University of Southern California Dissertations and Theses

Coherence generation, incompatibility, and symmetry in quantum processes

(USC Thesis Other)

Coherence generation, incompatibility, and symmetry in quantum processes

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Content C ohe r e nce ge ne r a tion, incomp a ti bi lit y ,
a nd s y mmetr y in q ua n tum pr o ces s es
b y
Georg ios St yl i aris
A Di s s e r t a t ion P r es e n t e d t o the
F a cu l t y of the U S C Gr adu a te Sc ho ol
Univer sit y of Sou thern C al iforni 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 g r e e
Do c t or of Philosophy
( Phy sics )
Au g ust 2020
D ed i c a ted t o my p a r ents
i i
Ackno w led g me n t s
I a m gr a t ef ul ; a nd thi s me a n s I w ould l i k e t o a ckno w le d g e m a n y pe op le w ho h a v e c on tr i but e d t o thi s
the si s .
I th a nk m y a dv i s or P a o lo Za n a r d i for hi s me n t or ship , the c oun tle s s d i s c us sion s in f r on t of the w hit e bo a r d ,
hi s effor ts t o sh a pe me as a p h ysic i s t , a nd for bein g the fir s t pe r s on t o sho w me th a t moun t ain s ca n be a lot of
f un . I a m gr a t ef ul t o L or e nz o C a mpos V e n ut i for hi s tr ip le r o le as a c o -a dv i s or , f r ie nd , a nd cl imb in g me n t or ;
th a nk y ou for bein g a v ai l a b le w he ne v e r I h a d a que s t ion .
I w ould al s o l i k e t o ex pr e s s m y gr a t itude t o the p h ysic s P r ofe s s or s a t U S C , e spe c i al ly t o D . L id a r , S . H a as,
a nd T . B r un their he lp , K . P i lch for do in g a n exc e l le n t j o b as the gr a dua t e a dv i s or , R . Di F e l ic e for thos e
s umme r s t a ys in I t aly a nd H. S ale ur in F r a nc e .
I a m v e r y a ppr e c i a t iv e of the p h ysic s a nd me t a p h ysic s d i s c us sion s w ith N a mit ( al s o for hi s c oo k in g ), t o
J eff for hi s a dv ic e , a nd Á lv a r o for in v it in g me t o P e r ime t e r I n s t itut e a nd bein g a v alua b le c o l l a bor a t or f r om
w hom I h a v e le a r ne d a lot.
I a m th a nk f ul for the e nc our a g e me n t a nd s u ppor t of f r ie nd s f r om home . I a m gr a t ef ul t o M a r i a for sh a p -
in g me , M e l in a for alw a ys bein g the r e , Di a m a n t i s for thos e d i s c us sion s o v e r c offe e , a nd C os t as w ho i s m y
o lde s t f r ie nd . I w ould not h a v e s ur v iv e d a P hD in L os A n g e le s w ithout the s u ppor t of m y local budd ie s: a
b i g th a nk y ou t o m y Gr e e k hous e m a t e s G ior g os a nd V a n g e l i s, the cl imb in g cr e w N ic o , M i s chi , a nd A r ash,
t o J osh for bein g a gr e a t f r ie nd sinc e m y fir s t d a y in L os A n g e le s, a nd al s o Z oe , S i a v ash a nd m y othe r ex -
i i i
hous e m a t e s in the B a r mor e .
Th a nk y ou N a y e l i for al l y our lo v e a nd c omp a nion ship , a nd for bein g p a t ie n t , s u ppor t iv e , a nd e v e n a
gr e a t c o l l a bor a t or .
F in al ly , I h a v e t o t ak e a s t e p b a ck a nd th a nk m y fa mi ly for r ai sin g me; I th a nk m y mom, m y d a d a nd m y
si s t e r for their unc ond it ion al lo v e a nd s u ppor t , for w hom I do not h a v e e nou gh w or d s .
iv
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 Fi g u r e s v iii
L is t o f Pu b li c a ti o n s i x
A bs tr a c t x i
Pr ef a c e x ii
1 Co h er en c e gener a ti o n b y u nit a r y a nd d eph a si n g pr o c e sse s 1
1.1 Qua n tum c o he r e nc e for the k i nde r g a r t e n . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 A bs tr a ct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 I n tr oduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 B asic m a the m a t ical definit io n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.5 Qua n tum c o he r e nc e as a r e s our c e the o r y . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.6 C o he r e nc e- g e ne r a t in g po w e r of q ua n tum ope r a t ion s . . . . . . . . . . . . . . . . . . . 7
1.7 M a x im al ly a nd p a r t i al ly d e p h asin g pr oc e s s e s . . . . . . . . . . . . . . . . . . . . . . . . 13
1.8 C o he r e nc e- g e ne r a t in g po w e r of m a x im al ly de p h asin g ch a nne l s . . . . . . . . . . . . . . 14
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1.9 C o he r e nc e- g e ne r a t in g po w e r of p a r t i al ly de p h asin g ch a nne l s . . . . . . . . . . . . . . . 22
1.10 D e p h asin g pr oc e s s e s a nd c o he r e nc e- g e ne r a t in g po w e r in M a rk o v i a n dy n a mic s . . . . . . 24
1.11 C o he r e nc e- g e ne r a t in g po w e r o f r a ndom qua n tum pr oc e s s e s . . . . . . . . . . . . . . . 36
1.12 C onclusion a nd outloo k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
A ppe nd i x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
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 49
2.1 L ocal i za t ion for th e k inde r g a r t e n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2 A bs tr a ct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.3 I n tr oduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4 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 . . . . . . . . . . . . . . . . . . . . . . . 53
2.5 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 . . . . . . . . 65
2.6 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 . . . . . . . . . . . . . . . . . 68
2.7 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 . . . . . . . . . . . . . . 72
2.8 C onclusion a nd outloo k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
A ppe nd ic e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3 In c omp a ti b i lit y o f me a s u r e ments , q u a nti fi ed 84
3.1 M e as ur e me n t inc omp a t i b i l it y for the k inde r g a r t e n . . . . . . . . . . . . . . . . . . . . 84
3.2 A bs tr a ct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.3 I n tr oduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4 P r e l imin a r ie s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.5 P r e or de r a nd monot one s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.6 A pr e or de r o v e r or th onor m al b as e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.7 The or de r in g f r om non s e le ct i v e or tho g on al me as ur e me n ts . . . . . . . . . . . . . . . . 91
3.8 M e as ur e s of r e l a t iv e ( in )c omp a t i b i l it y . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.9 I nc omp a t i b i l it y a nd c o he r e nc e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.10 I nc omp a t i b i l it y a nd unc e r t ain t y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
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3.11 Ge ne r al i z e d me as ur e me n ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.12 B asi s -inde pe nde n t inc omp a t i b i l it y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.13 C onclusion a nd outloo k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4 S y m me tr i e s a nd mo n o t o ne s i n Ma r k o v i a n q u a ntum d y n a mi c s 107
4.1 S y mme tr ie s a nd monot one s for the k i nde r g a r t e n . . . . . . . . . . . . . . . . . . . . . 107
4.2 A bs tr a ct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.3 I n tr oduct ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.4 S e tt in g the s t a g e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.5 A fir s t ex a mp le: D e p h asin g of a qub it . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.6 M onot one s imp ly N oe the r c on s e r v e d qua n t it ie s . . . . . . . . . . . . . . . . . . . . . . 119
4.7 S y mme tr ie s of the g e ne r a t or a nd the mo not one s . . . . . . . . . . . . . . . . . . . . . . 121
4.8 D e p h asin g g e ne r a t or s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.9 D a v ie s g e ne r a t or s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.10 C onclusion a nd outloo k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
A ppe nd ic e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Ou tlo o k 134
R efer en c e s 137
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Li st of F i g ur es
1.8.1 S in gle qub it c o he r e nc e- g e ne r a t in g po w e r of m a x im al ly de p h asin g ch a nne l s . . . . . . . . 19
1.9.1 C o he r e nc e- g e ne r a t in g po w e r o f E x a mp le 1.3 . . . . . . . . . . . . . . . . . . . . . . . . 25
1.10.1 C o he r e nc e- g e ne r a t in g po w e r f or a L ind b l a d i a n . . . . . . . . . . . . . . . . . . . . . . 36
1.10.2 C omp a r i s on be t w e e n the de p h asin g C o he r e nc e- g e ne r a t in g po w e r me a n a nd its u ppe r
bound f r om E q . ( 1.43 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.11.1 P r o b a b i l it y d i s tr i but ion for r a ndom m a x im al ly de p h asin g pr oc e s s e s . . . . . . . . . . . . 45
2.5.1 A v e r a g e r e tur n pr o b a b i l it y ( A nde r s on ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.5.2 R e l a t iv e e n tr op y C o he r e n c e- g e ne r a t in g po w e r ( A nde r s on ) . . . . . . . . . . . . . . . . 69
2.6.1 C o he r e nc e s cal in g a nd m a n y - body l ocal i za t ion . . . . . . . . . . . . . . . . . . . . . . 72
2.8.1 Es ca pe pr o b a b i l it y a nd d i s or de r in the L lo y d mode l . . . . . . . . . . . . . . . . . . . . 81
4.9.1 C on s tr ain ts impos e d on the t ime e v o lut ion b y c on side r in g as s y mme tr y the L ind b l a d i a n
its e l f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
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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:
• Ge or g ios S t y l i a r i s, 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 . 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 sic a l R ev iew A , 95:052306 (2017).
doi:10.1103/PhysRevA.95.052306 .
• Ge or g ios S t y l i a r i s, 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 . C o he r e nc e- g e ne r a t in g po w e r of
qua n tum de p h asin g pr oc e s s e s . Ph y sic a l R ev iew A , 97(3):032304 (2018).
doi:10.1103/PhysRevA.97.032304 .
• 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 a nd P a o lo Za n a r d i . Qua n t i f y in g the inc omp a t i b i l it y of qua n tum me as ur e me n ts
r e l a t iv e t o a b asi s . Ph y s . R ev . Lett . , 123:070401 (2019).
doi:10.1103/PhysRevLett.123.070401 .
• 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 . S y mme tr ie s a nd monot one s in M a rk o v i a n qua n tum dy n a m-
ic s . Q u a n t u m 4, 261 (2020)
doi:10.22331/q-2020-04-30-261 .
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Othe r pub l ica t ion s 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:
• 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 q ua n tum unit al ope r a t ion s . Ph y s . R ev . A , 95:052307 (2017).
doi:10.1103/PhysRevA.95.052307 .
• Ge or g ios S t y l i a r i s, Á lv a r o M . A l h a mbr a , a nd P a o lo Za n a r d i . M i x in g of qua n tum s t a t e s unde r M a rk o -
v i a n d i s sip a t ion a nd c o he r e n t c on tr o l . Ph y s . R ev . A , 99, 042333 (2019).
doi:10.1103/PhysRevA.99.042333 .
• Á lv a r o M . A l h a mbr a , Ge or g ios S t y l i a r i s, N a y e l i A . R odr í g uez - B r ione s, J a mie S i k or a , a nd E dua r do
M a r t ín- M a r t ínez . F und a me n t al L imit a t ion s t o L ocal E ne r g y E x tr a ct ion in Qua n tum S ys t e m s . Ph y s .
R ev . Lett . , 123, 190601 (2019).
doi:10.1103/PhysRevLett.123.190601 .
• A l ios c i a H a mm a , 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 . L ocal i za b le qua n tum c o he r e nc e .
arXiv:2005.02988 .
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A b str a ct W e ex p lor e f r om a n infor m a t ion-the or e t ic pe r spe ct iv e thr e e aspe cts of qua n tum pr oc e s s e s w ith no a p -
p a r e n t cl as sical c oun t e r p a r t : ( i) g e ne r a t ion of c o he r e nc e in qua n tum e v o lut ion s, ( ii) inc omp a t i b i l it y of
qua n tum me as ur e me n ts, a nd ( iii) imp l i ca t ion s of s y mme tr ie s in M a rk o v i a n qua n tum dy n a mic s .
W e de v e lop a fa mi ly of me as ur e s th a t qua n t i f y the a v e r a g e effe ct iv e ne s s of a qua n tum e v o lut ion t o g e n-
e r a t e qua n tum c o he r e nc e out of inc o he r e n t s t a t e s . W e ex a mine its t y p ical be h a v ior a nd e mp lo y it t o s tudy
local i za t ion in qua n tum me ch a nic s .
M e as ur e me n t inc omp a t i b i l it y i s fa mously ex pr e s s e d thr ou gh unc e r t ain t y r e l a t ion s . H e r e w e l a y do w n a
f r a me w ork t o ca ptur e inc omp a t i b i l it y b y me a n s of a n or de r in g. The m ain ide a be hind the definit ion r e l ie s
on me as ur e me n t e m ul a t ion ; mor e c omp a t i b le me as ur e me n ts ca n e m ul a t e the le s s c omp a t i b le one s .
F or M a rk o v i a n qua n tum dy n a mic s, w e e s t a b l i sh a d ir e ct c or r e sponde nc e be t w e e n s y mme tr ie s of the
e v o lut ion a nd monot one s, i . e ., qua n t it ie s th a t a r e non-incr e asin g unde r the e v o lut ion . W e find a simp le
r e c ipe t o as si gn a fa mi ly of monot one s for e a ch p air of s y mme tr ie s of the g e ne r a t or , pr o v id in g a pos si b le
g e ne r al i za t ion of N oe the r ’ s the or e m for M a rk o v i a n dy n a mic s .
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P r efa ce
Inf o rma ti o n is u l tima tel y e nc ode d , s t or e d a nd m a nipul a t e d in p h ysical s ys t e m s . “ I nf o r m a t io n is p h y s-
ic a l” , as R o l f L a nd a ue r put it [ 1 ]. A nd sinc e n a tur e ca n be qua n tum, al s o infor m a t ion pr oc e s sin g h as t o
be so m e h o w qua n tum . Qua n tum infor m a t ion s c ie nc e i s the c o l le ct iv e effor t t o tur n thi s “ s ome ho w ” in t o a
pr e c i s e h o w .
A t a fir s t gl a nc e , the s ubj e ct loo k s r a the r b i za r r e; it l ie s a t the in t e r s e ct ion of qua n tum p h ysic s, infor m a -
t ion the or y , a nd c omput e r s c ie nc e , thr e e d i s c ip l ine s th a t w e r e tr a d it ion al ly c on side r e d non- o v e rl a pp in g.
B ut , in fa ct , the in t e r p l a y be t w e e n the m h as f ue le d qua n tum infor m a t ion w ith no v e l ide as, a nd it i s thi s
de p a r tur e f r om the e s t a b l i she d p a r a d i gm s th a t m ak e s the fie ld fas c in a t in g.
I n p a r t ic ul a r , qua n tum infor m a t ion h as t ak e n a f r e sh a ppr o a ch on a n aly z in g the l a w s of qua n tum p h ysic s;
it se e k s t o e x p lo r e t h e sco p e of t h e p h y sic a l l a w s b y i n v e s t i g a t i n g w h a t a r e t h e p o ssi bi l it ie s a n d l i m it a t io n s t o t h e
t as k s t h a t t h e t h e o r y a llo w s us t o p er f o r m .
Thi s the si s i s the r efle ct ion of a n a tt e mpt t o push thi s l ine of r e s e a r ch f ur the r . I t ex a mine s thr e e aspe cts of
qua n tum pr oc e s s e s f r om a n infor m a t ion-the or e t ic pe r spe ct iv e: g e ne r a t ion of c o he r e nc e , inc omp a t i b i l it y
of me as ur e me n ts a nd c on s e que nc e s of s y mme tr ie s in M a rk o v i a n qua n tum dy n a mic s .
C o her enc e gener a t ion . Qua n tum c o he r e nc e [ 2 ] r efe r s t o the f und a me n t al a b i l it y of qua n tum s ys t e m s t o
ex i s t in s u pe r posit ion . I n C h a pt e r 1 w e e s t a b l i sh a g e ne r al , ope r a t ion al f r a me w ork t o qua n t i f y the
c o he r e nc e th a t qua n tum ope r a t ion s g e ne r a t e . I n v o k in g t y p ical it y t e chnique s for l a r g e s ys t e m s, w e
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pr o v e th a t r a ndom clos e d s ys t e m dy n a mic s almos t alw a ys g e ne r a t e s m a x im al c o he r e nc e , but de-
p h asin g e v o lut ion s, us ual ly e nc oun t e r e d in pr a ct ical situa t ion s, g e ne r a t e minim al . W e the n ut i l i z e
the f r a me w ork in C h a pt e r 2 t o s tudy the local i za t ion tr a n sit ion[ 3 ], both in one- a nd m a n y - body
s e tt in gs, de mon s tr a t i n g the v alue of c o he r e nc e me as ur e s in local i za t ion .
Qua n t u m i nc omp a t i b i l i t y . I nc omp a t i b i l it y of qua n tum me as ur e me n ts i s fa mously ca ptur e d thr ou gh un-
c e r t ain t y r e l a t ion s [ 4 ]. H o w e v e r , thi s i s not the w ho le s t or y; in C h a pt e r 3 w e a dopt the pe r spe ct i v e
th a t inc omp a t i b i l it y ca n be f und a me n t al ly unde r s t ood in t e r m s of e m ul a t ion of me as ur e me n ts b y
othe r me as ur e me n ts . F r om thi s v ie w po in t , w e e s t a b l i sh a c onne ct ion be t w e e n e n tr op ic unc e r t ain t y
r e l a t ion s, qua n tum s u pe r posit ion, a nd me as ur e me n t sim ul a t a b i l it y v i a the m a the m a t ical t oo l of m ul -
t iv a r i a t e m a j or i za t ion .
Sy m me t r ies i n M a rk o v i a n q ua n t u m d y n a m ic s . I t i s h a r d t o o v e r e mp h asi z e the r o le of s y mme tr ie s in the-
or e t ical p h ysic s . N oe the r ’ s fa mous the or e m e s t a b l i she s the c or r e sponde nc e be t w e e n c on t in uous
s y mme tr ie s a nd c on s e r v e d qua n t it ie s [ 5 ]. H o w e v e r , w he n it c ome s t o d i s sip a t iv e qua n tum s ys -
t e m s de s cr i be d b y a M a rk o v i a n m as t e r e qua t ion [ 6 ], c on s e r v e d qua n t it ie s h a v e a m uch mor e l imit e d
r o le [ 7 ]. The a bo v e r e m a rk s s e e m t o s u gg e s t th a t the r o le of s y mme tr ie s in ope n qua n tum s ys t e m s i s
r a the r l imit e d . I n C h a pt e r 4 w e de mon s tr a t e othe r w i s e b y c on side r in g a le s s s tr ict not ion th a n th a t
of a c on s e r v e d qua n t it y , n a me ly th a t of a monot one , i . e ., a qua n t it y th a t i s non-incr e asin g unde r the
t ime e v o lut ion . E x p lo it in g s y mme tr ie s in the g e ne r a t or of dy n a mic s, w e pr oduc e fa mi l ie s of mono -
t one s for a n y M a rk o v i a n m as t e r e qua t ion s, pr o v id in g a g e ne r al i za t ion of N oe the r ’ s the or e m for thi s
cl as s of d y n a mic s .
x i i i
1
C ohe r e nce ge ne r a tion b y unit a r y a nd deph a sin g
pr o ces s es
1.1 Qu an tum c oherenc e for the kindergar ten
In q u a ntum phy si c s, al l pr ope r t ie s of a p h ysical s ys t e m a r e e nc ode d in a n ( a dmitt e d ly , quit e a bs tr a ct )
m a the m a t ical o bj e ct dub be d the q u a n t u m s t a t e . A qua n tum s t a t e include s – a t le as t , in pr inc ip le – al l the
infor m a t ion r e quir e d t o a n s w e r que s t ion s c onc e r nin g p h ysical pr ope r t ie s of the s ys t e m .
H o w e v e r , the r e i s a s e r ious ca t ch w ith the s t a t e me n t a bo v e; que s t ion s in qua n tum me ch a nic s a dmit a n-
1
s w e r s th a t a r e pr o b a bi l is t ic . I n othe r w or d s, s ome t ime s y ou ca n ne v e r kno w for s ur e i f the a n s w e r t o y our
que s t ion “ w he r e i s m y e le ctr on r i gh t no w ?” i s “ he r e ” or “ o v e r the r e ” . Both ca n oc c ur , w ith a c e r t ain f r e-
que nc y th a t the m a chine r y of qua n tum me ch a nic s i s r e al ly s uc c e s sf ul a t pr e d ict in g.
N one the le s s, the r e a r e s ome pot e n t i al ly g ood ne w s; s u ppos e the qua n tum s t a t e i s s uch th a t y ou g e t a
w ho le spe ctr um of d i ffe r e n t a n s w e r s t o y our que s t ion ( e a ch w ith s ome fi xe d pr o b a b i l it y ). The r e i s a pr op -
e r t y cal le d q u a n t u m co h er en ce – w hich i s the c e n tr al the me of thi s ch a pt e r – th a t the qua n tum s t a t e mi gh t
pos s e s s . I f thi s i s s o , the n thi s me a n s th a t t h er e is a l w a y s t h e o p t io n t o i n s t e a d as k so m e o t h er q u e s t io n , a n d get
a le ss r a n do m d is tr i bu t io n f o r t h e p o ssi b le a n s w er s ¹ .
Thi s situa t ion, w he r e the d i ffe r e n t out c ome s a r e s aid t o be in co h er en t s u p er p o sit io n , i s t o be c on tr as t e d
w ith the one of i n co h er en t s u p er p o sit io n . I n the l a tt e r cas e , the d i ffe r e n t out c ome s oc c ur be ca us e the s ys t e m
bein g de s cr i be d b y the qua n tum s t a t e i s, in fa ct , a n e n s e mb le: r a ndomne s s a r i s e s in the s a me w a y as in the
out c ome of a d ic e bein g thr o w n – one kno w s befor e h a nd only p a r t i a ll y its de s cr ipt ion .
W h a t w e h a v e d i s c us s e d s o fa r a ctual ly jus t c or r e spond s t o the t w o ex tr e m al cas e s: f ul ly c o he r e n t , or
f ul ly inc o he r e n t s u pe r posit ion . I n r e al it y , the r e i s a r ainbo w of s t a t e s th a t l ie in- be t w e e n ; thi s i s the t op ic of
the q u a n t it a t i v e a ppr o a c h t o qua n tum c o he r e nc e .
The a moun t of c o he r e n t s u pe r posit ion in a qua n tum s t a t e ch a n g e s as the l a tt e r e v o lv e s in t ime . Thi s
ch a pt e r mos tly de al s w ith qua n t i f y in g ho w effe ct iv e a r e t w o t y pe s of qua n tum e v o lut ion s, on a v e r a g e , a t
g e ne r a t in g c o he r e n t s u pe r posit ion in s t a t e s th a t a r e t ot al ly inc o he r e n t t o be g in w ith . The fir s t t y pe , l a be le d
u n it a r y e v o lut ion s, i s a br o a d cl as s c or r e spond in g t o t ime e v o lut ion of qua n tum s ys t e m s ly in g in i s o l a t ion
f r om e v e r y thin g e l s e . The s e c ond t y pe , ca r r y the nickn a me dep h asi n g , a nd i s a p a r t ic ul a r cl as s of e v o lut ion s,
oft e n oc c ur r in g in the l a b ( the r e , no c o w i s sp he r ical ), w he r e qua n tum c o he r e nc e in s ome f r a me of r efe r e nc e
– cal le d a b asis – i s los t. N e v e r the le s s, e v e n i f c o he r e nc e i s de s tr o y e d w ith r e spe ct t o s ome b asi s, it mi gh t
be ( p a r t i al ly ) g aine d in s ome othe r one . Thi s ch a pt e r ex a mine s thi s, as w e l l as othe r que s t ion s r e l a t e d w ith
the qua n t it a t iv e a ppr o a ch t o qua n tum c o he r e nc e of qua n tum e v o lut ion s .
¹ I n l ine a r al g e br a a nd the the or y of m a j or i za t ion, thi s s t a t e me n t in the c on t e n t of m a tr ic e s i s kno w n as the S ch ur - H or n
the or e m [ 8 ].
2
One of the m ain r e s ult of thi s ch a pt e r ca n be infor m al ly s t a t e d as fo l lo w s:
A u n it a r y pr o ce sse s i n l a r ge q u a n t u m sy s t em s, pic k e d a t r a n do m , w i ll a l m o s t cer t a i n l y gen er a t e m ax i m a l co h er en ce .
On t h e o t h er h a n d , a dep h asi n g o n e w i ll a l m o s t cer t a i n l y gen er a t e a v a n is h i n g l y s m a ll a m o u n t of co h er en ce .
1.2 A bs tr a c t
W e pr o v ide a qua n t i fica t ion of the ca p a b i l it y of qua n tum unit a r y a nd de p h asin g pr oc e s s e s t o g e ne r a t e c o -
he r e nc e out of inc o he r e n t s t a t e s . The me as ur e s define d , a dmitt in g c omput a b le ex pr e s sion s for a n y finit e
H i l be r t sp a c e d ime n sion, a r e b as e d on pr o b a b i l i s t ic a v e r a g e s a nd a r i s e f r om the v ie w po in t of c o he r e nc e as
a r e s our c e . W e in v e s t i g a t e ho w the ca p a b i l it y of a de p h asin g pr oc e s s ( e . g., a non- s e le ct iv e or tho g on al me a -
s ur e me n t ) t o g e ne r a t e c o he r e nc e de pe nd s on the r e le v a n t b as e s of the H i l be r t sp a c e o v e r w hich c o he r e nc e
i s qua n t i fie d a nd the de p h asin g pr oc e s s oc c ur s, r e spe ct iv e ly .
W e ex t e nd our a n alysi s t o include thos e M a rk o v i a n e v o lut ion s w hich, in the infinit e t ime l imit , de p h as e
the s ys t e m unde r c on side r a t ion a nd calc ul a t e their c o he r e nc e- g e ne r a t in g po w e r as a f unct ion of t ime . W e
f ur the r ide n t i f y spe c i fic fa mi l ie s of s uch e v o lut i on s th a t , althou gh de p h asin g , h a v e opt im al ( o v e r al l qua n-
tum pr oc e s s e s ) c o he r e nc e- g e ne r a t in g po w e r for s ome in t e r me d i a t e t ime .
F in al ly , w e in v e s t i g a t e the c o he r e nc e- g e ne r a t in g ca p a b i l it y of r a ndom de p h asin g ch a nne l s . W e find th a t
c onc e n tr a t ion of me as ur e imp l ie s th a t , in l a r g e H i l be r t sp a c e d ime n sion s, unit a r y qua n tum pr oc e s s e s g e n-
e r a t e almos t c e r t ainly ne a rly m a x im al c o he r e nc e w hi le de p h asin g pr oc e s s e s ne a rly minim al .
C h a pt e r t ex t i s a d a pt e d f r om [ 9 ], includ in g r e s ults al s o f r om [ 10 ].
1.3 In troduc tion
One of the m ain d i s t inct iv e fe a tur e s of qua n tum the or y i s the s u p er p o sit io n pr i n cip le . Ac c or d in g t o it , p h ys -
ical s t a t e s of a qua n tum s ys t e m ca n be ex pr e s s e d as l ine a r c omb in a t ion s of othe r qua n tum s t a t e s a nd d i f -
fe r e n t b as e s, us ual ly as s oc i a t e d w ith ei g e n s t a t e s of o bs e r v a b le s, y ie ld d i ffe r e n t ex p a n sion s . The pr e s e nc e of
a c c e s si b le r e l a t iv e p h as e s be t w e e n the d i ffe r e n t br a nche s i s kno w n as q u a n t u m co h er en ce a nd g iv e s r i s e t o
qua n tum in t e r fe r e nc e p he nome n a , ly in g in the he a r t of the or y [ 11 ]. Qua n tum c o he r e nc e , be side s bein g
3
a in t e gr al p a r t of the qua n tum the or y , c on s t itut e s al s o a n impor t a n t in gr e d ie n t , for ex a mp le , in qua n tum
me tr o lo g y [ 12 – 14 ], qua n tum c omput a t ion [ 15 ] a nd qua n tum e r r or c or r e ct ion [ 16 ], qua n tum the r mody -
n a mic s [ 17 – 19 ] a nd qua n tum b io lo g ical pr oc e s s e s [ 20 – 22 ].
On the othe r h a nd , impor t a n t cl as s e s of dy n a mic s in ope n qua n tum s ys t e m s as w e l l as v a r ious me as ur e-
me n t pr oc e s s e s le a d t o dep h asi n g of the s ys t e m unde r c on side r a t ion ( s e e , e . g., [ 6 ] a nd [ 23 ]). D e p h asin g
pr oc e s s e s a r e l ink e d t o los s of infor m a t ion as s oc i a t e d w ith the r e l a t iv e p h as e s be t w e e n the br a nche s of the
w a v ef unct ion . N e v e r the le s s, de p h asin g of a qua n tum s t a t e doe s not ne c e s s a r i ly imp ly t ot al los s of its qua n-
tum c o he r e nc e , sinc e both de p h asin g a nd c o he r e nc e a r e not ion s w e l l - define d only w ith r e spe ct t o spe c i fic
b as e s in the H i l be r t sp a c e w hich ca n, in g e ne r al , be d i ffe r e n t.
The m ain aim of thi s ch a pt e r i s t o in tr oduc e a me thod t o qua n t i f y the ca p a b i l it y of unit a r y a nd de p h asin g
pr oc e s s e s t o g e ne r a t e qua n tum c o he r e nc e out of inc o he r e n t s t a t e s . W e in v e s t i g a t e ho w the effic ie nc y of a
de p h asin g pr oc e s s ( e . g., a non- s e le ct iv e or tho g on al me as ur e me n t ) t o g e ne r a t e c o he r e nc e de pe nd s on the
as s oc i a t e d b as e s in the H i l be r t sp a c e a nd s tudy its m a x imi za t ion . The situa t ion of unit a r y a nd de p h asin g
pr oc e s s e s oc c ur r in g o v e r a r a ndom b asi s i s al s o ex a mine d . W e f ur the r c on side r qua n tum e v o lut ion s de-
s cr i be d b y the L ind b l a d m as t e r e qua t ion w hich le a d t o de p h asin g of the s ys t e m unde r c on side r a t ion a nd
ex a mine ho w their ca p a b i l it y t o g e ne r a t e c o he r e nc e v a r ie s as a f unct ion of t ime . R e m a rk a b ly , w e find th a t
the r e ex i s t t ime in s t a nc e s o v e r w hich c e r t ain s uch de p h asin g e v o lut ion s ca n g e ne r a t e c o he r e nc e as w e l l as
the opt im al unit a r y pr oc e s s e s .
1.4 B a sic ma the ma tic al definit ion s
L e tfj i⟩g
d
i= 1
be a n or thonor m al b asi s of the H i l be r t sp a c e H

=C
d
a ndf P
i
:=j i⟩⟨ ijg
d
i= 1
be the as s oc i a t e d
fa mi ly of r a nk -1 or tho g on al pr oj e ct or s . W e c on side r the ope r a t or sp a c e B(H) o v e rH as a H i l be r t sp a c e
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) a nd nor m∥ X∥
2
:=
√
⟨ X; X⟩ . The
( S h a tt e n ) 1- nor m of ope r a t or X i s define d as∥ X∥
1
:= Tr
(
p
X
y
X
)
=
∑
d
i= 1
s
i
(w he r ef s
i
g
d
i= 1
a r e the
sin g ul a r v alue s of X ) w hi le∥ X∥
1
de not e s the ope r a t or ( spe ctr al ) nor m, i . e ., ∥ X∥
1
:= max
i
( s
i
) .
The a bo v e c on s tr uct ion ca n be ex t e nde d t o the s u pe r ope r a t or sp a c e B(B(H)) , w hich ca n be simi l a rly
e quippe d w ith a ( H i l be r t - S chmidt o v e r the H i l be r t sp a c e B(H) ) s cal a r pr oduct ⟨X;Y⟩ := Tr(X
y
Y)
4
a nd a 2- nor m ∥X∥
2
:=
√
⟨X;X⟩ (w he r eX;Y 2 B(B(H)) ) ² . The 1-1 induc e d nor m i s de not e d as
∥X∥
1; 1
:= sup
∥ A∥
1
= 1
(∥X A∥
1
) . The 1-1 nor m i s un s t a b le unde r t e n s or i za t ion [ 24 ], i . e ., in g e ne r al ∥X
Y∥
1; 1
̸=∥X∥
1; 1
∥Y∥
1; 1
. N e v e r the le s s the d i a mond nor m, w hich ca n be define d as ∥X∥
♢
:= ∥X
I
d
∥
1; 1
[ 25 ],
s a t i sfie s∥X
Y∥
♢
= ∥X∥
♢
∥Y∥
♢
( I
d
a bo v e de not e s the ide n t it y s u pe r ope r a t or o v e r H

