A lv e s, D a nie l KL Oi , M ich ał H or ode ck i , P a w e ł H or ode ck i , a nd
L e on g C h ua n K w e k . Dir e ct e s t im a t ion s of l ine a r a nd nonl ine a r f unct ion al s of a qua n tum s t a t e .
Ph y sic a l R ev iew Lett er s , 88(21):217901, 2002. doi: 10 . 1103 /PhysRevLett. 88 . 217901 .
[228] F a b io A n t onio Bo v ino , G ius e ppe C as t a gno l i , A r tur Ek e r t , P a w e ł H or ode ck i , C a r o l in a M our a A lv e s,
a nd A lex a nde r V l a d imir S e r g ie nk o . Dir e ct me as ur e me n t of nonl ine a r pr ope r t ie s of b ip a r t it e qua n-
tum s t a t e s . Ph y sic a l R ev iew Lett er s , 95(24):240407, 2005. doi: 10 . 1103 /PhysRevLett.95 .
240407 .
[229] C . M our a A lv e s a nd D . J ak s ch . M ult ip a r t it e e n t a n gle me n t de t e ct ion in bos on s . Ph y sic a l R ev iew
Lett er s , 93:110501, 2004. doi: 10 . 1103 /PhysRevLett. 93 . 110501 .
[230] A J D aley , H P ichle r , J S ch a che nm a y e r , a nd P Z o l le r . M e as ur in g e n t a n gle me n t gr o w th in que nch
dy n a mic s of bos on s in a n opt ical l a tt ic e . Ph y sic a l R ev iew Lett er s , 109(2):020505, 2012. doi: 10 .
1103 /PhysRevLett. 109 . 020505 .
[231] R a ji bul I sl a m, R uich a o M a , P hi l ipp M P r ei s s, M E r ic T ai , A lex a nde r L uk in, M a tthe w R i spo l i , a nd
M a rk us Gr eine r . M e as ur in g e n t a n gle me n t e n tr op y in a qua n tum m a n y - body s ys t e m . N a t u r e ,
528(7580):77–83, 2015. doi: 10 . 1038 /nature 15750 .
[232] T i ff B r y d g e s, A ndr e as El be n, P e t a r J ur c e v ic, Be no ît V e r me r s ch, C hr i s t ine M aie r , Be n P L a n y on, P e-
t e r Z o l le r , R aine r B l a tt , a nd C hr i s t i a n F R oos . P r o b in g R é n y i e n t a n gle me n t e n tr op y v i a r a ndomi z e d
me as ur e me n ts . Scien ce , 364(6437):260–263, 2019. doi: 10 . 1126 /science.aau 4963 .
[233] A ndr e as El be n, Be no ît V e r me r s ch, C hr i s t i a n F R oos, a nd P e t e r Z o l le r . S t a t i s t ical c or r e l a t ion s be-
t w e e n local ly r a ndomi z e d me as ur e me n ts: A t oo l bo x for pr o b in g e n t a n gle me n t in m a n y - body qua n-
tum s t a t e s . Ph y sic a l R ev iew A , 99(5):052323, 2019. doi: 10 . 1103 /PhysRevA. 99 . 052323 .
[234] H sin- Y ua n H ua n g , R ich a r d K ue n g , a nd J o hn P r e sk i l l . P r e d ict in g m a n y pr ope r t ie s of a qua n tum
s ys t e m f r om v e r y fe w me as ur e me n ts . N a t u r e Ph y sic s , 16(10):1050–1057, 2020. doi: 10 . 1038 /
s 41567 - 020 - 0932 - 7 .
[235] A ndr e as El be n, R ich a r d K ue n g , H sin- Y ua n ( R o be r t ) H ua n g , R ick v a n B i jne n, C hr i s t i a n Ko k ai l ,
M a r c e l lo D almon t e , P as quale C al a br e s e , B a r b a r a Kr a us, J o hn P r e sk i l l , P e t e r Z o l le r , a nd Be no ît V e r -
me r s ch . M i xe d - s t a t e e n t a n gle me n t f r om local r a ndomi z e d me as ur e me n ts . Ph y sic a l R ev iew Lett er s ,
125(20):200501, 2020. doi: 10 . 1103 /PhysRevLett. 125 . 200501 .
217
[236] Be no ît V e r me r s ch, A ndr e as El be n, L uk as M S ie be r e r , N or m a n Y Y a o , a nd P e t e r Z o l le r . P r o b in g
s cr a mb l in g usin g s t a t i s t ical c or r e l a t ion s be t w e e n r a ndomi z e d me as ur e me n ts . Ph y sic a l R ev iew X ,
9(2):021061, 2019. doi: 10 . 1103 /PhysRevX. 9 . 021061 .
[237] M a noj K . J oshi , A ndr e as El be n, Be no ît V e r me r s ch, T i ff B r y d g e s, C hr i s t ine M aie r , P e t e r Z o l le r ,
R aine r B l a tt , a nd C hr i s t i a n F . R oos . Qua n tum infor m a t ion s cr a mb l in g in a tr a ppe d -ion qua n-
tum sim ul a t or w ith tun a b le r a n g e in t e r a ct ion s . Ph y sic a l R ev iew Lett er s , 124(24):240505, J un
2020. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevLett. 124 . 240505 , doi: 10 .
1103 /PhysRevLett. 124 . 240505 .
[238] M ich a e l A N ie l s e n a nd I s a a c C h ua n g. Q u a n t u m co m pu t a t io n a n d q u a n t u m i nf o r m a t io n . C a mbr id g e
U niv e r sit y P r e s s, 2000.
[239] A Y u Kit a e v . Qua n tum c omput a t ion s: al g or ithm s a nd e r r or c or r e ct ion . R ussi a n M a t h em a t ic a l Su r-
v e y s , 52(6):1191–1 249, 1997. doi: 10 . 1070 /RM 1997 v 052 n 06 ABEH 002155 .
[240] M a rk M . W i lde . Q u a n t u m i nf o r m a t io n t h e o r y . C a mbr id g e U niv e r sit y P r e s s, 2013. doi: 10 . 1017 /
CBO 9781139525343 .
[241] M A k i l a , D W altne r , B G utk in, a nd T G uhr . P a r t icle-t ime dual it y in the k ick e d I sin g sp in ch ain .
J o u r n a l of Ph y sic s A : M a t h em a t ic a l a n d Th e o r et ic a l , 49(37):375101, 2016. doi: 10 . 1088 / 1751 - 8113 /
49 / 37 / 375101 .
[242] B r uno Be r t ini , P a v e l Kos, a nd T om a z P r os e n . E x a ct c or r e l a t ion f unct ion s for dual - unit a r y l a t -
t ic e mode l s in 1 + 1 d ime n sion s . Ph y sisc a l R ev iew Lett er s , 12 3:210601, 2019. doi: 10 . 1103 /
PhysRevLett. 123 . 210601 .
[243] L or e nz o P ir o l i , B r uno Be r t ini , J . I gn a c io C ir a c, a nd T om a z P r os e n . E x a ct dy n a mic s in dual - unit a r y
qua n tum c ir c uits . Ph y sisc a l R ev iew B , 101:094304, 2020. doi: 10 . 1103 /PhysRevB. 101 . 094304 .
[244] B r uno Be r t ini , P a v e l Kos, a nd T om a z P r os e n . O pe r a t or E n t a n gle me n t in L ocal Qua n tum C ir c uits I:
C h a ot ic Dual - U nit a r y C ir c uits . SciP o s t Ph y s . , 8:67, 2020. doi: 10 . 21468 /SciPostPhys. 8 . 4 . 067 .
[245] S uh ai l A hm a d R a the r , S . A r a v ind a , a nd A r ul L ak shmin a r a ya n . C r e a t in g e n s e mb le s of dual unit a r y
a nd m a x im al ly e n t a n gl in g qua n tum e v o lut ion s . Ph y sic a l R ev iew Lett er s , 125:070501, 2020. doi:
10 . 1103 /PhysRevLett. 125 . 070501 .