= C
d
). W e
define as p h ysical ly v al id qua n tum ope r a t ion s E o v e r the s e t of de n sit y ope r a t or s S(H) B(H) al l the
l ine a r , C omp le t e ly P osit iv e ( C P ) a nd T r a c e P r e s e r v in g ( TP ) m a ps E :S(H)!S(H) .
G iv e n a c omp le t e s e t of or tho g on al ( not ne c e s s a r i ly r a nk -1) pr oj e ct or s B = f Π
i
g
i
( i . e ., Π
i
= Π
y
i
,
Π
i
Π
j
= Π
i
δ
ij
,
∑
i
Π
i
= I ) w e define the B -dep h asi n g s u p er o p er a t o r as
D
B
() :=
∑
i
Π
i
() Π
i
; (1.1)
w hich i s a n or tho g on al pr oj e ct or o v e r B(H) . The c omp le me n t a r y pr oj e ct or i s de not e d Q
B
:= ID
B
.
E v e r y or thonor m al b asi s fj i⟩g
d
i= 1
ofH h as a n as s oc i a t e d de p h asin g s u pe r ope r a t or D
B
, w he r e B =f P
i
g
d
i= 1
.
The r a n g e of a l ine a r ope r a t or X i s de not e d as Ran( X) .
F or a d - d ime n sion al pr o b a b i l it y v e ct or p ( i . e ., p
i
0 ,
∑
d
i= 1
p
i
= 1 ) w e de not e its Sh a n n o n en tr o p y as
H( p) :=
∑
d
i= 1
p
i
ln( p
i
) . W e us e S( ρ) := Tr( ρ ln ρ) for the v o n N eu m a n n en tr o p y of ρ 2 S(H) .
F in al ly , w e h a v e s e t ℏ = 1 .
1.5 Qu an tum c oherenc e a s a re sou rc e theor y
Qua n tum r e s our c e the or ie s [ 26 ] i s a t e r m th a t r efe r s t o the e le g a n t m a the m a t ical f r a me w ork th a t h as be e n
de v e lope d in or de r t o for m al i z e the not ion of a r e so u r ce in qua n tum me ch a nic s ³ .
The m ain ide a be hind qua n tum r e s our c e the or ie s i s r a the r s tr ai gh tfor w a r d : a s ubs e t of the p h ysical s t a t e s
of the qua n tum s ys t e m unde r c on side r a t ion i s d i s t in g ui she d ( cal le d fr e e s t a t e s ), as w e l l as a s ubs e t of its pos -
si b le qua n tum e v o lut ion s ( cal le d fr e e o p er a t io n s ). I n tuit iv e ly , f r e e ope r a t ion s c or r e spond t o the qua n tum
e v o lut ion s th a t c on s ume the r e s our c e in c on side r a t ion, w hi le f r e e s t a t e s a r e r e s our c e le s s s t a t e s . F or the
² N ot ic e th a t w e us e the s a me s y mbo l jj()jj
2
both for the ope r a t or 2- nor m a nd the s u pe r ope r a t or 2- nor m . W hich definit ion
i s a pp l ica b le should be cle a r f r om the c on t ex t.
³ A mor e in- de pth d i s c us sion i s pos t pone d un t i l C h a pt e r 3 .
5
r e s our c e the or y t o be c on si s t e n t , f r e e ope r a t ion s should alw a ys m a p f r e e s t a t e s t o f r e e s t a t e s, i . e ., f r e e ope r -
a t ion s should not be a b le t o g e ne r a t e a n y r e s our c e out f r e e s t a t e s .
A r e s our c e qua n t i fi e r i s a f unct ion th a t qua n t i fie s the a moun t of r e s our c e in a s t a t e . M a the m a t ical ly ,
r e s our c e qua n t i fie r s a r e f unct ion s f r om qua n tum s t a t e s t o non- ne g a t iv e r e al n umbe r s w ith the pr ope r t ie s
th a t
( i ) the a moun t of r e s our c e of a f r e e s t a t e v a ni she s, a nd
( i i ) the a moun t of r e s our c e c on t aine d in a n y s t a t e ca nnot incr e as e unde r the a ct ion of f r e e ope r a t ion s .
S uch f unct ion s a r e he nc e al s o cal le d r e so u r ce m o n o t o n e s .
I n the r e s our c e the or y of c o he r e nc e the s e t of f r e e s t a t e s I
B
, w hich a r e cal le d i n co h er en t s t a t e s , i s define d
(w ith r e spe ct t o a b asi s ) as the im a g e ( o v e r qua n tum s t a t e s ) of the as s oc i a t e d B - de p h asin g s u pe r ope r a t or :
I
B
= Ran(D
B
) w he r e B =f P
i
g
d
i= 1
, i . e ., a s t a t e ρ i s inc o he r e n t i f a nd only i f ρ =
∑
i
p
i
P
i
w ithf p
i
g
d
i= 1
a n y
pr o b a b i l it y d i s tr i but ion .
The s e t of f r e e ope r a t ion s I
B
h as t o be c omp a t i b le w ith the s e t of f r e e s t a t e s, b y e n s ur in g no r e s our c e
ca n be g e ne r a t e d b y the a ct ion of f r e e ope r a t ion on f r e e s t a t e s . M a the m a t ical ly , i f W 2 I
B
i s f r e e the n
W( ρ)2 I
B
for a n y ρ2 I
B
. Thi s i s the minim al r e quir e me n t of the the or y for c on si s t e nc y a nd g iv e s r ai s e
t o the l a r g e s t cl as s of f r e e ope r a t ion s, kno w n as M a x im al ly I nc o he r e n t O pe r a t ion s ( MIO ) [ 27 ] w hich a r e
qua n tum ope r a t ion s W s uch th a t W( I
B
) I
B
. S e v e r al alt e r n a t iv e s ubcl as s e s of f r e e ope r a t ion s h a v e be e n
define d a nd in v e s t i g a t e d ( s e e , e . g., [ 28 ]), he r e w e me n t ion jus t a fe w :
• The s ubcl as s of I n co h er en t O p er a t o r s ( IO ) [ 29 ] c on si s ts of the C P TP m a ps a dmitt in g a s e t of Kr a us
ope r a t or s [ 30 ]f K
n
g
n
s uch th a t , for al l ρ2 I
B
, K
n
ρ K
y
n
= Tr( K
n
ρ K
y
n
)2 I
B
.
• The s ubcl as s of D ep h asi n g - co v a r i a n t I n co h er en t O p er a t o r s ( DIO ) [ 31 , 32 ] c on t ain s the ope r a t or s W
s uch th a t [W;D
B
] = 0 .
• The s ubcl as s of S tr ictly I nc o he r e n t O pe r a t ion s ( S IO ) [ 33 ] c on t ain s the ope r a t or s in IO th a t in a d -
d it ion f u l fi l l K
y
n
ρ K
n
= Tr( K
y
n
ρ K
n
)2 I
B
for a n y ρ2 I
B
.
6
• F in al ly , Ge n uine ly I nc o he r e n t O pe r a t or s ( GIO ) [ 34 ] a r e the ope r a t ion s W th a t le a v e al l inc o he r e n t
s t a t e s in v a r i a n t , i . e ., for a n y ρ 2 I
B
it ho ld s th a t W( ρ) = ρ . F r om the definit ion it imme d i a t e ly
fo l lo w s th a t al l GIO a r e in a dd it ion u n it a l .
A f unct ion al c
B
:S(H)!R
+
0
(w he r eR
+
0
:= [ 0;1) ) i s a c o he r e nc e monot one i f the fo l lo w in g t w o
pr ope r t ie s ho ld :
( i ) c
B
( ρ) = 0 for al l ρ2 I
B
, a nd
( i i ) c
B
(W ρ) c( ρ) for al l ρ2S(H) a ndW2I
B
.
C le a rly , w he the r s uch a f unct ion al i s a monot one or not de pe nd s on the s e t of f r e e ope r a t ion s, e . g., a mono -
t one of a s ubcl as s of MIO i s not ne c e s s a r i ly a monot one for MIO . The c on v e r s e , ho w e v e r , i s tr ue: MIO
monot one s a r e monot one s for al l the pos si b le s ubcl as s e s . M onot one s impos e ne c e s s a r y c ond it ion s for in-
t e r c on v e r sion of s t a t e s unde r f r e e ope r a t ion s, sinc e c
B
( ρ
1
) < c
B
( ρ
2
) s u gg e s ts th a t it i s impos si b le t o c on v e r t
ρ
1
7! ρ
2
unde r f r e e ope r a t ion s . M onot one s, the r efor e , q u a n t i f y ho w m uch r e s our c e a s t a t e c on t ain s – thi s
a moun t c a nnot incr e as e unde r f r e e ope r a t ion s .
1.6 Coherenc e - g ener a ting po wer of q u an tum oper a tion s
I n thi s s e ct ion w e in tr oduc e the definit ion of the Co h er en ce- Gen er a t i n g P o w er ( C GP ) of a qua n tum op -
e r a t ion, a qua n t it y th a t p l a ys a n impor t a n t r o le for thi s ch a pt e r . The C GP i s a n ope r a t ion al ly mot iv a t e d
definit ion aimin g t o ca ptur e ho w effic ie n t i s a qua n tum e v o lut ion a t g e ne r a t in g qua n tum c o he r e nc e out of
inc o he r e n t s t a t e s .
The ide a th a t al lo w s tr a n sit ionin g f r om qua n t i f y in g the a moun t of c o he r e nc e in a qua n tum s t a t e ρ t o
qua n t i f y in g the a b i l it y of a qua n tum ope r a t ion E t o g e ne r a t e c o he r e nc e , i s r a the r s tr ai gh tfor w a r d : one
im a g ine sE a ct in g on ρ
in
=
∑
i
p
i
P
i
, the l a tt e r bein g chos e n a t r a ndom f r om a uni for m e n s e mb le of “ input ”
s t a t e s, al l of w hich a r e inc o he r e n t w ith r e spe ct t o B =f P
i
g
d
i= 1
. The n one a v e r a g e s the a moun t of c o he r e nc e
c on t aine d in the pr oc e s s e d s t a t e E( ρ
in
) o v e r the e n s e mb le of input s t a t e s ( i . e . f p
i
g
d
i= 1
a r e tr e a t e d as r a ndom
v a r i a b le s ), o bt ainin g a qua n t i fie r ch a r a ct e r i z in g E . C le a rly , thi s qua n t i fica t ion of the a b i l it y of the qua n tum
7
ch a nne l t o g e ne r a t e c o he r e nc e de pe nd s on the cho ic e of the me as ur e of ( s t a t e ) c o he r e nc e c
B
. The cho ic e
of the c o he r e nc e monot one c
B
i s fa r f r om unique a nd d i ffe r e n t cho ic e s a r e pos si b le , de pe nd in g on the s e t
of f r e e ope r a t ion s .
Thi s a ppr o a ch i s e nca ps ule d in the fo l lo w in g definit ion .
D e finitio n 1.1
The C o he r e nc e- Ge ne r a t in g P o w e r ( C GP ) C
B
:E 7! C
B
(E)2 R
+
0
of a qua n tum ope r a t ion E w ith
r e spe ct t o B =f P
i
g
d
i= 1
a nd c o he r e nc e me as ur e c
B
i s define d as
C
B
(E) :=
∫
d μ
unif
( p) c
B
[E (
∑
i
p
i
P
i
)]; (1.2)
w he r e d μ
unif
( p) :=
1
( d 1)!
δ(
∑
i
p
i
1)
∏
i
dp
i
i s the uni for m me as ur e in the ( d 1) - d ime n sion al
simp lex .
The ( d 1) - d ime n sion al simp lex i s the sp a c e of al l pos si b le d -tu p le s p = ( p
1
;:::; p
d
) w ith p
i
0 a nd
∑
i
p
i
= 1 , the po in ts of w hich a r e in one t o one c or r e sponde nc e w ith the d i a g on al e le me n ts of the inc o -
he r e n t input s t a t e s ρ
in
=
∑
d
i= 1
p
i
P
i
( as s umin g a fi xe d B w ith r e spe ct t o w hich al l input s t a t e s a r e d i a g on al ).
Befor e pr oc e e d in g f ur the r b y spe c i f y in g the c o he r e nc e me as ur e c
B
, w e pr o v ide a n alt e r n a t iv e in t e r pr e-
t a t ion for the me a nin g of the C GP of a unit a r y qua n tum m a p C
B
(U) . S u ppos e w e a r e in t e r e s t e d in the
fo l lo w in g que s t ion : g iv e n a r a ndom pur e s t a t e j ψ⟩⟨ ψj 2 S(H) , w h a t i s the a v e r a g e c o he r e nc e c
B
of the
de p h as e d qua n tum s t a t e D
B
′ (j ψ⟩⟨ ψj) , for s ome fi xe d b as e s B a nd B
′
? F or w h a t fo l lo w s, w e as s ume the
r a ndom pur e s t a t e s a r e 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 . R e l a t e d m a tt e r s w e r e in v e s t i g a t e d in
[ 35 , 36 ].
A s w e w i l l sho w no w , the a v e r a g e c o he r e nc e pr e s e n t a ft e r the de p h asin g of a r a ndom pur e s t a t e s i s nothin g
e l s e th a n the C GP of a c or r e spond in g unit a r y ope r a t or c onne ct in g the b as e s B a nd B
′
.
8
P r opositio n 1.1
L e t B = fj i⟩⟨ ijg
d
i= 1
a nd B
′
= fj i
′
⟩⟨ i
′
jg
d
i= 1
be c omp le t e fa mi l ie s of r a nk -1 or tho g on al pr oj e ct or s a nd
U2 U( d) be a unit a r y ope r a t or s uch th a t j i
′
⟩ = Uj i⟩ for al l i = 1;:::; d . The n
∫
d μ
Haar
( ψ) c
B
(D
B
′j ψ⟩⟨ ψj) = C
B
(U) ; (1.3)
w he r eU() = U() U
y
.
P r o of . W e w i l l fir s t sho w th a t the uni for m e n s e mb le o v e r the simp lex in E q . ( 1.2 ) c o inc ide s, in fa ct ,
w ith the me as ur e o bt aine d b y H a a r d i s tr i but e d pur e s t a t e s j ψ⟩ th a t a r e de p h as e d in s ome ( a r b itr a r y )
b asi s B , c on si s t in g of e le me n ts fj i⟩⟨ ijg , i = 1;:::; d .
I nde e d , for a n y ( me as ur a b le ) f unct ion f the ex pe ct a t ion v alue o v e r the e n s e mb le i s g iv e n b y ⟨ f(D
B
(j ψ⟩⟨ ψj)⟩
ψ
.
C al l in g p
i
=j⟨ ij ψ⟩j
2
w e ca n w r it e it as
⟨ f(D
B
(j ψ⟩⟨ ψj)⟩
ψ
= M
∫
d ψ
1

∫
d ψ
d
f( p
1
;:::; p
d
) δ( 1
d
∑
i= 1
p
i
)
w he r e M i s a nor m al i za t ion c on s t a n t a nd d ψ
i
= dRe( ψ
i
) dIm( ψ
i
) . S w it chin g t o po l a r c oor d in a t e s one
h as d ψ
i
= r
i
dr
i
d ϑ
i
= dp
i
d ϑ
i
= 2 . P e r for min g the in t e gr a t ion o v e r the a n gle s ϑ
i
w e o bt ain
⟨ f(D
B
(j ψ⟩⟨ ψj)⟩
ψ
= M
′
∫
dp
1

∫
dp
d
f( p
1
;:::; p
d
) δ( 1
d
∑
i= 1
p
i
);
th a t i s, the uni fo r m me as ur e o v e r the simp lex ( M
′
i s a nothe r nor m al i za t ion c on s t a n t ).
A s a c on s e que nc e of the a bo v e , w e ca n ex pr e s s
C
B
(E) =
∫
d μ
Haar
( ψ) c
B
[ED
B
(j ψ⟩⟨ ψj)] : (1.4)
The r e s ult the r efor e fo l lo w s f r om D
B
′ =UD
B
U
y
a nd the fa ct th a t the H a a r me as ur e i s unit a r i ly in v a r i -
9
a n t. ■
N o w w e spe c i f y the c o he r e nc e me as ur e c
B
. W e ex a mine t w o pos si b le cho ic e s: the monot one a r i sin g
f r om the H i l be r t - S chmidt 2- nor m a nd the r e l a t iv e e n tr op y of c o he r e nc e .
Hi lb er t - Sc hmi d t 2- n o rm b a sed mo n o t o ne
The H i l be r t - S chmidt ope r a t or 2- nor m g iv e s r i s e t o the c o he r e nc e me as ur e
c
2; B
( ρ) := min
σ2 I B
∥ ρ σ∥
2
2
=∥Q
B
ρ∥
2
2
=
∑
i̸= j
j ρ
ij
j
2
: (1.5)
The f unct ion al c
2; B
i s a c o he r e nc e monot one in the ( r e s tr ict iv e ) cl as s of GIO . M or e g e ne r al ly , it c on s t itut e s
a monot one for a n y cl as s of f r e e ope r a t ion s i f one r e s tr icts t o u n it a l C P TP m a ps ⁴ ( s uch as de p h asin g pr o -
c e s s e s c on side r e d l a t e r ). The a for e me n t ione d cl aim fo l lo w s f r om the m a the m a t ical r e s ult in R ef . [ 37 ] for
the 2- nor m . A lthou gh spe c i fic t o unit al m a ps, the c
2; B
c o he r e nc e qua n t i fie r al lo w s for ex p l ic it c omput a b le
for m ul as for the C GP in a n y finit e H i l be r t sp a c e d ime n sion :
P r opositio n 1.2
L e t C
2; B
(E) be the c o he r e nc e- g e ne r a t in g po w e r ( E q . ( 1.2 )) of the unit al qua n tum ch a nne l E w ith
c o he r e nc e qua n t i fie r c
B
= c
2; B
. The n,
( i )
C
2; B
(E) =
1
d( d + 1)
∑
i
(
⟨E P
i
;E P
i
⟩⟨D
B
E P
i
;D
B
E P
i
⟩
)
(1.6)
⁴ One h as th a t ∥T( X)∥
2
2
= ⟨T

T( X); X⟩ λ
M
∥ X∥
2
2
w he r e λ
M
i s the l a r g e s t ei g e n v alue of T

T . The l a tt e r ope r a t or
i s a tr a c e-pr e s e r v in g C P - m a p for unit al T a nd the r efor e λ
M
1 . S inc e the a r g ume n t of the 2- nor m in E q . (( 1.5 )) i s alw a ys
tr a c e le s s, for the monot onic y pr ope r t y t o ho ld s u ffic e s th a t P
0
T

TP
0
(P
0
pr oj e ct ion o v e r the sp a c e of tr a c e le s s ope r a t or s ) h as
ei g e n v alue s s m al le r th a n one . Thi s pr ope r t y i s w e ak e r th a n unit al it y .
10
( i i ) I fE i s, in a dd it ion, nor m al ( [E;E
y
] = 0 ), the n
C
2; B
(E) =
1
2d( d + 1)
∥[E;D
B
]∥
2
2
(1.7)
P r o of . ( i ) E qua t ion ( 1.2 ) ca n be e quiv ale n tly w r itt e n as
C
2; B
(E) =
∫
d μ
unif
( p)
[
⟨Q
B
E
∑
i
p
i
P
i
;Q
B
E
∑
j
p
j
P
j
⟩
]
=
∑
i; j
∫
d μ
unif
( p)
[
p
i
p
j
]
⟨Q
B
E P
i
;Q
B
E P
j
⟩
=
∑
i; j
∫
d μ
unif
( p)
[
p
i
p
j
]
(
⟨E P
i
;E P
j
⟩⟨D
B
E P
i
;D
B
E P
j
⟩
)
;
w he r e the l as t e qual it y w as o bt aine d usin g the definit ion Q
B
= ID
B
a nd the fa ct th a t D
B
i s a
he r mit i a n or tho g on al pr oj e ct or ( a nd the r efor e ide mpot e n t ). A s s umin g th a t the qua n tum pr oc e s s e s i s
unit al ( E( I) = I ) a nd usin g the fa ct th a t for uni for m input e n s e mb le of s t a t e s
∫
d μ
unif
( p)
[
p
i
p
j
]
=
[ d( d + 1)]
1
( 1 + δ
ij
) ( s e e , e . g., [ 38 ] for a de r iv a t ion ) the r e s ult fo l lo w s .
( ii ) W e h a v e
∥[E;D
B
]∥
2
2
= Tr
(
E
y
ED
B
)
+ Tr
(
EE
y
D
B
)
2 Tr
(
D
B
E
y
D
B
E
)
:
F r om the nor m al it y as s umpt ion it fo l lo w s th a t the fir s t t w o t e r m s a r e e qual . The s u pe r ope r a t or tr a c e s
ca n the n be e v alua t e d usin g the H i l be r t - S chmidt ope r a t or inne r pr oduct as Tr(X) =
∑
i; j
⟨j i⟩⟨ jj;X (j i⟩⟨ jj)⟩
w hich y ie ld s Tr
(
E
y
ED
B
)
=
∑
i
⟨E P
i
;E P
i
⟩ . A simi l a r calc ul a t ion for the r e m ainin g t e r m g iv e s
Tr
(
D
B
E
y
D
B
E
)
=
∑
i
⟨D
B
E P
i
;D
B
E P
i
⟩
a nd he nc e the r e s ult fo l lo w s . ■
11
L e t us no w m ak e a c ou p le of r e m a rk s for C GP b as e d on the H i l be r t - S chmidt 2- nor m . The or i g in al a v e r -
a g in g definit ion for the C GP , E qua t ion ( 1.2 ), s ur pr i sin gly a dmits in thi s cas e the m uch simp le r for m g iv e n
b y E q . ( 1.7 ). The l as t e qua t ion al s o imp l ie s th a t the 2- nor m C GP for unit al qua n tum ch a nne l s define d or i g -
in al ly i s nothin g mor e th a n a m e as u r e of t h e de g r e e of n o n - co m m u t a t i v it y b et w e en E a n d t h e dep h asi n g c h a n n e l
D
B
( s e e al s o [ 39 ]).
R el a tiv e entr o p y b a sed mo n o t o ne
A c o he r e nc e monot one for MIO ( a nd the r efor e al l s ubcl as s e s of f r e e ope r a t ion s, s e e e . g. [ 31 ]) i s o bt aine d
usin g r e l a t iv e e n tr op y [ 29 ]:
c
r; B
( ρ) := min
σ2 I B
S( ρ∥ σ) = S(D
B
ρ) S( ρ) (1.8)
L e t C
r; B
(E) be the C GP E q . ( 1.2 ) of the qua n tum ch a nne l E w ith c
B
= c
r; B
. The n, f r om E q . ( 1.8 ) it i s
imme d i a t e th a t
C
r; B
(E) =
∫
d μ
unif
( p)
[
S(D
B
E ρ
in
( p)) S(E ρ
in
( p))
]
: (1.9)
U nde r the cl as s of IO the monot one c
r
( ρ) h as a n ope r a t ion al in t e r pr e t a t ion as the opt im al r a t e of as y mp -
t ot ic c o he r e nc e d i s t i l l a t ion, i . e ., c
r; B
( ρ) = sup R s uch th a t ρ

n
IO
7 ! Φ

nR
2
as n ! 1 , w he r e Φ
2
i s the
m a x im al c o he r e nc e qub it s t a t e [ 33 ]. The r e l a t iv e e n tr op y C GP of E the r efor e a dmits a n ope r a t ion al in t e r -
pr e t a t ion as the a v er a ge r a t e of d is t i ll a b le co h er en ce ( unde r IO ) of the pr oc e s s e d s t a t e E
(
ρ
in
)
.
Unit a r y pr o c e sse s
I n thi s s e ct ion w e br iefly d i s c us s ho w the pr e v ious r e s ults ch a n g e for spe c i al cas e of unit a r y qua n tum ch a n-
ne l s,E =U . The fo l lo w in g pr o v ide s the spe c i al i za t ion of P r oposit ion 1.2 t o unit a r y pr oc e s s e s .
12
P r opositio n 1.3
L e t C
2; B
(E) be the c o he r e nc e g e ne r a t in g po w e r ( E q . ( 1.2)) of the unit a r y qua n tum ch a nne l U() =
U() U
y
w ith c o he r e nc e qua n t i fie r c
B
= c
2; B
. The n,
C
2; B
(U) =
1
d( d + 1)
0
@
d
d
∑
i; j= 1
j⟨ ij Uj j⟩j
4
1
A
: (1.10)
The pr oof i s simi l a r t o the one of P r oposit ion 2.1 g iv e n l a t e r , a nd he nc e omitt e d .
The a bo v e ex pr e s sion, exc e pt pr o v id in g a simp le r c omput a b le v e r sion of the or i g in al C GP definit ion
for unit a r ie s, s u gg e s ts al s o a pr ot oc o l for d ir e ctly de t e ct in g the C GP of a unit a r y U , w ithout the ne e d t o
g e ne r a t e a n y uni for m e n s e mb le of input s t a t e s or a n y qua n tum s t a t e/pr oc e s s t omo gr a p h y [ 10 ].
Z h a n g et a l. in [ 40 ] h a v e o bt aine d ex p l ic it ex pr e s sion s for the r e l a t iv e e n tr op y C GP , E q . ( 1.9), w he nE i s
a unit a r y ch a nne l . L a t e r in thi s ch a pt e r w e ex p l ain ho w one ca n o bt ain ex pr e s sion s for de p h asin g ch a nne l s .
1.7 Maximal l y and p ar ti al l y deph a sing pro c e sse s
I n thi s a nd nex t s e ct ion s, w e in v e s t i g a t e ho w the pr e v ious c on side r a t ion s r e g a r d in g C GP spe c i al i z e t o the
spe c i fic cl as s of de p h asin g pr oc e s s e s . W e be g in b y fir s t d i s t in g ui shin g be t w e e n t w o fa mi l ie s of the m .
D e finitio n 1.2
W e ch a r a ct e r i z e a de p h asin g m a p D
B
: S(H) ! S(H) as a m a x im a lly de p h asin g ch a nne l i ff Rank[ Π
i
] = 1 for al l Π
i
2 B . Othe r w i s e the m a p i s a p a r ti a lly de p h asin g ch a nne l .
I n othe r w or d s, for a m ax i m a ll y de p h asin g pr oc e s s the r e ex i s ts a n or thonor m al b asi s s uch th a t al l out put
s t a t e s of the ch a nne l a r e d i a g on al w ith r e spe ct t o th a t b asi s . I f the out put s t a t e s a r e b lock d i a g on al (w ith
non-tr iv i al b lock s ) w e cal l the de p h asin g p a r t i a l ( a simi l a r not ion w as al s o define d in [ 41 , 42 ]). N ot ic e ,
ho w e v e r , th a t the b asi s o v e r w hich thi s i s a chie v e d i s not unique ly spe c i fie d : for ex a mp le , a n y pe r m ut a t ion
13
or p h as e shi ft on the b asi s e le me n ts w i l l s t i l l pr e s e r v e the d i a g on al out c ome s . N e v e r the le s s, the c omp le t e
s e t of pr oj e ct or s B c omp le t e ly ch a r a ct e r i z e s the ( m a x im al ly or p a r t i al ly ) de p h asin g ch a nne l :
L e mm a 1.1
The s e t of pr oj e ct or s B c or r e spond in g t o the ( m a x im al ly or p a r t i al ly ) de p h asin g ch a nne l D
B
i s unique .
P r o of . C le a rly , the s e t of pr oj e ct or s B =f Π
i
g i s a s e t of Kr a us ope r a t or s for the C P TP m a p D
B
. W e
ne e d t o sho w th a t i f B
′
=f Π
′
i
g
i
i s a c omp le t e fa mi ly of or tho g on al pr oj e ct or s w ith D
B
=D
B
′ , the n
B = B
′
. I nde e d , the t w o Kr a us de c omposit ion s de s cr i be the s a me ch a nne l i ff the r e ex i s ts a unit a r y U
s uch th a t Π
′
i
=
∑
j
U
ij
Π
j
( s e e , e . g. [ 43 ]). B ut
∑
i
Π
′
i
= I w hich i s tr ue only i f
∑
i
U
ij
= 18 j . On the
othe r h a nd , U i s a unit a r y m a tr i x s o the c o lumn s for m or thonor m al v e ct or s . The l as t t w o pr ope r t ie s ca n
ho ld t o g e the r only w he n U i s a pe r m ut a t ion m a t r i x . B ut a pe r m ut a t ion of the Kr a us ope r a t or ind ic e s
doe s n ’t a ffe ct the s e t B , the r efor e B = B
′
. ■
D e p h asin g pr oc e s s e s ca n be v ie w e d , for ex a mp le , as non- s e le ct iv e or tho g on al me as ur e me n ts . I n the cas e
of m a x im al ly de p h asin g the me as ur e d o bs e r v a b le i s non- de g e ne r a t e a nd th us al l pr oj e ct or s B
′
=f P
′
i
g
d
i= 1
of
D
B
′ a r e r a nk -1, c or r e spond in g t o the d i s t inct ei g e n v alue s of the me as ur e d o bs e r v a b le . On the othe r h a nd ,
in the cas e of a de g e ne r a t e o bs e r v a b le non-tr iv i al s ubsp a c e s B
′
=f Π
′
i
g oc c ur w ith Rank( Π
i
) e qual t o the
de g e ne r a c y of the i -th ei g e n v alue , i . e ., the de p h asin g i s p a r t i al .
1.8 Coherenc e - g ener a ting po wer of m aximal l y deph a sing c h annel s
L e t us be g in w ith a simp le r e m a rk . F or a n y m a x im al ly de p h asin g pr oc e s s the r e ex i s ts a b asi s fj i⟩g
d
i= 1
s uch
th a t the “ c o he r e nc e s ” of the out put s t a t e ( i . e ., e le me n ts ⟨ ijD
B
( ρ)j j⟩ for i̸= j ) v a ni sh . N e v e r the le s s, thi s
doe s not ne c e s s a r i ly imp ly a n inc o he r e n t out put s t a t e sinc e no s t a t e me n t h as be e n m a de r e g a r d in g the
r efe r e nc e b asi s w ith r e spe ct t o w hich c o he r e nc e i s qua n t i fie d . C on side r for ex a mp le the qub it cas e , m a x -
14
im al ly de p h asin g D
B
′ w he r e B
′
= fj+⟩⟨+j;j⟩⟨jg , a nd c o he r e nc e qua n t i fie d w ith r e spe ct t o B =
fj 0⟩⟨ 0j;j 1⟩⟨ 1jg . The n cle a rly Ran(D
B
)̸= Ran(D
B
′) : not al l B
′
de p h as e d s t a t e s a r e inc o he r e n t in B .
I n w h a t fo l lo w s w e ex p a nd on thi s simp le o bs e r v a t ion, qua n t i f y in g thr ou gh E q . ( 1.2 ) ho w “ v alua b le ” a
m a x im al ly de p h asin g ch a nne l i s a t cr e a t in g c o he r e nc e or , in othe r w or d s, w e calc ul a t e ho w m uch r e s our c e
a m a x im al ly de p h asin g ch a nne l g e ne r a t e s on a v e r a g e a ft e r a ct in g on inc o he r e n t s t a t e s . The 2- nor m a nd
r e l a t iv e e n tr op y c o he r e nc e qua n t i fie r s a r e a dopt e d , e a ch r e le v a n t in a d i ffe r e n t cl as s of f r e e ope r a t ion s of
the the or y .
Hi lb er t - Sc hmi d t 2- n o rm c o h er en c e
W e a r e no w r e a dy t o pr o v ide a n ex p l ic it ex pr e s sion of the C GP of m a x im al ly de p h asin g pr oc e s s e s, w he n
the c o he r e nc e me as ur e i s the 2- nor m c o he r e nc e .
P r opositio n 1.4: 2-n o r m C GP of m a x im a lly deph a sin g
L e t B = fj i⟩⟨ ijg
d
i= 1
a nd B
′
= fj i
′
⟩⟨ i
′
jg
d
i= 1
be c omp le t e fa mi l ie s of r a nk -1 or tho g on al pr oj e ct or s a nd
U2 U( d) be a unit a r y ope r a t or s uch th a t j i
′
⟩ = Uj i⟩ for al l i = 1;:::; d . The n
( i ) The 2- nor m C GP of the m a x im al ly de p h asin g ch a nne l D
B
′ i s g iv e n b y
C
2; B
(D
B
′) =
1
d( d + 1)
Tr
[
X
U
X
T
U
(
I X
U
X
T
U
)]
(1.11)
w he r e X
U
2R
d d
i s b i s t och as t ic w ith ( X
U
)
ij
=j⟨ ij Uj j⟩j
2
.
( i i ) A lt e r n a t iv e ly , on the s u pe r ope r a t or le v e l ,
C
2; B
(D
B
′) =
1
2d( d + 1)
∥[D
B
′;D
B
]∥
2
2
(1.12)
15
( i i i )
0 C
2; B
(D
B
′) C
max
2; B
( d) :=
d 1
4d( d + 1)
; (1.13)
w he r e the l o w e r bound i s a chie v e d i f a nd only i f [D
B
;D
B
′] = 0 .
P r o of . ( i ) F ir s t w e not ic e th a t D
B
′ =UD
B
U
y
, w he r eU() = U() U
y
. N ex t w e calc ul a t e the qua n t i -
t ie s
∑
i
⟨D
B
′ P
i
;D
B
′ P
i
⟩ a nd
∑
i
⟨D
B
D
B
′ P
i
;D
B
D
B
′ P
i
⟩ , a ppe a r in g in e qua t ion ( 1.6 ). A s tr ai gh tfor w a r d
calc ul a t ion g iv e s
∑
i
⟨D
B
′ P
i
;D
B
′ P
i
⟩ =
∑
i
Tr
(
P
i
UD
B
U
y
P
i
)
= Tr
[
X
U
X
T
U
]
:
A simi l a r calc ul a t ion for the othe r t e r m g iv e s
∑
i
⟨D
B
D
B
′ P
i
;D
B
D
B
′ P
i
⟩ =
∑
i
Tr( P
i
D
B
′D
B
D
B
′ P
i
)
=
∑
i
Tr
(
P
i
UD
B
U
y
D
B
UD
B
U
y
P
i
)
= Tr
[(
X
U
X
T
U
)
2
]
:
The r e s ult for C
2; B
(D
B
′) fo l lo w s . The b i s t och as t ic it y of the m a tr i x ( X
U
)
ij
i s a d ir e ct c on s e que nc e of
the unit a r it y of U .
( ii ) W e h a v eD
B
= D
y
B
, i . e ., the m a x im al ly de p h asin g ch a nne l s a r e he r mit i a n the r efor e nor m al , a nd
al s o unit al . The cl aim he nc e fo l lo w s b y s e tt in g E =D
B
′ in e qua t ion ( 1.7 ).
( iii ) The lo w e r bound pr ope r t ie s fo l lo w imme d i a t e ly f r om e qua t ion ( 1.12 ). F or the u ppe r bound o b -
s e r v e th a t the m a tr i x X
U
X
T
U
i s posit iv e s e mi - definit e a nd al s o b i s t och as t ic ( as pr oduct of b i s t och as t ic
m a tr ic e s ). N o w , sinc e b i s t och as t ic m a tr ic e s h a v e a t le as t one ei g e n v alue e qual t o one , the d i ffe r e nc e
( Tr[ X
U
X
T
U
] Tr[( X
U
X
T
U
)
2
]) i s bounde d f r om a bo v e b y ( d 1)[ 1= 2 ( 1= 2)
2
] = ( d 1)= 4 . The
n u me r ical fa ct or of 1= 2 c or r e spond s t o the ( ( d 1) - fo ld de g e ne r a t e ) ei g e n v alue of X
U
X
T
U
w hich m a x -
16
imi z e s the d i ffe r e nc e ( sinc e 0 λ 1 for al l ei g e n v alue s ). N ot ic e th a t it i s not a pr io r i g ua r a n t e e d
th a t a unit a r y m a tr i x U ( c or r e spond in g t o s uch a n X
U
) ex i s ts . W e t a ck le thi s que s t ion in s ubs e ct ion A
of the A ppe nd i x . ■
F or a t w o -le v e l s ys t e m, the a bo v e ex pr e s sion s simp l i f y c on side r a b ly .
E x a mple 1.1: S in g le qubit m a x im a l deph a sin g – 2-n o r m
C on side r a qub it ( H = Spanfj 0⟩;j 1⟩g ) w ith its c o he r e nc e qua n t i fie d w ith r e spe ct t o B =fj 0⟩⟨ 0j;j 1⟩⟨ 1jg
a nd a n y m a x im al ly de p h asin g ch a nne l D
B
′ , w he r e B
′
=fj ψ
0
⟩⟨ ψ
0
j;j ψ
1
⟩⟨ ψ
1
jg (⟨ ψ
0
j ψ
1
⟩ = 0 ).
F or the qub it cas e D
B
′ ca n be p a r a me tr i z e d thr ou gh the ( B loch sp he r e ) a n gle s θ a nd φ , w he r e
j ψ
0
( θ; φ)⟩⟨ ψ
0
( θ; φ)j = 1= 2( I + v σ) ;
w i th v = ( sin θ cos φ; sin θ sin φ; cos θ) . F r om P r op . 1.4 w e h a v e
X
U
=
0
B
@
cos
2
( θ= 2) sin
2
( θ= 2)
sin
2
( θ= 2) cos
2
( θ= 2)
1
C
A
( inde pe n de n t o f φ ) a nd he nc e C
2; B
= 1= 24 sin
2
( 2θ) , θ2 [ 0; π] .
Obs e r v e th a t the u ppe r bound f r om E q . ( 1.13 ) i s a chie v e d for θ 2 f π= 4; 3 π= 4g . On the othe r
h a nd , for θ 2 f 0; π= 2; πg the C GP v a ni she s . The cas e s θ = 0 a nd θ = π g iv e B = B
′
but the
cas e θ = π= 2 ( for al l φ ) c or r e spond s t o B
′
bein g a m utual ly unb i as e d b asi s of B . I n al l s uch cas e s
[D
B
;D
B
′] = 0 .
I n R ef . [ 10 ] it w as sho w n th a t , for unit al qua n tum ch a nne l s E ,
C
2; B
(E)
d 1
d( d + 1)
= 4C
max
2; B
( d) ; (1.14)
17
w he r e the m a x im um i s a chie v e d o v e r unit a r y U() = U() U
y
w ithj⟨ ij Uj j⟩j
2
= 1= d for al l i; j ( e . g. the
qua n tum F our ie r tr a n sfor m ). F or m a x im al ly de p h asin g pr oc e s s e s o bs e r v e th a t the opt im al 2- nor m C GP
ca n be a t mos t one qua r t e r of the m a x im um v alue for the C GP ( o v e r al l unit al C P TP m a ps ). I n s ubs e ct ion A
of the A ppe nd i x w e pr o v ide a n ex p l ic it c on s tr uct ion t o sho w th a t the u ppe r bound for m a x im al ly de p h asin g
pr oc e s s e s C
max
2; B
( d) i s a chie v a b le for H i l be r t sp a c e d ime n sion s d 13 w hi le w e al s o g iv e a s e t of s u ffic ie n t
c ond it ion s for the bound t o be a chie v a b le in a n y d ime n sion .
I n E x a mp le 1.1 it w as not ic e d th a t for a qub it s ys t e m C
2; B
(D
B
′) v a ni she s w he n B
′
i s a m utual ly unb i as e d
b a si s of B . Thi s o bs e r v a t ion ho ld s tr ue for a n y finit e d ime n sion al s ys t e m . C on side r a m utual ly unb i as e d
b a si sfj i
′
⟩g
d
i= 1
s uch th a t j⟨ jj i
′
⟩j
2
= 1= d . The n the m a tr i x X
U
f r om E q . ( 1.11 ) h as m a tr i x e le me n ts ( X
U
)
ij
=
1= d a nd the r efor e C
2; B
(D
B
′) = 0 .
R e l a tiv e entr o p y o f c o h er en c e
W e w i l l no w g iv e a s e t of r e s ults “ p a r al le l ” t o P r op . 1.4 usin g as c o he r e nc e qua n t i fie r , in s t e a d of c
2; B
, the
r e l a t iv e e n tr op y of c o he r e nc e c
r; B
. G iv e n a d - d ime n sion al pr o b a b i l it y v e ct or p w e de not e its s u b en tr o p y
[ 44 ] as Q( p) , define d as
Q( p) :=
d
∑
i= 1
p
d
i
ln p
i
∏
j̸= i
( p
i
p
j
)
; (1.15)
w he r e i f t w o or mor e of the p
i
’ s a r e e qual the n the l imit should be t ak e n as they be c ome e qual . The definit ion
i s ex t e nde d t o c o lumn- s t och as t ic m a tr ic e s X2 (R
+
0
)
d d
as Q( X) := 1= d
∑
j
Q( p
j
) , w he r e ( p
j
)
i
= ( X)
ij
.
P r opositio n 1.5: R e l a tiv e e n tr op y C GP of m a x im a l deph a sin g
L e t B = fj i⟩⟨ ijg
d
i= 1
a nd B
′
= fj i
′
⟩⟨ i
′
jg
d
i= 1
be c omp le t e fa mi l ie s of r a nk -1 or tho g on al pr oj e ct or s a nd
U2 U( d) be a unit a r y ope r a t or s uch th a t j i
′
⟩ = Uj i⟩ for al l i = 1;:::; d . The n
18
0.70 0.75 0.80 0.85
0.038
0.042
0.046
θ
CGP
C
2,B
(θ)
C
r,B
(θ)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.00
0.01
0.02
0.03
0.04
θ
CGP
Figure 1.8.1: Single qubit coherence-generating p o w er of maximally dephasing channels as a function
of the angle θ (see Ex. 1.1 & Ex. 1.2 ), fo r the relative entrop y c
r; B
and 2-no rm c
2; B
coherence quanti-
fiers. Notice that the maximum is obtained fo r slightly different values of the angle θ .
19
( i ) The r e l a t iv e e n tr op y C GP of the m a x im al ly de p h asin g ch a nne l D
B
′ i s g iv e n b y
C
r; B
(D
B
′) = Q
(
X
U
X
T
U
)
Q( X
U
) (1.16)
w he r e X
U
2 R
d d
i s b i s t och as t ic w ith ( X
U
)
ij
= j⟨ ij Uj j⟩j
2
a nd Q( X) de not e s the s ube n tr op y
of X .
( i i ) C
r; B
(D
B
′) = 0 i f a nd only i f [D
B
;D
B
′] = 0 .
P r o of . ( i ) The pr oof i s b as e d o n a le mm a f r om [ 40 ], s t a t in g th a t
∫
d μ
unif
( p) H
0
@
∑
j
B
ij
p
j
1
A
= H
d
1 + Q
(
B
T
)
; (1.17)
w he r e B i s a d d b i s t och as t ic m a tr i x a nd H
d
i s the d -th h a r monic n umbe r . H e r e w e w a n t t o calc ul a t e
the qua n t it y in E q . ( 1.9 ) forE =D
B
′ =UD
B
U
y
, w he r eU() = U() U
y
. Obs e r v e th a t
S(D
B
′ ρ
in
) = S
(
UD
B
U
y
∑
i
( p
i
P
i
)
)
= S
(
D
B
U
y
∑
i
( p
i
P
i
)
)
= H
0
@
∑
j
( X
U
y)
ij
p
j
1
A
= H
0
@
∑
j
( X
U
)
T
ij
p
j
1
A
;
w he r e w e us e d the unit a r y in v a r i a nc e of the v on N e um a nn e n tr op y a nd the fa ct th a t D
B
U
∑
i
( p
i
P
i
) =
∑
i; j
( X
U
)
ij
p
j
P
j
. F r om the le mm a E q . ( 1.17 ) w e the r efor e g e t
∫
d μ
unif
( p) S(D
B
′ ρ
in
) = H
d
1 + Q( X
U
) :
20
I n a simi l a r fashion,
S(D
B
D
B
′ ρ
in
) = S
(
D
B
UD
B
U
y
∑
i
( p
i
P
i
)
)
= S
0
@
∑
i; j
( X
U
X
T
U
)
ij
p
j
P
j
1
A
;
the r efor e
∫
d μ
unif
( p) S(D
B
D
B
′ ρ
in
) = H
d
1 + Q( X
U
X
T
U
) :
C omb inin g th e t w o ex pr e s sion s w e g e t the de sir e d r e s ult.
( ii ) The r e l a t iv e e n tr op y of c o he r e nc e i s a faithf ul me as ur e ( c
r; B
( σ) = 0 i ff σ 2 I
B
) a nd , the r efor e ,
C
r; B
(D
B
′) = 0 i ff D
B
′( σ) 2 I
B
for al l σ 2 I
B
. I n othe r w or d s, C
r; B
(D
B
′) = 0 i s e quiv ale n t t o
Ran(D
B
) bein g in v a r i a n t unde r the a ct ion of D
B
′ . B utD
B
′ i s he r mit i a n a nd the r efor e nor m al a nd as a
r e s ult the v a ni shin g C GP c ond it ion ho ld s i ff both Ran(D
B
) a nd Ker( D
B
) a r e in v a r i a n t o v e r the a ct ion
ofD
B
′ . W e w i l l no w a r g ue th a t thi s c ond it ion i s e quiv ale n t t o [D
B
;D
B
′] = 0 .
S u ppos e[D
B
;D
B
′] = 0 . The n cle a rly Ker(D
B
) i s a n in v a r i a n t s ubsp a c e of D
B
′ a nd s o i s the r efor e
Ran(D
B
) ( f r om nor m al it y of D
B
′ ).
C on v e r s e ly , s u ppos e th a t both Ker( D
B
) a nd Ran(D
B
) a r e in v a r i a n t o v e r the a ct ion of D
B
′ . The n
for a n y ope r a t or X2 Ker(D
B
) w e h a v eD
B
′D
B
X =D
B
D
B
′ X = 0 . B ut al s o for a n y Y2 Ran(D
B
) w e
h a v eD
B
′D
B
Y =D
B
D
B
′ Y ( sinc eD
B
Y = Y ). A s a r e s ult [D
B
;D
B
′] = 0 .
N ot ic e th a t the pr oof r e l ie s only on the faithf ulne s s of the c o he r e nc e me as ur e c
r; B
a nd th us the r e s u lt
ho ld s tr ue for a n y faithf ul c o he r e nc e me as ur e c
B
. ■
L e t us kno w a g ain spe c i al i z e the a bo v e t o the cas e of a t w o -le v e l s ys t e m .
21
E x a mple 1.2: S in g le qubit m a x im a l deph a sin g – r e l a tiv e e n tr op y
A s s ume a sin gle qub it as in E x . 1.1 . U sin g E q . ( 1.16 ) w e ca n calc ul a t e the r e l a t iv e e n tr op y C GP for
m a x im al ly de p h asin g D
B
′ . S e tt in g c := cos
2
( θ= 2) a nd s := sin
2
( θ= 2) , w e g e t
C
r; B
(D
B
′) =
1
( c s)
2
(
[
c
2
( c s) log c + 2s
2
c
2
log( 2cs)