[246] Gr e g W A nde r s on, A l ic e G uionne t , a nd O fe r Z eit ouni . A n in tr oduct ion t o r a ndom m a tr ic e s, 2010.
doi: 10 . 1017 /CBO 9780511801334 .
[247] J e n s E i s e r t , M a r c us C r a me r , a nd M a r t in B P le nio . C o l loquium : A r e a l a w s for the e n t a n gle me n t
e n tr op y . R ev iew s of M o der n Ph y sic s , 8 2(1):277, 2010. doi: 10 . 1103 /RevModPhys. 82 . 277 .
[248] R oe Goodm a n a nd N o l a n R W al l a ch . S y m m etr y , r epr e sen t a t io n s, a n d i n v a r i a n ts . S pr in g e r , 2009.
doi: 10 . 1007 / 978 - 0 - 387 - 79852 - 3 .
[249] J o hn W a tr ous . I s the r e a n y c onne ct ion be t w e e n the d i a mond nor m a nd the d i s t a nc e of the as s oc i -
a t e d s t a t e s? The or e t ical C omput e r S c ie nc e S t a ck E xch a n g e , 2011. UR L : https://cstheory.
stackexchange.com/q/ 4920 .
218
[250] Zak W e b b . The C l i ffor d gr ou p for m s a unit a r y 3- de si gn . Q u a n t u m I nf . Co m pu t . , 16(15&16):1379–
1400, 2016. UR L : https://arxiv.org/abs/ 1510 . 02769 .
[251] P a o lo Za n a r d i a nd N a mit A n a nd . I nfor m a t ion s cr a mb l in g a nd ch a os in ope n qua n tum s ys t e m s .
Ph y sic a l R ev iew A , 103(6):062214, J une 2021. doi: 10 . 1103 /PhysRevA. 103 . 062214 .
[252] Y iche n H ua n g , Y on g - L i a n g Z h a n g , a nd X ie C he n . Out - of -t ime- or de r e d c or r e l a t or s in m a n y - body
local i z e d s ys t e m s . A n n a len der Ph y si k , 529(7):1600318, D e c 2016. UR L : http://dx.doi.org/
10 . 1002 /andp. 201600318 , doi: 10 . 1002 /andp. 201600318.
[253] Y u C he n . U niv e r s al lo g a r ithmic s cr a mb l in g in m a n y body local i za t ion, 2016. arXiv: 1608 . 02765 .
[254] X i a o C he n, T i a nc i Z hou , D a v id A . H us e , a nd E dua r do F r a d k in . Out - of -t ime- or de r c or r e l a t ion s in
m a n y - body local i z e d a nd the r m al p h as e s . A n n a l en der Ph y si k , 529(7):1600332, D e c 2016. UR L :
http://dx.doi.org/ 10 . 1002 /andp. 201600332 , doi: 10 . 1002 /andp. 201600332 .
[255] R on g - Qi a n g H e a nd Z hon g - Y i L u . C h a r a ct e r i z in g m a n y - body local i za t ion b y out - of -t ime- or de r e d
c or r e l a t ion . Ph y sic a l R ev iew B , 95(5), F e b 2017. UR L : http://dx.doi.org/ 10 . 1103 /
PhysRevB. 95 . 054201 , doi: 10 . 1103 /physrevb. 95 . 054201 .
[256] B r i a n S w in gle a nd D e b a n j a n C ho w d h ur y . S lo w s cr a mb l in g in d i s or de r e d qua n tum s ys t e m s . Ph y sic a l
R ev iew B , 95(6), F e b 2017. UR L : http://dx.doi.org/ 10 . 1103 /PhysRevB. 95 . 060201 , doi:
10 . 1103 /physrevb. 95 . 060201 .
[257] N ic o le Y un g e r H alpe r n, A n thon y B a r t o lott a , a nd J as on P o l l a ck . E n tr op ic unc e r t ain t y r e l a t ion s for
qua n tum infor m a t ion s cr a mb l in g. Co m m u n ic a t io n s Ph y sic s , 2(1), Au g 2019. UR L : http://dx.
doi.org/ 10 . 1038 /s 42005 - 019 - 0179 - 8 , doi: 10 . 1038 /s 42005 - 019 - 0179 - 8.
[258] L or e nz o L e one , S alv a t or e F . E . Ol iv ie r o , a nd A l ios c i a H a mm a . I s ospe ctr al t w irl in g a nd qua n tum
ch a os, 2020. arXiv: 2011 . 06011 .
[259] S alv a t or e F . E . Ol iv ie r o , L or e nz o L e one , F r a nc e s c o C a r a v e l l i , a nd A l ios c i a H a mm a . R a ndom m a tr i x
the or y of the i s ospe ctr al t w irl in g , 2020. arXiv: 2012 . 07681 .
[260] X i a o M i , P e dr a m R oush a n, C hr i s Quin t a n a , S alv a t or e M a ndr a , J eff r ey M a r sh al l , C h a rle s N ei l l ,
F r a nk A r ut e , K un al A r ya , J ua n A t al a ya , R ya n B a b bush, J os e p h C . B a r d in, R a mi B a r e nd s, A ndr e as
Be n g ts s on, S e r g io Bo i xo , A lex a ndr e Bour as s a , M ich a e l B r ou gh t on, Bo b B . B uck ley , D a v id A . B ue l l ,
B r i a n B urk e tt , N icho l as B ushne l l , Z i jun C he n, Be n j a min C hi a r o , R o be r t o C o l l in s, W i l l i a m C our t -
ney , S e a n D e m ur a , A l a n R . D e rk , A ndr e w Dun s w or th, D a nie l E ppe n s, C a the r ine E r ick s on, E d -
w a r d F a rhi , Aus t in G . F o w le r , B r oo k s F o xe n, C r ai g G idney , M a r i s s a G ius t in a , J on a th a n A . Gr os s,
M a tthe w P . H a r r i g a n, S e a n D . H a r r in g t on, J e r e m y H i lt on, A l a n H o , S a br in a H on g , T r e n t H ua n g ,
W i l l i a m J . H u gg in s, L . B . I offe , S e r g ei V . I s ak o v , E v a n J eff r ey , Z h a n g J i a n g , C ody J one s, D v ir K a f r i , J u-
l i a n Ke l ly , S e on Kim, A lexei Kit a e v , P a ul V . K l imo v , A lex a nde r N . Kor otk o v , F e dor Kos tr its a , D a v id
L a nd h ui s, P a v e l L a pt e v , E r i k L uc e r o , Or ion M a r t in, J a r r od R . M c C le a n, T r e v or M c C our t , M a tt M cE w e n, A n thon y M e gr a n t , Ke v in C . M i a o , M as oud M o h s e ni , W oj c ie ch M r ucz k ie w icz , J osh M u-
tus, O fe r N a a m a n, M a tthe w N e e ley , M ich a e l N e w m a n, M ur p h y Y uez he n N iu , Thom as E . O ’ B r ie n,
A lex O pr e mcak , E r ic O s t b y , B al in t P a t o , A ndr e P e tuk ho v , N icho l as R e dd , N icho l as C . R ub in,
219
D a nie l S a nk , Ke v in J . S a tz in g e r , V l a d imir S h v a r ts, D ou g S tr ain, M a r c o S zal a y , M a tthe w D . T r e-
v ithick , Be n j a min V i l l alon g a , The odor e W hit e , Z. J a mie Y a o , P in g Y e h, Ad a m Zalcm a n, H a r tm ut
N e v e n, I g or A leine r , Kos t ya n t y n Ke che d z hi , V a d im S me lya n sk iy , a nd Y u C he n . I nfor m a t ion s cr a m-
b l in g in c omput a t ion al ly c omp lex qua n tum c ir c uits, 2021. arXiv: 2101 . 08870 .