1
2
( c
2
+ s
2
)
2
log( c
2
+ s
2
)
]
+ [ s$ c]
)
: (1.18)
The r e s ult in g f unct ion i s c omp a r e d w ith the c or r e spond in g one f r om E x . 1.1 in F i g ur e 1.8.1 .
N ot ic e th a t e qua t ion s ( 1.11 ) a nd ( 1.16 ) a r e f unct ion s of the m a x im al ly de p h asin g s u pe r ope r a t or s D
B
a ndD
B
′ , w hich a r e ch a r a ct e r i z e d ( a c c or d in g t o P r op . 1.1 ) b y the s e ts of r a nk -1 pr oj e ct or s B a nd B
′
, r e spe c -
t iv e ly . I t i s the r efor e ex pe ct e d th a t a n y cho ic e of a unit a r y U in thos e e qua t ion s s uch th a t Uj i⟩ = e
iθ i
j σ( i)
′
⟩ ,
w i th θ
i
2R a nd σ2S
d
( pe r m ut a t ion ) le a v e s both C
2; B
(D
B
′) a nd C
r; B
(D
B
′) un a ffe ct e d .
1.9 Coherenc e - g ener a ting po wer of p a r ti al l y deph a sing c h annel s
I n thi s s e ct ion w e ex t e nd the pr e v ious r e s ults for the 2- nor m C GP t o p a r t i al ly de p h asin g ch a nne l s .
P r opositio n 1.6: 2-n o r m C GP of p a r ti a l deph a sin g
L e t B =f P
i
=j i⟩⟨ ijg
d
i= 1
a nd B
′
=f Π
k
=
∑
d
k
α= 1
j k; α⟩⟨ k; αjg
r
k= 1
be c omp le t e fa mi l ie s of or tho g on al
pr oj e ct or s, w he r e fj k; α⟩g
k; α
i s a n or t honor m al b asi s a nd d
k
:= Rank( Π
k
) . The n
C
2; B
(D
B
′) =
1
d( d + 1)
Tr
[
Z( I Z)
]
; (1.19)
22
w he r e Z2R
d d
w ith
Z
ij
= Tr
(
∑
k
P
i
Π
k
P
j
Π
k
)
(1.20)
=
r
∑
k= 1
d
k
∑
α; β= 1
⟨ ij k; α⟩⟨ k; αj j⟩⟨ jj k; β⟩⟨ k; βj i⟩ : (1.21)
P r o of . Our s t a r t in g po i n t i s E q . ( 1.6 ) w ithE() =D
B
′() =
∑
k
Π
k
() Π
k
. W e h a v e
∑
i
⟨D
B
′ P
i
;D
B
′ P
i
⟩ =
∑
i
⟨D
B
′ P
i
; P
i
⟩
=
∑
i; k
Tr( Π
k
P
i
Π
k
P
i
) = Tr( Z) :
The mor e de t ai le d ex pr e s sion for Z in t e r m s of the b asi s e le me n ts fj k; α⟩g fo l lo w s b y w r it in g Π
k
=
∑
d
k
α= 1
j k; α⟩⟨ k; αj a nd pe r for min g the tr a c e . The s e c ond t e r m be c ome s
∑
i
⟨D
B
D
B
′ P
i
;D
B
D
B
′ P
i
⟩ =
∑
i
⟨D
B
′D
B
D
B
′ P
i
; P
i
⟩
=
∑
i; j; k; l
Tr
(
Π
l
P
j
Π
k
P
i
Π
k
P
j
Π
l
P
i
)
=
∑
i; j
Tr
[(
∑
l
Π
l
P
i
Π
l
P
j
)(
∑
k
Π
k
P
i
Π
k
P
j
)]
= Tr( Z
2
) :
B y the c y cl ic pr ope r t y of the tr a c e , the m a tr i x Z i s s y mme tr ic a nd al s o r e a l . ■
One o bs e r v e s th a t the Z m a tr i x f r om E q . ( 1.21 ) ca n be al s o ex pr e s s e d as a f unct ion of the unit a r y U c on-
ne ct in g B a nd B
′
( as in P r op . 1.4 ). I nde e d , for a n y U s uch th a t Uj j⟩ =j k( j); α( j)⟩ the n⟨ ij k( j); α( j)⟩ = U
ij
.
O f c our s e , s uch a U i s fa r f r om unique sinc e it de pe nd s on the cho ic e of b asi s for e a ch s ubsp a c e c or r e spond -
in g t o a Π
l
. I n the cas e w he r e the de p h asin g i s m a x im al , r a the r th a t p a r t i al , E q . ( 1.19 ) r e duc e s t o E q . ( 1.11 ).
I nde e d , sinc e d
k
= 1 for al l k = 1;:::; d , the l a be l α inj k; α⟩ be c ome s r e dund a n t ( α = 1 ) a nd w e ca n us e
the n ot a t ion j k; α⟩!j k
′
⟩ . A s a r e s ult Z = X
U
X
T
U
, w he r e U i s a unit a r y s uch th a t j i
′
⟩ = Uj i⟩ .
23
E x a mple 1.3: 2- qubit p a r ti a l a n d m a x im a l deph a sin g
C on side r a 2- qub it H i l be r t sp a c e H

=C
2

C
2
w ith
H = Spanfj 00⟩;j 01⟩;j 10⟩;j 11⟩g
a n d c o he r e nc e qua n t i fie d w ith r e spe ct t o B =f P
00
; P
01
; P
10
; P
11
g ( P
ij
=j ij⟩⟨ ijj ). D efine the c omp le t e
s e t of r a nk -1 pr oj e ct or s B
′
=f P
′
00
; P
′
01
; P
′
10
; P
′
11
g w he r e P
′
ij
=j ψ
i
( θ
1
; φ
1
) ψ
j
( θ
2
; φ
2
)⟩⟨ ψ
i
( θ
1
; φ
1
) ψ
j
( θ
2
; φ
2
)j
( not a t ion as in E x a mp le 1.1).
W e loo k a t de p h asin g D
B
′′ w ith r e spe ct t o B
′′
= f Π
1
= P
′
00
+ P
′
01
; Π
2
= P
′
10
+ P
′
11
g . A pp ly in g
E q . ( 1.19 ) w e g e t
C
2; B
(D
B
′′) =
1
40
sin
2
( 2θ
1
) :
C le a rly , the m a x im um v alue o v e r the de p h asin g ch a nne l s ex a mine d i s M
p
= 1= 40 .
The C GP of the m a x im al ly de p h asin g C
2; B
(D
B
′) i s p lott e d i n F i g ur e 1.9.1 .
1.10 D eph a sing pro c e sse s and c oherenc e - g ener a ting po wer in Mark o vi an d y -
n a mics
I n thi s s e ct ion w e c on side r qua n tum M a rk o v i a n dy n a mical pr oc e s s e s th a t le a d in the lon g t ime l imit t o
de p h asin g of the s ys t e m unde r c on side r a t ion . M or e spe c i fical ly , w e ex a mine M a rk o v i a n dy n a mic s of L ind -
b l a d for m ( s e e , e . g., [ 6 ])
_ ρ =L ρ := i[ H; ρ] +
∑
α
(
L
α
ρ L
y
α

1
2
{
L
y
α
L
α
; ρ
}
)
; (1.22)
24
Figure 1.9.1: C
2; B
(D
B
′) of Ex. 1.3 is a (symmetric) function of b oth θ
1
and θ
2
. The difference
C
2; B
(D
B
′) C
2; B
(D
B
′′) can b e p ositive o r negative, de p ending on the values of θ
1
and θ
2
. The max-
imum value M
m
of C
2; B
(D
B
′) satisfies M
p
< M
m
< C
max
2; B
( d = 4) demonstrating th at fo r a 2-
qubit system the maximally dephasingD
B
′ such that C
2; B
(D
B
′) = C
max
2; B
( d = 4) is not of the fo rm
D
B
′ =D
B
′
1

D
B
′
2
(fo r any qubit bases B
′
1
; B
′
2
).
25
w he r e H i s the s ys t e m H a mi lt oni a n a nd f L
α
g
α
a r e the L ind b l a d ope r a t or s . W e fir s t d i s t in g ui sh thos e L ind -
b l a d t ime e v o lut ion s th a t m a x im al ly de p h as e in the lon g t ime l imit.
D e finitio n 1.3
W e ch a r a ct e r i z e a n ope r a t or L2B(B(H)) of the L ind b l a d for m E q . ( 1.22 ) as a m a x im a lly deph a s -
in g L in d bl a di a n i ff lim
t!1
exp(L t) =D
B
, for s ome m a x im al ly de p h asin g ch a nne l D
B
.
Our nex t s t e ps w i l l be t o ch a r a ct e r i z e the m a x im al ly de p h asin g L ind b l a d i a n s a nd the n calc ul a t e the 2-
nor m C GP for al l s uch t ime e v o lut ion s as a f unct ion of t ime . N a tur al ly , w e w i l l r e c o v e r p a r t of our pr e v ious
r e s ults for the C GP of m a x im al ly de p h asin g ch a nne l s in the l imit t!1 .
W e be g in w ith a L e mm a , w hich ch a r a ct e r i z e s the cl as s of M a rk o v i a n m as t e r e qua t ion s th a t le a d t o m a x -
im al de p h asin g in the lon g t ime l imit.
L e mm a 1.2
L e tL be a L ind b l a d i a n of the g e ne r al for m E q . ( 1.22 ). The n j ψ⟩⟨ ψj 2 Ker(L) i f a nd only i f the
fo l l o w in g c ond it ion s ho ld sim ult a ne ously:
( a ) j ψ⟩ i s a n ei g e n v e ct or of L
α
for al l α , a nd
( b ) j ψ⟩ i s a n ei g e n v e ct or of iH +
1
2
∑
α
⟨ ψj L
α
j ψ⟩ L
y
α
.
P r o of . L e tfj ψ
?
j
⟩g
d 1
j= 1
be a s e t of or thonor m al v e ct or s w ith ⟨ ψj ψ
?
j
⟩ = 0 for al l j = 1;:::; d 1 . W e
h a v eL(j ψ⟩⟨ ψj) = 0 i ff the fo l lo w in g ho ld tr ue:
( a
′
) ⟨ ψjL(j ψ⟩⟨ ψj)j ψ⟩ = 0 ,
( b
′
) ⟨ ψjL(j ψ⟩⟨ ψj)j ψ
?
j
⟩ = 0 for al l j ,
( c
′
) ⟨ ψ
?
k
jL(j ψ⟩⟨ ψj)j ψ
?
j
⟩ = 0 for al l j; k .
26
B y p lu gg in g in the L ind b l a d for m for L ( e qua t ion ( 1.22 )), it fo l lo w s th a t c ond it ion ( a
′
) ho ld s tr ue i ff c ond it ion ( a) it tr ue . The n, g iv e n( a) , c ond it ion ( b
′
) i s tr iv i al ly s a t i sfie d . F in al ly c ond it ion ( c
′
) , a g ain
g iv e n ( a) , r e duc e s t o ( b) . ■
W e no w e s t a b l i sh the ne c e s s a r y a nd s u ffic ie n t c ond it ion s t o h a v e m a x im al de p h asin g unde r L ind b l a d i a n
dy n a mic s a nd the n w e calc ul a t e the 2- nor m C GP for al l s uch pr oc e s s e s .
P r opositio n 1.7
L e t B
′
=f P
′
i
:=j i
′
⟩⟨ i
′
jg
d
i= 1
a ndD
B
′ be the as s oc i a t e d m a x im al ly de p h asin g ch a nne l . The n for L ind -
b l a d i a n dy n a mic s:
( i ) lim
t!1
exp(L t) =D
B
′ i f a nd only i f the fo l lo w in g c ond it ion s ho ld sim ult a ne ously:
( a ) The H a mi lt oni a n H i s d i a g on al i n B
′
.
( b ) A l l L ind b l a d ope r a t or s L
α
a r e d i a g on al in B
′
.
( c ) F or e v e r y i̸= j the r e ex i s ts a n α s uch th a t ⟨ i
′
j L
α
j i
′
⟩̸=⟨ j
′
j L
α
j j
′
⟩ .
( i i ) I f lim
t!1
exp(L t) =D
B
′ the n exp(L t) i s unit al for t 0 a ndL(j i
′
⟩⟨ j
′
j) = λ
ij
j i
′
⟩⟨ j
′
j , w ith
λ
ij
= i( E
i
E
j
) +
∑
α
(
( L
α
)
ii
( L
α
)

jj
1= 2j( L
α
)
ii
j
2
1= 2

( L
α
)
jj

2
)
;
w he r e E
i
=⟨ i
′
j Hj i
′
⟩ a nd( L
α
)
ii
=⟨ i
′
j L
α
j i
′
⟩ .
( i i i ) L e t in a dd it ion B =f P
i
g
d
i= 1
a nd U2 U( d) be a unit a r y ope r a t or s uch th a t j i
′
⟩ = Uj i⟩ for al l
i = 1;:::; d . I f lim
t!1
exp(L t) =D
B
′ , the n
C
2; B
[ exp(L t)] =
1
d( d + 1)
[
Tr
(
X
U
Λ( t) X
T
U
)
Tr
(
Y
U
( t) Y
T
U
( t)
)]
; (1.23)
27
w he r e X
U
2R
d d
i s b i s t och as t ic w ith ( X
U
)
ij
=j⟨ ij Uj j⟩j
2
, a nd
[ Λ( t)]
ij
= exp
(
2 Re( λ
ij
) t
)
[ Y
U
( t)]
ij
=
∑
k; l
exp( λ
kl
t) U
il
U

ik
U
jk
U

jl
:
P r o of . ( i ) & ( ii ) The c ond it ion lim
t!1
exp(L t) =D
B
′ i s e quiv ale n t t o
( a
′
) P
′
i
2 Ker(L) for al l i ( i . e ., al l P
′
i
be lon g t o the s e t of s t e a dy s t a t e s ) a nd
( b
′
) al l m a tr i x e le me n ts ⟨ i
′
j exp(L t) ρ
0
j j
′
⟩ ( i̸= j ) v a ni sh for t!1 for al l init i al s t a t e s ρ
0
.
W e w i l l fir s t sho w th a t ( a
′
) i s e quiv ale n t t o ( a) &( b) ho ld i n g tr ue . I nde e d , f r om the L e mm a it fo l lo w s
th a t L P
′
i
= 0 for al l i i ff H a ndf L
α
g
α
a r e al l d i a g on al w ith r e spe ct t o B , i . e ., c ond it ion s ( a) a nd( b)
a r e tr ue . N o w w e ne e d t o m ak e s ur e th a t ( b
′
) ho ld s, i . e ., non- d i a g on al e le me n ts de ca y for t!1 . B y
p lu gg in g in the for m of the L ind b l a d i a n E q . ( 1.22 ) it fo l lo w s th a t L(j i
′
⟩⟨ j
′
j) = λ
ij
j i
′
⟩⟨ j
′
j , w ith
λ
ij
= i( E
i
E
j
) +
∑
α
(
( L
α
)
ii
( L
α
)

jj
1= 2j( L
α
)
ii
j
2
1= 2

( L
α
)
jj

2
)
;
sinc e al l ope r a t or s a r e d i a g on al in the B
′
b asi s . Th us the s e e le me n ts de ca y i ff Re( λ
ij
) < 0 for i ̸= j ,
w hi ch i s e quiv ale n t t o c ond it ion ( c) , sinc e Re
(
λ
ij
)
= 1= 2
∑
α

( L
α
)
ii
( L
α
)
jj

2
. F in al ly , the ch a n-
ne l i s unit a l f or al l t 0 sinc e I= d =
∑
i
P
′
i
= d2 KerL as c on v ex c omb in a t ion of e le me n ts of B
′
.
( iii ) Our s t a r t in g po in t i s E q . ( 1.6 ) ( sinc eE( t) = exp(L t) i s unit al 8 t 0 ). W e fir s t calc ul a t e
∑
i
⟨E( t) P
i
;E( t) P
i
⟩ . F r om ( i i ),E( t)() =
∑
k; l
P
′
k
() P
′
l
exp( λ
kl
t) . U sin g th a t , it i s d ir e ct t o sho w th a t
∑
i
⟨E( t) P
i
;E( t) P
i
⟩ =
∑
i; k; l
( X
U
)
ik
exp[( λ
kl
+ λ
lk
) t]( X
U
)
il
= Tr
(
X
U
Λ( t) X
T
U
)
:
28
I n a simi l a r w a y it fo l lo w s th a t
∑
i
⟨D
B
E( t) P
i
;D
B
E( t) P
i
⟩ = Tr
(
Y
U
( t) Y
T
U
( t)
)
:
C o mb i nin g the t w o calc ul a t ion s the cl aime d r e s ult fo l lo w s . ■
Obs e r v e th a t for t!1 ex pr e s sion ( 1.23 ) r e duc e s t o E q . ( 1.11 ), sinc e lim
t!1
exp(L t) =D
B
′ . I nde e d ,
unde r the c ond it ion s s t a t e d in P r op . 1.7 , Λ( t!1) = I a nd Y( t!1) = X
U
X
T
U
.
Maxim um c o h er en c e - gener a ti n g po wer o f d eph a si n g L i nd b l a d i a n s
A n a tur al que s t ion t o be ask e d i s w he the r or not the r e ex i s t m a x im al ly de p h asin g L ind b l a d i a n s w hich, al -
thou gh de p h asin g in the lon g t ime l imit , h a v e be tt e r ca p a b i l it y t o pr oduc e c o he r e nc e for finit e t ime s th a n
a n y m a x im al ly de p h asin g ch a nne l . A s w e w i l l sho w mome n t a r i ly , s uch de p h asin g t ime e v o lut ion s ex i s t w ith
2- nor m C GP th a t ca n g e t a r b itr a r i ly clos e t o the m a x im um pos si b le ( o v e r al l unit al qua n tum ope r a t ion s )
C
2; B
.
C on side r for simp l ic it y L ind b l a d i a n dy n a mic s w ith H a mi lt oni a n H = 0 a nd a sin gle unit a r y L ind b l a d
ope r a t or V , ex pr e s s e d as V = e
iH V
. The e v o lut ion e qua t ion the n t ak e s the simp le for m L
V
ρ = V ρ V
y
ρ .
I n the cas e w he r e H
V
i s non- de g e ne r a t e , al l c ond it ion s of pr op . 1.7( i ) a r e me t s o in the lon g t ime l imit
m a x im al de p h asin g oc c ur s (w ith r e spe ct t o the ei g e nb asi s of the ope r a t or H
V
).
W ithout los s of g e ne r al it y , w e as s ume th a t al l ei g e n v alue s of H
V
a r e non- ne g a t iv e w ith ∥ H
V
∥
1
< 2 π .
N o w le t us ex a mine w h a t h a ppe n s w he n ∥ H
V
∥
1
≪ 1 . B y ex p a nd in g , w e g e t V = I iH
V
+O(∥ H
V
∥
2
1
) ,
the r efor e
L
V
ρ = i[ H
V
; ρ] +O(∥ H
V
∥
2
1
) ; (1.24)
A s a r e s ult , for t ime s cale s s uch th a t t∥ H
V
∥
2
1
≪ 1 the s ys t e m unde r g oe s “ almos t ” unit a r y e v o lut ion
unde r effe ct iv e H a mi lt oni a n H
V
a nd e r r or O( t∥ H
V
∥
2
1
) , w hich ca n be m a de a r b itr a r i ly s m al l as s umin g
∥ H
V
∥
1
≪ 1 . The oc c ur r in g effe ct iv e unit a r y e v o lut ion i s the k ey aspe ct th a t al lo w s a chie v in g a l a r g e C GP
29
v alue , w hi le de p h asin g i s s t i l l the domin a n t pr oc e s s for l a r g e t ime s cale s t∥ H
V
∥
2
1
≫ 1 . L e t us no w m ak e
the a bo v e r e m a rk s pr e c i s e . F or the fo l lo w in g , w e nor m al i z e
~
C
2; B
(E) :=
C
2; B
(E)
4C
max
2; B
( d)
(1.25)
s o th a t 0
~
C
2; B
(E) 1 for al l unit a l c h a nne l s E .
P r opositio n 1.8
L e t B =f P
i
:=j i⟩⟨ ijg
d
i= 1
be a c omp le t e fa mi ly of r a nk -1 or tho g on al pr oj e ct or s a nd V = e
iH V
be a
non- de g e ne r a t e unit a r y w ith L
V
() = V() V
y
() de not in g the as s oc i a t e d ( m a x im al ly ) de p h asin g
L ind b l a d i a n a nd H
V
() := i[ H
V
;()] de not in g the r e le v a n t H a mi lt oni a n g e ne r a t or . The n,
( i ) The d i ffe r e nc e of the C GP be t w e e n the L ind b l a d i a n a nd the H a mi lt oni a n e v o lut ion i s bounde d
b y

~
C
2; B
(
e
L V t
)

~
C
2; B
(
e
H V t
)

64
d
d 1
∥ H
V
∥
2
1
t (1.26)
for a n y ∥ H
V
∥
1
1= 2 a nd t 0 .
( i i ) L e t W de not e a non- de g e ne r a t e unit a r y c onne ct in g B w ith a m utual ly unb i as e d b asi s, i . e .,
j⟨ ij Wj j⟩j =
1
p
d
8 i; j :
The n for V = W
1= t
a n d a n y t

4 π ,

~
C
2; B
(
e
L V t
)
1

256 π
2
d
d 1
1
t

: (1.27)
30
( i i i ) L e t F de not e the qua n tum F our ie r tr a n sfor m m a tr i x , i . e .,
⟨ jj Fj k⟩ =
1
p
d
exp
(
i
2 π
d
( j 1)( k 1)
)
: (1.28)
I f H
V
( θ
d
) =
∑
d
k= 1
θ
k
P
′
k
, w he r e P
′
k
= FP
k
F
y
, θ
k
= θ
d
f
k
f
d
w ith f
k
= ( k 1)( k 2) ( d odd ) a nd
f
k
= ( k 1)
2
( d e v e n ), the n

~
C
2; B
(
e
L t
)
1

64 π( d 1) θ
d
; (1.29)
for t

( θ
d
) =
π f
d
dθ
d
a nd θ
d

1
2
.
P r o of . ( i ) W e sp l it the pr oof in t o thr e e p a r ts ( a ) – ( c ) w hich ca n be c omb ine d t o sho w the de sir e d
ine qual it y .
( a ) F or unit al C P TP m a ps E
1
;E
2
the fo l lo w i n g ine qual it y ho ld s:
j C
2; B
(E
1
) C
2; B
(E
2
)j
8
d + 1
∥E
1
E
2
∥
♢
: (1.30)
T o sho w thi s, w e s t a r t f r om E q . ( 1.6 ). U sin g the tr i a n gle ine qual it y , w e g e t

C
2; B
(E
1
) C
2; B
(E
2
)

1
d( d + 1)
( T
1
T
2
) ; w he r e
T
1
:=

∑
i
⟨E
1
P
i
;E
1
P
i
⟩
∑
i
⟨E
2
P
i
;E
2
P
i
⟩

T
2
:=

∑
i
⟨D
B
E
1
P
i
;D
B
E
1
P
i
⟩
∑
i
⟨D
B
E
2
P
i
;D
B
E
2
P
i
⟩

:
D e not in g ρ
B
:= 1= d
∑
d
i= 1
P
i

P
i
a nd usin g the ide n t it y
Tr( AB) = Tr
(
P
( 12)
A
B
)
; (1.31)
31
w he r e P
( 12)
:=
∑
i; j
j ij⟩⟨ jij i s the S W A P ope r a t or , w e g e t
T
1
= d

Tr
(
P
( 12)
E

2
1
ρ
B
)
Tr
(
P
( 12)
E

2
2
ρ
B
)

= d

Tr
[
P
( 12)
(
E

2
1
E

2
2
)
ρ
B
]

d

(
E

2
1
E

2
2
)
ρ
B

1
;
w he r e in the thir d l ine w e us e d the fa ct th a t j Tr( AB)j ∥ A∥
1
∥ B∥
1
a nd th a t

P
( 12)

1
= 1 . N o w ,
sinc e

ρ
B

1
= 1 , w e h a v e
T
1
d

E

2
1
E

2
2

1; 1
d

E

2
1
E

2
2

♢
:
S e tt in g M :=E
1
E
2
, w e h a v e
T
1
d

(M +E
2
)

2
E

2
2

♢
d
(
∥M
M∥
♢
+∥M
E
2
∥
♢
+∥E
2

M∥
♢
)
d
(
∥M∥
2
♢
+ 2∥M∥
♢
)
d∥M∥
♢
(
∥M∥
♢
+ 2
)
4d∥M∥
♢
:
N o w w e w i l l bound the t e r m T
2
. The pr oof pr oc e e d s in a simi l a r w a y . S inc e ∥D
B
∥
1; 1
= 1 , w e al s o g e t
T
2
4d∥M∥
♢
:
The de sir e d ine qual it y fo l lo w s .
( b ) W e a r e g o in g t o pr o v e th a t the fo l lo w in g ine qual it y ho ld s:

e
L t
e
H t

♢
t∥LH∥
♢
(1.32)
32
for al l t 0 . D e not in g U
t
:= exp(H t) w e h a v e

e
L t
e
H t

♢
=

U
y
t
e
L t
I

♢
=∥K
t
I∥
♢
w he r eK
t
:= U
y
t
exp(L t) . W e al s o h a v e
_
K( t) = V
t
K( t) w ithV
t
= U
y
t
(LH)U
t
( in t e r a ct ion
p ictur e ). N o w w e ca n bound the qua n t it y of in t e r e s t :
∥K
t
I∥
♢
=

∫
t
0
ds
_
K
s

♢
=

∫
t
0
dsV
s
K
s

♢
t sup
s2[ 0; t]
∥V
s
K
s
∥
♢
t∥LH∥
♢
;
w he r e in the l as t ine qual it y w e us e d s ub - m ult ip l ica t iv it y a nd the unit a r y in v a r i a nc e of the d i a mond
nor m t o g e the r w ith the fa ct th a t ∥ exp(L t)∥
♢
= 1 ( C P TP m a p for al l t 0 ).
( c ) F or∥ H
V
∥
1
1= 2 , w e w i l l sho w th a t
∥L
V
H
V
∥
♢
8∥ H
V
∥
2
1
: (1.33)
N ot ic e th a t H
V
ca n alw a ys be chos e n s o th a t al l ei g e n v alue s a r e non- ne g a t iv e a nd ∥ H
V
∥
1
< 2 π . N o w
w e define the s u pe r ope r a t or s
L
A
( ρ) = A ρ (1.34)
R
B
( ρ) = ρ B ; (1.35)
for w hich it ho ld s th a t
∥L
A
∥
♢
=∥L
A

I
d
∥
1; 1
=∥L
A
I
d
∥
1; 1
∥ A∥
1
;
∥R
B
∥
♢
=∥R
B

I
d
∥
1; 1
=∥R
B
I
d
∥
1; 1
∥ B∥
1
33
a nd the r efor e
∥L
A
R
B
∥
♢
∥ A∥
1
∥ B∥
1
: (1.36)
S e tt in g Δ := V ( I iH
V
) w e ca n ex p r e s s the a ct ion of (L
V
H
V
) on s ome ρ as
(L
V
H
V
) ρ = Δ ρ + ρ Δ
y
+ Δ ρ Δ
y
+ H
V
ρ H
V
iH
V
ρ Δ
y
+ i Δ ρ H
V
;
a n d t he r efor e
∥L
V
H
V
∥
♢
∥L
Δ
∥
♢
+∥R
Δ
y∥
♢
+∥L
Δ
R
Δ
y∥
♢
+∥L
H V
R
H V
∥
♢
+∥L
H V
R
Δ
y∥
♢
+∥L
Δ
R
H V
∥
♢
:
U sin g E q . ( 1.36 ), the a bo v e r e duc e s t o
∥L
V
H
V
∥
♢
2∥ Δ∥
1
+∥ Δ∥
2
1
+∥ H
V
∥
2
1
+ 2∥ Δ∥
1
∥ H
V
∥
1
W e w i l l e s t im a t e ∥ Δ∥
1
. W e h a v e
∥ Δ∥
1

1
∑
n= 2
∥ H
V
∥
n
1
n!
∥ H
V
∥
2
1
1
∑
n= 0
∥ H
V
∥
n
1
( n + 2)!
:
N o w w e m ak e the as s umpt ion th a t ∥ H
V
∥
1
1= 2 . U nde r thi s as s umpt ion,
∥ Δ∥
1
∥ H
V
∥
2
1
1
1∥ H
V
∥
1
2∥ H
V
∥
2
1
:
U s in g th i s u ppe r bound w e g e t e qua t ion ( 1.33 ).
F in al ly , w e ca n c omb ine t o g e the r p a r ts ( a ) – ( c ) ( a nd the n nor m al i z e ) t o g e t the de sir e d ine qual it y
34
for p a r t ( i ) of the pr oposit ion .
( ii ) W e w i l l fir s t sho w th a t the unit a r y e v o lut ion h as opt im al C GP a t t = t

, n a me ly
~
C
2; B
(
e
H V t
)
= 1 .
I n [ 10 ] it w as sho w n th a t the m a x im um v alue of
~
C
2; B
i s 1 a nd i s a tt aine d b y unit a r y ch a nne l s U () =
U() U
y
i ff ( X
U
)
ij
:=j⟨ ij Uj j⟩j
2
=
1
d
8 i; j : (1.37)
H e r e U( t) = exp( iH
V
t) . F or t = t

, w e h a v e U( t

) = V
t
= W a nd th us, inde e d , the a bo v e
c ond it ion i s s a t i sfie d .
N o w w e ca n a pp ly p a r t ( i) of the pr oposit ion t o g e t the de sir e d bound . The m a tr i x W i s unit a r y s o
∥ W∥
1
2 π , he nc e a n H
V
th a t s a t i sfie s the e qua t ion ( H
V
)
t
= W ca n be chos e n w ith ∥ H
V
∥
1
t

2 π . A s a r e s ult , ∥ H
V
∥
1
1= 2 f r om p a r t ( i) imp l ie s t

4 π .
( iii ) W e w i l l fir s t sho w th a t
~
C
2; B
(
e
H V t
)
= 1 . A s in the pr e v ious p a r t , w e ne e d t o pr o v e th a t for the
unit a r y ope r a t or U( t

) = exp( iH
V
t

) E q . ( 1.37 ) i s s a t i sfie d . W e h a v e
( X
U
)
ij
=

d
∑
k= 1
e
iθ
k
t
F
ik
F

jk

2
(1.38)
=
1
d
2

d
∑
k= 1
exp
(
i
2 π
d
( k 1)( j i)
)
exp
(
i
π
d
f
k
)

2
: (1.39)
N o w c on side r the odd d cas e . S ubs t itut in g for f
k
, w e g e t
( X
U
)
ij
=

d 1
∑
k= 0
exp
[
i
π
d
( k
2
+ k[ 2( j i) 3])
]

2
;
w he r e the s um in side the modulus i s a (g e ne r al i z e d ) qua dr a t ic G a us s s um ( s e e , e . g., [ 45 ]) a nd for d
odd a nd ( j i) in t e g e r , e v alua t e s t o
p
d ( i gnor in g s ome ir r e le v a n t p h as e fa ct or ). The r efor e , inde e d ,
( X
U
)
ij
= 1= d . The e v e n d cas e pr oc e e d s simi l a rly .
35
0 20 40 60 80 100 120
0.0
0.2
0.4
0.6
0.8
1.0
t
C
2,B
e
ℒt

4C
2,B
max
t
*
=5
t
*
=20
t
*
=50
Figure 1.10.1: Plot of C
2; B
(
e
L V t
)
fo r a 2-level system (no rmalized such that 4C
max
2; B
= 1 ). The
Lindbladian is chosen as in pa rt (iii) of Prop. 1.8 : B = f P
0
; P
1
g and V = P
+
+ e
i
π
2t
P

(where
P

= j⟩⟨j ). Observe ho w, as t

increases ( o r, equivalently ,∥ H
V
∥
1
decreases) the p eak moves
higher up, app roaching unit y . Time should b e interp reted in units fo r which the (so fa r explicitly omit-
ted) rate of the Lindbladian tak es the value γ = 1 .
N o w w e ca n us e p a r t ( i) of the pr oposit ion for t = t

, w he r e∥ H
V
∥
1
= θ
d
. S ubs t itut in g for t

( θ
d
)
a nd usin g the fa ct th a t f
d
( d 1)
2
( for al l d ) w e g e t th e de sir e d ine qual it y . ■
P r op . 1.8 pr o v ide s t w o d i ffe r e n t “ r e c ipe s ” t o c on s tr uct de p h asin g L ind b l a d i a n s s uch th a t for t = t

the 2-
nor m C GP i s ne a rly m a x im al a nd f ur the r pr o v ide s a n u ppe r bound for the d i ffe r e nc e be t w e e n the oc c ur r in g
C GP a nd the opt im al one a t t = t

. A n ex a mp le de mon s tr a t in g the c on s tr uct ion de s cr i be d in p a r t ( i i i ) of
the pr oposit ion i s p lott e d in F i g ur e 1.10.1 . N ot ic e th a t thi s fa mi ly of L ind b l a d i a n s ca n g e t a r b itr a r i ly clos e
t o the m a x im um v alue of C GP for t = t

but , ne v e r the le s s, h as v a ni shin g C GP for t!1 . Thi s i s be ca us e
lim
t!1
exp(L
V
t) =D
B
′ w ith B a nd B
′
bein g m utual ly unb i as e d b as e s .
1.11 Coherenc e - g ener a ting po wer of r and om qu an tum p ro c e sse s
I n thi s s e ct ion, w e ex a mine the situa t ion w he n the oc c ur r in g pr oc e s s i s r a n do m .
36
2 4 6 8 10 12 14 16
0.2
0.3
0.4
0.5
0.6
0.7
d
〈C
2,B
( 
B
′
) 〉
U
C
2,B
max
(d)
Figure 1.10.2: Compa rison b et w een the numerically computed dephasing CGP mean⟨ C
2; B
(D
B
′)⟩
U
(individual p oints) and its upp er b ound from Eq. ( 1.43 ) (solid line), as a function of the Hilb ert space
dimension d . Both quantities a re no rmalized b y dividing with the upp er b ound C
max
2; B
( d) .
37
R a nd om u nit a r y pr o c e sse s
I n v ie w of E q . ( 1.11 ), the C GP ca n al s o be tr e a t e d as a r a ndom v a r i a b le o v e r the unit a r y gr o u p U( d) , w hich
w e c on side r e quippe d w ith the H a a r me as ur e . H e r e w e s t a t e the r e le v a n t r e s ults f r om [ 10 ], w he r e the
c or r e spond in g pr oof ca n be found .
P r opositio n 1.9
( i ) The pr o b a b i l it y d i s tr i but ion de n sit y P
CGP
( c) :=
∫
d μ
Haar
( U) δ( c C
B
( U)) for the C GP
E q . (( 1.11 )) i s inde pe nde n t of B .
( i i ) The fir s t mome n t i s g iv e n b y
⟨ C
B
( U)⟩
U
=
∫
dcP
CGP
( c) =
d 1
( d + 1)
2
: (1.40)
( i i i ) L e t us define the nor m al i z e d C GP
~
C
B
( U) := C
2; B
( U)= C
d
1 the n⟨
~
C
2; B
( U)⟩
U
= ( 1+ 1= d)
1
.
U sin g L e v y ’ s le mm a f or unit a r ie s, one o bt ain s
Prob
(
~
C
2; B
( U) 1 2= d
1= 3
)
1 exp
(
d
1= 3
= 256
)
: (1.41)
R a nd om d eph a si n g pr o c e sse s
N o w w e in v e s t i g a t e the situa t ion w he r e the m a x im al ly de p h asin g ch a nne l s a r e r a n do m . M or e spe c i fical ly ,
in v ie w of E q . ( 1.11 ), the C GP of a m a x im al ly de p h asin g ch a nne l ca n al s o be tr e a t e d as a r a ndom v a r i a b le
o v e r the unit a r y gr ou p U( d) , w hich w e c on side r e quippe d w ith the H a a r me as ur e . I n othe r w or d s, the b asi s
o v e r w hich the qua n tum s ys t e m i s bein g de p h as e d ca n be r e g a r de d as r a ndom v a r i a b le as, for ex a mp le , in
the oc c ur r e nc e of a n or tho g on al me as ur e me n t of s ome ( a pr io r i unkno w n ) non- de g e ne r a t e o bs e r v a b le . F or
the fo l lo w in g , w e us e the nor m al i za t ion
~
C
2; B
(D
B
′) := C
2; B
(D
B
′)= C
max
2; B
( d) s o th a t 0
~
C
2; B
(D
B
′) 1 ⁵ .
⁵ F or in t e r pr e t in g the r e s ults th a t fo l lo w w e c on j e ctur e th a t the nor m al i za t ion f unct ion C
max
2; B
( d) i s not only a n u ppe r bound
of C
2; B
(D
B
′) ( o v e r al l m a x im al ly de p h asin g ch a nne l s ), but al s o a chie v a b le . I n the A ppe nd i x , w e a r e a b le t o pr o v e th a t the u ppe r
38
P r opositio n 1.10: C GP of r a n do m deph a sin g
L e t P
CGP
( c) :=
∫
d μ
Haar
( U) δ
(
c
~
C
2; B
(D
B
′)
)
be the pr o b a b i l it y de n sit y f unct ion of the m a x im al
de p h asin g (2- nor m ) c o he r e nc e- g e ne r a t in g po w e r o v e r H a a r d i s tr i but e d U2 U( d) . The n
( i ) F or a qub i t ( d = 2 ) the pr o b a b i l it y de n sit y f unct ion i s
P
CGP
( c) =
1
√
32 c( 1 c)
(
√
1 +
p
1 c +
√
1
p
1 c
)
: (1.42)
( i i ) The me a n v alue ⟨
~
C
2; B
(D
B
′)⟩
U
:=
∫
dc cP
CGP
( c) i s bounde d f r om a bo v e b y
⟨
~
C
2; B
(D
B
′)⟩
U
M( d) ; (1.43)
w he r e
M( d) :=
4d[ d( d + 5) + 2]
( d + 1)
2
( d + 2)( d + 3)
: (1.44)
( i i i ) U sin g L e v y ’ s l e mm a for H a a r d i s tr i but e d unit a r y m a tr ic e s, w e o bt ain
Prob
{
~
C
2; B
(D
B
′)
1
d
1= 4
+ M( d)
}
exp
(

p
d
640
2
)
: (1.45)
P r o of . ( i ) I n E x . 1.1 the 2- nor m C GP for a qub it ( in the B loch sp he r e p a r a me tr i za t ion ) w as found t o be
~
C
2; B
( θ) = sin
2
( 2θ) . I n thi s p a r a me tr i za t ion, H a a r d i s tr i but e d unit a r y m a tr ic e s U2 U( 2) c or r e spond
t o the me as ur e
μ = 1=( 4 π)
∫
2 π
0
d φ
∫
π
0
dθ sin θ :
bound i s a chie v a b le a t le as t for al l H i l be r t sp a c e d ime n sion s d 13 .
39
The r efor e P
CGP
( c) = 1=( 4 π)
∫
2 π
0
d φ
∫
π
0
dθ sin θ δ( c sin
2
( 2θ)) . The r e s ult fo l lo w s d ir e ctly b y pe r -
for min g the in t e gr al ( e . g., b y ch a n g in g v a r i a b le s ).
( ii ) W e h a v e t o a v e r a g e E q . ( 1.11 ) o v e r U . B y l ine a r it y , w e ca n calc ul a t e ⟨ Tr( X
U
X
T
U
)⟩
U
a nd⟨ Tr[( X
U
X
T
U
)
2
]⟩
U
s e p a r a t e ly .
The fir s t qua n t it y bein g a v e r a g e d i s e qual t o Tr( X
U
X
T
U
) =
∑
i; j
j⟨ ij Uj j⟩j
4
. B y s e tt in gj j
U
⟩ := Uj j⟩ ,
w e h a v ej⟨ ij j
U
⟩j
4
= Tr
(
(j i⟩⟨ ij)

2
(j j
U
⟩⟨ j
U
j)

2
)
. A g ain b y l ine a r it y , it fo l lo w s th a t
⟨
∑
i; j
Tr
(
(j i⟩⟨ ij)

2
(j j
U
⟩⟨ j
U
j)

2
)
⟩
U
=
∑
i; j
Tr
(
(j i⟩⟨ ij)

2
⟨
(j j
U
⟩⟨ j
U
j)

2
⟩
U
)
:
N o w w e ca n e mp lo y the w e l l -kno w n g e ne r al r e s ult ( for a pr oof s e e , e . g., [ 46 ])
⟨
(j j
U
⟩⟨ j
U
j)

n
⟩
U
=
1
n!
1
(
d+ n 1
n
)
∑
π2 S n
P
π
; (1.46)
w he r e S
n
i s the s y mme tr ic gr ou p of n - o bj e cts a nd P
π
i s the ope r a t or th a t e n a cts the pe r m ut a t ion π in
H

n
. F or n = 2 , w e h a v e
⟨
(j j
U
⟩⟨ j
U
j)

2
⟩
U
= [ d( d + 1)]
1
(
I + P
( 12)
)
; (1.47)
w he r e P
( 12)
i s the ( 12) c y cle ( i . e ., P
( 12)
i s jus t the S W A P ope r a t or ). P lu gg in g thi s in a nd pe r for min g the
tr a c e , w e g e t ⟨ Tr( X
U
X
T
U
)⟩
U
= 2d=( d + 1) .
W e w i l l fo l lo w a simi l a r s tr a t e g y for the s e c ond qua n t it y ⟨ Tr[( X
U
X
T
U
)
2
]⟩
U
. The qua n t it y bein g a v e r -
a g e d ca n be r e ex pr e s s e d as
Tr
[
( X
U
X
T
U
)
2
]
=
∑
i; j; k; l
j⟨ ij k
U
⟩j
2
j⟨ jj k
U
⟩j
2
j⟨ ij l
U
⟩j
2
j⟨ jj l
U
⟩j
2
=
∑
i; j; k; l
Tr
(
j iijj⟩⟨ iijjj(j k
U
l
U
⟩⟨ k
U
l
U
j)

2
)
:
40
N o w w e sp l it the s um t o t w o p a r ts: k = l a nd k ̸= l , w hich w e cal l Σ
1
a nd Σ
2
, r e spe ct iv e ly . F or Σ
1
w e g e t
∑
i; j; k
Tr
(
j iijj⟩⟨ iijjj(j k
U
⟩⟨ k
U
j)

4
)
. T ak in g the a v e r a g e w e ca n us e the for m ul a as befor e w ith
n = 4 . The r efor e w e no w h a v e t o e v alua t e
∑
i; j; π
Tr(j iijj⟩⟨ iijjj P
π
) for al l pe r m ut a t ion s π 2 S
4
. Out
of the 4! = 24 e le me n ts, a ft e r pe r for min g the i; j s um, 4 of the m g iv e d
2
[ the pe r m ut a t ion s w ith c y cle
de c omposit ion ( 12) , ( 34) , ( 12)( 34) a nd the ide n t it y pe r m ut a t ion ] w hi le the r e s t g iv e d . A s a r e s ult ,
Σ
1
= ( 4d + 20)=[( d + 1)( d + 2)( d + 3)] .
S o fa r e v e r y thin g i s ex a ct. F or the Σ
2
t e r m w e not ic e w e ca n w r it e
⟨ Σ
2
⟩
U
=
∑
k̸= l
⟨(
Tr
[
(
∑
i

ii⟩⟨ ii

)

k
U
l
U
⟩⟨ k
U
l
U

]
)
2
⟩
U
:
W e kno w a ppr o x im a t e the me a n ⟨ Σ
2
⟩
U
usin g the i ne qual it y ⟨ A
2
⟩⟨ A⟩
2
, w hich y ie ld s
⟨ Σ
2
⟩
U

∑
k̸= l
(
Tr
[(
∑
i
j ii⟩⟨ iij
)
⟨j k
U
l
U
⟩⟨ k
U
l
U
j⟩
U
])
2
:
N o w w e ca nnot us e the for m ul a as befor e t o calc ul a t e the a v e r a g e , sinc e j k
U
⟩ a ndj l
U
⟩ a r e c or r e l a t e d .
N e v e r the le s s, w e ca n us e the sl i gh tly mor e g e ne r al r e s ult ( s e e , e . g., [ 47 ])
⟨ U

2
AU
y
2
⟩
U
=
(
Tr A
d
2
1

Tr
[
P
( 12)
A
]
d( d
2
1)
)
I

(
Tr A
d( d
2
1)

Tr
[
P
( 12)
A
]
d
2
1
)
P
( 12)
: (1.48)
E v alua t in g for A =j kl⟩⟨ klj (w ith k̸= l ) w e g e t Σ
2
d( d 1)=( d + 1)
2
.
P utt in g e v e r y thin g t o g e the r w e g e t
M( d) =
4d
( d 1)( d + 1)
(
d + 3
d + 1

4d + 20
( d + 2)( d + 3)
)
; (1.49)
w hich simp l i fie s t o the ex pr e s sion cl aime d .
41
iii ) I n or de r t o pr o v e the de sir e d ine qual it y w e a r e g o in g t o us e the fo l lo w in g for m of L e v y ’ s le mm a
for H a a r d i s tr i but e d U2 U( d) ( s e e , e . g., [ 48 ]):
Prob
{
j f( U)⟨ f( U)⟩
U
j ε
}
exp
(

d ε
2
4K
2
)
; (1.50)
w he r e K i s a L ips chitz c on s t a n t of f : U( d) ! R , i . e .,j f( U) f( V)j K∥ U V∥
2
. H e r e our
f unct ion f( U) i s g o in g t o be
~
C
2; B
(D
B
′) ( v ie w e d as a f unct ion of the unit a r y U c onne ct in g B a nd B
′
),
i . e .,
f( U) :=
4
d 1
(
Tr
(
X
U
X
T
U
)
Tr
[(
X
U
X
T
U
)
2
])
:
A lthou gh the ex a ct ex pr e s sion for the me a n v alue ⟨ f( U)⟩
U
h as not be e n calc ul a t e d , the u ppe r bound
f r om E q . ( 1.44 ) al lo w s a ppr o x im a t in g the de sir e d pr o b a b i l it y , sinc e
Prob
{
j f( U)⟨ f( U)⟩
U
j ε
}

Prob
{
f( U) ε +⟨ f( U)⟩
U
}

Prob
{
f( U) ε + M( d)
}
:
T o c omp le t e the pr oof w e ne e d t o e s t im a t e a L ips chitz c on s t a n t for the f unct ion f( U) . W e h a v e
j f( U) f( V)j
4
d 1
( T
1
+ T
2
) ;
w he r e w e s e t
T
1
:=

Tr
(
X
U
X
T
U
)
Tr
(
X
V
X
T
V
)

T
2
:=

Tr
[(
X
U
X
T
U
)
2
]
Tr
[(
X
V
X
T
V
)
2
]

:
42
F r om the pr oof of P r op . 1.4 , w e ca n e quiv ale n tly w r it e
T
1
=

∑
i
⟨D
B
′
( U)
P
i
;D
B
′
( U)
P
i
⟩
∑
j
⟨D
B
′
( V)
P
j
;D
B
′
( V)
P
j
⟩

=

∑
i
⟨D
B
U
y
P
i
;D
B
U
y
P
i
⟩
∑
j
⟨D
B
V
y
P
j
;D
B
V
y
P
j
⟩

=

∑
i; j
Tr
(
P
( 12)
D

2
B
[
U
y
2
P

2
i
V
y
2
P

2
j
])

;
w he r e in the l as t s t e p w e us e d E q .( 1.31 ). A n u ppe r bound for thi s qua n t it y w as calc ul a t e d in [ 10 ],
n a me ly
T
1
8d∥ U V∥
2
:
Befor e pr oc e e d in g w ith calc ul a t ion of a n u ppe r bound for T
2
, le t us fir s t pr o v e the fo l lo w in g ine qual it y
w hich w i l l be ne e de d mome n t a r i ly:

U ρ U
y
V ρ V
y

1
4∥ U V∥
1
; (1.51)
w he r e U; V a r e unit a r y a nd ρ i s a n ope r a t or w ith ∥ ρ∥
1
= 1 . S e tt in g Δ := U V w e h a v e

U ρ U
y
V ρ V
y

1
=

Δ ρ Δ
y
+ Δ ρ V
y
+ V ρ Δ
y

1

Δ ρ Δ
y

1
+

Δ ρ V
y

1
+

V ρ Δ
y

1
. U sin g the fa cts th a t the nor m i s
unit a r i ly in v a r i a n t , ∥ Δ∥
1
2 a nd th a t ∥ AB∥
1
∥ A∥
1
∥ B∥
1
the a for e me n t ione d ine qual it y fo l lo w s .
W e ca n ex pr e s s T
2
, in the s a me sp ir it as befor e , as
T
2
=

∑
i
⟨D
B
D
B
′
( U)
P
i
;D
B
D
B
′
( U)
P
i
⟩

∑
j
⟨D
B
D
B
′
( V)
P
j
;D
B
D
B
′
( V)
P
j
⟩

=

Tr
(
P
( 12)
D

2
B
[
(
UD
B
U
y
)

2
(∑
i
P

2
i
)

(
VD
B
V
y
)

2
(
∑
j
P

2
j
)])

= d

Tr
(
P
( 12)
D

2
B
[
(
UD
B
U
y
)

2
ρ
B

(
VD
B
V
y
)

2
ρ
B
])

;
43
w he r e in the l as t s t e p w e s e t ρ
B
:= 1= d
∑
i
P

2
i
( not ic e

ρ
B

1
= 1 ). N o w usin g the ine qual it y
Tr( AB)∥ A∥
1
∥ B∥
1
( he r e

P
( 12)

1
= 1 ) a nd the fa ct th a t D
B
i s a C P TP m a p , w e g e t
T
2
d

(
UD
B
U
y
)

2
ρ
B

(
VD
B
V
y
)

2
ρ
B

1
= d

U

2
ρ
1
V

2
ρ
2

1
= d

U

2
ρ
1
V

2
ρ
1
+V

2
( ρ
1
ρ
2
)

1
d
(

U

2
ρ
1
V

2
ρ
1

1
+

V

2
( ρ
1
ρ
2
)

1
)
;
w he r e w e s e t ρ
1
:=
(
D
B
U
y
)

2
ρ
B
a nd ρ
2
:=
(
D
B
V
y
)

2
ρ
B
. N o w the ine qual it y f r om E q . ( 1.51 )
a pp l ie s t o both t e r m s, y ie ld in g
T
2
8d

U

2
V

2

1
= 8d

(
V
y
Δ + I
)