[261] J oche n B r a um ül le r , A mir H. K a r a mlou , Y a r iv Y a n a y , B h a r a th K a nn a n, D a v id Kim, M or t e n K j a e r -
g a a r d , A lex a nde r M e lv i l le , Be th a n y M . N ie d z ie l sk i , Y oun gk y u S un g , A n tt i V e ps äl äine n, R oni W ini k ,
J oni ly n L . Y ode r , T e r r y P . Orl a ndo , S imon G us t a v s s on, C h a rle s T ah a n, a nd W i l l i a m D . Ol iv e r .
P r o b in g qua n tum infor m a t ion pr op a g a t ion w ith out - of -t ime- or de r e d c or r e l a t or s, 2021. arXiv:
2102 . 11751 .
[262] Ke n X ua n W ei , C h a ndr as e k h a r R a m a n a th a n, a nd P a o l a C a ppe l l a r o . E x p lor in g local i za t ion in n u-
cle a r sp in ch ain s . Ph y sic a l R ev iew Lett er s , 120(7), F e b 2018. UR L : http://dx.doi.org/ 10 .
1103 /PhysRevLett. 120 . 070501 , doi: 10 . 1103 /physrevlett. 120. 070501 .
[263] J un L i , R ui h ua F a n, H e n g ya n W a n g , B in g t i a n Y e , Bei Z e n g , H ui Z h ai , X inh ua P e n g , a nd J i a n g fe n g
Du . M e as ur in g out - of -t ime- or de r c or r e l a t or s on a n ucle a r m a gne t ic r e s on a nc e qua n tum sim ul a t or .
Ph y s . R ev . X , 7:031011, J ul 2017. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevX. 7 .
031011 , doi: 10 . 1103 /PhysRevX. 7 . 031011 .
[264] X infa n g N ie , Z e Z h a n g , X iuz h u Z h a o , T a o X in, D a w ei L u , a nd J un L i . D e t e ct in g s cr a mb l in g v i a
s t a t i s t ical c or r e l a t ion s be t w e e n r a ndomi z e d me as ur e me n ts on a n nmr qua n tum sim ul a t or , 2019.
arXiv: 1903 . 12237 .
[265] X infa n g N ie , Bo - Bo W ei , X i C he n, Z e Z h a n g , X iuz h u Z h a o , C h ud a n Qiu , Y u T i a n, Y unl a n J i ,
T a o X in, D a w ei L u , a nd e t al . E x pe r ime n t al o bs e r v a t ion of e qui l i br ium a nd dy n a mical qua n-
tum p h as e tr a n sit ion s v i a out - of -t ime- or de r e d c or r e l a t or s . Ph y sic a l R ev iew Lett er s , 124(25),
J un 2020. UR L : http://dx.doi.org/ 10 . 1103 /PhysRevLett. 124 . 250601 , doi: 10 . 1103 /
physrevlett. 124 . 250601 .
[266] M a r t in G ä r ttne r , J us t in G . Bo hne t , A r gh a v a n S a fa v i - N aini , M ich a e l L . W al l , J o hn J . Bo l l in g e r , a nd
A n a M a r i a R ey . M e as ur in g out - of -t ime- or de r c or r e l a t ion s a nd m ult ip le qua n tum spe ctr a in a
tr a ppe d -ion qua n tum m a gne t. N a t u r e Ph y sic s , 13(8):781–786, M a y 2017. UR L : http://dx.
doi.org/ 10 . 1038 /nphys 4119 , doi: 10 . 1038 /nphys 4119 .
[267] M a noj K . J oshi , A ndr e as El be n, Be no ît V e r me r s ch, T i ff B r y d g e s, C hr i s t ine M aie r , P e t e r
Z o l le r , R aine r B l a tt , a nd C hr i s t i a n F . R oos . Qua n tum infor m a t ion s cr a mb l in g in a tr a ppe d -
ion qua n tum sim ul a t or w ith tun a b le r a n g e in t e r a ct ion s . Ph y sic a l R ev iew Lett er s , 124(24),
J un 2020. UR L : http://dx.doi.org/ 10 . 1103 /PhysRevLett. 124 . 240505 , doi: 10 . 1103 /
physrevlett. 124 . 240505 .
[268] E r ic J . M eie r , J a ck s on A n g ’ on g ’ a , F a n g z h a o A lex A n, a nd B r y c e G a dw a y . E x p lor in g qua n tum
si gn a tur e s of ch a os on a floque t s y n the t ic l a tt ic e . Ph y sic a l R ev iew A , 100(1), J ul 2019. UR L :
http://dx.doi.org/ 10 . 1103 /PhysRevA. 100 . 013623 , doi: 10 . 1103 /physreva. 100 . 013623 .
[269] B in g C he n, X i a nfei H ou , F ei fei Z hou , P e n g Qi a n, H e n g S he n, a nd N a n ya n g X u . D e t e ct in g the out -
of -t ime- or de r c or r e l a t ion s of dy n a mical qua n tum p h as e tr a n sit ion s in a s o l id - s t a t e qua n tum sim u-
l a t or . App l ie d Ph y sic s Lett er s , 116(19):194002, M a y 2020. UR L : http://dx.doi.org/ 10 . 1063 /
5 . 0004152 , doi: 10 . 1063 / 5 . 0004152 .
220
[270] X i a o g ua n g W a n g a nd P a o lo Za n a r d i . Qua n tum e n t a n gle me n t of unit a r y ope r a t or s on b ip a r t it e
s ys t e m s . Ph y s . R ev . A , 66:044303, O ct 2002. UR L : https://link.aps.org/doi/ 10 . 1103 /
PhysRevA. 66 . 044303 , doi: 10 . 1103 /PhysRevA. 66 . 044303 .
[271] K . A . L a nd s m a n, C . F i gg a tt , T . S ch us t e r , N . M . L ink e , B . Y oshid a , N . Y . Y a o , a nd C . M onr oe . V e r i fie d
qua n tum infor m a t ion s cr a mb l in g. N a t u r e , 567(7746):61–65, M a r 2019. UR L : http://dx.doi.
org/ 10 . 1038 /s 41586 - 019 - 0952 - 6 , doi: 10 . 1038 /s 41586 - 019 - 0952 - 6 .
[272] M . S . B lo k , V . V . R a m as e sh, T . S ch us t e r , K . O ’ B r ie n, J . M . Kr ei k e b a um, D . D ahle n, A . M or v a n,
B . Y oshid a , N . Y . Y a o , a nd I. S idd iqi . Qua n tum infor m a t ion s cr a mb l in g in a s u pe r c onduct in g qutr it
pr oc e s s or , 2021. arXiv: 2003 . 03307 .
[273] B r i a n S w in gle a nd N ic o le Y un g e r H alpe r n . R e si l ie nc e of s cr a mb l in g me as ur e me n ts . Ph y s . R ev .
A , 97:062113, J un 2018. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevA. 97 . 062113 ,
doi: 10 . 1103 /PhysRevA. 97 . 062113 .
[274] Y on g - L i a n g Z h a n g , Y iche n H ua n g , a nd X ie C he n . I nfor m a t ion s cr a mb l in g in ch a ot ic s ys t e m s w ith
d i s sip a t ion . Ph y s . R ev . B , 99:014303, J a n 2019. UR L : https://link.aps.org/doi/ 10 . 1103 /
PhysRevB. 99 . 014303 , doi: 10 . 1103 /PhysRevB. 99 . 014303 .
[275] Be ni Y oshid a a nd N or m a n Y . Y a o . Di s e n t a n gl in g s cr a mb l in g a nd de c o he r e nc e v i a qua n tum t e le-
por t a t ion . Ph y s . R ev . X , 9:011006, J a n 2019. UR L : https://link.aps.org/doi/ 10 . 1103 /
PhysRevX. 9 . 011006 , doi: 10 . 1103 /PhysRevX. 9 . 011006 .