2
I

1
8d
(

V
y
Δ

2
1
+ 2

V
y
Δ

1
)
8d∥ Δ∥
1
( 2 +∥ Δ∥
1
)
32d∥ U V∥
1
32d∥ U V∥
2
:
F in al ly , w e o bt ain
j f( U) f( V)j 160
d
d 1
∥ U V∥
2
;
the r efor e the L ips chitz c on s t a n t ca n be t ak e n t o be K = 320 . The p a r a me t e r ε , in or de r t o g iv e a
me a nin g f ul r e s ult for l a r g e H i l be r t sp a c e d ime n sion d , ca n be t ak e n t o be ε = d
α
, w ith α2 ( 0; 1= 2) .
H e r e w e choos e α = 1= 4 . The ine qual it y fo l lo w s . ■
P r oposit ion 1.10 de mon s tr a t e s th a t , in a qua n tum s ys t e m de s cr i be d b y a l a r g e H i l be r t sp a c e , a m a x i -
m al ly de p h asin g pr oc e s s o v e r a r a ndom b asi s h as v a ni shin gly s m al l ca p a b i l it y t o pr oduc e c o he r e nc e out of
44
0.05 0.10 0.15 0.20 0.25
0
20
40
60
80
100
120
140
c
P
CGP
(c)
d=20
d=30
d=40
d=50
Figure 1.11.1: Numerically computed p robabilit y distribution functions P
CGP
( c) fo r th e CGP of ran-
dom maximally dephasing p ro cesses fo r different Hilb ert space dimensions d . Last pa rt of Prop. 1.10
gua rantees that, fo r sufficiently la rge d , the p robabilit y distribution function is concentrated a round
the mean value, which decreases as d gets la rger, as is indeed observe d. Notice that in p ractice the
concentration o ccurs fo r smaller d that what is gua ranteed b y the p rop osition.
inc o he r e n t s t a t e s . P a r t ( i i ) of the a bo v e pr oposit ion s e ts a n u ppe r bound the ( pr ope rly nor m al i z e d ) C GP ,
e s t a b l i shin g th a t for l a r g e H i l be r t sp a c e d ime n sion d , the qua n t it y ⟨
~
C
2; B
(D
B
′)⟩
U
dr ops a t le as t as fas t as
1= d . A gr a p hical c omp a r i s on be t w e e n the u ppe r bound f r om E q . ( 1.43 ) a nd the n ume r ical ly c omput e d
me a n i s pr e s e n t e d in F i g ur e 1.10.2 . The l as t p a r t of P r op . 1.10 sho w s th a t , as the H i l be r t sp a c e si z e gr o w s,
a m a x im al ly de p h asin g pr oc e s s oc c ur r in g o v e r a r a ndom b asi s h as (w ith a n ex pone n t i al ly incr e asin g pr o b -
a b i l it y ) C GP w hich i s t i gh tly d i s tr i but e d a r ound the ( de cr e asin g ) me a n v alue . Thi s c onc e n tr a t ion of the
pr o b a b i l it y d i s tr i but ion f unct ion a r ound the me a n i s de p ict e d in F i g ur e 1.11.1 .
1.12 Conc l u sion and ou tlo ok
I n thi s ch a pt e r w e h a v e in v e s t i g a t e d the a b i l it y of v a r ious de p h asin g pr oc e s s e s t o g e ne r a t e c o he r e nc e . F or
thi s pur pos e , w e a dopt e d v a r ious me as ur e s for the co h er en ce-gen er a t i n g p o w er of qua n tum ch a nne l s, al l b as e d
on pr o b a b i l i s t ic a v e r a g e s a nd a r i sin g f r om the v ie w po in t of c o he r e nc e as a r e s our c e the or y . W e pr o v ide d
45
ex p l ic it for m ul as for unit a r y a nd m a x im al ly de p h asin g pr oc e s s e s, v al id for al l finit e H i l be r t sp a c e d ime n-
sion s, me as ur in g ho w m uch c o he r e nc e in g e ne r a t e d on a v e r a g e f r om inc o he r e n t s t a t e s . I n al l cas e s, the
c o he r e nc e- g e ne r a t in g po w e r of the de p h asin g pr oc e s s de pe nd s on the in t e r p l a y be t w e e n the b as e s o v e r
w hich c o he r e nc e i s qua n t i fie d a nd de p h asin g oc c ur s . Thi s ca p a b i l it y cle a rly v a ni she s w he n the t w o b as e s
c o inc ide w hi le the m a x im um ca p a b i l it y oc c ur s for a b asi s w hich de pe nd s on the me as ur e of s t a t e c o he r -
e nc e , as sho w n for de p h asin g pr oc e s s e s . I f the b asi s o v e r w hich de p h asin g oc c ur s i s chos e n a t r a ndom in a
uni for m w a y , the a v e r a g e c o he r e nc e- g e ne r a t in g po w e r dr ops fas t as the H i l be r t sp a c e d ime n sion incr e as e s .
On the othe r h a nd , for r a ndom unit a r y pr oc e s s e s, i s ne a rly m a x im al .
W e the n ex t e nde d the a n alysi s t o al l L ind b l a d t y pe qua n tum e v o lut ion s th a t m a x im al ly de p h as e in the
infinit e t ime l imit b y calc ul a t in g the r e le v a n t H i l be r t - S chmidt 2- nor m c o he r e nc e- g e ne r a t in g po w e r of the
as s oc i a t e d t ime e v o lut ion for al l in t e r me d i a t e t ime s . A lthou gh m a x im al ly de p h asin g pr oc e s s e s ca n g e ne r -
a t e finit e a moun ts of c o he r e nc e ( de pe nd in g on the as s oc i a t e d b as e s ), c o he r e nc e g e ne r a t ion ca nnot be as
po w e r f ul as for s ome unit a r y pr oc e s s e s . Thi s i s not alw a ys the cas e , ho w e v e r , for L ind b l a d e v o lut ion s th a t
le a d t o de p h asin g. F or the l a tt e r , w e ide n t i fie d fa mi l ie s of t ime pr op a g a t or s th a t h a v e v a ni shin g c o he r e nc e-
g e ne r a t in g po w e r in the lon g t ime l imit but ne v e r the le s s ca n g e t a r b itr a r i ly clos e t o h a v in g opt im al one for
in t e r me d i a t e t ime s .
The co h er en ce-gen er a t i n g p o w er of a qua n tum ope r a t ion a dmits, d ir e ctly b y its definit ion, a n in t e r pr e t a t ion
as the a v e r a g e c o he r e nc e c on t aine d in the pos t -pr oc e s s e d s t a t e s, as qua n t i fie d b y the r e le v a n t c o he r e nc e
me as ur e . A n clos e r c onne ct ion t o o p er a t io n a l i n t er pr et a t io n of the co h er en ce-gen er a t i n g p o w er , r e le v a n t t o
pr a ct ical t ask s for w hich c o he r e nc e i s kno w n t o be a cr it ical in gr e d ie n t ( s uch as thos e me n t ione d in the
s e ct ion 1.3 ), i s mi s sin g a nd c ould r e pr e s e n t a ch al le n g e for f utur e in v e s t i g a t ion .
46
A ppendix
A A tt a i nt ment o f th e u pper bo u nd C
max
2; B
( d)
I n thi s s e ct ion w e ex a mine i f the u ppe r bound
C
max
2; B
( d) :=
d 1
4d( d + 1)
(1.52)
of the m a x im al ly de p h asin g 2- nor m C GP C
2; B
(D
B
′) ( E q . ( 1.11 ) of the m ain t ex t ) i s a tt aine d o v e r s ome
b asi s B
′
. F or ex a mp le , for a qub it it ca n be ex p l ic itly v e r i fie d ( as in E x . 1.1 ) th a t the u ppe r bound C
max
2; B
( d) i s
a chie v a b le . H e r e w e t a ck le the g e ne r al d - d ime n sion al cas e .
F r om the pr oof of P r op . 1.4 it fo l lo w s th a t the m a x im um v alue C
max
2; B
( d) for s ome ( fi xe d ) d i s a tt aine d i f
a nd only i f the r e ex i s ts a unit a r y m a tr i x U s uch th a t σ ( X
U
X
T
U
) =f 1; 1= 2g w ith 1 bein g a simp le ei g e n v alue
( σ ( A) de not e s the spe ctr um of the ope r a t or A ). S uch a d - d ime n sion al m a tr i x i s not g ua r a n t e e d t o ex i s t a
pr ior i , sinc e the d - d ime n sion al uni s t och as t ic m a tr ic e s a r e a pr ope r s ubs e t of the d - d ime n sion al b i s t och as t ic
m a tr ic e s for d 3 ( s e e , e . g., [ 49 ]).
F or w h a t fo l lo w s, w e f ur the r r e s tr ict t o thos e b i s t och as t ic m a tr ic e s X
U
s uch th a t
( a ) a r e s y mme tr ic a nd
( b ) h a v e spe ctr u m σ( X
U
) =f 1; 1=
p
2g .
S uch a m a tr i x h as the for m
X
U
=j ψ
1
⟩⟨ ψ
1
j +
1
p
2
d
∑
i= 2
j ψ
i
⟩⟨ ψ
i
j =
(
1
1
p
2
)
j ψ
1
⟩⟨ ψ
1
j +
1
p
2
I;
w he r e
{
j ψ
i
⟩
}
d
i= 1
i s the ei g e nb asi s of X
U
. H o w e v e r , ( X
U
)
ij
should al s o be a b i s t och as t ic m a tr i x w he n ex -
pr e s s e d in the B =fj i⟩g
d
i= 1
b asi s . Thi s fi xe s the c ompone n ts t o
( X
U
)
ij
=
1
p
2
δ
ij
+
1
d
(
1
1
p
2
)
: (1.53)
47
The a bo v e X
U
m a tr i x i s c ir c ul a n t a nd the r efor e d i a g on al i za b le b y the d i s cr e t e F our ie r tr a n sfor m [ 50 ]
W
lm
=
1
p
d
exp
(
i
2 π
d
( l 1)( m 1)
)
. N o w w e f ur the r r e s tr ict t o c ir c ul a n t U w hich i s he nc e al s o d i a g on al -
i z e d b y W . I f s uch a unit a r y U ex i s ts, i s g iv e n b y U = WDW
y
, w he r e D := diag( e
i α 0
;:::; e
i α
d 1
) . A s a r e s ult ,
E q . ( 1.53 ), a ft e r s ome calc ul a t ion s, r e duc e s t o th e fo l lo w in g ( d 1) e qua t ion s in v o lv in g the ei g e n v alue s of
U :
d 1
∑
m= 0
exp[ i( α
m+ r
α
m
)] =
d
p
2
; r = 1;:::; d 1 ; (1.54)
w he r e the index a dd it ion i s unde r s t ood Mod( d) . The a bo v e s e t of e qua t ion s for the ei g e n v alue s of U c on-
s t i tut e s a s u ffic ie n t c ond it ion for the a tt ain tme n t of C
max
2; B
( d) .
A fa mi ly of s o lut ion t o the a bo v e e qua t ion s, v al id for d = 2;:::; 13 , i s g iv e n b y α
m
= φ
0
for m =
0;:::; k 1; k + 1;:::; d 1 w ith φ
0
2 [ 0; 2 π) a n d α
k
= φ
0
+ φ , w he r e cos φ =
1
2
(
d
p
2
d + 2
)
. The
r e s tr ict ion d 13 c ome s f r om cos φ 1 .
48
2
Qua n tum cohe r e nce a nd the lo c ali z a tion tr a nsition
2.1 L o c al iz a tion for t he kindergar ten
Qu a ntum phy si c is ts spec i a lizi n g i n ma ter i a l s h a v e a fa v or it e s ys t e m they l i k e t o s tudy: th a t of a
p er io d ic cr y s t a l . P e r iod ic cr ys t al s a r e m a t e r i al s w hos e c on s t itue n ts ( us ual ly mo le c ul a r c omp lexe s ) a r e a r -
r a n g e d in a n or de r e d micr os c op ic s tr uctur e th a t r e pe a ts a d infinitum in al l d ir e ct ion s . I f y ou h a ppe n t o
be l ie v e th a t s uch o bj e cts a r e pr o b a b ly t oo dul l t o be a b i g de al , thi s i s w h a t E r w in S chöd in g e r think s a bout
pe r iod ic cr ys t al s: “ T o a h umb le p h ysic i s t ’ s mind , the s e a r e v e r y in t e r e s t in g a nd c omp l ica t e d o bj e cts; they
c on s t itut e one of the mos t fas c in a t in g a nd c omp lex m a t e r i al s tr uctur e s b y w hich in a nim a t e n a tur e puz z le s
hi s w its . ”
49
W h a t i s mor e , p h ysic i s ts l i k e t o s tudy t o y pr o b le m s in or de r t o g ain in si gh t of ho w the micr os c op ic
s tr uctur e of m a t e r i al s a ffe cts their m a cr os c op ic pr ope r t ie s . One s uch pr o b le m r e l a t e d w ith p h ysical p h ysi -
cal pr ope r t ie s of cr ys t al -l i k e m a t e r i al s i s the fo l lo w in g. S u ppos e y ou in j e ct a sin gle p a r t icle ( s a y , e le ctr on )
s ome w he r e in the cr ys t al , a nd th a t the p a r t icle i s w e ak ly a ttr a ct e d t o al l the cr ys t al c e n t e r s; w h a t i s g o in g
t o h a ppe n a ft e r s ome t ime , i f the p a r t icle i s left alone? U nde r c e r t ain simp l i f y in g as s umpt ion s, the a n s w e r
tur n s out t o be pr e tt y simp le: it w i l l d i ff use . M or e c oncr e t e ly , the r et u r n pr o b a bi l it y ( i . e ., the pr o b a b i l it y of
the p a r t icle r e tur nin g t o the sit e w he r e it s t a r t e d ) be c ome s a ft e r a lon g e nou gh t ime e qual t o 1= N , w he r e N
i s the t ot al n umbe r of sit e s . F or l a r g e s ys t e m s, thi s i s pr a ct ical ly z e r o .
S o fa r , no s ur pr i s e s . H o w e v e r , the as t oni shme n t ca me in 1958 a nd w as init i a t e d b y P . W . A nde r s on ( al -
thou gh hi s ide as t oo k s ome t ime t o be a ppr e c i a t e d b y the c omm unit y , a nd e v e n b y A nde r s on him s e l f , a nd
e v e n tual ly le d t o a N o be l pr i z e ). A nde r s on loo k e d a t a v a r i a n t of the pr o b le m of a sin gle p a r t icle hopp in g
a r ound on a l a tt ic e; he as s ume d th a t the l a tt ic e a ttr a cts the p a r t icle w ith s tr e n g th s th a t a r e r a n do m . H o w -
e v e r , the pr o b le m the n be c ome s n as t y t o tr e a t in f ul l m a the m a t ical de t ai l , a nd one ne e d s t o r e side in s ome
k ind of sem i- c l assic a l a ppr o x i m a t io n . A ft e r r o l l in g - u p the sle e v e s a nd pe r for min g s ome calc ul a t ion s, one
g e ts the ( pr o b a b ly r e a s on a b le a t fir s t si gh t ) r e s ult th a t it i s d i ff usion th a t s t i l l t ak e s p l a c e ( in sp it e of the
r a ndomne s s ), but w ith a s ome w h a t slo w e r r a t e .
The br e akthr ou gh of A nde r s on w as t o sho w th a t the a bo v e in tuit ion i s co m p let e l y f a lse . B y th a t , I do
not me a n th a t he found a w r on g s t e p in a le n g th y calc ul a t ion, but in s t e a d he a r g ue d th a t the s e mi - cl as sical
a ppr o x im a t ion i s utt e rly in s u ffic ie n t t o ca ptur e the p h ysic s of the a ctual pr o b le m . I n s t e a d , he pr e d ict e d th a t ,
a ft e r s ome cr it ical a moun t of r a ndomne s s (w hich p h ysic i s ts l i k e t o cal l d iso r der ) the s ys t e m w i l l lo c a l i ze .
Thi s me a n s th a t the r e tur n pr o b a b i l it y w i l l be finit e , e v e n a ft e r infinit e t ime p as s e s a nd for a r b itr a r i ly l a r g e
s ys t e m s: the p a r t icle e s s e n t i al ly ne v e r g oe s a w a y .
Y ou mi gh t w onde r a t thi s po in t ho w i s al l thi s busine s s of local i za t ion i s c onne ct e d t o qua n tum c o he r e nc e
d i s c us s e d in C h a pt e r 1 . W e a ctual ly h a ppe ne d t o w onde r the ex a ct s a me thin g s ome t ime a g o , a nd the
c onne ct ion be t w e e n the t w o t op ic s i s the the me of the pr e s e n t C h a pt e r . L e t me no w pr o v ide s ome in tuit ion
w h y a c onne ct ion i s t o be ex pe ct e d in the fir s t p l a c e .
I n in tr oduct or y c our s e s on qua n tum me ch a nic s, a c e n tr al t op ic i s the t ime e v o lut ion of qua n tum s t a t e s .
50
H o w s t a t e s of a qua n tum s ys t e m e v o lv e in t ime i s c omp a ctly ca ptur e d b y a n o bj e ct cal le d the H a m i l t o n i a n ,
w hich al s o define s w h a t e ne r g y me a n s for the s ys t e m in c on side r a t ion . I n mor e t e chnical t e r m s, a H a mi l -
t oni a n define s a n as s oc i a t e d en er g y b asis . One al s o le a r n s e a rly - on in s uch c our s e s th a t for a qua n tum s t a t e
t o ch a n g e in t ime , it should be in c o he r e n t s u pe r posit ion w ith r e spe ct t o the e ne r g y b asi s ( othe r w i s e , i s
s t a ys mot ionle s s for e v e r ). W e l l , he r e i s the c onne ct ion : i f a p a r t icle pe r for m s d i ff usiv e mot ion or i s in s t e a d
local i z e d , should s ome ho w be g o v e r ne d b y the a moun t of c o he r e nc e the init i al s t a t e h as w ith r e spe ct t o the
e ne r g y b asi s . Thi s C h a pt e r m ak e s thi s c onne ct ion m a the m a t ical ly pr e c i s e , b y in v o k in g s ome of the m a chin-
e r y de v e lope d in C h a pt e r 1 . I n fa ct , it al s o t ouche s u pon local i za t ion in the cas e of m a n y p a r t icle-w al k e r s .
A fe w of the m ain find in gs ca n be infor m al ly s t a t e d as fo l lo w s:
Th e r et u r n pr o b a bi l it y , w h ic h is a fi g u r e of m er it i n det e c t i n g lo c a l i za t io n , c a n b e e q u i v a len t l y u n der s t o o d as a m e a-
s u r e of q u a n t u m co h er en ce . W h a t is m o r e, m a n y o t h er q u a n t i fier s of q u a n t u m co h er en ce c a n b e u t i l i ze d t o det e c t
lo c a l i za t io n , b o t h i n t h e o n e- a n d m a n y- b o d y sett i n g s . C h a n ge i n t h e a bi l it y t o gen er a t e q u a n t u m co h er en ce c a n
b e d i r e c t l y r e l a t e d t o so m e f o r m of co n d u c t i v it y , w h ic h is kn o w n t o b e h a v e d i ff er en t l y i n t h e d i ff usi v e a n d lo c a l i ze d
r e g i m e s .
2.2 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 -
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-
51
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 [ 51 ].
2.3 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 [ 11 ]. 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 omp 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 [ 27 – 29 ]. 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 [ 3 , 52 ] a nd m a n y - body local i za t ion ( MBL ) tr a n sit ion s [ 53 – 55 ]. The s e “ infinit e t e mpe 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
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 [ 39 ] ( s e e al s o C h a pt e r 1 ).
52
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 [ 56 ]. 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 .
2.4 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
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 [ 29 ]. 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 )
53
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 [ 33 , 58 ]. 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 1 , w e in tr oduc e the
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
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 [ 31 ] 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 [ 33 , 57 ].
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 .
54
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)
55
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 1 . 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
2B(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-
56
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
57
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 .
58
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 [ 60 ] 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 [ 59 ] 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
59
( 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 [ 38 ] 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 ult iv a r i -
a t e m a j or i za t ion [ 61 ]. 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
60
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
j7!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 . [ 60 ],
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 . [ 60 ]
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 [ 61 ]. 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 [ 61 ] 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
61
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)⟩ [ 62 – 65 ]. 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 [ 62 , 63 ], 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 ).
62
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 -
63
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)
64
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 [ 3 , 66 ] 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 .
2.5 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 [ 3 ] 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 [ 67 ].
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
65
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 2.5.1 a nd F i g ur e 2.5.2 ).
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 2.5.1 . 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 ,
66
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 [ 68 ]). W e foc us
67
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 2.5.1: 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 [ 69 ] 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.
2.6 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
68
5 10 50 100 500 1000
0
2
4
6
8
L
C
B
(rel)
( 
W
)
W
5
2
1
1/2
0
Figure 2.5.2: 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.
69
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 . [ 66 , 70 ] 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 . [ 71 – 76 ].
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 2.6.1 , 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)
70
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 2.6.1 . 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 1 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)
71
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 2.6.1: Asymptotic b ehavio r fo r the slop e of the 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 st rength 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 the W = 3: 7 , where the sample sizes w ere doubled.
The e rro r ba rs rep resent 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 .
2.7 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
72
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 lt o -
ni 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 [ 77 – 79 ] 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 [ 80 ].
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 . [ 81 ] 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) ,
73
w he r e V = @
λ
H( λ) . M or e pr e c i s e ly ( s e e R ef . [ 81 ]),
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 [ 82 – 84 ]. 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 [ 56 , 85 ].
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 [ 56 ].
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 .
2.8 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 [ 28 , 29 ], a nd
local i za t ion [ 3 , 53 – 55 ].
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 .
74
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
.
75
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 [ 62 ,
76
63 ]. 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 [ 86 ], 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
77
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 [ 87 ]). 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
;
78
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
)
:
79
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 [ 69 ], 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 . [ 68 ], 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 2.8.1 .
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)
80
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Γ
C
B
(2)
( 
Γ
)
Figure 2.8.1: 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 2.7 .
81
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 [ 39 ]. 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 . [ 39 ] 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)
82
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 [ 39 ]. 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 . [ 39 ].
83
3
I ncomp a ti bi lit y of me a s ur e me n t s, qua n ti fied
3.1 Me a s u re men t in c omp a tib il it y for the kindergar ten
In o r d i n a r y li fe, w e a r e us e d t o ask in g que s t ion s in or de r t o g a the r infor m a t ion a bout a situa t ion w e
w a n t t o kno w mor e . W e r a r e ly ( i f e v e r ) p a us e t o w onde r w he the r the infor m a t ion w e a r e loo k in g for i s in
s ome s e n s e inc omp a t i b le . F or in s t a nc e , one mi gh t s ur e ly ex pe ct t o be a b le t o kno w w ith c e r t ain t y ( a t le as t
in pr inc ip le ) a n y as s or tme n t of p h ysical pr ope r t ie s ch a r a ct e r i z in g a s ys t e m .
W e l l , not alw a ys w he n it c ome s t o qua n tum me ch a nic s .
L e t us c on side r the fo l lo w in g ide al i z e d s c e n a r io . S u ppos e y ou h a v e a c c e s s t o a b l a ck bo x th a t , w ith the
simp le pr e s s of a butt on, ca n out put a t w i l l a f r e sh c op y of a “ qua n tum s ome thin g ” , a nd th a t y ou a r e al s o
84
g ua r a n t e e d th a t its out put i s c on si s t e n tly de s cr i be d b y the s a me qua n tum s t a t e j ψ⟩ . M or e o v e r , s u ppos e y ou
ca n in t e r r o g a t e a n y sin gle c op y of the out put b y pe r for min g one out of t w o t y pe s of me as ur e me n t , w hich
I w i l l de not e as A a nd B . A ft e r e a ch me as ur e me n t pr oc e s s of eithe r A or B , a n as s oc i a t e d r e s ult i s o bt aine d
( i . e ., a n umbe r ), w hi le the c op y of the “ qua n tum s ome thin g ” i s d i s ca r de d t o the g a r b a g e a nd r e p l a c e d w ith
a ne w one .
The r e s ult of a me a s ur e me n t in qua n tum the or y i s inhe r e n tly pr o b a b i l i s t ic ; ne v e r the le s s, one ca n in pr in-
c ip le c on s tr uct the pr o b a b i l it y d i s tr i but ion s p
A
(j ψ⟩) a nd p
B
(j ψ⟩) as s oc i a t e d w ith A a nd B b y pe r for min g
mor e a nd mor e me as ur e me n ts ¹ . H e r e c ome s the inc omp a t i b i l it y s t a t e me n t :
T w o me as ur e me n ts A a nd B a r e cal le d i n co m p a t i b le i f the r e ex i s ts a qua n tum s t a t e j ψ⟩ s uch th a t the
me as ur e me n t out c ome for only one of the o bs e r v a b le s ca n be pr e d ict e d w ith a bs o lut e c e r t ain t y .
I n othe r w or d s, i f the t w o me as ur e me n ts a r e inc omp a t i b le the n the r e ex i s ts a cho ic e of the s t a t e j ψ⟩
th a t the b l a ck bo x out puts, s uch th a t only one 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 p
A
, p
B
h as s ome
r a ndomne s s ( i . e ., i s not pe ak e d t o a sin gle pos si b le r e s ult ), w hi le the othe r doe s not.
The nex t o b v ious ( a nd h a r d ly or i g in al ) que s t ion th a t a r i s e s on inc omp a t i b le me as ur e me n ts i s: h o w m u c h
i n co m p a t i b le a r e t h e y? F or in s t a nc e , in s ome s e n s e posit ion a nd mome n tum a r e m a x im al ly inc omp a t i b le:
ne a r pe r fe ct kno w le d g e of one imp l ie s c omp le t e i gnor a nc e of the othe r , no m a tt e r w h a t the s t a t e j ψ⟩ i s .
Thin gs a r e not alw a ys as b a d , thou gh ; the r e a r e cas e s w he r e t w o t y pe s of me as ur e me n ts a r e inc omp a t i b le
but jus t for c e r t ain s t a t e s j ψ⟩ a nd 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 mi gh t h a ppe n t o be ne a rly ( but
not quit e ) p e ak e d sim ult a ne ously .
Thi s C h a pt e r a tt e mpts t o g iv e a n a n s w e r t o thi s que s t ion . Befor e br iefly s a y in g in w h a t s e n s e thi s i s done ,
le t me fir s t s a y in w h a t s e n s e it i s n o t done: W e a r e not g o in g t o c on s tr uct a nothe r nic e f unct ion o v e r me a -
s ur e me n ts th a t sp its out z e r o for c omp a t i b le me as ur e me n ts a nd s ome l a r g e v alue for r e al ly inc omp a t i b le
one s . I n s t e a d , the m ain ide a i s t o in tr oduc e a n o r der i n g o v e r al l pos si b le me as ur e me n ts, w ith r e spe ct t o
a r efe r e nc e o bs e r v a b le . I n the or de r in g , me as ur e me n ts th a t l ie “ lo w e r ” h a v e a simp le in t e r pr e t a t ion ; they
ca n be e m ul a t e d b y me as ur e me n ts th a t l ie “ hi ghe r ” jus t b y pos t -pr oc e s sin g of the as s oc i a t e d pr o b a b i l it y
¹ W e w i l l al s o as s ume th a t the t w o me as ur e me n ts c or r e spond t o w h a t i s kno w n as n o n -de gen er a t e o b ser v a b le s , w hich e s s e n t i al ly
me a n s t h a t they ca n d i s t in g ui sh al l pos si b le me as ur e me n t out c ome s of the p h ysical qua n t it y bein g me as ur e d .
85
d i s tr i but ion s .
A t thi s po in t , y ou a r e pr o b a b ly w onde r in g w h y I c on side r th a t t o be a q u a n t i fic a t io n of me as ur e me n t
inc omp a t i b i l it y . W e l l , b y its definit ion, the or de r in g t ur n s out t o be e quiv ale n t t o i nfi n it e l y m a n y s cal a r
f unct ion s, l i k e the one s de s cr i be d a bo v e . H e nc e it m ak e s s e n s e t o c on side r the or de r in g its e l f as a qua n t i fie r s
of inc omp a t i b i l it y .
The m ain ide a a nd find in gs of the C h a pt e r ca n be infor m al ly s t a t e d as fo l lo w s:
A do p t i n g t h e p er s p e c t i v e t h a t i n co m p a t i bi l it y c a n b e f u n d a m en t a ll y u n der s t o o d i n t er m s of si m u l a t io n of m e a-
s u r em en ts b y o t h er m e as u r em en ts, w e e s t a b l is h a q u a n t it a t i v e co n n e c t io n b et w e en en tr o pic u n cer t a i n t y r e l a t io n s,
q u a n t u m co h er en ce, a n d m e as u r em en t si m u l a t a bi l it y v i a t h e m a t h em a t ic a l t o o l of m u l t i v a r i a t e m aj o r i za t io n.
3.2 A bs tr a c t
M ot iv a t e d b y qua n tum r e s our c e the or ie s, w e in tr oduc e a not ion of inc omp a t i b i l it y for qua n tum me as ur e-
me n ts r e l a t iv e t o a r efe r e nc e b asi s . The not ion a r i s e s b y c on side r in g s t a t e s d i a g on al in th a t b asi s a nd in v e s -
t i g a t in g w he the r pr o b a b i l it y d i s tr i but ion s as s oc i a t e d w ith d i ffe r e n t qua n tum me as ur e me n ts ca n be c on-
v e r t e d in t o one a nothe r b y pr o b a b i l i s t ic pos t pr oc e s sin g. The induc e d pr e or de r o v e r qua n tum me as ur e-
me n ts i s d ir e ctly r e l a t e d t o m ult iv a r i a t e m a j or i za t ion a nd g iv e s r i s e t o fa mi l ie s of monot one s, i . e ., s cal a r
qua n t i fie r s th a t pr e s e r v e the or de r in g. F or the cas e of or tho g on al me as ur e me n t w e e s t a b l i sh a qua n t it a t iv e
c onne ct ion be t w e e n inc omp a t i b i l it y , qua n tum c o he r e nc e a nd e n tr op ic unc e r t ain t y r e l a t ion s . W e g e ne r al -
i z e the c on s tr uct ion t o include a r b itr a r y PO VM me as ur e me n ts a nd r e por t c omp le t e fa mi l ie s of monot one s .
T ex t for thi s C h a pt e r i s a d a pt e d f r om [ 88 ].
3.3 In troduc tion
One of the c or ne r s t one s of qua n tum the or y i s the c onc e pt of 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 [ 11 ].
A p air of qua n tum o bs e r v a b le s i s de e me d inc omp a t i b le i f the c or r e spond in g s e l f -a d j o in t ope r a t or s fai l t o
c omm ut e . O pe r a t ion al ly , inc omp a t i b i l it y imp l ie s th a t the r e ex i s t pur e qua n tum s t a t e s for w hich it i s impos -
si b le t o sim ult a ne ously pr e d ict w ith c e r t ain t y the me as ur e me n t out c ome s of t w o inc omp a t i b le o bs e r v a b le s .
86
F init e- d ime n sion al o bs e r v a b le s th a t sh a r e the s a me ei g e nb asi s a r e f ul ly c omp a t i b le , w hi le a n y p air of o b -
s e r v a b le s as s oc i a t e d w ith b as e s th a t a r e m utual ly unb i as e d a r e m a x im al ly inc omp a t i b le: c e r t ain kno w le d g e
for the out c ome of one as s ur e s c omp le t e r a ndomne s s for the pos si b le out c ome s of the othe r .
I nc omp a t i b i l it y i s fa mously ca ptur e d thr ou gh unc e r t ain t y r e l a t ion s, th a t m a y in v o lv e v a r i a nc e s [ 4 , 89 ,
90 ], e n tr op ie s [ 91 – 97 ] or othe r infor m a t ion-the or e t ic qua n t it ie s [ 98 – 103 ]. A qua n t it a t iv e de s cr ipt ion of
inc omp a t i b i l it y in qua n tum me ch a nic s w as pur s ue d r e c e n tly , f r om the pe r spe ct iv e of s t a t e d i s cr imin a t ion
a nd qua n tum s t e e r in g [ 104 – 112 ]. I n thi s a ppr o a ch, one of the c e n tr al not ion s i s th a t of a p a r en t me as ur e-
me n t , i . e ., one th a t ca n sim ul a t e the or i g in al one thr ou gh pr o b a b i l i s t ic pos t pr oc e s sin g.
Qua n tum r e s our c e the or ie s pr o v ide a f r a me w ork t o s ys t e m a t ical ly ch a r a ct e r i z e a nd qua n t i f y qua n tum
pr ope r t ie s ( for ex a mp le , e n t a n gle me n t ). The r e , s uch a pr ope r t y i s f ul ly de s cr i be d b y the c on v e r sion r e l a -
t ion s a mon g s t a t e s unde r a cl as s of qua n tum pr oc e s s e s th a t , s uit a b ly chos e n, ca nnot e nh a nc e it [ 26 ]. The
tr a n sfor m a t ion r e l a t ion s a mon g qua n tum s t a t e s ca n be m a the m a t ical ly de s cr i be d b y a pr e o r der : i f a s t a t e
ca n be tr a n sfor me d in t o a nothe r unde r the d i s t in g ui she d cl as s of pr oc e s s e s, the n it l ie s “ hi ghe r ” in t he or -
de r in g ² . I n tur n, the pr e or de r induc e s a fa mi ly of s cal a r f unct ion s, cal le d m o n o t o n e s , th a t ca nnot incr e as e
unde r the al lo w e d s t a t e tr a n sit ion s a nd the r efor e j o in tly qua n t i f y the r e s our c ef ulne s s of s t a t e s .
I n thi s ch a pt e r , w e in tr oduc e a not ion of inc omp a t i b i l it y of qua n tum me as ur e me n ts r e l a t iv e t o a r efe r -
e nc e b asi s b y me a n s of a pr e or de r . M or e spe c i fical ly , c on side r in g s t a t e s th a t a r e d i a g on al in the r efe r e nc e
b asi s, w e in v e s t i g a t e w he the r the pr o b a b i l it y d i s tr i but ion s as s oc i a t e d w ith d i ffe r e n t me as ur e me n ts ca n be
tr a n sfor me d in t o one a nothe r , b y me a n s of pr o b a b i l i s t ic pos t pr oc e s sin g. The a for e me n t ione d que s t ion of
c on v e r t i b i l it y g e ne r a t e s a pr e or de r o v e r qua n tum me as ur e me n ts w hich, in tur n, g iv e s r i s e t o fa mi l ie s of
s cal a r f unct ion s th a t j o in tly qua n t i f y the in tr oduc e d not ion of inc omp a t i b i l it y r e l a t iv e t o a b asi s . W e fir s t
c on side r the spe c i al cas e of or tho g on al me as ur e me n ts in w hich the or de r in g pr o v ide s a qua n t it a t iv e , as
w e l l as c onc e ptual , c onne ct ion be t w e e n inc omp a t i b i l it y , qua n tum c o he r e nc e a nd e n tr op ic unc e r t ain t y r e-
l a t ion s . W e the n ex t e nd t o include g e ne r al i z e d me as ur e me n ts a nd w e r e l a t e the r e s ult in g not ion t o p a r e n t
me as ur e me n ts .
² M a the m a t ical ly , pr e or de r ( or quasior de r ) i s a b in a r y r e l a t ion t h a t i s r eflex iv e a nd tr a n sit iv e .
87
3.4 Prel imin arie s
C on side r a nonde g e ne r a t e o bs e r v a b le A o v e r a finit e d ime n sion al H i l be r t sp a c e H

=C
d
w ith spe ctr al de-
c omposit ion A =
∑
d
i= 1
a
i
P
i
(w e de not e P
i
:=j i⟩⟨ ij ). The r o le of the ei g e n v alue s a
i
i s t o l a be l the pos si b le
out c ome s a nd , as lon g as they a r e d i s t inct , thi s r o le i s unimpor t a n t f r om the po in t of v ie w of the me as ur e-
me n t pr oc e s s, sinc e the pr o b a b i l it y d i s tr i but ion p
B
( ρ) w ith c ompone n ts [ p
B
( ρ)]
i
:= Tr( P
i
ρ) ( r e pr e s e n t -
in g a me as ur e me n t of A in s t a t e ρ ) only de pe nd s on the s e t of pr oj e ct or s f P
i
g
i
³ . W e w i l l he nc efor th us e
the t e r m b asis ( alw a ys me a nin g or thonor m al ) t o r efe r t o a s e t of r a nk -1 or tho g on al pr oj e ct or s B =f P
i
g
d
i= 1
,
w ith
∑
i
P
i
= I ⁴ . A g e ne r al i z e d me as ur e me n t ( PO VM ) i s r e pr e s e n t e d b y a s e t of ope r a t or s F = f F
i
g
i
s uch th a t F
i
0 a nd
∑
i
F
i
= I . W e as s oc i a t e w ith e v e r y b asi s B the r e al A be l i a n al g e br a of o bs e r v a b le s
A
B
g e ne r a t e d b y f P
i
g
i
. The s e t of b as e s o v e r the H i l be r t sp a c e i s de not e d b y M(H) .
3.5 Preorder and m o not one s
The ide a of de r iv in g fa mi l ie s of s cal a r f unct ion s th a t qua n t i f y s ome fe a tur e ( for in s t a nc e , the de gr e e of uni -
for mit y of a pr o b a b i l it y d i s tr i but ion ) b y in v o k in g a pr e or de r h as its r oots in the m a the m a t ical the or y of m a -
j or i za t ion [ 113 ]. S uch a p a r a d i gm h as be e n ex t e n siv e ly e mp lo y e d in qua n tum infor m a t ion in the c on t ex t
of r e s our c e the or ie s for qua n t i f y in g fe a tur e s of qua n tum s ys t e m s, s uch as e n t a n gle me n t [ 114 ], c o he r e nc e
[ 29 ] a nd out - of - e qui l i br ium the r mody n a mic s [ 17 ].
I n thi s a ppr o a ch, one d i s t in g ui she s a cl as s of qua n tum ope r a t ion s, de e me d as “ e as y ” , mot iv a t e d b y s ome
pr a ct ical c on side r a t ion . F or ex a mp le , in the cas e of e n t a n gle me n t , the e as y ope r a t ion s a r e local qua n tum
ope r a t ion s be t w e e n t w o p a r t ie s t o g e the r w ith cl as sical c omm unica t ion . Thi s s e t of m a ps induc e s a pr e or de r
” ” in the s e t of qua n tum s t a t e s, define d b y the al lo w e d tr a n sit ion s unde r e as y ope r a t ion s, n a me ly ρ σ i f
a nd only i f the r e ex i s ts a n e as y ope r a t ion E s uch th a t σ =E( ρ) . The b in a r y r e l a t ion induc e d i s a pr e or de r
sinc e , b y definit ion, the ide n t it y qua n tum ch a nne l i s alw a ys a n e as y ope r a t ion a nd al s o the c omposit ion of
e as y ope r a t ion s i s a g ain a n e as y ope r a t ion . M or e o v e r , ρ σ should in tuit iv e ly c or r e spond in our ex a mp le
³ I n thi s ch a pt e r , w e w i l l not d i s t in g ui sh a mon g pr o b a b i l i t y d i s tr i but ion s th a t d i ffe r s o le ly b y pe r m ut a t ion s .
⁴ N ot ic e th a t s uch a definit ion doe s not d i s t in g ui sh be t w e e n or thonor m al s e que nc e s of k e ts th a t d i ffe r s o le ly b y r e or de r in g
of e le me n ts o r b y p h as e fa ct or s, i . e ., b y a tr a n sfor m a t ion j j⟩7! e
iθ j
j σ( j)⟩ ( σ2P
d
i s a pe r m ut a t ion ).
88
t o a s t a t e me n t l i k e “ ρ i s mor e e n t a n gle d th a n σ . ” Thi s qua n t i fica t ion i s r i g or ously ca ptur e d b y the not ion of
m o n o t o n e s .
D e finitio n 3.1
M onot one s a r e s cal a r f unct ion s f o v e r s t a t e s, nonincr e asin g unde r al lo w e d s t a t e tr a n sit ion s
ρ σ =) f( ρ) f( σ) : (3.1)
F a mi l ie s of monot one s f f
a
g
α
a r e s aid t o for m a co m p let e set , i f they s a t i sf y
f
α
( ρ) f
α
( σ)8 α () ρ σ : (3.2)
3.6 A preorder o ver o r thonormal ba se s
Our g o al i s t o define a not ion of inc omp a t i b i l it y r e l a t iv e t o a b asi s . L e t us be g in w ith the cas e of or -
tho g on al me as ur e me n ts . C on side r a b asi s B
0
= f P
( 0)
i
g
i
a nd a s t a t e ρ
0
=
∑
i
p
i
P
( 0)
i
2 A
B 0
d i a g o -
n al o v e r it , de s cr i be d b y the pr o b a b i l it y d i s tr i but ion p . G iv e n a nothe r b asi s B
1
= f P
( 1)
i
g
i
, one ca n al s o
as s oc i a t e w ith ρ
0
the pr o b a b i l it y d i s tr i but ion p
B 1
( ρ
0
) c or r e spond in g t o a me as ur e me n t o v e r B
1
. I n fa ct ,
p
B 1
( ρ
0
) = X(B
1
;B
0
) p , w he r e X(B
1
;B
0
) de not e s the b i s t och as t ic m a tr i x ⁵ w ith e le me n ts
[ X(B
1
;B
0
)]
ij
:= Tr
(
P
( 1)
i
P
( 0)
j
)
: (3.3)
M or e o v e r , the pr o b a b i l it y d i s tr i but ion p
B 1
( ρ
0
) i s alw a ys “ mor e uni for m ” th a n p . Thi s i s pr e c i s e ly ca ptur e d
b y the m a j or i za t ion s t a t e me n t p≻ p
B 1
( ρ
0
) th a t i s tr ue for a n y b asi s B
1
a nd fo l lo w s d ir e ctly f r om the b i s -
t och as t ic it y of X [ 8 ].
L e t us no w in tr oduc e a nothe r me as ur e me n t , o v e r a b asi s B
2
, s uch th a t the r e ex i s ts s ome b i s t och as t ic
⁵ N ot ic e th a t the or de r in g of the pr oj e ct or s in a b asi s i s a r b itr a r y , he nc e the X m a tr i x i s non unique u p t o pe r m ut a t ion s .
89
m a tr i x M w ith
X(B
2
;B
0
) = MX(B
1
;B
0
) : (3.4)
Thi s r e l a t ion h as a r a the r s tr on g imp l ica t ion : for al l s t a t e s ρ
0
d i a g on al in B
0
, the d i s tr i but ion p
B 2
( ρ
0
) ca n be
o bt a ine d f r om p
B 1
( ρ
0
) thr ou gh “ uni for min g ” cl as sical pos t pr oc e s sin g , r e pr e s e n t e d b y s ome b i s t och as t ic M
w hich i s inde pe nde n t of the s t a t e .
M ot iv a t e d b y the a bo v e , i f E q . ( 3.4 ) ho ld s, w e de cl a r e th a t “ a n or tho g on al me as ur e me n t o v e r B
1
i s mor e
c omp a t i b le th a n o v e r B
2
, r e l a t iv e t o s t a t e s d i a g on al in B
0
” . W e in tr oduc e the fo l lo w in g not a t ion .
D e finitio n 3.2
W e de not eB
1
≻
B 0
B
2
i f a nd only i f the r e ex i s ts a b i s t och as t ic m a tr i x M s uch th a t X(B
2
;B
0
) =
MX(B
1
;B
0
) .
The fo l lo w in g fo l lo w d ir e ctly f r om the definit ion .
P r opositio n 3.1
( i ) The b in a r y r e l a t ion “≻
B 0
” o v e rM(H) i s a pr e or de r , i . e ., B≻
B 0
B8B ( r eflex iv it y ) a nd B
1
≻
B 0
B
2
,B
2
≻
B 0
B
3
=)B
1
≻
B 0
B
3
( tr a n sit iv it y ).
( i i ) B
0
≻
B 0
B for al l b as e s B (“ me as ur e me n t o v e r B
0
i s mor e c omp a t i b le th a n o v e r a n y othe r
b asi s ”)
( i i i )B ≻
B 0
B
MU
for al l b as e s B , w he r eB
MU
i s a n y b asi s m utual ly unb i as e d t o B
0
(“ me as ur e me n t
o v e r a n y b asi s i s mo r e c omp a t i b le th a n o v e r a m utual ly unb i as e d one ”).
90
P r o of . ( i ) R eflex iv it y fo l lo w s sinc e the ide n t it y m a tr i x i s b i s t och as t ic a nd tr a n sit iv it y f r om the fa ct th a t
a pr oduct of b i s t och as t ic m a tr ic e s i s al s o b i s t och as t ic .
( ii ) S inc e X(B
0
;B
0
) = I , fo l lo w s b y s e tt i n g M = X(B;B
0
)
( iii ) B y definit ion,[ X(B
MU
;B
0
)]
ij
= 1= d , he nc e fo l lo w s b y s e tt in g M
ij
= 1= d . ■
The pr e or de r “ ≻
B 0
” i s not in g e ne r al a p a r t i al or de r , i . e ., B
1
≻
B 0
B
2
a ndB
2
≻
B 0
B
1
do not ne c e s s a r i ly
imp lyB
1
= B
2
. F or ex a mp le , a n y B
1
a ndB
2
th a t a r e unb i as e d r e l a t iv e t o B
0
s a t i sf y the a for e me n t ione d
r e l a t ion s bu t ca n be t ak e n t o be d i s t inct.
The or de r in g ( 3.4 ) o v e r m a tr ic e s h as be e n s tud ie d in the c on t ex t of m ult iv a r i a t e m a j or i za t ion, cal le d
m a tr i x m aj o r i za t io n [ 61 ]. The r e , A ≻ C for m a tr ic e s A a nd C i f the r e ex i s ts a b i s t och as t ic B s uch th a t
C = BA . W e no w c onne ct the a for e me n t ione d pr e or de r w ith qua n tum me as ur e me n ts .
3.7 T he ordering from n on sel ec tive or tho gon al me a s u re men ts
P r oposit ion 3.2 ca n be ope r a t ion al ly unde r s t ood in t e r m s of cl as sical pos t pr oc e s sin g of pr o b a b i l it y d i s tr i -
but ion s . H e r e w e sho w th a t the or de r in g “ ≻
B 0
” al s o a dmits a qua n tum ope r a t ion al in t e r pr e t a t ion in t e r m s
of e m ul a t ion of a non s e le ct iv e me as ur e me n t v i a a dd it ion al s uch me as ur e me n ts .
A n y b asi sB g iv e s r i s e t o a c or r e spond in g dep h asi n g or m e as u r em en t qua n tum m a p
D
B
( X) :=
∑
i
P
i
XP
i
: (3.5)
The l a tt e r ca n be thou gh t of as a non s e le ct iv e or tho g on al me as ur e me n t of a n y nonde g e ne r a t e o bs e r v a b le
be lon g in g in A
B
, w hi le a c omposit ion D
B n
:::D
B 1
r e pr e s e n ts the qua n tum ope r a t ion as s oc i a t e d w ith n
s uch s uc c e s siv e me as ur e me n ts ⁶ .
⁶ I n fa ct , the b asi s B c or r e spond in g t o a de p h asin g m a p D
B
i s unique , i . e ., the m a pp in g B 7! D
B
i s in j e ct iv e [ 115 ], a nd
simi l a rly for B7!A
B
[ 39 ].
91
W e a r e no w r e a dy t o s t a t e the r e s ult. The or de r in g B
1
≻
B 0
B
2
ho ld s i f a nd only i f , for a n y init i al s t a t e
d i a g on al in B
0
, the out put of a non s e le ct iv e B
2
me as ur e me n t ca n be e m ul a t e d b y a non s e le ct iv e B
1
me as ur e-
me n t , fo l lo w e d pos si b ly b y a n a dd it ion al s e que nc e of me as ur e me n ts a nd a unit a r y r ot a t ion . M or e spe c i fi -
cal ly:
P r opositio n 3.2
B
1
≻
B 0
B
2
i f a nd only i f the r e ex i s t a unit a r y s u pe r ope r a t or U a nd a ( pos si b ly tr iv i al ) s e que nc e of
me as ur e me n ts fD
B
′
α
g
α
s uch th a t
D
B 2
D
B 0
=U
[
∏
α
D
B
′
α
]
D
B 1
D
B 0
: (3.6)
The a u x i l i a r y s e que nc e of me as ur e me n ts ne e de d mi gh t be , in fa ct , infinit e . E q . ( 3.6 ) should be unde r -
s t ood as “

D
B 2
D
B 0
U
[∏
α
D
B
′
α
]
D
B 1
D
B 0

ca n be m a de a r b itr a r i ly s m al l ” , i . e ., the s t a t e tr a n sfor m a t ion
of the rh s ca n a ppr o x im a t e a r b itr a r i ly w e l l the one of the l h s .
Befor e pr oc e e d in g w ith the pr oof , le t us fir s t e s t a b l i sh a le mm a . W e r e mind the r e a de r th a t a b i s t och as t ic
m a tr i x A
ij
i s u n is t o c h as t ic [ 59 ] 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 A
ij
=j U
ij
j
2
.
L e mm a 3.1
E v e r y b i s t och as t ic m a tr i x ca n be a ppr o x im a t e d a r b itr a r i ly w e l l b y a pr oduct of uni s t och as t ic m a tr ic e s .
P r o of . A s s ume M i s a b i s t och as t ic m a tr i x s uch th a t M
ij
> 0 for al l i; j . The n, M ca n be ex p a nde d in t o
a finit e pr oduct of T -tr a n sfor m [ 8 ], w hich a r e uni s t och as t ic m a tr ic e s . Thi s i s be ca us e T -tr a n sfor m s a ct
non tr iv i al ly only on a 2- d ime n sion al s ubsp a c e a nd al l b i s t och as t ic m a tr ic e s in d = 2 a r e uni s t och as t ic .
The s e t of b i s t och as t ic m a tr ic e s for m s a c on v ex po ly t ope a nd he nc e in a n y ε - nei ghbourhood ( as
92
define d , e . g., b y the l
1
nor m ) of a m a tr i x M th a t fai l s the e le me n t -w i s e posit iv it y c ond it ion, the r e ex i s ts
s ome M
′
th a t f ul fi l l s it. ■
W e a r e no w r e a dy for the pr oof of P r oposit ion 3.2.
P r o of . E q . ( 3.6 ) ho ld s i f a nd only i f the a ct ion of the l h s a nd the rh s on a n y P
( 0)
i
c o inc ide . Thi s i s be ca us e
D
B 0
i s a pr oj e ct or a nd he nc e the a ct ion i s non tr iv i al only o v e r the im a g e Im(D
B 0
) = Span
{
P
( 0)
i
}
i
.
W e h a v e ,
l h s: D
B 2
D
B 0
P
( 0)
i
=
∑
j
X
ji
(B
2
;B
0
) P
( 2)
j
(3.7a)
rh s: U
[
∏
α
D
B
′
α
]
D
B 1
D
B 0
P
( 0)
i
=
∑
f j αg
[
α max 1
∏
α= 1
X
j
α+ 1
j α
(B
′
α+ 1
;B
′
α
)
]
X
j 1 i
(B
1
;B
0
)U
(
P
( α max)
j α max
)
:
(3.7b)
N ot ic e , in a dd it ion, th a t a n a ppr opr i a t e U for the t w o ex pr e s sion s t o be e qual should s a t i sf y B
2
=
U(B
′
α max
) .
L e t us fir s t sho w s u ffic ie nc y . I f E q . ( 3.6) ho ld s, the n the ex pr e s sion s ( 3.7 ) a r e e qual a nd the r efor e one
ca n d ir e ctly s e e th a t E q . ( 3.4) al s o ho ld s for b i s t och as t ic M =
∏
α
A
( α)
, w he r e A
( α)
= X(B
′
α+ 1
;B
′
α
) .
W e no w pr o v e ne c e s sit y . A s s ume B
1
≻
B 0
B
2
, he nc e the r e ex i s ts a b i s t och as t ic M s uch th a t E q . ( 3.4 )
ho ld s . N o w , w ith us e of the L e mm a , w e de c ompos e M =
∏
α
A
( α)
in t o a pr oduct of uni s t och as t ic m a -
tr ic e s . F or al l A
( α)
the r e ex i s t , b y definit ion, unit a r y ope r a t or s U
( α)
s uch th a t A
( α)
ij
= Tr
(
P
( 0)
j
U
( α)
( P
( 0)
i
)
)
for al l i; j , or e quiv ale n tly , A
( α)
= X(U
( α)
(B
0
);B
0
) . N o w w e sho w th a t E q . ( 3.6 ) al s o ho ld s for a s e-
que nc e of de p h asin g s u pe r ope r a t or s fD
B
′
α
g
α max
α= 1
o v e r the b as e s
B
′
1
=W(B
0
) (3.8)
B
′
α
=WU
( 1)
U
( 2)
:::U
( α)
(B
0
) for al l 1 α α
max
; (3.9)
93
w he r eW(B
0
) = B
1
. T o s e e th a t , fir s t not ic e th a t for a n y unit a r y s u pe r ope r a t or V it ho ld s th a t
X(B
α
;B
β
) = X(V(B
α
);V(B
β
)) . A s a r e s ult , w e ca n w r it e
X(B
2
;B
0
) =
[
∏
α
A
( α)
]
X(B
1
;B
0
) =
[
∏
α
X(U
( α)
(B
0
);B
0
)
]
X(B
1
;B
0
)
= ::: X(WU
( 1)
U
( 2)
(B
0
); WU
( 1)
(B
0
)) X(WU
( 1)
(B
0
);W(B
0
)) X(W(B
0
);B
0
)
= X(B
′
α max
;B
′
α max 1
) ::: X(B
′
2
;B
′
1
) X(B
′
1
;B
1
) X(B
1
;B
0
) :
C hoosin g U s uch th a t B
2
=U(B
′
α max
) , the a bo v e e qua t ion imp l ie s th a t the ex pr e s sion s ( 3.7 ) a r e e qual
a nd he nc e E q . ( 3.6) al s o ho ld s for the de s cr i be d s e que nc e of de p h asin g s u p e r ope r a t or s .
■
W e no w a n aly z e the d = 2 cas e , b y in v o k in g P r oposit ion 3.2 t o g e the r the us ual B loch b al l r e pr e s e n t a t ion
of qua n tum s t a t e s ρ =
1
2
( I + v σ) , w he r e d i ffe r e n t b as e s a r e in one t o one c or r e sponde nc e w ith l ine s
p as sin g f r om the c e n t e r . I n thi s r e pr e s e n t a t ion, the a ct ion of D
B 1
on a s t a t e ρ c o inc ide s w ith pr oj e ct in g v
on t o the B
1
l ine w hi le the a ct ion of U i s tr a n sl a t e d in t o a n SO( 3) r ot a t ion . C le a rly , E q . ( 3.6) ca n be s a t i sfie d
( in fa ct , b y me a n s of a sin gle D
B
′
1
) i f a nd only i f θ
1
θ
2
; he r e θ
i
i s the ( a c ut e ) a n gle be t w e e n the l ine s
c or r e spond in g t o B
0
a ndB
i
. I n p a r t ic ul a r , for d = 2 the or de r in g “ ≻
B 0
” i s a t ot al pr e or de r , but not for
d > 2 .
3.8 Me a s u re s of rel a tive ( in )c omp a tib il it y
A pr e or de r g iv e s r i s e t o a d i s t in g ui she d cl as s of s cal a r f unct ion s, i . e ., monot one s . W e a dopt the fo l lo w in g
definit ion .
D e finitio n 3.3
A f unct ion f
B 0
:M(H)!R
+
0
i s m e as u r e of co m p a t i bi l it y ( i n co m p a t i bi l it y ) r e l a t iv e t o B
0
i f it c on v ex
( c onca v e ) w ith r e spe ct t o the pr e or de r “ ≻
B 0
” , i . e .,B
1
≻
B 0
B
2
=) f
B 0
(B
1
) f
B 0
(B
2
)
(
B
1
≻
B 0
94
B
2
=) f
B 0
(B
1
) f
B 0
(B
2
)
)
. M or e o v e r , i f f
B 0
(B
1
) = f
B 1
(B
0
) , w e cal l it a sy m m etr ic me as ur e of
r e l a t iv e c omp a t i b i l it y ( inc omp a t i b i l it y ).
The fo l lo w in g P r oposit ion g iv e s a c on s tr uct ion for me as ur e s of r e l a t iv e c omp a t i b i l it y a r i sin g f r om c on v ex
f unct ion s . I t i s a d ir e ct c on s e que nc e of a r e s ult f r om [ 116 ], de r iv e d in the c on t ex t of m a tr i x m a j or i za t ion .
P r opositio n 3.3
L e t φ :R
d
!R be a c on t in uous c on v ex f unct ion . The n,
f
φ
B 0
(B
1
) :=
∑
i
φ( X
R
i
(B
1
;B
0
)) (3.10)
i s a me as ur e of r e l a t iv e c omp a t i b i l it y; he r e , X
R
i
s t a nd s for the r o w v e ct or s of the m a tr i x X
ij
.
P r o of . L e tB
1
≻
B 0
B
2
. The n, the r e ex i s ts a b i s t och as t ic m a tr i x M s uch th a t X(B
2
;B
0
) = MX(B
1
;B
0
) .
F or a n y c on t in uous c on v ex f unct ion φ :R
d
!R ,
d
∑
i= 1
φ
(
X
R
i
(B
2
;B
0
)
)
=
∑
i
φ
(
∑
k
M
ik
X
R
k
(B
1
;B
0
)
)

∑
i; k
M
ik
φ
(
X
R
k
(B
1
;B
0
)
)
=
∑
i
φ
(
X
R
i
(B
1
;B
0
)
)
:
■
A n alo g ous cl aim s ho ld for the inc omp a t i b i l it y cas e in t e r m s of c onca v e f unct ion s .
I n fa ct , the fa mi ly f f
φ
B 0
(B
1
)g
φ
for al l c on t in uous c on v ex φ i s kno w n t o be a c omp le t e fa mi ly of monot one s
for m a tr i x m a j or i za t ion [ 116 ]; i . e ., j o in t monot onic it y f
φ
B 0
(B
1
) f
φ
B 0
(B
2
) for al l s uch f unct ion s i s e nou gh
t o imp lyB
1
≻
B 0
B
2
. I n th a t s e n s e , the ex i s t e nc e of a pr o b a b i l i s t ic uni for min g pr oc e s s M s uch th a t E q . ( 3.4 )
ho ld s i s f ul ly ca ptur e d b y thi s fa mi ly of f unct ion s .
95
3.9 Inc omp a tib il it y and c oherenc e
A s d i s c us s e d in C h a pt e r 1 , qua n tum c o he r e nc e r efe r s t o the pr ope r t y of qua n tum s ys t e m s t o ex i s t in a
l ine a r s u pe r posit ion of d i ffe r e n t p h ysical s t a t e s . I t i s a not ion define d w ith r e spe ct t o s ome pr efe r r e d ,
p h ysical ly r e le v a n t b asi s, w hich w e w i l l de not e as B
0
. A s t a t e ρ i s s aid t o be c o he r e n t i f the r e ex i s t non-
v a ni shin g off - d i a g on al e le me n ts w he n ρ i s ex pr e s s e d as a m a tr i x in B
0
. R e c e n tly , c o he r e nc e w as for m u-
l a t e d as a r e s our c e the or y [ 28 ]. One of the c e n tr al me as ur e s in the the or y i s r e l a t iv e e n tr op y of c o he r -
e nc e , c
( r e l)
B 0
( ρ) := S( ρ∥D
B 0
ρ) th a t a dmits 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 c on v e r sion r a t e s
[ 33 , 58 ]. L a t e r , w e w i l l al s o in v o k e the 2- c o he r e nc e c
( 2)
B 0
:=
∑
i̸= j

ρ
ij

2
.
A lthou gh qua n tum c o he r e nc e r efe r s t o s t a t e s a nd r e l a t iv e inc omp a t i b i l it y t o me as ur e me n ts, the t w o no -
t ion s a r e clos e ly c onne ct e d . I n fa ct , the or de r in g “ ≻
B 0
” h as r a the r s tr on g imp l ica t ion s in t e r m s of qua n tum
c o he r e nc e , both for s t a t e c on v e r sion unde r inc o he r e n t ope r a t ion s [ 60 ] ( i . e ., e as y ope r a t ion in the r e s our c e
the or y of c o he r e nc e [ 29 ]) a nd c o he r e nc e monot one s . W e define the a ct ion of a unit a r y s u pe r ope r a t or o v e r
a b asi s as U(B) :=fU( P
i
)g
i
.
P r opositio n 3.4
L e tB
1
≻
B 0
B
2
.
( i ) C on side r a p air of unit a r y qua n tum m a ps U;V s uch th a t U(B
1
) = B
0
a ndV(B
2
) = B
0
a nd
a pur e s t a t e P
j
2B
0
. The n, V( P
j
) ca n be tr a n sfor me d t o U( P
j
) v i a inc o he r e n t ope r a t ion s o v e r
B
0
. C on s e que n tly , al l c o he r e nc e me as ur e s o v e r s uch s t a t e s a r e nonincr e asin g.
( i i ) c
B 1
( ρ
0
) c
B 2
( ρ
0
) for al l ρ
0
d i a g on al in B
0
, w he r e c
B
de not e s eithe r the r e l a t iv e e n tr op y of
c o he r e nc e or the 2- c o he r e nc e o v e r B .
The pr oof r e l ie s in the c on v e r t i b i l it y r e s ult of R ef . [ 60 ].
96
P r o of . ( i ) I n [ 60 ] ( s e e al s o [ 33 ]) it w as sho w n th a t j ψ⟩⟨ ψj ca n be tr a n sfor me d t o j φ⟩⟨ φj v i a inc o he r e n t
ope r a t ion s ( in fa ct , s tr ictly inc o he r e n t ope r a t ion s ) w ith r e spe ct t o B
0
i fD
B 0
(j φ⟩⟨ φj)≻D
B 0
(j ψ⟩⟨ ψj) .
F r om the as s umpt ion B
1
≻
B 0
B
2
, w e h a v e th a t X
C
j
(B
1
;B
0
) ≻ X
C
j
(B
2
;B
0
)8 j , w he r e X
C
j
de not e s
the j th c o lumn v e ct or of X . W e ca n r e w r it e
[ X
C
j
(B
1
;B
0
)]
i
= Tr( P
( 1)
i
P
( 0)
j
) = Tr(U
y
( P
( 0)
i
) P
( 0)
j
)
= Tr( P
( 0)
i
U( P
( 0)
j
))
a nd simi l a rly
[ X
C
j
(B
2
;B
0
)]
i
= Tr( P
( 0)
i
V( P
( 0)
j
)) :
N o w , w e ca n w r it e the r e l a t io n X
C
j
(B
1
;B
0
)≻ X
C
j
(B
2
;B
0
)8 j in ope r a t or not a t ion as
D
B 0
(U( P
( 0)
j
))≻D
B 0
(V( P
( 0)
j
)) 8 j
f r om w hich c on v e r t i b i l it y fo l lo w s .
( ii ) L e t us be g in w ith the r e l a t iv e e n tr op y of c o he r e nc e . W e h a v e
c
( r e l)
B 1
= S( ρ
0
∥D
B 1
ρ
0
) = S( ρ
0
) Tr
(
ρ
0
log[D
B 1
( ρ
0
)]
)
= S( ρ
0
) Tr
(
D
B 1
( ρ
0
) log[D
B 1
( ρ
0
)]
)
= S(D
B 1
( ρ
0
)) S( ρ
0
) :
S inc e v on N e um a nn e n tr op y i s a S ch ur - c onca v e f unct ion, the as s umpt ion B
1
≻
B 0
B
2
imp l ie s S(D
B 1
( ρ
0
))
S(D
B 2
( ρ
0
)) f r om w hich the cl aim fo l lo w s .
97
I n the fo l lo w in g , w e us e the ope r a t or 2- nor m ∥ X∥
2
:=
√
Tr( X
y
X) . W e h a v e ,
c
( 2)
B 2
( ρ
0
) =

(ID
B 2
) ρ
0

2
2
=

ρ
0

2
2

D
B 2
ρ
0

2
2
=

ρ
0

2
2

U
(
∏
α
D
B α
)
D
B 1
ρ
0

2
2

ρ
0

2
2

D
B 1
ρ
0

2
2
= c
( 2)
B 1
( ρ
0
) :
The ine qual it y fo l lo w s sinc e the 2- nor m i s s ubm ult ip l ica t iv e o v e r unit al C P TP m a ps . ■
I n a dd it ion t o the in t e r pr e t a t ion of P r oposit ion 3.4 in the f r a me w ork of c o he r e nc e , one ca n al s o infe r
f r om ( i i ) a bo v e th a t a D
B 1
me as ur e me n t d i s tur bs le s s ρ
0
c omp a r e d t o a D
B 2
me as ur e me n t , i f B
1
≻
B 0
B
2
, as
it i s pr e c i s e ly ca ptur e d b y s t a t i s t ical me a nin g of the r e l a t iv e e n tr op y [ 117 ].
I n the l i gh t of the in t e r pr e t a t ion of c
( r e l)
B
as d i s t i l l a b le c o he r e nc e [ 33 ], ( i i ) a bo v e de mon s tr a t e s a qua n-
t it a t iv e tr a de- off be t w e e n c omp a t i b i l it y a nd c o he r e nc e . M or e o v e r , a n y c o he r e nc e a v e r a g e C
B 0
(B) :=
∫
d μ( ρ
0
) c
B
( ρ
0
) i s a me as ur e of inc omp a t i b i l it y of B r e l a t iv e t o B
0
. I n fa ct , the s e a v e r a g e s o v e r the uni -
for m d i s tr i but ion h a v e be e n pe r for me d ( s e e C h a pt e r 1 ), v e r i f y in g ex p l ic itly th a t C
B 0
(B) = f
φ
B 0
(B
1
) i s of
the for m ind ica t e d in ( the c onca v e a n alo g ue of ) P r oposit ion 3.3 . I nde e d , φ c o inc ide s w ith the s ube n tr op y
[ 44 ] for the cas e of the r e l a t iv e e n tr op y of c o he r e nc e [ 40 ], w hi le φ( p
1
;:::; p
d
)/
∑
i
([ 1= d] p
2
i
) for the
2- c o he r e nc e [ 118 ].
F in al ly , w e not e th a t in [ 119 ], the a uthor s c on side r e d a g e ome tr ical ly mot iv a t e d me as ur e of “ m utual un-
b i as e dne s s ” be t w e e n p air s of or thonor m al b as e s . Their me as ur e i s pr opor t ion al t o the 2- c o he r e nc e a v e r a g e
a bo v e a nd he nc e i s al s o a s y mme tr ic me as ur e of r e l a t iv e inc omp a t i b i l it y .
3.10 Inc omp a tib il it y and u nc er t ain t y
W e no w c on side r imp l ica t ion of the pr e or de r “ ≻
B 0
” in t e r m s of unc e r t ain t y a nd fluctua t ion s .
B y its definit ion, the or de r in g B
1
≻
B 0
B
2
as s ur e s th a t the d i s tr i but ion p
B 2
( ρ
0
) i s “ mor e uni for m ” th a n
p
B 1
( ρ
0
) , for a n y s t a t e ρ
0
d i a g on al in B
0
. A n imme d i a t e c on s e que nc e i s th a t al l S ch ur - c onca v e f unct ion s [ 59 ],
w hich for in s t a nc e include α - R é n y i e n tr op ie s f or ( α = 1 c or r e spond s t o the us ual S h a nnon e n tr op y ), s a t i sf y
S
α
( p
B 1
( ρ
0
)) S
α
( p
B 2
( ρ
0
)) . The c on v e r s e s t a t e me n t in t e r m s of m a j or i za t ion, ho w e v e r , doe s not ho ld , i . e .,
98
p
B 1
( ρ
0
)≻ p
B 2
( ρ
0
) for al l d i a g on al ρ
0
i s not e nou gh t o as s ur e B
1
≻
B 0
B
2
. A spe c i fic c oun t e r ex a mp le w as
c on s tr uct e d in the c on t ex t of m ult iv a r i a t e m a j or i za t ion b y H or n in R ef . [ 120 ].
Qua n tum fluctua t ion s o v e r d i ffe r e n t b as e s ca n be qua n t i fie d v i a e n tr op ic unc e r t ain t y r e l a t ion s [ 96 ].
The r e , one tr ie s t o impos e bound s o v e r e n tr op ic qua n t it ie s, s uch as S
α
( p
B 1
( ρ
0
))+ S
β
( p
B 2
( ρ
0
)) r
B 0
(B
2
;B
1
)
( α = β = 1 c or r e spond s t o the us ual S h a nnon e n tr op y ), as a f unct ion of the b as e s . The mos t w e l l -
kno w n ine qual it y i s due t o M a as s e n a nd U ffink [ 92 ] a nd s t a t e s th a t a ( B
0
inde pe nde n t ) cho ic e for the
a bo v e bound i s r
( M U)
(B
2
;B
1
) := log( max
i; j
X
ij
(B
2
;B
1
)) for a n y α; β 1= 2 w ith 1= α + 1= β = 2 . The
bound h as r e c e n tly be e n impr o v e d b y C o le s et a l. [ 94 ] for the cas e of S h a nnon- v on N e um a nn e n tr op y , as
S( p
B 1
( ρ
0
)) + S( p
B 2
( ρ
0
)) S( ρ
0
) + r
( M U)
(B
2
;B
1
) .
L e t us al s o c on side r the qua n t it y
Q
B 0
(B
1
) := sup
A2A
B 1
;∥ A∥
2
= 1
max
i= 1;:::; d
V a r
i
( A) ; (3.11)
w he r e V a r
i
( A) := Tr
(
P
( 0)
i
A
2
)

[
Tr
(
P
( 0)
i
A
)]
2
;
th a t ca ptur e s the s tr e n g th of the fluctua t ion s o f a pur e s t a t e d i a g on al in B
0
o v e r a B
1
me as ur e me n t. W e no w
de r iv e the u ppe r bound
P r opositio n 3.5
I t ho ld s
Q
B 0
(B
1
) 1 λ
min
(
X(B
1
;B
0
) X
T
(B
1
;B
0
)
)
:= q(B
1
;B
0
) ; (3.12)
w he r e λ
min
( X) s t a nd s for the minim um ei g e n v alue of X .
The bound i s s y mme tr ic a nd s a t i sfie s q(B
1
;B
0
) = 0 i f a nd only i f B
1
=B
0
; he nc e it v a ni she s i f a nd
only i f Q
B 0
(B
1
) v a ni she s .
99
P r o of . One h as for A =
∑
k
a
k
P
( 1)
k
,
V a r
i
( A) = Tr
(
P
( 0)
i
A
2
)

[
Tr
(
P
( 0)
i
A
)]
2
=
∑
k
a
2
k
Tr
(
P
( 0)
i
P
( 1)
k
)

∑
k; l
a
k
a
l
Tr
(
P
( 0)
i
P
( 1)
k
)
Tr
(
P
( 0)
i
P
( 1)
l
)
;
he n c e
Q
B 0
(B
1
) sup
A2A
B 1
;∥ A∥
2
= 1
∑
i
V a r
i
( A)
= sup
A2A
B 1
;∥ A∥
2
= 1
(
1

X
T
(B
1
;B
0
) a

2
)
1 λ
min
(
X(B
1
;B
0
) X
T
(B
1
;B
0
)
)
w hich i s the de sir e d bound . ■
I n w or d s, r
( M U)
a nd q pr o v ide bound s on unc e r t ain t y a nd fluctua t ion s th a t a r i s e due t o the inc omp a t i b i l -
it y be t w e e n the b as e s of me as ur e me n t ( for r
( M U)
) or s t a t e pr e p a r a t ion a nd me as ur e me n t ( for q ), a nd ca n
be thou gh t of as p l a y in g a r o le a n alo g ous t o the c omm ut a t or t e r m in the us ual unc e r t ain t y r e l a t ion s for o b -
s e r v a b le s . A s s uch, they both tur n out t o be ( s y mme tr ic ) me as ur e s of r e l a t iv e inc omp a t i b i l it y , monot onic
r e l a t iv e t o the or de r in g “ ≻
B 0
” .
P r opositio n 3.6
L e tB
1
≻
B 0
B
2
. The n, q(B
1
;B
0
) q(B
2
;B
0
) a nd r
( M U)
(B
1
;B
0
) r
( M U)
(B
2
;B
0
) .
P r o of . W e be g in w ith the fir s t ine qual it y . I f B
1
≻
B 0
B
2
, the n the r e ex i s ts a b i s t och as t ic m a tr i x M s uch
th a t X(B
2
;B
0
) = MX(B
1
;B
0
) . W e ne e d t o sho w th a t thi s imp l ie s λ
min
( X(B
1
;B
0
) X
T
(B
1
;B
0
)) :=
s
2
d
( X(B
1
;B
0
)) ( s
d
de not e s the minim um sin g ul a r v alue ) s a t i sfie s s
d
( X(B
1
;B
0
)) s
d
( X(B
2
;B
0
)) .
100
I nde e d , thi s i s g ua r a n t e e d b y the Ge l ’ fa nd - N aim a rk ine qual it y w hich s t a t e s th a t ( for the sin g ul a r v al -
ue s s or t e d in de cr e asin g or de r )
∏
k
j= 1
s
i j
( AB)
∏
k
j= 1
s
j
( A)
∏
k
j= 1
s
i j
( B) for al l 1 i
1
::: i
k
n
a nd k = 1;:::; n ( in our cas e w e s e t k = 1 a nd i
1
= n ) [ 8 ]. N ot ic e th a t s
1
( M) = 1 sinc e M i s b i s -
t och as t ic .
F or the s e c ond one , sinc e X(B
2
;B
0
) = MX(B
1
;B
0
) for b i s t och as t ic M , w e h a v e th a t max
i; j
X
ij
(B
2
;B
0
)
max
i; j
X
ij
(B
1
;B
0
) . The r e s ult fo l lo w s f r om the monot onic it y of the log f unct ion . S y mme tr y fo l lo w s
f r om X(B
2
;B
1
) = X
T
(B
1
;B
2
) . ■
3.11 Gener al ized me a s u re men ts
The or de r in g “ ≻
B 0
” ca n be d ir e ctly ex t e nde d t o include g e ne r al i z e d me as ur e me n ts de s cr i be d b y PO VM s .
C on side r a s t a t e ρ
0
=
∑
i
p
i
P
( 0)
i
2 A
B 0
a nd a me as ur e me n t F = f F
i
g
i
. The pr o b a b i l it y d i s tr i bu-
t ion of pos si b le out c ome s i s p
F
( ρ
0
) = X(F;B
0
) p , w he r e no w [ X(F;B
0
)]
ij
:= Tr( F
i
P
( 0)
j
) i s jus t c o l -
umn s t och as t ic ⁷ . The a n alo g ous or de r in g o v e r PO VM s F a ndG r e l a t iv e t o a b asi s B
0
ca n be define d as
F≻
B 0
G i f a nd only i f the r e ex i s ts a b i s t och as t ic M s uch th a t X(G;B
0
) = MX(F;B
0
) . I n fa ct , the fa mi ly
f f
φ
B 0
(F) :=
∑
d
i= 1
φ( X
R
i
(F;B
0
))g
φ
for al l c on t in uous c on v ex φ s t i l l for m s a c omp le t e fa mi ly of monot one s
for th e or de r in g “ ≻
B 0
” , no w c on side r e d o v e r P O VM s .
H o w e v e r , in c on tr as t w ith the or tho g on al me as ur e me n t cas e , no w it doe s not ho ld th a t p
B 0
( ρ
0
) ≻
p
F
( ρ
0
) for al l F , n a me ly g e ne r al i z e d me as ur e me n ts ca n “ pur i f y ” the init i al pr o b a b i l it y d i s tr i but ion ( as, for
ex a mp le , w ith F =f I; 0;:::; 0g ). F or thi s r e as on, w e c on side r as the a ppr opr i a t e me a nin g f ul g e ne r al i za -
t ion of “ inc omp a t i b i l it y r e l a t iv e t o a b asi s ” t o PO VM s the le s s r e s tr ainin g or de r in g th a t oc c ur s b y r e l a x in g
the c on s tr ain t of b i s t och as t ic it y on the m a tr i x M , a nd in s t e a d r e quir in g only c o lumn s t och as t ic it y . I n thi s
cas e , i f F l ie s “ hi ghe r ” in the or de r in g th a n G , the n p
G
( ρ
0
) ca n be o bt aine d b y pr o b a b i l i s t ic pos t pr oc e s sin g
( not ne c e s s a r i ly a uni for min g one ) f r om p
F
( ρ
0
) , inde pe nde n tly of ρ
0
2A
B 0
.
⁷ The PO VM s a r e al lo w e d h a v e a r b itr a r y n umbe r of e le me n ts . I f thi s n umbe r i s d i ffe r e n t , it i s unde r s t ood th a t the PO VM
w ith the le as t n umbe r of e le me n ts i s p a dde d w ith z e r os un t i l the ca r d in al it y of the s e ts be c ome s e qual , s o th a t the ( r e ct a n g ul a r )
m a tr ic e s X(F;B
0
) a nd X(G;B
0
) h a v e e qual d ime n sion s .
101
D e finitio n 3.4
W e de not e F≻≻
B 0
G i f a nd only i f the r e ex i s ts a s t och as t ic m a tr i x M s uch th a t X(G;B
0
) = MX(F;B
0
) .
The or de r in g i s a pr e or de r a nd cle a rly F≻
B 0
G =)F≻≻
B 0
G . A s s uch, the c or r e spond in g monot one s
for “≻≻
B 0
” a r e r e l a t e d t o E q . ( 3.10 ). The fo l lo w in g i s a d ir e ct imp l ica t ion o f a r e s ult b y A l be r t i et a l. [ 121 ]
( s e e al s o [ 122 ]).
P r opositio n 3.7
L e t ψ :R
d
!R be a f unct ion th a t i s sim ult a ne ously c on v ex a nd homo g e ne ous in al l its a r g ume n ts .
The n,
g
ψ
B 0
(F) :=
∑
i
ψ( X
R
i
(F;B
0
)) (3.13)
i s a monot one o v e r “ ≻≻
B 0
” , i . e .,F≻≻
B 0
G =) g
ψ
B 0
(F) g
ψ
B 0
(G) ; he r e , X
R
i
s t a nd s for the r o w
v e ct or s of the m a tr i x X
ij
. M or e o v e r , the fa mi ly f g
ψ
B 0
(F)g
ψ
for m s a c omp le t e s e t of monot one s for
“≻≻
B 0
” .
W e no w fo r m ul a t e a n a n alo g ue of 3.2 for the or de r in g “ ≻≻
B 0
” . W e fir s t define , g iv e nB
0
= f P
( 0)
i
g
d
i= 1
,
the fo l lo w in g fa mi ly of qua n tum ch a nne l s (w ith input a nd out put s t a t e s o v e r the H i l be r t sp a c e H