[276] J os é R a úl González A lon s o , N ic o le Y un g e r H alpe r n, a nd J us t in Dr e s s e l . Out - of -t ime- or de r e d -
c or r e l a t or quasipr o b a b i l it ie s r o bus tly w itne s s s cr a mb l in g. Ph y s . R ev . Lett . , 122:040404, F e b
2019. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevLett. 122 . 040404 , doi: 10 .
1103 /PhysRevLett. 122 . 040404.
[277] F e de r ic o D . D omin g uez , M a r i a C r i s t in a R odr i g uez , R o b in K ai s e r , Die t e r S ut e r , a nd Gonzalo A .
A lv a r ez . D e c o he r e nc e s cal in g tr a n sit ion in the dy n a mic s of qua n tum infor m a t ion s cr a mb l in g , 2020.
arXiv: 2005 . 12361 .
[278] Z he n y u X u , Aur e l i a C he n u , T om a ž P r os e n, a nd Ado l fo de l C a mpo . The r mofie ld dy n a mic s: Qua n-
tum ch a os v e r s us de c o he r e nc e , 2020. arXiv: 2008 . 06444 .
[279] Z he n y u X u , L ui s P e dr o G a r c í a - P in t os, Aur é l i a C he n u , a nd Ado l fo de l C a mpo . E x tr e me de c o he r -
e nc e a nd qua n tum ch a os . Ph y s . R ev . Lett . , 122:014103, J a n 2019. UR L : https://link.aps.
org/doi/ 10 . 1103 /PhysRevLett. 122 . 014103 , doi: 10 . 1103 /PhysRevLett. 122 . 014103 .
[280] A kr a m T oui l a nd S e b as t i a n D eff ne r . I nfor m a t ion s cr a mb l in g v s . de c o he r e nc e – t w o c ompe t in g sink s
for e n tr op y , 2020. arXiv: 2008 . 05559 .
[281] S . V . S y zr a no v , A . V . Gor shk o v , a nd V . G al itsk i . Out - of -t ime- or de r c or r e l a t or s in finit e ope n s ys t e m s .
Ph y s . R ev . B , 97:161114, A pr 2018. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevB.
97 . 161114 , doi: 10 . 1103 /PhysRevB. 97 . 161114 .
[282] H einz - P e t e r B r e ue r a nd F . P e tr uc c ione . Th e Th e o r y of O p en Q u a n t u m S y s t em s . Ox for d U niv e r sit y
P r e s s, Ox for d ; N e w Y ork , 2002.
221
[283] R odney J B a x t e r . Exa c t l y so l v e d m o de ls i n s t a t is t ic a l m e c h a n ic s . El s e v ie r , 2016.
[284] Be r i sl a v B uča a nd T om a ž P r os e n . A not e on s y mme tr y r e duct ion s of the l ind b l a d e qua t ion : tr a n s -
por t in c on s tr aine d ope n sp in ch ain s . N ew J o u r n a l of Ph y sic s , 14(7):073007, J ul 2012. UR L : http:
//dx.doi.org/ 10 . 1088 / 1367 - 2630/ 14 / 7 / 073007 , doi: 10 . 1088 / 1367 - 2630/ 14 / 7 / 073007 .
[285] T om a ž P r os e n . E x a ct none qui l i br ium s t e a dy s t a t e of a s tr on gly dr iv e n ope n xxz ch ain . Ph y s . R ev .
Lett . , 107:137201, S e p 2011. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevLett.
107 . 137201 , doi: 10 . 1103 /PhysRevLett. 107 . 137201 .
[286] M a r iya V . M e dv e dy e v a , F a b i a n H. L . Es sle r , a nd T om a ž P r os e n . E x a ct be the a n s a tz spe c -
tr um of a t i gh t - b ind in g ch ain w ith de p h asin g no i s e . Ph y s . R ev . Lett . , 117:137202, S e p
2016. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevLett. 117 . 137202 , doi: 10 .
1103 /PhysRevLett. 117 . 137202 .
[287] L ucas S á , P e dr o R i beir o , a nd T om a ž P r os e n . C omp lex sp a c in g r a t ios: A si gn a tur e of d i s sip a t iv e
qua n tum ch a os . Ph y s . R ev . X , 10:021019, A pr 2020. UR L : https://link.aps.org/doi/ 10 .
1103 /PhysRevX. 10 . 021019 , doi: 10 . 1103 /PhysRevX. 10 . 021019 .
[288] N a mit A n a nd a nd P a o lo Za n a r d i . T o a ppe a r .
[289] N a mit A n a nd a nd P a o lo Za n a r d i . B r ot oc s a nd qua n tum infor m a t ion s cr a mb l in g a t finit e t e mpe r a -
tur e , 2021. UR L : https://arxiv.org/abs/ 2111 . 07086 , doi: 10 . 48550 /ARXIV. 2111 . 07086 .
[290] M a rk S r e dnick i . C h a os a nd qua n tum the r m al i za t ion . Ph y sic a l R ev iew E , 50(2):888–901, Au g us t
1994. doi: 10 . 1103 /PhysRevE. 50 . 888 .
[291] F . Bor g ono v i , F . M . I zr ai le v , L .F . S a n t os, a nd V . G . Z e le v in sk y . Qua n tum ch a os a nd the r m al i za t ion in
i s o l a t e d s ys t e m s of in t e r a ct in g p a r t icle s . Ph y sic s R ep o r ts , 626:1–58, A pr i l 2016. doi: 10 . 1016/j.
physrep. 2016 . 02 . 005 .
[292] S a r a n g Gop al akr i shn a n, D a v id A . H us e , V e d i k a K he m a ni , a nd R om ain V as s e ur . H y dr ody n a mic s
of ope r a t or spr e a d in g a nd quasip a r t icle d i ff usion in in t e r a ct in g in t e gr a b le s ys t e m s . Ph y s . R ev . B ,
98:220303, D e c 2018. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevB. 98 . 220303 ,
doi: 10 . 1103 /PhysRevB. 98 . 220303 .
[293] A mos C h a n, A ndr e a D e L uca , a nd J . T . C h al k e r . S o lut ion of a minim al mode l for m a n y - body qua n-
tum ch a os . Ph y s . R ev . X , 8:041019, N o v 2018. UR L : https://link.aps.org/doi/ 10 . 1103 /
PhysRevX. 8 . 041019 , doi: 10 . 1103 /PhysRevX. 8 . 041019 .
[294] D a nie l E . P a rk e r , X i a n g y u C a o , A lex a nde r A v doshk in, Thom as S ca ffid i , a nd Eh ud A ltm a n . A
U niv e r s al O pe r a t or Gr o w th H y pothe si s . Ph y s ic a l R ev iew X , 9(4):041017, O ct o be r 2019. doi:
10 . 1103 /PhysRevX. 9 . 041017 .
[295] C h ait a n ya M ur th y a nd M a rk S r e dnick i . Bound s on C h a os f r om the E i g e n s t a t e The r m al i za t ion H y -
pothe si s . Ph y sic a l R ev iew Lett er s , 123(23):230606, D e c e mbe r 2019. doi: 10 . 1103 /PhysRevLett.
123 . 230606 .
222
[296] J ua n M ald a c e n a , S t e p he n H. S he nk e r , a nd D ou gl as S t a nfor d . A bound on ch a os . J o u r n a l of H i g h
E n er g y Ph y sic s , 2016(8):106, Au g us t 2016. arXiv: 1503 . 01409 , doi: 10 . 1007 /JHEP 08 ( 2016 ) 106 .
[297] T i a nr ui X u , Thom as S ca ffid i , a nd X i a n g y u C a o . D oe s S cr a mb l in g E qual C h a os? Ph y sic a l R ev iew
Lett er s , 124(14):140602, A pr i l 2020. doi: 10 . 1103 /PhysRevLett. 124 . 140602 .