=C
d
):
E
H
( ρ) :=
∑
i
Tr( H
i
ρ) P
( 0)
i
; w he r eH =f H
i
g
d
i= 1
i s a PO VM . (3.14)
The a bo v e ch a nne l s ca n be in t e r pr e t e d as c on si s t in g of a me as ur e me n t H a nd a s t a t e pr e p a r a t ion (w hich
i s a n e le me n t of B
0
), c ond it ione d on the me as ur e me n t out c ome . N ot ic e th a t for the ch a nne l E
H
( ρ) t o
be w e l l - define d , the e le me n ts of the s e ts B
0
a ndH a r e c on side r e d indexe d . The fo l lo w in g P r oposit ion,
ne v e r the le s s, ho ld s inde pe nde n tly of the p a r t ic ul a r index in g of the b asi s a nd the PO VM s in v o lv e d .
102
P r opositio n 3.8
L e tF =f F
i
g
d
i= 1
a ndG =f G
i
g
d
i= 1
be PO VM s . The n, F≻≻
B 0
G i f a nd only i f the r e ex i s ts a PO VM
H =f H
i
g
d
i= 1
s uch th a t
E
G
D
B 0
=E
H
E
F
D
B 0
(3.15)
P r o of . W e fir s t r e w r it e E q . ( 3.15 ) in the e quiv ale n t for m
E
G
D
B 0
=E
H
E
F
D
B 0
() E
G
(
P
( 0)
i
)
=E
H
E
F
(
P
( 0)
i
)
8 i : (3.16)
L e t us pr o v e s u ffic ie nc y . A s s ume the r e ex i s ts a n H s uch th a t E q . ( 3.15 ) ho ld s . The n, f r om the definit ion
( 3.14 ) one g e ts
E
G
(
P
( 0)
i
)
=
∑
k
[ X(G;B
0
)]
ki
(
P
( 0)
k
)
E
H
E
F
(
P
( 0)
i
)
=E
H
0
@
∑
j
[ X(F;B
0
)]
ji
P
( 0)
j
1
A
=
∑
j; k
[ X(H;B
0
)]
kj
[ X(F;B
0
)]
ji
P
( 0)
k
:
A s a r e s ult , one h as
[ X(G;B
0
)]
ki
=
∑
j
[ X(H;B
0
)]
kj
[ X(F;B
0
)]
ji
8 i; k ;
the r efor e F≻≻
B 0
G w ith M th a t ca n be chos e n as M
ij
= [ X(H;B
0
)]
ij
. The s t och as t ic it y fo l lo w s f r om
the fa ct th a t H i s a PO VM .
W e no w tur n t o ne c e s sit y . A s s ume F≻≻
B 0
G , or e quiv ale n tly , th a t the r e ex i s ts a s t och as t ic M s uch
103
th a t
X(G;B
0
) = MX(F;B
0
) :
W e w i l l sho w th a t E q . ( 3.16 ) ho ld s for H =f H
i
g
d
i= 1
, s uch th a t al l ope r a t or s H
i
a r e d i a g on al in the B
0
b a si s, w ith their d i a g on al e le me n ts g iv e n b y
Tr
(
H
i
P
( 0)
j
)
= M
ij
:
N ot ic e th a t the spe c i fie d H i s a v al id PO VM sinc e M i s a s t och as t ic m a tr i x . B y as s umpt ion, w e h a v e in
c ompone n t for m
[ X(G;B
0
)]
ki
=
∑
j
M
kj
[ X(F;B
0
)]
ji
8 i; k
he nc e al s o
∑
k
[ X(G;B
0
)]
ki
P
( 0)
k
=
∑
j; k
M
kj
[ X(F;B
0
)]
ji
P
( 0)
k
8 i :
The a bo v e ca n be r e w r i tt e n as
LHS :
∑
k
[ X(G;B
0
)]
ki
P
( 0)
k
=E
G
(
P
( 0)
i
)
rh s:
∑
j; k
[ X(H;B
0
)]
kj
[ X(F;B
0
)]
ji
P
( 0)
k
=E
H
0
@
∑
j
[ X(F;B
0
)]
ji
P
( 0)
j
1
A
=E
H
E
F
(
P
( 0)
i
)
;
w hich i s the de sir e d r e s ult.
■
104
3.12 B a sis - independen t inc omp a tib il it y
F in al ly , w e c onne ct the or de r in gs de s cr i b in g me as ur e me n t inc omp a t i b i l it y r e l a t iv e t o a b asi s w ith the no -
t ion of a p a r en t m e as u r em en t [ 112 , 123 ]. I n thi s c on t ex t , F i s cal le d a p a r e n t of G i f the r e ex i s ts a s t och as t ic M
s uch th a t G
i
=
∑
j
M
ij
F
j
8 i , w hi le a fa mi ly of me as ur e me n ts a r e j o in tly me as ur a b le i f they a dmit a c ommon
p a r e n t.
P r opositio n 3.9
F i s a p a r e n t of G i f a nd only i f F≻≻
B 0
G for al l B
0
2M(H) a nd the pos t pr oc e s sin g m a tr i x M ca n
be chos e n t o be the s a me for al l B
0
.
P r o of . W e fir s t r e w r it e th e c ond it ion for F≻≻
B 0
G in the fo l lo w i n g e quiv ale n t for m .
X(G;B
0
) = MX(F;B
0
) () (3.17a)
Tr
(
G
i
P
( 0)
j
)
= Tr
(
∑
k
M
ik
F
k
P
( 0)
j
)
8 i; j () (3.17b)
D
B 0
( G
i
) =D
B 0
(
∑
k
M
ik
F
k
)
8 i : (3.17c)
I fF i s a p a r e n t of G the n E q . ( 3.17c ) ho ld s for al l B
0
w ith M th a t i s inde pe nde n t of B
0
.
F or the c on v e r s e , le t D
B 0
( G
i
) = D
B 0
(
∑
k
M
ik
F
k
)8 i a nd8B
0
2 M(H) w ith M th a t i s inde pe n-
de n t of B
0
. S inc e for a n y t w o b as e s the r e i s alw a ys a unit a r y s u pe r ope r a t or c onne ct in g the m, the B
0
f r e e dom a moun ts t o in s e r t in g a n a r b itr a r y unit a r y in E q .( 3.17b ) as
Tr
[
G
i
U
(
P
( 0)
j
)]
= Tr
[
∑
k
M
ik
F
k
U
(
P
( 0)
j
)]
8 i; j; U :
N o w w e sho w th a t the a bo v e imp l ie s G
i
=
∑
k
M
ik
F
k
. N ot ic e th a t both G
i
a nd
∑
k
M
ik
F
k
a r e non-
105
ne g a t iv e ope r a t or s, he nc e al s o H e r mit i a n . The a bo v e e qua t ion for c e s the ( H e r mit i a n ) G
i
a nd
∑
k
M
ik
F
k
t o h a v e the s a me ex pe ct a t ion v alue o v e r al l pur e s t a t e s . Thi s ca n only h a ppe n i f the the t w o ope r a t or s
a r e e qual , i . e ., G
i
=
∑
k
M
ik
F
k
8 i . The l as t cl aim fo l lo w s f r om the fa ct th a t , i f ⟨ ψj Aj ψ⟩ = ⟨ ψj Bj ψ⟩
for H e r mit i a n A; B a nd8j ψ⟩ , the n the H e r mit i a n ope r a t or A B h as al l its ei g e n v alue s e qual t o z e r o ,
he n c e ne c e s s a r i ly A = B .
■
3.13 Conc l u sion and ou tlo ok
Qua n tum r e s our c e the or ie s s e e m t o s u gg e s t th a t a n a ppr opr i a t e qua n t i fica t ion of qua n tum pr ope r t ie s, e v e n
c onc e ptual ly simp le one s s uch as the “ uni for mit y ” of a s t a t e [ 18 ], ca nnot be a chie v e d b y me a n s of a sin gle
s cal a r qua n t i fie r . I n s t e a d , only a n infinit e s e t of f unct ion s i s a b le t o ca ptur e s uch pr ope r t ie s in their w ho le-
ne s s, as they n a tur al ly r e s ult out of pr e or de r s . I n thi s ch a pt e r , w e define d ope r a t ion al ly mot iv a t e d pr e or de r s
o v e r qua n tum me as ur e me n ts th a t ca ptur e a not ion of inc omp a t i b i l it y r e l a t iv e t o a b asi s . Our a ppr o a ch un-
c o v e r s a qua n t it a t iv e , as w e l l as c onc e ptual , c onne ct ion be t w e e n inc omp a t i b i l it y , unc e r t ain t y a nd qua n tum
c o he r e nc e uni fie d unde r the pr i s m of m ult iv a r i a t e m a j or i za t ion .
106
4
S y mmetr ies a nd monot ones in M a rk o v i a n qua n tum
dy n a mic s
4.1 S ym me trie s and m o not one s for the kindergar ten
L e t ’ s ima gi ne a c u b e sl id in g do w n a n incl ine d p l a ne (w a r nin g : thi s mi gh t br in g b a ck me mor ie s f r om
the hi gh s choo l d a ys ). P h ysic i s ts l i k e t o t al k a lot a bout thi s qua n t it y cal le d e ne r g y , a nd e mp h asi z e th a t i s
c on s e r v e d . I n thi s ex a mp le the po in t i s th a t , in the a bs e nc e of a n y f r ict ion, althou gh the s t a t e of the c ube
w i l l ch a n g e in t ime ( it ’ s v e loc it y v a nd hei gh t h f r om a le v e le d s ur fa c e a r e c on s t a n tly e v o lv in g ), one ca n
107
c on s tr uct a f unct ion of the s e v a r i a b le s th a t i s c on s t a n t in t ime . I n our simp le ex a mp le , the e ne r g y f unct ion
ca n be t ak e n t o be
E( h; t) = mgh +
1
2
mv
2
= c on s t: (4.1)
W h a t i f the r e i s f r ict ion, i . e ., the s ys t e m i s d issip a t i v e ? The n e ne r g y e s ca pe s, s o E( h; t) i s not c on s e r v e d
in t ime , but the r e i s s ome thin g w e ca n s t i l l cl aim a bout it : it i s monot onical ly non-incr e asin g. The r o le of
e ne r g y mi gh t s e e m m uch mor e r e s tr ict e d no w ( but le t ’ s think t w ic e befor e w e h as t i ly de cl a r e it us e le s s ).
I n fa ct , it ca n s t i l l be us e d t o dr a w c onclusion s a bout the mot ion, as it c a n r u le o u t co nfi g u r a t io n s t h a t a r e
i m p o ssi b le t o h a pp en . F or ex a mp le , the mot ion ca nnot spon t a ne ously t ak e the c ube t o hei gh t mor e th a t
h
max
= E=( mg) . W e w i l l cal l m o n o t o n e s a n y s uch qua n t it ie s th a t a r e non-incr e asin g unde r the e v o lut ion .
C on s e r v e d qua n t it ie s h a v e a v e r y e le g a n t c onne ct ion w ith s y mme tr ie s of the p h ysical s ys t e m, or i g in al ly
due t o E mm y N oe the r . F or in s t a nc e , c on s e r v a t ion of e ne r g y i s a c on s e que nc e of the fa ct th a t the p h ysical
l a w s a r e , in fa ct , l a w s , n a me ly they a r e not ch a n g in g f r om one mome n t t o a nothe r . Thi s be c ome s a pp a r e n t
i f one r e side s t o the L a gr a n g i a n for m ul a t ion, a nd e v e n mor e o b v ious in the H a mi lt oni a n one .
L e t t al k qua n tum no w . The H a mi lt oni a n for m ul a t ion i s al s o a pp l ica b le for qua n tum me ch a nic s (w ith
the a ppr opr i a t e a d a pt ion s of c our s e ), a nd he nc e it should be no s ur pr i s e th a t the N oe the r c or r e sponde nc e
ho ld s, thou gh w ith a sl i gh tly d i ffe r e n t fl a v or . I n qua n tum p h ysic s, i f a me as ur e me n t i s a s y mme tr y of the
dy n a mic s, the n its pr o b a b i l it y d i s tr i but ion ( as s oc i a t e d w ith the pos si b le out c ome s ) doe s not ch a n g e in
t i me .
W h a t a bout c on s e r v e d qua n t it ie s in d i s sip a t iv e qua n tum s ys t e m s? L e t us c on side r the br o a d cl as s of
M a r k o v i a n d y n a m ic s , i . e ., thos e t ime e v o lut ion s th a t g iv e n the pr e s e n t , the p as t i s ir r e le v a n t t o pr e d ict the
f utur e . I n th a t t y pe of qua n tum pr oc e s s e s, c on s e r v e d qua n t it ie s h a v e t y p ical ly a v e r y l imit e d r o le . F or
in s t a nc e , in the cas e of M a rk o v i a n dy n a mic s de s cr i b in g the pr oc e s s of r e a chin g a ( unique ) the r m al e qui -
l i br ium s t a t e , al l c on s e r v e d qua n t it ie s a r e tr iv i al ( i . e ., they ca n a t be s t t e l l y ou th a t pr o b a b i l it y i s c on s e r v e d ,
but the r e i s nothin g infor m a t iv e a bout th a t ). I n tuit iv e ly , thi s i s t o be ex pe ct e d : sinc e al l init i al s t a t e s r e a ch
the one a nd only e qui l i br ium s t a t e , they h a v e t o for g e t al l the infor m a t ion th a t d i s t in g ui she s the m!
108
T ime for the l as t ( a nd mos t impor t a n t for thi s C h a pt e r ) que s t ion : W h a t a bout monot one s in M a rk o v i a n
dy n a mic s of qua n tum s ys t e m s? A s d i s c us s e d , s uch f unct ion s of the qua n tum s t a t e ca n be us e d t o exclude
s t a t e s th a t a r e impos si b le t o r e a ch b y the t ime e v o lut ion ( l i k e in the ex a mp le of the bo x sl id in g unde r f r ic -
t ion ).
I n fa ct , it tur n s out one ca n in v o k e s y mme tr ie s of the dy n a mic s à l a N o et h er t o al s o c on s tr uct monot one s
( a nd not jus t c on s e r v e d qua n t it ie s ). H e r e i s w h a t w e fi g ur e d out for M a rk o v i a n dy n a mic s:
Th er e is a f a m i l y of m o n o t o n e s co r r e s p o n d i n g t o a n y p a i r of sy m m etr ie s of t h e ev o l u t io n. Th e m o n o t o n e s a r e m o r e
gen er a l t h a n co n ser v e d q u a n t it ie s; o n e c a n i nf er fr o m t h e l a tt er a ll co n ser v e d q u a n t it ie s . I n t h e c ase s w h er e a ll
co n ser v e d q u a n t it ie s a r e tr i v i a l , m o n o t o n e s s t i ll pr o v i de usef u l co n s tr a i n ts f o r t h e ev o l u t io n. F i n a ll y , e xa c t l y as
t h e H a m i l t o n i a n c a n a l w a y s b e co n si der e d a sy m m etr y of itse lf y ie l d i n g en er g y co n ser v a t io n , t h e gen er a t o r of t h e
d y n a m ic s is a lso a sy m m etr y of itse lf, y ie l d i n g a n o n -tr i v i a l m o n o t o n e of t h e ev o l u t io n.
4.2 A bs tr a c t
W h a t ca n one infe r a bout the dy n a mical e v o lut ion of qua n tum s ys t e m s jus t b y s y mme tr y c on side r a t ion s?
F or M a rk o v i a n dy n a mic s in finit e d ime n sion s, w e pr e s e n t a simp le c on s tr uct ion th a t as si gn s t o e a ch s y m-
me tr y of the g e ne r a t or a fa mi ly of s cal a r f unct ion s o v e r qua n tum s t a t e s th a t a r e monot onic unde r the t ime
e v o lut ion . The a for e me n t ione d monot one s ca n be ut i l i z e d t o ide n t i f y s t a t e s th a t a r e non- r e a ch a b le f r om
a n init i al s t a t e b y the t ime e v o lut ion a nd include al l c on s tr ain ts impos e d b y c on s e r v e d qua n t it ie s, pr o v id -
in g a g e ne r al i za t ion of N oe the r ’ s the or e m for thi s cl as s of dy n a mic s . A s a spe c i al cas e , the g e ne r a t or its e l f
ca n be c on side r e d a s y mme tr y , r e s ult in g in non-tr iv i al c on s tr ain ts o v e r the t ime e v o lut ion, e v e n i f al l c on-
s e r v e d qua n t it ie s tr iv i al i z e . The c on s tr uct ion ut i l i z e s t oo l s f r om qua n tum infor m a t ion- g e ome tr y , m ainly
the the or y of monot one R ie m a nni a n me tr ic s . W e a n aly z e the pr ot ot y p ical cas e s of de p h asin g a nd D a v ie s
g e ne r a t or s .
T ex t i s a d a pt e d f r om [ 124 ].
109
4.3 In troduc tion
One of the m ain t ask s in the s tudy of non- r e l a t iv i s t ic qua n tum dy n a mical s ys t e m s i s pr e d ict in g ho w qua n-
tum s t a t e s a nd o bs e r v a b le s e v o lv e o v e r t ime g iv e n s ome dy n a mical l a w , for in s t a nc e , a H a mi lt oni a n ope r -
a t or a nd the as s oc i a t e d e qua t ion s of mot ion . I t i s oft e n the cas e , ho w e v e r , th a t in pr a ct ic e the tr a j e ct or y
eithe r doe s not a dmit a n ex p l ic it clos e d for m, or e v e n i f it doe s, it ca n be c omp l ica t e d t o dr a w c onclusion s
t o p h ysical que s t ion s of in t e r e s t f r om it. I t i s the r efor e impor t a n t t o h a v e a c c e s si b le me thod s a nd t oo l s th a t
al lo w one t o ex tr a ct the e s s e n t i al fe a tur e s of the e v o lut ion d ir e ctly f r om the dy n a mical l a w s .
A t the he a r t of s uch a ppr o a che s l ie s the fa r - r e a chin g ide a of s y mme tr y . The mos t pr omine n t ex a mp le i s
pe rh a ps pr o v ide d b y N oe the r ’ s the or e m w hich, in the c on t ex t of L a gr a n g i a n me ch a nic s, y ie ld s a c on s e r v e d
qua n t it y for e a ch d i ffe r e n t i a b le s y mme tr y of the g e ne r a t or [ 5 ]. I n the qua n tum r e alm, the the or e m ind ica t e s
th a t al l mome n ts of a n o bs e r v a b le th a t i s a s y mme tr y of the t ime e v o lut ion a r e c on s e r v e d [ 125 ].
I n thi s C h a pt e r w e c on side r ope n qua n tum s ys t e m s in finit e d ime n sion s e v o lv in g unde r M a rk o v i a n dy -
n a mic s [ 6 ]. W e a ddr e s s the que s t ion : G i v en t h e gen er a t o r of t h e d y n a m ic s a n d a n i n it i a l s t a t e, w h a t co n s tr a i n ts
c a n sy m m etr y co n si der a t io n s i m p o se o n t h e set of s t a t e s t h a t a r e r e a c h a b le u n der t h e t i m e ev o l u t io n ?
O pe n qua n tum dy n a mic s i s, in g e ne r al , d i s sip a t iv e . A s s uch, the r o le of c on s e r v e d qua n t it ie s ca n be
r a the r l imit e d . F or in s t a nc e , M a rk o v i a n dy n a mic s w ith a unique s t e a dy s t a t e doe s not a dmit a n y non-tr iv i al
c on s e r v e d qua n t it ie s [ 126 ]. F or thi s r e as on, w e a ppr o a ch the pr o b le m b y in s t e a d s e e k in g t o ut i l i z e s y m-
me tr ie s t o o bt ain monot one s, i . e ., f unct ion s of the qua n tum s t a t e th a t a r e monot onic ( in our cas e , non-
incr e asin g ) unde r the t ime e v o lut ion . S imi l a rly t o c on s e r v e d qua n t it ie s, monot one s ca n be ut i l i z e d t o ex -
clude s t a t e tr a n sit ion s th a t a r e impos si b le unde r the dy n a mic s, i . e ., s t a t e s th a t do not be lon g in the tr a j e ct or y
of a g iv e n init i al s t a t e .
A n a c c e s si b le d i s c us sion r e g a r d in g the c onne ct ion be t w e e n s y mme tr ie s a nd c on s e r v e d qua n t it ie s for
M a rk o v i a n dy n a mic s ca n be found in the p a pe r s b y B a um g a r tne r a nd N a r nhofe r [ 7 ], a nd b y A l be r t a nd
J i a n g [ 126 ]. S y mme tr ie s h a v e al s o be e n d i s c us s e d for the clos e ly r e l a t e d cas e of it e r a t e d qua n tum ch a n-
ne l s [ 127 ]. F r om the v ie w po in t of qua n tum r e s our c e the or ie s [ 26 ], c on s e que nc e s of s y mme tr ie s in qua n-
tum dy n a mic s h a v e be e n s ys t e m a t ical ly c on side r e d in the the or y of r efe r e nc e f r a me s a nd as y mme tr y [ 128 ,
110
129 ]. The r e , one in v e s t i g a t e s the al lo w e d s t a t e tr a n sit ion s unde r qua n tum dy n a mic s th a t r e spe cts a spe c -
i fie d s y mme tr y ¹ . I n p a r t ic ul a r , M a r v i a n a nd S pe k k e n s e s t a b l i she d the fa ct th a t , for the cas e of H a mi lt o -
ni a n dy n a mic s, as y mme tr y monot one s y ie ld c on s e r v e d qua n t it ie s th a t ca n be inde pe nde n t of the N oe the r
one s [ 130 ]. A s y mme tr y monot one s h a v e al s o be e n ut i l i z e d t o put c on s tr ain ts on the e v o lut ion of qua n-
tum c o he r e nc e s b y L os t a gl io e t al . [ 131 ]. S y mme tr y c on side r a t ion s h a v e mor e o v e r be e n in v o k e d t o s tudy
tr a n spor t in a M a rk o v i a n mode l of ope n sp in ch ain s b y B uča a nd P r os e n [ 132 ].
M onot one s g e ne r al i z e the c onc e pt of a c on s e r v e d qua n t it y a nd , as w e sho w in our c on s tr uct ion, in fa ct
one ca n de duc e f r om the m the f ul l s e t of c on s e r v e d qua n t it ie s . Our m ain r e s ult c on si s ts of a me thod t o
as si gn t o e a ch p air of s y mme tr ie s of the g e ne r a t or a one-p a r a me t e r fa mi ly of monot one s for the t ime e v o -
lut ion . A s a spe c i al cas e , the g e ne r a t or its e l f ca n be c on side r e d a s y mme tr y of the dy n a mic s, r e s ult in g in
non-tr iv i al c on s tr ain ts o v e r the t ime e v o lut ion, e v e n i f al l c on s e r v e d qua n t it ie s tr iv i al i z e .
The b asic ide a w e in v o k e t o o bt ain monot one s of the t ime e v o lut ion r e l ie s on the “ infinit e sim al v e r -
sion ” of qua n tum d a t a -pr oc e s sin g ine qual it ie s . Di s t in g ui sh a b i l it y me as ur e s D( ρ; σ) 0 , define d o v e r
p air s of qua n tum s t a t e s s uch th a t D( ρ; ρ) = 0 , a r e s aid t o o bey the d a t a -pr oc e s sin g ine qual it y i f they a r e
non-incr e asin g unde r the j o in t a ct ion of a qua n tum ope r a t ion on both a r g ume n ts a nd p l a y a c e n tr al r o le in
qua n tum infor m a t ion the or y [ 117 , 133 ]. S inc e M a rk o v i a n dy n a mic s i s f ul ly ch a r a ct e r i z e d b y its g e ne r a t or ,
w e in v o k e a n infinit e sim al v e r sion of d i s t in g ui sh a b i l it y me as ur e s, c onne ct in g w ith monot one R ie m a nni a n
me tr ic s in the sp a c e of qua n tum s t a t e s . S uch me tr ic s h a v e pr o v e n us ef ul for the s tudy of M a rk o v i a n qua n-
tum dy n a mic s, as for in s t a nc e for the s tudy of the mi x in g t ime a nd c on v e r g e nc e r a t e s [ 134 – 136 ].
¹ M or e spe c i fical ly , g iv e n a gr ou p G , one in v e s t i g a t e s the pos si b i l it y of s t a t e tr a n sit ion s unde r the cl as s of qua n tum ope r a t ion s
th a t a r e s y mme tr ic w ith r e spe ct t o a unit a r y r e pr e s e n t a t ion of G , n a me ly the ope r a t ion s s uch th a t
[
E;U
g
]
= 08 g2 G .
111
4.4 Se tting the s t a g e
W e c on side r qua n tum s ys t e m s de s cr i be d b y a s t a t e ρ2S(H)B(H) , w he r e the H i l be r t sp a c e H

=C
d
i s finit e d ime n sion al ² . W e as s ume th a t the dy n a mic s i s M a rk o v i a n a nd t ime-homo g e ne ous, i . e .,
d
dt
ρ
t
=L( ρ
t
) (4.2)
w he r e the g e ne r a t or of the dy n a mic s L , al s o kno w n as the L ind b l a d i a n, i s a t ime inde pe nde n t s u pe r ope r a t or
a nd he nc e ca n be ex pr e s s e d in the s t a nd a r d for m
L( X) = i[ H; X] +
∑
i
(
L
i
XL
y
i

1
2
{
L
y
i
L
i
; X
})
(4.3)
( s e e , e . g., R ef . [ 6 ] for mor e de t ai l s ). E qua t ion ( 4.2 ) 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
E
t
= exp( tL) ; t 0 ; (4.4)
be lon g in g in the sp a c e of 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, al s o kno w n as qua n tum
ch a nne l s . W e define as s y mme tr ie s of a L ind b l a d i a n thos e s u pe r ope r a t or s th a t c omm ut e w ith L , i . e .,
Sym(L) :=fM2B(B(H))j [M;L] = 0g : (4.5)
W e fir s t infor m al ly pr e s e n t the b asic ide a of ho w one ca n o bt ain monot one s of the e v o lut ion f r om s y m-
me tr ie s of the g e ne r a t or . L e t us c on side r a d i s t in g ui sh a b i l it y me as ur e , i . e ., a f unct ion D( ρ; σ) o v e r p air s of
s t a t e s w i th
D( ρ; σ) 0 a nd D( ρ; ρ) = 0; (4.6a)
² W e us eB(H) t o de not e the sp a c e of ( bounde d ) l ine a r ope r a t or s . S(H) de not e s the sp a c e of non- ne g a t iv e l ine a r ope r a t or s
w it h unit tr a c e .
112
s uch th a t it r e spe cts the d a t a -pr oc e s sin g ine qual it y
D( ρ; σ) D(E( ρ);E( σ)) (4.6b)
for al l s t a t e s ρ , σ a nd qua n tum ch a nne l s E . L e t us al s o define
f
s
( ρ) = D( ρ; e
sM
ρ) ; w ithM2 Sym(L) ; (4.7)
w hich i s a me as ur e of the d i ffe r e nc e be t w e e n a s t a t e ρ a nd its v a r i a t ion ³ g e ne r a t e d b y the s y mme tr y M .
The d a t a -pr oc e s sin g ine qual it y , t o g e the r w ith the fa ct th a t M i s a s y mme tr y of the t ime e v o lut ion, im-
p l ie s th a t
f
s
( ρ) f
s
( e
tL
ρ) ; 8 t 0 ; (4.8)
i . e ., th a t f
s
i s a monot one of the dy n a mic s . H o w e v e r , t o calc ul a t e f
s
for a finit e v alue of the p a r a me t e r s one
ne e d s t o fir s t calc ul a t e the a ct ion of the ex pone n t i al exp( sM) on ρ , w hich in g e ne r al i s impr a ct ical . I n s t e a d ,
i f the f unct ion f
s
i s s uit a b ly w e l l - be h a v e d , one ca n ex a mine its ex p a n sion
f
s
( ρ) = s
k
h
M
( ρ) + O( s
k+ 1
); k2N ; (4.9)
w he r e the for m of h
M
de pe nd s on D a ndM . S inc e the ine qual it y ( 4.8 ) i s v al id for al l s 0 , no m a tt e r
ho w s m al l , it fo l lo w s th a t al s o h
M
( ρ) i s a monot one of the e v o lut ion
h
M
( ρ) h
M
(E
t
( ρ)) ; t 0 : (4.10)
Befor e pr oc e e d in g t o for m al i z e the pr e v ious o bs e r v a t ion v i a b asic t oo l s f r om qua n tum infor m a t ion-
g e ome tr y , le t us c on side r a n ex a mp le . I f the d i s t in g ui sh a b i l it y me as ur e i s t ak e n t o be the r e l a t iv e α - R é n y i
³ N ot ic e th a t for exp( sM)( ρ) t o be a v al id qua n tum s t a t e for s2 [ 0; T)R
0
, the s u pe r ope r a t or M should o bey c e r t ain
c on s tr ain ts, as for in s t a nc e th a t M( ρ) i s he r mit i a n a nd tr a c e le s s . W e w i l l l a t e r g e ne r al i z e thi s c on s tr uct ion t o include a n y M2
Sym(L) .
113
e n tr op y w ith α = 1= 2 , i . e .,
D( ρ; σ) = S
1
2
( ρ; σ) = 2 log
(
Tr
[
p
ρ
p
σ
])
(4.11)
the n it i s not h a r d t o sho w th a t the ex p a n sion ( 4.9) y ie ld s k = 2 a nd
h
M
( ρ) =
∑
ij
j⟨ ijM( ρ)j j⟩j
2
(
p
p
i
+
p
p
j
)
2
; (4.12)
w he r e w e spe ctr al ly de c ompos e d ρ =
∑
i
p
i
j i⟩⟨ ij . A de t ai le d de r iv a t ion of E q . ( 4.12 ) v i a e le me n tr a y me th-
od s ca n be found in A ppe nd i x A . The a bo v e qua n t it y i s w e l l -kno w n in qua n tum infor m a t ion- g e ome tr y as
the W i gne r - Y a n as e me tr ic [ 137 ], he r e e v alua t e d on M( ρ) .
Mo n o t o ne s o f th e ev o l u ti o n a nd mo n o t o ne R i e ma nni a n me tr i c s
I n the pr e v ious s e ct ion w e d i s c us s e d a w a y of o bt ainin g monot one s for a M a rk o v i a n e v o lut ion g iv e n a d i s -
t i n g ui sh a b i l it y me as ur e a nd a s y mme tr y M of the g e ne r a t or , as in the ex a mp le of E q . ( 4.12 ). Thi s r ai s e s the
que s t ion of ho w t o s ys t e m a t ical ly pe r for m the ex p a n sion ( 4.9) for a s uit a b ly w ide cl as s of d i s t in g ui sh a b i l it y
me as ur e s, a que s t ion th a t n a tur al ly le a d s t o the c on side r a t ion of monot one R ie m a nni a n me tr ic s .
T o s e e th a t , c on side r t o the cas e w he r e
√
D( ρ; σ) i s a d i s t a nc e f unct ion o v e r the m a ni fo ld S
> 0
(H) of
posit iv e definit e s t a t e s s uch th a t
p
D a r i s e s f r om a R ie m a nni a n me tr ic ⁴ g . I n th a t cas e , the ex p a n sion ( 4.9 )
y ie ld s k = 2 a nd the f unct ion h
M
( ρ) i s nothin g e l s e th a t ( s qua r e d ) le n g th of the t a n g e n t v e ct or M( ρ) o v e r
the t a n g e n t sp a c e T
ρ
S
> 0
(H) , i . e .,
h
M
( ρ) = g
ρ
(M( ρ);M( ρ)) : (4.13)
M or e impor t a n tly , the monot onic it y pr ope r t y ( 4.10 ) i s s a t i sfie d i f the me tr ic i s c on tr a ct iv e unde r the a ct ion
⁴ I. e ., the d i s t a nc e be t w e e n t w o s t a t e s i s the le n g th of the g e ode sic c onne ct in g the m .
114
of qua n tum ch a nne l s T , n a me ly
g
ρ
( X; X) g
T ( ρ)
(T ( X);T ( X)) : (4.14)
The a for e me n t ione d cl as s of R ie m a nni a n me tr ic s, cal le d monot one me tr ic s [ 138 – 141 ], c on s t itut e the
qua n tum a n alo g of the F i she r me tr ic for pr o b a b i l it y d i s tr i but ion s ( s e e al s o R ef . [ 59 ] for a n a c c e s si b le in-
tr oduct ion t o the s ubj e ct ). W hi le the l a tt e r i s unique ly de t e r mine d f r om the monot onic it y pr ope r t y un-
de r cl as sical s t och as t ic m a ps ( u p t o a nor m al i za t ion c on s t a n t ), the s a me doe s not ho ld for its qua n tum
c oun t e r p a r ts, w hich sho w a r ich v a r ie t y . M onot one me tr ic s a r e al s o clos e ly r e l a t e d t o g e ne r al i z e d r e l a t iv e
e n tr op ie s [ 142 ], as e v e r y monot one R ie m a nni a n me tr ic a r i s e s f r om a g e ne r al i z e d r e l a t iv e e n tr op y [ 143 ].
W e include for c omp le t e ne s s a n e le me n t a r y d i s c us sion of monot one R ie m a nni a n me tr ic s in A ppe nd i x B .
F or the pur pos e s of thi s w ork , the cr uc i al pr ope r t y of s uch qua n t it ie s i s th a t they s a t i sf y the d a t a pr oc e s s -
in g ine qual it y . Thi s monot onic it y pr ope r t y ca n be unde r s t ood as a c on s e que nc e of a k ey ope r a t or ine qual -
it y , fir s t pr o v e d in R ef . [ 143 ], w hich w e r e pe a t he r e due t o its c e n tr al impor t a nc e for w h a t fo l lo w s .
Th e o r e m: L e s nie ws k i a n d R us k a i [ 143 ]
Tr
(
A
y
1
R
σ
+ λL
τ
A
)
Tr
(
E( A)
y
1
R
E( σ)
+ λL
E( τ)
E( A)
)
; (4.15)
w he r e A2B(H) i s a l ine a r ope r a t or , L
σ
( X) := σ X(R
τ
( X) := X τ) i s the s u pe r ope r a t or r e pr e s e n t in g
left ( r i gh t ) m ult ip l ica t ion, E i s a qua n tum ch a nne l a nd λ 2 R
0
. The ope r a t or s σ; τ 2B
> 0
(H) a r e
posit iv e definit e , as s ur in g th a t the s u pe r ope r a t or in v e r s e s e n t e r in g the ine qual it y a r e w e l l - define d , as
w e l l as th a t the r e s ult in g tr a c e s a r e non- ne g a t iv e .
L e t us no w r e tur n t o c on side r in g qua n tum M a rk o v i a n dy n a mic s g e ne r a t e d b y s ome t ime-inde pe nde n t
L ind b l a d i a n L . F or our pur pos e s, the qua n tum ch a nne l in the ine qual it y ( 4.15 ) i s spe c i al i z e d t o the t ime
115
e v o lut ion s u pe r ope r a t or E
t
:= exp( tL) for t 0 . L e tt in g ρ2S
> 0
(H) be a f ul l - r a nk s t a t e , w e t ak e
A =M( ρ); w he r eM2 Sym(L) (4.16)
σ = τ =N( ρ); w he r eN 2 Sym(L) (4.17)
s uch th a t N( ρ)2B
> 0
(H) i s posit iv e definit e for al l ρ2S
> 0
(H) . W e w i l l r efe r t o a n y s uch s u pe r ope r a t or s
M a ndN as s y mme tr ie s of the dy n a mic s .
The r e s ult in g ine qual it y , due t o the c omm ut a t ion r e l a t ion s [M;E
t
] = [N;E
t
] = 0 , ex pr e s s e s the fa ct
th a t the q ua n t it y
J ρK
( λ)
M;N
:= Tr
(
M( ρ)
y
1
R
N( ρ)
+ λL
N( ρ)
M( ρ)
)
; (4.18)
s a t i sfie s
J ρK
( λ)
M;N
JE
t
( ρ)K
( λ)
M;N
for λ; t 0: (4.19)
W e h a v e sho w n the fo l lo w in g.
P r opositio n 4.1
I fM;N a r e s y mme tr ie s of the dy n a mic s a nd λ 0 , the n th e f unct ion
J ρK
( λ)
M;N
:S
> 0
(H)!R
0
define d in E q . ( 4.18 ) i s non-incr e asin g unde r the t ime e v o lut ion g e ne r a t e d b y L .
I n the r e s t of thi s C h a pt e r , w e w i l l m ainly foc us on t w o cas e s .
116
( i ) N =I , in w hich cas e for simp l ic it y w e de not e the r e s ult in g fa mi ly of monot one s as
J ρK
( λ)
M
= Tr
(
M( ρ)
y
1
R + λL
M( ρ)
)
: (4.20)
( i i ) I f the dy n a mic s a dmits a f ul l - r a nk s t a t ion a r y s t a t e ω , the nN( X) = Tr( X) ω i s a s y mme tr y . I n th a t
cas e , it i s c on v e nie n t t o de not e d ir e ctly the fi xe d s t a t e ω in s t e a d of the s y mme tr y , i . e ., w r it e
J ρK
( λ)
M; ω
= Tr
(
M( ρ)
y
1
R + λL
M( ρ)
)
: (4.21)
The monot one s J ρK
( λ)
M
ca n be al s o ex pr e s s e d in c oor d in a t e s v i a spe ctr al ly de c omposin g the a r g ume n t
ρ =
∑
i
p
i
j i⟩⟨ ij . S ubs t itut in g , one g e ts
J ρK
( λ)
M
=
∑
ij
1
λ p
i
+ p
j
j⟨ ijM( ρ)j j⟩j
2
: (4.22)
S imi l a rly , for the cas e of J ρK
( λ)
M; ω
one ca n de c ompos e ω =
∑
i
q
i
j i
ω
⟩⟨ i
ω
j r e s ult in g in
J ρK
( λ)
M; ω
=
∑
ij
1
λ q
i
+ q
j
j⟨ i
ω
jM( ρ)j j
ω
⟩j
2
: (4.23)
F or H a mi lt oni a n dy n a mic s, L( X) =K
H
( X) := i[ H; X] , the monot one s ( 4.18 ) a r e , in fa ct , c on s e r v e d .
Thi s fo l lo w s be ca us e unit a r y ch a nne l s a r e in v e r t i b le , he nc e for thi s cas e E q . ( 4.19 ) ho ld s tr ue for t 2 R .
Thi s for c e s monot one s t o m ain t ain a c on s t a n t v alue alon g the or b it.
F or e v e r y fa mi ly fJ ρK
( λ)
M;N
g
λ
, one ca n c on side r c on v ex c omb in a t ion s a c c or d in g t o the me as ur e μ( λ) ,
n a m e ly
fJ ρK
( λ)
M;N
g
λ
7!
∫
d μ( λ)J ρK
( λ)
M;N
: (4.24)
F r om thi s c on s tr uct ion one o bt ain s v al id monot one s, pos si b ly a dmitt in g a c on v e nie n t m a the m a t ical for m
for a n a ppr opr i a t e cho ic e of the me as ur e . H o w e v e r , not ic e th a t the r e s ult in g f unct ion s do not impos e a n y
117
a dd it ion al c on s tr ain ts c omp a r e d t o the one s f r om the p a r e n t f unct ion s ⁵ .
4.5 A fir s t e xa m pl e : D eph a sing of a qu b it
I t mi gh t be us ef ul a t thi s s t a g e t o c on side r a simp le ex a mp le t o i l lus tr a t e the for m al i s m . L e t us a n aly z e a
t w o -le v e l s ys t e m w ith de p h asin g dy n a mic s de s cr i be d b y the L ind b l a d i a n
L( X) = ig[ σ
z
; X] + σ
z
X σ
z
X; g2R : (4.25)
The t ime e v o lut ion in t e r m s of the B loch v e ct or r e pr e s e n t a t ion of the s t a t e ρ( t) =
1
2
( I + v σ) i s, in c y l in-
dr ical c oor d in a t e s,
r
t
= r
0
e
2t
; φ
t
= φ
0
+ 2gt; z
t
= z
0
: (4.26)
Th a t i s, the B loch v e ct or l ie s on t o a hor i z on t al p l a ne e v o lv in g in w a r d s in a sp ir al mot ion .
L e t us no w i l lus tr a t e ho w one ca n de duc e the qual it a t iv e fe a tur e s of the e v o lut ion jus t b y s y mme tr y
c on side r a t ion s, b y us e of E q . ( 4.20 ). S inc e the H a mi lt oni a n H = g σ
z
a nd the sin gle L ind b l a d ope r a t or L =
σ
z
a r e both d i a g on al in the σ
z
:=j 0⟩⟨ 0jj 1⟩⟨ 1j ei g e nb asi s, cle a rly the left m ult ip l ica t ion s u pe r ope r a t or s
L
j 0⟩⟨ 0j
a ndL
j 1⟩⟨ 1j
a r e s y mme tr ie s of the L ind b l a d i a n . One imme d i a t e ly g e ts th a t the popul a t ion s
J ρK
( 0)
L
j i⟩⟨ ij
= Tr( ρj i⟩⟨ ij) ; i = 0; 1 (4.27)
a r e non-incr e asin g. H o w e v e r , sinc e the e v o lut ion i s tr a c e pr e s e r v in g , e a ch of the popul a t ion s i s s e p a r a t e ly
c on s e r v e d . N ot ic e th a t the s e a r e ex a ctly the t w o l ine a rly inde pe nde n t c on s e r v e d qua n t it ie s of the e v o lut ion
pr e d ict e d b y N oe the r ’ s the or e m for L ind b l a d i a n s, w hich w e d i s c us s mome n t a r i ly .
N o w w e c on side r a g ain E q . ( 4.20 ) but for the s y mme tr y M( X) = K
σ z
a nd λ = 1 . S pe ctr al ly de c om-
⁵ I. e ., i f the pos si b i l it y of a tr a n sit ion ρ7! σ i s r ule d out b y the ine qual it y
∫
d μ( λ)J ρK
( λ)
M;N
<
∫
d μ( λ)J σK
( λ)
M;N
, the n the r e
ex i s ts a ( non- z e r o me as ur e ) s e t of λ ’ s for w hich al s o J ρK
( λ)
M;N
< J σK
( λ)
M;N
.
118
posin g ρ = p
+
P
+
+ p

P

a nd in v o k in g E q . ( 4.22 ), w e h a v e
J ρK
( 1)
K σ z
= 2 Tr( P
+
σ
z
P

σ
z
)( p
+
p

)
2
= 2j⟨ 0j ρj 1⟩j
2
; (4.28)
th a t i s, c o he r e nc e s a r e non-incr e asin g unde r the t ime e v o lut ion .
The monot one s f r om E qs . ( 4.27 ) a nd ( 4.28 ) j o in tly l imit the s e t of s t a t e s R th a t a r e ( pos si b ly ) r e a ch a b le
unde r t ime e v o lut ion . The a for e me n t ione d s e t c on si s ts of al l B loch v e ct or s ly in g on hor i z on t al d i sk w ith
r a d ius e qual t o r
0
( the r a d i al d i s t a nc e of the init i al B loch v e ct or f r om the z -a x i s .)
I n fa ct , the s e t R of our ex a mp le i s as c on s tr aine d as pos si b le , in the s e n s e th a t a dd it ion al s y mme tr y a r g u-
me n ts ( r e ly in g on the s e t Sym(L) ) ca nnot r e s tr ict it mor e . T o s e e thi s, fir s t not ic e th a t the s e t of s y mme tr ie s
of our de p h asin g L ind b l a d i a n doe s not de pe nd on the v alue of the p a r a me t e r g , as lon g as g̸= 0 . On the
othe r h a nd , b y v a r y in g g , al l po in ts in the bul k of R ca n be r e a che d b y the tr a j e ct or ie s of E q . ( 4.26 ) ( for
a n y fi xe d init i al c ond it ion ). Thi s i s be ca us e g c on tr o l s the f r e que nc y of the r ot a t ion al mot ion, w hich ca n
be m a de a r b itr a r i ly hi gh . The r efor e no mor e po in ts in R ca n be exclude d b y s y mme tr y a r g ume n ts .
4.6 Monot one s impl y Noe ther c on ser ved qu an titie s
W e w i l l s a y th a t a n ope r a t or Y2B(H) i s c on s e r v e d i f
Tr
[
Y
y
exp( tL)( ρ)
]
= c on s t 8 ρ a nd t 0 : (4.29)
W e w i l l al s o r efe r t o s uch f unct ion s of the s t a t e as N oe the r c on s e r v e d qua n t it ie s, sinc e they g e ne r al i z e the
c or r e spond in g H a mi lt oni a n c on s tr uct ion . I t i s w e l l -kno w n th a t
Y i s c on s e r v e d () Y2 KerL