[298] B r i a n S w in gle . U n s cr a mb l in g the p h ysic s of out - of -t ime- or de r c or r e l a t or s . N a t u r e Ph y sic s ,
14(10):988–990, O ct o be r 2018. doi: 10 . 1038 /s 41567 - 018 - 0295 - 5 .
[299] S he n glon g X u a nd B r i a n S w in gle . S cr a mb l in g dy n a mic s a nd out - of -t ime or de r e d c or r e l a t or s in
qua n tum m a n y - body s ys t e m s: a tut or i al , 2022. UR L : https://arxiv.org/abs/ 2202 . 07060 ,
doi: 10 . 48550 /ARXIV. 2202 . 07060 .
[300] El l iott H L ie b . The finit e gr ou p v e loc it y of qua n tum sp in s ys t e m s . p a g e 7.
[301] S . B r a v y i , M . B . H as t in gs, a nd F . V e r s tr a e t e . L ie b - R o b in s on Bound s a nd the Ge ne r a t ion of C or -
r e l a t ion s a nd T opo lo g ical Qua n tum Or de r . Ph y sic a l R ev iew Lett er s , 97(5), J uly 2006. doi:
10 . 1103 /PhysRevLett. 97 . 050401 .
[302] J or d a n S . C otle r , G u y G ur - A r i , M as a nor i H a n a d a , J os e p h P o lchin sk i , P hi l S a a d , S t e p he n H. S he nk e r ,
D ou gl as S t a nfor d , A lex a ndr e S tr eiche r , a nd M as ak i T ez uk a . B l a ck ho le s a nd r a ndom m a tr i -
c e s . J o u r n a l of H i g h E n er g y Ph y sic s , 2017(5):118, M a y 2017. arXiv: 1611 . 04650 , doi: 10 . 1007 /
JHEP 05 ( 2017 ) 118 .
[303] J un L i , R ui h ua F a n, H e n g ya n W a n g , B in g t i a n Y e , Bei Z e n g , H ui Z h ai , X inh ua P e n g , a nd J i a n g fe n g
Du . M e as ur in g Out - of - T ime- Or de r C or r e l a t or s on a N ucle a r M a gne t ic R e s on a nc e Qua n tum S im-
ul a t or . Ph y sic a l R ev iew X , 7(3):031011, J uly 2017. doi: 10 . 1103 /PhysRevX. 7 . 031011 .
[304] M a r t in G ä r ttne r , J us t in G . Bo hne t , A r gh a v a n S a fa v i - N aini , M ich a e l L . W al l , J o hn J . Bo l l in g e r ,
a nd A n a M a r i a R ey . M e as ur in g out - of -t ime- or de r c or r e l a t ion s a nd m ult ip le qua n tum spe ctr a in
a tr a ppe d -ion qua n tum m a gne t. N a t u r e Ph y sic s , 13(8):781–786, Au g us t 2017. doi: 10 . 1038 /
nphys 4119 .
[305] Ge or g ios S t y l i a r i s, N a mit A n a nd , a nd P a o lo Za n a r d i . I nfor m a t ion S cr a mb l in g o v e r B ip a r t it ion s:
E qui l i br a t ion, E n tr op y P r oduct ion, a nd T y p ical it y . Ph y sic a l R ev iew Lett er s , 126(3):030601, J a n-
ua r y 2021. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevLett. 126. 030601 , doi:
10 . 1103 /PhysRevLett. 126. 030601 .
[306] P a o lo Za n a r d i a nd N a mit A n a nd . I nfor m a t ion s cr a mb l in g a nd ch a os in ope n qua n tum s ys t e m s .
Ph y sic a l R ev iew A , 103(6):062214, J une 2021. doi: 10 . 1103 /PhysRevA. 103 . 062214 .
[307] N a mit A n a nd , Ge or g ios S t y l i a r i s, M e e n u K um a r i , a nd P a o lo Za n a r d i . Qua n tum c o he r e nc e as a
si gn a tur e of ch a os . a r X i v:2009.02760 [ co n d-m a t, p h y sic s:h ep -t h , p h y sic s: q u a n t-p h ] , N o v e mbe r 2020.
arXiv: 2009 . 02760 .
[308] N ic o le Y un g e r H alpe r n, A n thon y B a r t o lott a , a nd J as on P o l l a ck . E n tr op ic unc e r t ain t y r e l a t ion s for
qua n tum infor m a t ion s cr a mb l in g. Co m m u n ic a t io n s Ph y sic s , 2(1):92, D e c e mbe r 2019. arXiv:
1806 . 04147 , doi: 10 . 1038 /s 42005 - 019 - 0179 - 8 .
223
[309] P a o lo Za n a r d i . E n t a n gle me n t of qua n tum e v o lut ion s . Ph y sic a l R ev iew A , 63(4), M a r ch 2001. doi:
10 . 1103 /PhysRevA. 63 . 040304 .
[310] X i a o g ua n g W a n g a nd P a o lo Za n a r d i . Qua n tum e n t a n gle me n t of unit a r y ope r a t or s on b ip a r t it e
s ys t e m s . Ph y s . R ev . A , 66:044303, O ct 2002. UR L : https://link.aps.org/doi/ 10 . 1103 /
PhysRevA. 66 . 044303 , doi: 10 . 1103 /PhysRevA. 66 . 044303 .
[311] A kr a m T oui l a nd S e b as t i a n D eff ne r . I nfor m a t ion s cr a mb l in g v e r s us de c o he r e nc e—t w o c ompe t in g
sink s for e n tr op y . PRX Q u a n t u m , 2:010306, J a n 2021. UR L : https://link.aps.org/doi/ 10 .
1103 /PRXQuantum. 2 . 010306 , doi: 10 . 1103 /PRXQuantum. 2 . 010306 .
[312] J o hn W a tr ous . Th e Th e o r y of Q u a n t u m I nf o r m a t io n . C a mbr id g e U niv e r sit y P r e s s, fir s t e d it ion, A pr i l
2018. doi: 10 . 1017 / 9781316848142 .
[313] N a ot o T s uji , T omo hir o S hit a r a , a nd M as ahit o U e d a . Bound on the ex pone n t i al gr o w th r a t e of
out - of -t ime- or de r e d c or r e l a t or s . Ph y sic a l R ev iew E , 98(1):012216, J uly 2018. doi: 10 . 1103 /
PhysRevE. 98 . 012216 .
[314] L a ur a F o ini a nd J or g e K ur ch a n . E i g e n s t a t e the r m al i za t ion h y pothe si s a nd out of t ime or de r c or r e-
l a t or s . Ph y sic a l R ev iew E , 99(4):042139, A p r i l 2019. doi: 10 . 1103 /PhysRevE. 99 . 042139 .
[315] S a g a r V i j a y a nd A sh v in V i sh w a n a th . F init e- T e mpe r a tur e S cr a mb l in g of a R a ndom H a mi lt oni a n .
a r X i v:1803.08483 [ co n d-m a t, p h y sic s:h ep -t h , p h y sic s: q u a n t-p h ] , M a r ch 2018. arXiv: 1803 . 08483 .
[316] S ub h a ya n S ah u a nd B r i a n S w in gle . I nfor m a t ion s cr a mb l in g a t finit e t e mpe r a tur e in local qua n-
tum s ys t e m s . Ph y sic a l R ev iew B , 102(18):184303, N o v e mbe r 2020. doi: 10 . 1103 /PhysRevB.
102 . 184303 .
[317] Y unx i a n g L i a o a nd V ict or G al itsk i . N onl ine a r si gm a mode l a ppr o a ch t o m a n y - body qua n tum
ch a os: R e g ul a r i z e d a nd unr e g ul a r i z e d out - of -t ime- or de r e d c or r e l a t or s . Ph y s . R ev . B , 98:205124,
N o v 2018. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevB. 98 . 205124 , doi: 10 .
1103 /PhysRevB. 98 . 205124 .
[318] M a r t in B . P le nio a nd S . V ir m a ni . A n in tr oduct ion t o e n t a n gle me n t me as ur e s . a r X i v: q u a n t-
p h/0504163 , A pr i l 2005. arXiv:quant- ph/ 0504163 .
[319] C os mo L u po , P a o lo A nie l lo , a nd A n t one l lo S ca r d ic chio . B ip a r t it e qua n tum s ys t e m s: on the r e al i gn-
me n t cr it e r ion a nd bey ond . J o u r n a l of Ph y sic s A : M a t h em a t ic a l a n d Th e o r et ic a l , 41(41):415301, S e p
2008. U R L : http://dx.doi.org/ 10 . 1088 / 1751 - 8113 / 41 / 41 / 415301 , doi: 10 . 1088 / 1751 - 8113 /
41 / 41 / 415301 .
[320] P a o lo A nie l lo a nd C os mo L u po . On the r e l a t ion be t w e e n s chmidt c oeffic ie n ts a nd e n t a n gle me n t.
O p en S y s t em s a n d I nf o r m a t io n D y n a m ic s , 16(02n03):127–143, S e p 2009. UR L : http://dx.doi.
org/ 10 . 1142 /S 1230161209000104 , doi: 10 . 1142 /s 1230161209000104 .
[321] T i a nc i Z hou a nd D a v id J . L uitz . O pe r a t or e n t a n gle me n t e n tr op y of the t ime e v o lut ion ope r a t or in
ch a ot ic s ys t e m s . Ph y sic a l R ev iew B , 95(9), M a r ch 2017. doi: 10 . 1103 /PhysRevB. 95 . 094206 .
224
[322] I v a n K uk ul j a n, S aš o Gr o z d a no v , a nd T om a ž P r os e n . W e ak qua n tum ch a os . Ph y sic a l R ev iew B , 96(6),
Au g us t 2017. doi: 10 . 1103 /PhysRevB. 96 . 060301 .
[323] C he n g - J u L in a nd Olexei I. M otr unich . Out - of -t ime- or de r e d c or r e l a t or s in a qua n tum I sin g ch ain .
Ph y sic a l R ev iew B , 97(14), A pr i l 2018. doi: 10 . 1103 /PhysRevB. 97 . 144304 .
[324] X i a o C he n a nd T i a nc i Z hou . Qua n tum ch a os dy n a mic s in lon g - r a n g e po w e r l a w in t e r a ct ion s ys t e m s .
Ph y s . R ev . B , 100:064305, Au g 2019. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevB.
100 . 064305 , doi: 10 . 1103 /PhysRevB. 100 . 064305 .
[325] V e d i k a K he m a ni , D a v id A . H us e , a nd Ad a m N ah um . V e loc it y - de pe nde n t lya puno v ex po -
ne n ts in m a n y - body qua n tum, s e micl as sical , a nd cl as sical ch a os . Ph y s . R ev . B , 98:144304, O ct
2018. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevB. 98 . 144304 , doi: 10 . 1103 /
PhysRevB. 98 . 144304 .
[326] A lex a nde r A v doshk in a nd A n a t o ly D y m a r sk y . E ucl ide a n ope r a t or gr o w th a nd qua n tum ch a os .
Ph y s . R ev . R e se a r c h , 2:043234, N o v 2020. UR L : https://link.aps.org/doi/ 10 . 1103 /
PhysRevResearch. 2 . 043234 , doi: 10 . 1103 /PhysRevResearch. 2 . 043234 .
[327] M a d a n L al M e h t a . R a n do m M a tr ice s . N umbe r 142 in P ur e a nd A pp l ie d M a the m a t ic s S e r ie s . El s e-
v ie r , A m s t e r d a m, 3. e d e d it ion, 2004.
[328] Y as ushi T ak ah ashi a nd H ir oomi U meza w a . The r mo fie ld dy n a mic s . I n t er n a t io n a l J o u r n a l of M o der n
Ph y sic s B , 10(13n14):1755–1805, J une 1996. doi: 10 . 1142 /S 0217979296000817 .
[329] E th a n D y e r a nd G u y G ur - A r i . 2d cft p a r t it ion f unct ion s a t l a t e t ime s . J o u r n a l of H i g h E n er g y Ph y sic s ,
2017(8), Au g 2017. UR L : http://dx.doi.org/ 10 . 1007 /JHEP 08 ( 2017 ) 075 , doi: 10 . 1007 /
jhep 08 ( 2017 ) 075 .
[330] A . de l C a mpo , J . M o l in a - V i l a p l a n a , a nd J . S onne r . S cr a mb l in g the spe ctr al for m fa ct or : U nit a r -
it y c on s tr ain ts a nd ex a ct r e s ults . Ph y sic a l R ev iew D , 95(12):126008, J une 2017. doi: 10 . 1103 /
PhysRevD. 95 . 126008 .
[331] K y r i ak os P a p a dod im as a nd S uv r a t R a ju . L ocal ope r a t or s in the e t e r n al b l a ck ho le . Ph y s . R ev .
Lett . , 115:211601, N o v 2015. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevLett.
115 . 211601 , doi: 10 . 1103 /PhysRevLett. 115 . 211601 .
[332] J in g x i a n g W u a nd T imoth y H. H sie h . V a r i a t ion al the r m al qua n tum sim ul a t ion v i a the r mofie ld dou-
b le s t a t e s . Ph y s . R ev . Lett . , 123:220502, N o v 2019. UR L : https://link.aps.org/doi/ 10 .
1103 /PhysRevLett. 123 . 220502 , doi: 10 . 1103 /PhysRevLett. 123 . 220502 .
[333] J o hn M a r t y n a nd B r i a n S w in gle . P r oduct spe ctr um a n s a tz a nd the simp l ic it y of the r m al s t a t e s . Ph y s .
R ev . A , 100:032107, S e p 2019. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevA. 100 .
032107 , doi: 10 . 1103 /PhysRevA. 100 . 032107 .
[334] D . Z h u , S . J o hr i , N . M . L ink e , K . A . L a nd s m a n, N . H. N g u y e n, C . H. A lde r e t e , A . Y . M a ts uur a , T . H.
H sie h, a nd C . M onr oe . Ge ne r a t ion of the r mofie ld doub le s t a t e s a nd cr it ical gr ound s t a t e s w ith a
qua n tum c omput e r , 2020. arXiv: 1906 . 02699 .
225
[335] W i l l i a m C ottr e l l , Be n F r eiv o g e l , Die g o M . H of m a n, a nd S a g a r F . L o k h a nde . H o w t o bui ld the the r -
mofie ld doub le s t a t e . J o u r n a l of H i g h E n er g y Ph y sic s , 2019(2), F e b 2019. UR L : http://dx.doi.
org/ 10 . 1007 /JHEP 02 ( 2019 ) 058 , doi: 10 . 1007 /jhep 02 ( 2019) 058 .
[336] É t ie nne L a n t a gne- H ur tub i s e , S t e p h a n P lu gg e , O g uz h a n C a n, a nd M a r c e l F r a nz . Di a gnosin g qua n-
tum ch a os in m a n y - body s ys t e m s usin g e n t a n gle me n t as a r e s our c e . Ph y s . R ev . R e se a r c h , 2:013254,
M a r 2020. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevResearch. 2 . 013254 , doi:
10 . 1103 /PhysRevResearch. 2 . 013254 .
[337] M a rk M W i lde . F r o m C l assic a l t o Q u a n t u m Sh a n n o n Th e o r y . C a mbr id g e univ e r sit y pr e s s e d it ion,
2016.
[338] Y as uhir o S e k ino a nd L S us sk ind . F as t s cr a mb le r s . J o u r n a l of H i g h E n er g y Ph y sic s , 2008(10):065–
065, O ct 2008. UR L : http://dx.doi.org/ 10 . 1088 / 1126 - 6708 / 2008 / 10 / 065 , doi: 10 . 1088 /
1126 - 6708 / 2008 / 10 / 065 .
[339] L ucas S á , P e dr o R i beir o , a nd T om a ž P r os e n . S pe ctr al a nd s t e a dy - s t a t e pr ope r t ie s of r a ndom L i -
ouv i l l i a n s . J o u r n a l of Ph y sic s A : M a t h em a t ic a l a n d Th e o r et ic a l , 53(30):305303, J uly 2020. doi:
10 . 1088 / 1751 - 8121 /ab 9337 .
[340] S e r g ey D e ni s o v , T e t ya n a L a pt y e v a , W oj c ie ch T a r no w sk i , D a r ius z C hr uś c iń sk i , a nd K a r o l Ż y -
cz k o w sk i . U niv e r s al S pe ctr a of R a ndom L ind b l a d O pe r a t or s . Ph y sic a l R ev iew Lett er s , 123(14),
O ct o be r 2019. doi: 10 . 1103 /PhysRevLett. 123 . 140403 .
[341] T a nk ut C a n . R a ndom L ind b l a d dy n a mic s . J o u r n a l of Ph y sic s A : M a t h em a t ic a l a n d Th e o r et ic a l ,
52(48):485302, N o v e mbe r 2019. doi: 10 . 1088 / 1751 - 8121 /ab 4 d 26 .
[342] T a nk ut C a n, V a d im O g a ne s ya n, Dr or Or g a d , a nd S a r a n g Gop al akr i shn a n . S pe ctr al g a ps
a nd mid g a p s t a t e s in r a ndom qua n tum m as t e r e qua t ion s . Ph y s . R ev . Lett . , 123:234103, D e c
2019. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevLett. 123 . 234103 , doi: 10 .
1103 /PhysRevLett. 123 . 234103 .
[343] R aine r Gr o be , F r itz H a ak e , a nd H a n s - J ür g e n S omme r s . Qua n tum d i s t inct ion of r e g ul a r a nd ch a ot ic
d i s sip a t iv e mot ion . Ph y s . R ev . Lett . , 61:1899–1902, O ct 1988. UR L : https://link.aps.org/
doi/ 10 . 1103 /PhysRevLett. 61 . 1899 , doi: 10 . 1103 /PhysRevLett. 61 . 1899 .
[344] Ge r not A k e m a nn, M a r io Kie bur g , Ad a m M ie l k e , a nd T om a ž P r os e n . U niv e r s al si gn a tur e
f r om in t e gr a b i l it y t o ch a os in d i s sip a t iv e ope n qua n tum s ys t e m s . Ph y s . R ev . Lett . , 123:254101,
D e c 2019. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevLett. 123 . 254101 , doi:
10 . 1103 /PhysRevLett. 123 . 254101 .
[345] L ucas S á , P e dr o R i beir o , a nd T om a ž P r os e n . C omp lex S p a c in g R a t ios: A S i gn a tur e of Di s sip a t iv e
Qua n tum C h a os . Ph y sic a l R ev iew X , 10(2):021019, A pr i l 2020. UR L : https://link.aps.org/
doi/ 10 . 1103 /PhysRevX. 10 . 021019 , doi: 10 . 1103 /PhysRevX. 10 . 021019 .
[346] W oj c ie ch H ube r t Z ur e k a nd J ua n P a b lo P a z . D e c o he r e nc e , ch a os, a nd the s e c ond l a w . Ph y sic a l
R ev iew Lett er s , 72(16):2508–2511, A pr i l 1994. doi: 10 . 1103 /PhysRevLett. 72 . 2508 .
226
[347] S ub ir S a chde v . Q u a n t u m Ph ase T r a n sit io n s . C a mbr id g e U niv e r sit y P r e s s, C a mbr id g e ; N e w Y ork ,
s e c ond e d it ion e d it ion, 2011.
[348] M a rk us H ey l , F r a nk P o l lm a nn, a nd B al á z s D ór a . D e t e ct in g e qui l i br ium a nd dy n a mical qua n tum
p h as e tr a n sit ion s in i sin g ch ain s v i a out - of -t ime- or de r e d c or r e l a t or s . Ph y s . R ev . Lett . , 121:016801,
J ul 2018. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevLett. 121 . 016801 , doi: 10 .
1103 /PhysRevLett. 121 . 016801 .
[349] L or e nz o C a mpos V e n ut i , N . T o b i as J a c o bs on, S idd h a r th a S a n tr a , a nd P a o lo Za n a r d i . E x a ct I nfinit e-
T ime S t a t i s t ic s of the L os chmidt E cho for a Qua n tum Que nch . Ph y sic a l R ev iew Lett er s , 107(1), J uly
2011. doi: 10 . 1103 /PhysRevLett. 107 . 010403 .
[350] Á lv a r o M . A l h a mbr a , J on a thon R idde l l , a nd L ui s P e dr o G a r c í a - P in t os . T ime E v o lut ion of C or r e-
l a t ion F unct ion s in Qua n tum M a n y - Body S ys t e m s . Ph y sic a l R ev iew Lett er s , 124(11), M a r ch 2020.
doi: 10 . 1103 /PhysRevLett. 124 . 110605 .
[351] I gn a c io G a r c í a - M a t a , M a r c os S a r a c e no , R odo l fo A . J al a be r t , Au g us t o J . R onca gl i a , a nd Die g o A .
W i s ni a ck i . C h a os S i gn a tur e s in the S hor t a nd L on g T ime Be h a v ior of the Out - of - T ime Or de r e d
C or r e l a t or . Ph y sic a l R ev iew Lett er s , 121(21):210601, N o v e mbe r 2018. UR L : https://link.
aps.org/doi/ 10 . 1103 /PhysRevLett. 121 . 210601 , doi: 10 . 1103 /PhysRevLett. 121 . 210601.
[352] E mi l i a no M . F or t e s, I gn a c io G a r c í a - M a t a , R odo l fo A . J al a be r t , a nd Die g o A . W i s ni a ck i . G a u g -
in g cl as sical a nd qua n tum in t e gr a b i l it y thr ou gh out - of -t ime- or de r e d c or r e l a t or s . Ph y sic a l R ev iew
E , 100(4):042201, O ct o be r 2019. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevE.
100 . 042201 , doi: 10 . 1103 /PhysRevE. 100 . 042201 .
[353] Y iche n H ua n g , F e r n a ndo G . S . L . B r a nd ã o , a nd Y on g - L i a n g Z h a n g. F init e- S i z e S cal in g of Out - of -
T ime- Or de r e d C or r e l a t or s a t L a t e T ime s . Ph y sic a l R ev iew Lett er s , 123(1):010601, J uly 2019. doi:
10 . 1103 /PhysRevLett. 123 . 010601 .
[354] Z he n y u X u , Aur e l i a C he n u , T om a ž P r os e n, a nd Ado l fo de l C a mpo . The r mofie ld dy n a mic s: Qua n-
tum C h a os v e r s us D e c o he r e nc e . a r X i v:2008.06444 [ co n d-m a t, p h y sic s:h ep -t h , p h y sic s: q u a n t-p h ] , Au-
g us t 2020. arXiv: 2008 . 06444 .
[355] E mi l i a no M . F or t e s, I gn a c io G a r c í a - M a t a , R odo l fo A . J al a be r t , a nd Die g o A . W i s ni a ck i . S i gn a tur e s
of qua n tum ch a os tr a n sit ion in shor t sp in ch ain s . E PL ( E u r o p h y sic s Lett er s ) , 130(6):60001, J uly
2020. doi: 10 . 1209 / 0295 - 5075 / 130 / 60001 .
[356] V inith a B al a ch a ndr a n, G iul i a no Be ne n t i , G iul io C as a t i , a nd D a r io P o le tt i . F r om the ei g e n s t a t e the r -
m al i za t ion h y pothe si s t o al g e br aic r e l a x a t ion of ot oc s in s ys t e m s w ith c on s e r v e d qua n t it ie s . Ph y s .
R ev . B , 104:104306, S e p 2021. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevB. 104 .
104306 , doi: 10 . 1103 /PhysRevB. 104 . 104306 .
[357] M . C . B a ñ ul s, J . I. C ir a c, a nd M . B . H as t in gs . S tr on g a nd w e ak the r m al i za t ion of infinit e nonin-
t e gr a b le qua n tum s ys t e m s . Ph y s . R ev . Lett . , 106:050405, F e b 2011. UR L : https://link.aps.
org/doi/ 10 . 1103 /PhysRevLett. 106 . 050405 , doi: 10 . 1103 /PhysRevLett. 106 . 050405 .
227
[358] H y un g w on Kim a nd D a v id A . H us e . B al l i s t ic spr e a d in g of e n t a n gle me n t in a d i ff usiv e nonin t e gr a b le
s ys t e m . Ph y s . R ev . Lett . , 111:127205, S e p 2013. UR L : https://link.aps.org/doi/ 10 . 1103 /
PhysRevLett. 111 . 127205 , doi: 10 . 1103 /PhysRevLett. 111 . 127205 .
[359] M ich a e l M . W o l f , F r a nk V e r s tr a e t e , M a tthe w B . H as t in gs, a nd J . I gn a c io C ir a c . A r e a L a w s in Qua n-
tum S ys t e m s: M utual I nfor m a t ion a nd C or r e l a t ion s . Ph y sic a l R ev iew Lett er s , 100(7), F e br ua r y 2008.
doi: 10 . 1103 /PhysRevLett. 100 . 070502 .
[360] R ah ul N a nd k i shor e a nd D a v id A . H us e . M a n y - Body L ocal i za t ion a nd The r m al i za t ion in Qua n tum
S t a t i s t ical M e ch a nic s . A n n u a l R ev iew of Co n den se d M a tt er Ph y sic s , 6(1):15–38, M a r ch 2015. doi:
10 . 1146 /annurev- conmatphys- 031214 - 014726 .
[361] X i a o C he n, T i a nc i Z hou , D a v id A . H us e , a nd E dua r do F r a d k in . Out - of -t ime- or de r c or r e l a t ion s in
m a n y - body local i z e d a nd the r m al p h as e s . A n n a len der Ph y si k , 529(7):1600332, J uly 2017. arXiv:
1610 . 00220 , doi: 10 . 1002 /andp. 201600332 .
[362] M ich a e l V ict or Be r r y a nd M ich a e l T a bor . L e v e l clus t e r in g in the r e g ul a r spe ctr um . P r o ce e d i n g s of
t h e R oy a l So ciet y of Lo n do n. A . M a t h em a t ic a l a n d Ph y sic a l Scien ce s , 356( 1686):375–394, 1977.
[363] F e l i x A . P o l lock , C é s a r R odr í g uez - R os a r io , Thom as F r a ue nheim, M a ur o P a t e r nos tr o , a nd K a v a n
M od i . N on- m a rk o v i a n qua n tum pr oc e s s e s: C omp le t e f r a me w ork a nd effic ie n t ch a r a ct e r i za t ion .
Ph y s . R ev . A , 97:012127, J a n 2018. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevA.
97 . 012127 , doi: 10 . 1103 /PhysRevA. 97 . 012127 .
[364] F e l i x A . P o l lock , C é s a r R odr í g uez - R os a r io , Thom as F r a ue nheim, M a ur o P a t e r nos tr o , a nd K a v a n
M od i . O pe r a t ion al m a rk o v c ond it ion for qua n tum pr oc e s s e s . Ph y s . R ev . Lett . , 120:040405,
J a n 2018. UR L : https://link.aps.org/doi/ 10 . 1103 /PhysRevLett. 120. 040405 , doi:
10 . 1103 /PhysRevLett. 120 . 040405 .
228
Abstract (if available)
Abstract
This thesis explores from the lens of quantum information theory, three distinct quantum many-body phenomena: (i) quantum chaos, (ii) Anderson and many-body localization, and (iii) information scrambling.
Coherence and chaos: We show that the quantum coherence content of both states and dynamics provides a diagnostic tool for the onset of quantum chaos in many-body systems. In particular, the average coherence of Hamiltonian eigenstates in the middle of the spectrum can distinguish integrable and chaotic models. Moreover, chaotic systems are known to be fast scramblers of information and we show that the so-called out-of-time-ordered correlator (OTOC), a quantifier of information scrambling, is in fact intimately related to the coherence-generating power of the dynamics.
Coherence and localization: The absence of transport in disordered systems, termed localization, is usually quantified by the (asymptotic) return probability of a particle. We show that for non-degenerate Hamiltonians, this is exactly equal to the 2-norm coherence of the quantum state, with respect to the Hamiltonian eigenbasis. This allows us to obtain a wealth of results for various notions of coherence-generating power and employ them as a signature of the ergodic-to-localization transition.
Scrambling and operator entanglement: We show that a special class of averaged OTOCs quantify exactly the operator entanglement and entangling power of unitary dynamics, in turn providing operational meaning to OTOCs themselves. An evolution is termed scrambling if it quickly disseminates local information throughout the nonlocal degrees of freedom of the system. For unitary scrambling evolutions, the reduced dynamics is expected to be mixing and we quantify this by (i) relating the (average) local entropy production to (averaged) OTOCs and (ii) showing that the (average) OTOC quantifies how far the reduced dynamics is from a completely depolarizing channel.
Scrambling in open quantum systems: Almost every physical system one encounters is ultimately open and here we study the role of these open system effects on scrambling. We find that averaged OTOCs capture two distinct contributions: one from global environmental decoherence and the other from genuine information scrambling within the system's degrees of freedom. Surprisingly, these two contributions compete with each other and the OTOC can vanish even for systems with nontrivial amounts of scrambling and decoherence. Using our analytical results, we show how to disentangle these contributions and recover signatures of scrambling in the presence of environmental decoherence.
Scrambling at finite temperature: The now famous, bound on chaos as introduced by Maldacena and Shenkar studies information scrambling in quantum systems at finite temperature, by introducing a new class of regularized OTOCs. We show that suitable averages of these regularized OTOCs are related to the purity of the time-evolving thermofield double state (which simply refers to a canonical purification of the Gibbs state). Moreover, we relate these regularized OTOCs to a key quantifier of quantum chaos, the (analytically continued) spectral form factor.
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University of Southern California Dissertations and Theses
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Asset Metadata
Creator
Anand, Namit
(author)
Core Title
Quantum information-theoretic aspects of chaos, localization, and scrambling
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Physics
Degree Conferral Date
2022-08
Publication Date
07/25/2022
Defense Date
04/29/2022
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
information scrambling,localization,OAI-PMH Harvest,open quantum systems,out-of-time-ordered correlators,quantum chaos,quantum coherence,quantum entanglement,quantum information theory
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application/pdf
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Language
English
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Electronically uploaded by the author
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Advisor
Zanardi, Paolo (
committee chair
), Brun, Todd (
committee member
), Felice, Rosa Di (
committee member
), Lauda, Aaron (
committee member
), Lidar, Daniel (
committee member
)
Creator Email
namitana@usc.edu,namitniser11@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-oUC111374347
Unique identifier
UC111374347
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etd-AnandNamit-10961
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Dissertation
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application/pdf (imt)
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Anand, Namit
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20220728-usctheses-batch-962
(batch),
University of Southern California
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University of Southern California Dissertations and Theses
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Tags
information scrambling
localization
open quantum systems
out-of-time-ordered correlators
quantum chaos
quantum coherence
quantum entanglement
quantum information theory