(4.30)
( the a d j o in t i s w ith r e spe ct t o the H i l be r t - S chmidt inne r pr oduct ). The cl aim fo l lo w s b y not ic in g th a t the
a bo v e tr a c e ca n be ex pr e s s e d v i a the H i l be r t - S chmidt inne r pr oduct ( ⟨ A; B⟩ := Tr
(
A
y
B
)
for A; B 2
119
B(H) ) as
Tr
[
Y
y
exp( tL)( ρ)
]
=⟨ Y; exp( tL)( ρ)⟩
=⟨ exp( tL

)( Y); ρ⟩:
A c on s e que nc e i s th a t the n umbe r of l ine a rly inde pe nde n t c on s e r v e d qua n t it ie s e qual s the d ime n sion
of the s ubsp a c e sp a nne d b y the s t e a dy s t a t e s of the e v o lut ion, a fa ct al s o not e d in [ 126 ]. Thi s o bs e r v a t ion
fo l lo w s f r om the ide n t it y dim Ker(L) = dim Ker(L

) . I n p a r t ic ul a r , i f the r e i s a unique fi xe d po in t of the
e v o lut ion the n al l c on s e r v e d qua n t it ie s tr iv i al i z e , in the s e n s e th a t they a r e ne c e s s a r i ly pr opor t ion al t o the
tr a c e of the ( t ime- e v o lv e d ) s t a t e , sinc e alw a ys I2 KerL

.
On the othe r h a nd , the monot one s J ρK
( λ)
M
impos e non-tr iv i al , in g e ne r al , c on s tr ain ts on the r e a ch a b le
s t a t e s of the e v o lut ion, for in s t a nc e b y t ak in g M = L ( a spe c i fic ex a mp le i s g iv e n l a t e r in F i g ur e 4.9.1 ).
W h a t i s mor e , i f a s t e a dy s t a t e ω i s kno w n ( r e g a r d le s s w he the r it i s unique or not ), the n al s o monot one s of
the for m J ρK
( λ)
M; ω
ca n be ut i l i z e d , imposin g a dd it ion al c on s tr ain ts .
A n a tur al que s t ion t o be ask e d i s w he the r the c on s tr ain ts impos e d b y the N oe the r c on s e r v e d qua n t it ie s
a r e include d in the monot one s a r i sin g f r om 4.1 or not. The a n s w e r i s a ffir m a t iv e .
P r opositio n 4.2
I f the L ind b l a d i a n a dmits a f ul l - r a nk s t a t ion a r y s t a t e , the n for e a ch c on s e r v e d ope r a t or Y the r e i s a n
a ppr opr i a t e cho ic e of the s y mme tr ie s M;N in the fa mi ly of monot one s J ρK
( 0)
M;N
th a t imp l ie s the
c on s e r v a t ion .
N ot ic e , ho w e v e r , the c on v e r s e i s not tr ue; c on s tr ain ts impos e d b y the monot one s J ρK
( 0)
M;N
ca nnot ne c -
e s s a r i ly be infe r r e d f r om c on s e r v a t ion l a w s .
P r o of . L e t us c on side r s ome c on s e r v e d Y > 0 . The s u pe r ope r a t orM( ρ) = Tr( Y ρ) ω , for ω a f ul l -
r a nk s t a t ion a r y s t a t e of the e v o lut ion, i s a s y mme tr y of the dy n a mic s . The r efor e J ρK
( 0)
M; ω
= Tr( Y ρ)
2
i s non-incr e asin g unde r the t ime e v o lut ion, he nc e al s o Tr( Y ρ) i s non-incr e asin g.
120
On the othe r h a nd , sinc e ω i s a f ul l - r a nk s t e a dy s t a t e a nd Tr( Y ρ) > 0 , al s o J ρK
( 0)
ω;M
= [ Tr( Y ρ)]
1
i s a v al i d non-incr e asin g monot one . A s a r e s ult , Tr( Y ρ) h as t o be a c on s t a n t of the e v o lut ion .
T o c omp le t e the pr oof , w e ne e d t o sho w th a t the c on s tr ain ts impos e d b y c on s e r v e d Y2B
> 0
(H)
a r e the s a me as the one s impos e d b y a r b itr a r y Y2B(H) . Thi s i s tr ue be ca us eL

pr e s e r v e s he r mit ic it y
a nd the r efor e Y 2 Ker(L

) al s o imp l ie s Y
y
2 Ker(L

) , al lo w in g t o tr a de a pos si b ly non-he r mit i a n
Y for its ( s e p a r a t e ly c on s e r v e d ) he r mit i a n a nd a n t i -he r mit i a n p a r ts . F in al ly , sinc e L

i s unit al , i f Y i s
c on s e r v e d the n al s o Y+ aI i s c on s e r v e d , w hich ca n alw a ys be m a de posit iv e for l a r g e e nou gh a . N ot ic e
th a t thi s a moun ts t o jus t a dd in g the c on s t a n t a t o the v alue of the N oe the r c on s e r v e d qua n t it y .
The a bo v e de mon s tr a t e th a t the monot one fa mi ly J ρK
( 0)
M;N
impos e s a t le as t as m a n y c on s tr ain ts as
c on s e r v e d qua n t it ie s . ■
4.7 S ym me trie s of the g ener a t or and the m onot one s
W e no w v e r y br iefly a ddr e s s the que s t ion of ho w one ca n find s y mme tr ie s of a L ind b l a d i a n, bey ond thos e
g ue s s e d b y in spe ct i on . D e t e r minin g ex p l ic itly the e n t ir e s e t of s y mme tr ie s Sym(L) of a L ind b l a d i a n i s
us ual ly t oo c omp l ica t e d t o be of pr a ct ical impor t a nc e . I n s t e a d , w he n a r e pr e s e n t a t ion of the L ind b l a d i a n
in L ind b l a d ope r a t or s i s kno w n ( i . e ., a de c omposit ion as in E q . ( 4.3 )), it mi gh t be c on v e nie n t t o loo k a t
ope r a t or s sim ult a ne ously c omm ut in g w ith al l e le me n ts of the s e t S =f Hg[
{
L
i
; L
y
i
}
i
. I fA i s the al g e br a
g e ne r a t e d b y S , the n the e le me n t of the al g e br a g e ne r a t e d b y fL
X
;R
X
g
X2A
′ be lon g t o Sym(L) ( he r eA
′
de not e s the c omm ut a n t of the al g e br a ⁶ ). W e w i l l us e thi s fa ct t o a n aly z e de p h asin g g e ne r a t or s l a t e r in the
C h a pt e r .
S t e a dy s t a t e s ω g iv e r i s e t o s y mme tr ie s, for in s t a nc e v i a m a ps N( X) = Tr( X) ω , as in E q . ( 4.21 ). A l -
thou gh Ker(L) i s not in g e ne r al clos e d w ith r e spe ct t o ope r a t or m ult ip l ica t ion, it w as sho w n in R ef . [ 144 ]
th a t it for m s a n E ucl ide a n J or d a n A l g e br a . M or e spe c i fical ly , i f the l imit F := lim
t!1
exp( tL) I ex i s ts a nd
⁶ I. e ., the ope r a t or s ub al g e br a of B(H) c on si s t in g of al l ( a nd only ) the e le me n ts th a t c omm ut e w ith e v e r y e le me n t of A .
121
i s f ul l - r a nk , the n for A; B fi xe d po in ts of the e v o lut ion, it ho ld s th a t
A
F
B :=
1
2
( A
F
B
F
+ B
F
A
F
) (4.31)
i s al s o a fi xe d po in t ⁷ , w he r e X
F
:= F
1= 2
XF
1= 2
.
I n a dd it ion, in R ef . [ 145 ] the a uthor s c on s tr uct e d a fa mi ly of m a ps th a t a r e b i j e ct ion s o v e r the s ubsp a c e
of fi xe d po in ts of the dy n a mic s . M or e spe c i fical ly , the m a pp in gs ca n be c on s tr uct e d w ith the input of a n
ope r a t or monot one f unct ion [ 8 ] a nd t w o f ul l - r a nk s t e a dy s t a t e s . The a bo v e fa cts ca n be of us e t o g e ne r a t e
mor e fi xe d po in ts, a nd he nc e s y mme tr ie s, out of a s e t of kno w n one s .
F in al ly , w e a ddr e s s the que s t ion of ho w t o c on s tr uct monot one s th a t the m s e lv e s pos s e s s s ome s y mme tr y .
L e t us c on side r a unit a r y r e pr e s e n t a t ion U
g
of a gr ou p G s uch th a t U
g
= U
g
() U
y
g
i s a s y mme tr y of the
L ind b l a d i a n for al l g2 G . The n one ca n define a ( left ) gr ou p a ct ion on the sp a c e of monot one s v i a
JK
( λ)
M;N
g
7 ! JK
( λ)
U gM;N
= JK
( λ)
M;U
g
1
N
: (4.32)
F r om the l a tt e r , it i s s tr ai gh tfor w a r d t o c on s tr uct monot one s th a t a r e in v a r i a n t unde r the a ct ion b y in v o k in g
the ( H a a r ) gr ou p a v e r a g e , i . e ., c on side r
Q
( λ)
M;N
( ρ) :=
∫
d μ
H a a r
( g)J ρK
( λ)
U gM;N
: (4.33)
F or in s t a nc e ,
Q
( 0)
M; ω
( ρ) = Tr
[
P
(
M( ρ)
y
M( ρ)
)
ω
1
]
; (4.34)
w he r eP( X) :=
∫
d μ
H a a r
( g)U
g
( X) i s the pr oj e ct or s u pe r ope r a t or on t o the G -in v a r i a n t s ubsp a c e , i . e ., for
a n y ope r a t or X it ho ld s U
g
P( X) = P( X) U
g
8 g . The pr oj e ct orP ca n be c on s tr uct e d ex p l ic itly in t e r m s
of the ir r e duc i b le r e pr e s e n t a t ion s of f U
g
g
g
usin g s t a nd a r d t e chnique s f r om the or y of no i s e le s s s ubs ys -
⁷ N ot ic e th a t a n y ope r a t or X 2 Ker(L) ca n be de c ompos e d in t o ( u p t o four ) s t e a dy s t a t e s b y c on side r in g the posi -
t iv e/ne g a t iv e p a r t of
1
2
( X + X
y
) a nd
1
2i
( X X
y
) . Ea ch of the l a tt e r , a ft e r pr ope r nor m al i za t ion, for m s a s t e a dy s t a t e . Thi s
fo l lo w s f r om the fa ct th a t the c or r e spond in g qua n tum ch a nne l E
t
pr e s e r v e s posit i v it y .
122
t e m s [ 146 , 147 ].
4.8 D eph a sing g ener a t or s
H e r e w e c on side r L ind b l a d g e ne r a t or s of the g e ne r al for m ( 4.3) s uch th a t al l e le me n ts of the s e t S =f Hg[
{
L
i
; L
y
i
}
i
m utual ly c omm ut e w ith e a ch othe r . W e w i l l r efe r t o thi s cl as s as de p h asin g g e ne r a t or s . N ot ic e
th a t thi s fa mi ly i s a g e ne r al i za t ion of the qub it ex a mp le of s e ct ion 4.5 .
D e p h asin g L ind b l a d i a n s a r e both unit al a nd H i l be r t - S chmidt nor m al , as ca n be infe r r e d w ith a d ir e ct
calc ul a t ion in v o k in g the s t a nd a r d for m ( 4.3) of the g e ne r a t or . H e nc e b y the spe ctr al the or e m they a dmit
a c omp le t e fa mi ly of ei g e nope r a t or s . The c or r e spond in g t ime e v o lut ion ca n be for m al ly ex pr e s s e d v i a r e-
s or t in g t o the ( m a x im al ) pr oj e ct or s on t o the j o in t ei g e n sp a c e s of al l e le me n ts in S . L e t us de not e as f P
i
g
n
i= 1
( n d ) thi s c omp le t e fa mi ly of or tho g on al pr oj e ct or s, a nd al s o as P
ij
( X) := P
i
XP
j
the c or r e spond -
in g fa mi ly of ( H i l be r t - S chmidt or tho g on al ) pr oj e ct or s u pe r ope r a t or s . The n, for de p h asin g g e ne r a t or s, the
t ime e v o lut ion ca n be w r itt e n as
exp( tL) =
∑
ij
e
λ ij t
P
ij
: (4.35)
I n v o k in g onc e a g ain the s t a nd a r d for m of the g e ne r a t or ( 4.3 ), it fo l lo w s th a t the ei g e n v alue s s a t i sf y λ
ij
= λ

ji
a n d λ
ii
= 08 i , w hi le Re
(
λ
ij
)
< 0 for i̸= j .
The l as t ine qual it y ex pr e s s e s th a t fa ct th a t the r e a r e no s t e a dy s t a t e s w ith s u ppor t o v e r a n y of the ei g e n sp a c e s
P
ij
w ith i̸= j . Thi s i s tr ue sinc e , for unit al L ind b l a d i a n s, Ker(L) c o inc ide s w ith A
′
[ 148 ], w he r eA i s the
( a be l i a n ) al g e br a g e ne r a t e d b y the s e t S . I n our cas e , thi s c omm ut a n t i s ex a ctly g iv e n b y ope r a t or s in the
r a n g e of
∑
i
P
ii
.
Qual it a t iv e ly , the e v o lut ion d ict a t e s th a t the d i a g on al e le me n ts
∑
i
P
ii
( ρ) a r e c on s e r v e d w hi le the off -
d i a g on al p a r ts
∑
i
P
ij
( ρ) ( i ̸= j ) de ca y , pos si b ly w ith os c i l l a t ion s, w hich i s the definin g ch a r a ct e r i s t ic of
de p h asin g.
L e t us no w ex tr a ct the s a me qual it a t iv e infor m a t ion f r om a s y mme tr y a n alysi s . C le a rly , [L
P i
;L] = 08 i
123
he nc e
J ρK
( 0)
L P
i
= Tr( ρ P
i
) ; 8 i (4.36)
a r e non-incr e asin g. On the othe r h a nd , the e v o lut ion i s tr a c e-pr e s e r v in g , the r efor e al l P
i
a r e , in fa ct , c on-
s e r v e d . The s e a r e al l the l ine a rly inde pe nde n t N oe the r c on s e r v e d qua n t it ie s .
I n a dd it ion, L
P i
R
P j
i s a s y mme tr y of the e v o lut ion sinc e al l e le me n ts in S m utual ly c omm ut e . C omb inin g
thi s w ith the fa ct th a t the e v o lut ion i s unit al , he nc e ω = I i s a fi xe d po in t , w e d ir e ctly g e t f r om E q . ( 4.21 )
th a t
J ρK
( 0)
L P
i
R P
j
; I
= Tr
(
P
i
ρ P
j
ρ
)
(4.37)
i s non-incr e asin g. Thi s c or r e spond s t o the de ca y of the ( i; j) - c o he r e n c e s of the s t a t e ρ a nd i s a g e ne r al i za t ion
of E q . ( 4.28 ). W e h a v e sho w n the fo l lo w in g g e ne r al fa ct.
P r opositio n 4.3
L e tL be a L ind b l a d i a n . The n,
( i ) I fL
P
i s a s y mme tr y of L , the n Tr
(
ρ
t
P
y
P
)
i s a non-incr e asin g f unct ion of t ime .
( i i ) I fL i s unit al a nd L
P
R
Q
i s a s y mme tr y , the n Tr
(
P
y
P ρ
t
Q
y
Q ρ
t
)
i s a non-incr e asin g f unct ion of
t ime .
F or P; Q or tho g on al pr oj e ct or s, the n ( i) c or r e spond s t o de ca y in g popul a t ion a nd ( ii) t o de ca y in g c o -
he r e nc e .
124
4.9 D a vie s g ener a t or s
Pr elimi n a r i e s
A p h ysical ly si gni fica n t cl as s of L ind b l a d i a n s i s pr o v ide d b y the D a v ie s g e ne r a t or s [ 149 ] ( for a mor e mode r n
tr e a tme n t , s e e al s o R ef s . [ 150 – 152 ]). Thi s fa mi ly a r i s e s f r om a fir s t -pr inc ip le s de r iv a t ion unde r the m ain
as s umpt ion th a t the in t e r a ct ion be t w e e n the s ys t e m a nd its e n v ir onme n t i s w e ak a nd g iv e s r i s e t o s t e a dy
s t a t e s th a t a r e the r m al . D a v ie s g e ne r a t or s h a v e the fo l lo w in g pr ope r t ie s:
( i ) The H a mi l t oni a n p a r t c omm ut e s w ith the d i s sip a t iv e p a r t , i . e ., L =K
H
+D w ith
[K
H
;D] = 0 : (4.38a)
( i i ) I t s a t i sfie s the q ua n tum de t ai le d b al a nc e c ond it ion, i . e ., the r e ex i s ts a s t a t e τ s uch th a t
DR
τ
=R
τ
D

; (4.38b)
K
H
( τ) = 0 : (4.38c)
F or the s c ope of thi s C h a pt e r , w e w i l l r efe r t o a n y L ind b l a d i a n th a t s a t i sfie s the c ond it ion s of E qs . ( 4.38 ) as
a D a v ie s g e ne r a t or .
L e t us r e cal l a fe w us ef ul m a the m a t ical fa cts r e g a r d in g the de t ai le d b al a nc e c ond it ion [ 150 , 153 ]. F ir s t of
al l , sinc e D

i s unit al , E q . ( 4.38b ) imp l ie s th a t τ i s a fi xe d po in t of the e v o lut ion . The de t ai le d b al a nc e c on-
d it ion ca n be e quiv ale n tly unde r s t ood as the r e quir e me n t th a t D

i s he r mit i a n w ith r e spe ct t o the ( pos si b ly
de g e ne r a t e ) s cal a r pr oduct
⟨ A; B⟩
τ
:= Tr
(
τ A
y
B
)
: (4.39)
A s s umin g th a t τ i s f ul l r a nk , the de t ai le d b al a nc e c ond it ion E q . ( 4.38b ) ca n be r e cas t in a y e t a nothe r for m,
125
n a me lyD bein g he r mit i a n w ith r e spe ct t o the s cal a r pr oduct ⟨;⟩
τ
1
. The l as t cl aim fo l lo w s b y m ult ip ly in g
both f r om the left a nd the r i gh t E q . ( 4.38b ) w ithR
1
τ
.
L e t us no w ex pr e s s the g e ne r al for m of the t ime e v o lut ion of a D a v ie s g e ne r a t or s . N ot ic e th a t D a v ie s
g e ne r a t or s a r e alw a ys nor m al ope r a t or s, he nc e a dmit a c omp le t e fa mi ly of ei g e nope r a t or s . Thi s i s be ca us e ,
b y as s umpt ion, D i s he r mit i a n w ith r e spe ct t o the s cal a r pr oduct ( 4.39 ) ( for τ
1
) a nd it c omm ut e s w ith the
H a mi lt oni a n p a r t (w hich i s al s o a n t i -he r mit i a n w ith r e spe ct t o the s a me s cal a r pr oduct ). L e t us for m al ly
de not e as fP
i
g
i
the c omp le t e fa mi ly of he r mit i a n (w ith r e spe ct t o ⟨;⟩
τ
1
) s u pe r ope r a t or pr oj e ct or s . The n
the g e ne r al s o lut ion h as the for m
exp( tL) =
∑
i
e
λ i t
P
i
(4.40)
w ithP
i
P
j
= δ
ij
P
i
a nd
∑
i
P
i
=I .
D a v i e s gener a t o r : Si n gle q u b it
L e t us c on side r a qub it s ys t e m w hos e L ind b l a d i a n i s de s cr i be d b y H = σ
z
t o g e the r w ith t w o L ind b l a d
ope r a t or s f a σ
+
; b σ

g . The unique s t e a dy s t a t e τ =
1
2
( I + w σ) h as a c or r e spond in g B loch v e ct or ly in g
alon g the z -a x i s w ith w
z
=
a
2
b
2
a
2
+ b
2
. One ca n e asi ly che ck d ir e ctly th a t E qs . ( 4.38 ) a r e s a t i sfie d , he nc e L i s
inde e d a v al id D a v ie s g e ne r a t or . S e tt in g g =j aj
2
+j bj
2
, the dy n a mical e v o lut ion of the B loch v e ct or i s, in
c y l indr ical c oor d in a t e s,
r
t
= r
0
e
gt= 2
; φ
t
= φ
0
+ 2t; z
t
= ( 1 e
gt
) w
z
+ e
gt
z
0
: (4.41)
L e t us a n aly z e s ome of its s y mme tr ie s no w . B y definit ion of the D a v ie s g e ne r a t or , K
H
=K
σ z
i s a s y mme-
tr y , the r efor e E q . ( 4.28 ) i s a v al id monot one , ind ica t in g th a t c o he r e nc e s ca n only de ca y . Thi s c or r e spond s
t o the in w a r d sp ir al mot ion of the tr a j e ct or y .
R e g a r d in g the z - e v o lut ion of the B loch v e ct or t o w a r d s the fi xe d po in t , le t us as s ume th a t τ i s kno w n t o
be a fi xe d s t a t e of the e v o lut ion ( but w e do not ex p l ic itly as s ume th a t it i s the unique fi xe d s t a t e ). The n b y
126
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
0.5
1.0
v
x
v
z
Figure 4.9.1: Constraints imp osed on the time evolution b y considering as symmetry the Lindbla-
dianL itself. In this exampleL is a single qubit Davies generato r with H = σ
z
and jump op e rato rs
L
1
=
p
2 σ
+
, L
2
= σ

. W e depict in Blo ch co o rdinates contours sepa rating the allo w ed (inside) from
the fo rbidden (outside) regions as p redicted b y the monotone J ρK
( 1= 2)
L
fo r va rious ρ . The monotone is
symmetric under rotations a round the z -axis, so w e only plot an x z slice of the Blo ch ball. The co r-
resp onding states ρ a re ma rk ed with a p oint and a re chosen to co rresp ond to inst ances of the actual
trajecto ry (as p rojected onto the x z plane) of the initially pure st ate v = ( 1= 2; 0;
p
3= 2) .
127
in spe ct ion one ca n v e r i f y th a t M( X) =⟨ σ
z
; X⟩
τ
1
σ
z
i s a s y mme tr y of th e e v o lut ion . The r efor e
J ρK
( 0)
M; τ
=

Tr
(
1
τ
σ
z
ρ
)

2
Tr
(
τ
1
)
(4.42)
i s non-incr e asin g , imp ly in g th a t al s o j Tr( τ
1
σ
z
ρ)j i s a monot one . I n the B loch r e pr e s e n t a t ion, the l as t
qua n t it y e quiv ale n tly ex pr e s s e s the fa ct th a t j v
z
w
z
j i s non-incr e asin g , i . e ., the z - c ompone n t of the B loch
v e ct or monot onical ly a ppr o a che s the s t e a dy s t a t e .
F in al ly , w e n ume r ical ly in v e s t i g a t e in F i g ur e 4.9.1 a ex a mp le for the c on s tr ain ts impos e d on the qub it
t ime e v o lut ion jus t b y c on side r in g as s y mme tr y the L ind b l a d i a n its e l f .
D a v i e s gener a t o r s: Gener a l r e ma r ks
L e t us m ak e a fe w o bs e r v a t ion s r e g a r d in g monot one s a nd D a v ie s g e ne r a t or s . F ir s t of al l , not ic e th a t mono -
t one s ( 4.21 ) for ω = τ ( the de t ai le d b al a nc e fi xe d po in t ) a nd λ = 0 a r e d ir e ctly r e l a t e d t o the D a v ie s inne r
pr oduct
J ρK
( 0)
M; τ
=⟨M( ρ);M( ρ)⟩
τ
1
: (4.43)
I n p a r t ic ul a r , the cas e M =I ex pr e s s e s the fa ct th a t
J ρK
( 0)
I; τ
= Tr
(
1
τ
ρ
2
)
(4.44)
i s non-incr e asin g unde r the t ime e v o lut ion, w hich i s a g e ne r al i za t ion of the w e l l -kno w n fa ct th a t the pur it y
Pur( ρ) := Tr( ρ
2
) i s non-incr e asin g unde r unit al dy n a mic s . E quiv ale n tly , the d i s t a nc e

ρ
t
τ

2
τ
1
=⟨ ρ
t
τ; ρ
t
τ⟩
τ
1
(4.45)
i s non-incr e asin g.
One ca n e asi ly g e ne r al i z e the c on s tr uct ion of the monot one ( 4.42 ). Ea ch of the pr oj e ct or s in E q . ( 4.40 )
ca n be w r itt e n asP
i
( X) =
∑
d i
k= 1
⟨ Y
i; k
; X⟩
τ
1
Y
i; k
, w he r ef Y
i; k
g
i; k
i s a c omp le t e or thonor m al fa mi ly of ei g e n-
128
ope r a t or s . H e nc e , for a n y s uch ope r a t or , M( X) =⟨ Y; X⟩
τ
1
Y i s cle a rly a s y mme tr y of the dy n a mic s, a nd
be ca us e for the s e s y mme tr ie s
J ρK
( 0)
M; τ
=

Tr
(
1
τ
Y
y
ρ
)

2
(4.46)
the f unct ion s

Tr
(
τ
1
Y
y
ρ
)

a r e monot one s, g e ne r al i z in g E q . ( 4.42 ).
4.10 Conc l u sion and ou tlo ok
I de n t i f y in g s y mme tr ie s a nd their c on s e que nc e s in p h ysical s ys t e m s i s a n impor t a n t t ask for v a r ious fie ld s of
p h ysic s . The pr e s e n t ch a pt e r c on s t itut e s a n a tt e mpt t o pr o v ide a simp le c or r e sponde nc e be t w e e n s y mme-
tr ie s in the g e ne r a t or s of qua n tum M a rk o v i a n dy n a mic s a nd monot one s of the c or r e spond in g e v o lut ion .
M or e spe c i fical ly , w e pr e s e n t e d a c on s tr uct ion th a t as si gn s t o e v e r y p air of s y mme tr ie s of the g e ne r a t or a
one-p a r a me t e r fa mi ly of f unct ion s o v e r qua n tum s t a t e s th a t a r e non-incr e asin g unde r the t ime e v o lut ion .
S uch monot onic f unct ion s ca n be e mp lo y e d in or de r t o ide n t i f y s t a t e s th a t a r e non- r e a ch a b le b y the e v o -
lut ion . The c on s tr uct ion ut i l i z e s po w e r f ul t oo l s f r om qua n tum infor m a t ion- g e ome tr y , m ainly f r om the
the or y of monot one R ie m a nni a n me tr ic s . F in al ly , w e h a v e de mon s tr a t e d ho w one ca n de duc e f r om s y m-
me tr ie s the qual it a t iv e fe a tur e s of the e v o lut ion for the pr ot ot y p ical cas e s of de p h asin g a nd D a v ie s g e ne r a -
t or s .
One ca n e asi ly r e p hr as e the que s t ion a ddr e s s e d in the pr e s e n t C h a pt e r t o fit in the f r a me w ork of qua n-
tum r e s our c e the or ie s . G iv e n s ome fi xe d L ind b l a d i a n, one ca n define as f r e e ope r a t ion s thos e as s oc i a t e d
w ith the t ime e v o lut ion fE
t
:= exp( tL)g
t 0
. Our m ain r e s ult , 4.1, pr o v ide s a fa mi ly of monot one s for
e a ch p air of s y mme tr ie s of the g e ne r a t or . H o w e v e r , e v e n i f al l pos si b le s y mme tr ie s a r e c on side r e d , the
r e s ult in g fa mi l ie s of monot one s a r e not in g e ne r al c omp le t e , i . e ., they do not exclude al l s t a t e s th a t a r e non-
r e a ch a b le b y the e v o lut ion . I t r e m ain s a n ope n que s t ion t o find a g e ne r al c on s tr uct ion for a c omp le t e fa mi ly
of monot one s .
129
A ppendic e s
A D er iv a ti o n o f Eq . ( 4.12 )
W e include be lo w a n ex p l ic it de r iv a t ion of E q . ( 4.12 ), i . e ., w e calc ul a t e the ex p a n sion of the
1
2
- R é n y i e n-
tr op y S
1
2
( ρ
0
; ρ
s
) = 2 log
(
Tr
[
p
ρ
0
p
ρ
s
])
for ρ
s
= exp( sM)( ρ
0
) t o the fir s t non- v a ni shin g or de r in s ,
as s umin g th a t the s t a t e ρ
0
i s f ul l - r a nk .
S e tt in g χ
s
=
p
ρ
s
, w e h a v e
Tr
(
p
ρ
0
p
ρ
s
)
=⟨ χ
0
; χ
s
⟩ = 1 + s⟨ χ
0
; _ χ
0
⟩ +
s
2
2
⟨ χ
0
; χ
0
⟩ + O
(
s
3
)
: (4.47)
M or e o v e r , f r om ρ
s
= χ
2
s
one find s
_ ρ
s
= χ
s
_ χ
s
+ _ χ
s
χ
s
=
(
L
χ
s
+R
χ
s
)
_ χ
s
:=A
χ
s
(
_ χ
s
)
;
he nc e al s o
_ χ
s
=A
1
χ
s
(
_ ρ
s
)
;
χ
s
=A
1
χ
s
(
ρ
s
)
+
dA
1
χ
s
ds
(
_ ρ
s
)
:
U sin g the a bo v e ex pr e s sion s w e ca n calc ul a t e the t e r m s in the ex p a n sion ( 4.47 ). W e h a v e
⟨ χ
s
; _ χ
s
⟩ =
1
2
⟨A
χ
s
( I);A
1
χ
s
(
_ ρ
s
)
⟩ =
1
2
Tr
(
_ ρ
s
)
= 0
he nc e the t e r m l ine a r in s doe s not c on tr i but e . The qua dr a t ic t e r m h as t w o p a r ts,
⟨ χ
s
; χ
s
⟩ =⟨ χ
s
;A
1
χ
s
(
ρ
s
)
⟩ +⟨ χ
s
;
dA
1
χ
s
ds
(
_ ρ
s
)
⟩
130
but one of the m v a ni she s
⟨ χ
s
;A
1
χ
s
(
ρ
s
)
⟩ =
1
2
Tr
(
ρ
s
)
= 0 :
T o calc ul a t e the r e m ainin g t e r m not ic e th a t
A
χ
s
=A
χ
0
+ s
(
L
_ χ
0
+R
_ χ
0
)
+ O
(
s
2
)
=
(
I + sA
_ χ
0
A
1
χ
0
)
A
χ
0
+ O
(
s
2
)
a n d t he r efor e
A
1
χ
s
=A
1
χ
0
(
I + sA
_ χ
0
A
1
χ
0
)
1
+ O
(
s
2
)
;
w hich g iv e s
dA
1
χ
s
ds

s= 0
=A
1
χ
0
A
_ χ
0
A
1
χ
0
:
F in al ly , thi s e qua t ion imp l ie s
⟨ χ
0
; χ
0
⟩ =
1
2
Tr
[
A
_ χ
0
(
_ χ
0
)]
= Tr
[(
_ χ
0
)
2
]
w hich c omp le t e s the ex p a n sion s
Tr
(
p
ρ
0
p
ρ
s
)
= 1
s
2
2
Tr
[(
_ χ
0
)
2
]
a nd he nc e
S
1
2
( ρ
0
; ρ
s
) = s
2
Tr
[(
_ χ
0
)
2
]
+ O
(
s
3
)
:
The for m ( 4.12 ) fo l lo w s b y usin g onc e a g ain the fa ct th a t _ χ
0
=A
1
χ
0
(
_ ρ
0
)
a nd v i a ex pr e s sin g ρ
0
=
∑
i
p
i
j i⟩⟨ ij .
131
B Mo n o t o ne R i e ma nni a n me tr i c s a nd Eq . ( 4.15 )
H e r e w e r e cal l for c omp le t e ne s s s ome w e l l -kno w n fa cts a bout monot one R ie m a nni a n me tr ic s a nd d i s c us s
ho w the k ey ine qual it y ( 4.15 ), o bt aine d b y L e s nie w sk i a nd R usk ai in [ 143 ], i s r e l a t e d t o the monot onic it y
pr ope r t y E q . ( 4.14 ).
M onot one R ie m a nni a n me tr ic s o v e r S
> 0
(H) , w hos e definin g pr ope r t y i s the d a t a -pr oc e s sin g ine qual it y
E q . ( 4.14 ), w e r e fir s t cl as si fie d c omp le t e ly b y P e tz [ 139 , 140 ], w ho ex t e nde d a n e a rl ie r r e s ult b y M or o z o v a
a nd C he n ts o v [ 138 ]. I n shor t , e v e r y s uch me tr ic a dmits the for m
g
ρ
( X; Y) =⟨ X;K
1
ρ
( Y)⟩ (4.48a)
for X; Y2 T
ρ
S
> 0
(H) ( i . e ., a r e he r mit i a n a nd tr a c e le s s ope r a t or s ) a nd
K
ρ
:= k
(
L
ρ
R
1
ρ
)
R
ρ
(4.48b)
w he r e k :R
0
!R i s a n ope r a t or monot one f unct ion th a t s a t i sfie s
k( t) = tk
(
t
1
)
8 t > 0 : (4.48c)
I t mi gh t be in s tr uct iv e a t thi s po in t t o che ck th a t the b i l ine a r for m g
ρ
inde e d define s a R ie m a nni a n me tr ic .
O pe r a t or monot one f unct ion s a r e alw a ys a n aly t ic [ 8 ], he nc e the b i l ine a r for m ( 4.48a ) i s s mooth w ith
r e spe ct t o ρ . E qua t ion ( 4.48c ) g ua r a n t e e s th a t K
1
ρ
pr e s e r v e s he r mit ic it y , m ak in g g
ρ
r e al v alue d . S inc e ρ i s
a f ul l - r a nk s t a t e , K
1
ρ
i s alw a ys w e l l - define d a nd posit iv e definit e , imp ly in g th a t the b i l ine a r for m ( 4.48a ) i s
non- de g e ne r a t e , a nd the r efor e g
ρ
i s inde e d a R ie m a nni a n me tr ic .
One ca n e asi ly ex pr e s s g
ρ
( X; X) ex p l ic itly b y in v o k in g the spe ctr al de c omposit ion ρ =
∑
i
p
i
j i⟩⟨ ij ,
w hich y ie ld s
g
ρ
( X; X) =
∑
ij
[
p
i
k
(
p
j
p
i
)]
1
j⟨ ij Xj j⟩j
2
: (4.49)
132
N ot ic e th a t , for al l v al id k , the d i a g on al p a r t i = j c or r e spond s t o the F i she r me tr ic ( u p t o pr opor t ion al -
it y c on s t a n t ). C on s e que n tly one r e c o v e r s the cl as sical a n alo g ue r e s ult [ 59 ] o v e r the ( d 1) - d ime n sion al
pr o b a b i l it y simp lex s t a t in g th a t i f a R ie m a nni a n me tr ic g
p
s a t i sfie s the monot onic it y pr ope r t y
g
p
( x; x) g
T( p)
( T( x); T( x)) (4.50)
unde r the a ct ion of s t och as t ic m a tr ic e s T the n it i s ne c e s s a r i ly pr opor t ion al t o the F i she r me tr ic, i . e .,
(
g
p
)
ij
/
δ
ij
p
i
: (4.51)
A c on s e que nc e of the r ichne s s of the monot one me tr ic s in the qua n tum r e g ime i s the pr e s e nc e of the f r e e
p a r a me t e r λ in the monot one s J ρK
( λ)
M;N
in tr oduc e d in the m ain t ex t.
L e t us fin al ly r e cal l ho w ine qual it y ( 4.15 ) imp l ie s the monot onic it y pr ope r t y E q . ( 4.14 ). I n [ 143 ], the
a uthor s sho w e d th a t for the cl as s of f unct ion s k define d a bo v e s a t i sf y in g k( 1) = 1 (w hich a moun t t o a me r e
nor m al i za t ion c ond it ion ), s u pe r ope r a t or s K
1
ρ
a dmit th e in t e gr al r e pr e s e n t a t ion
K
1
ρ
=
∫
1
0
(
[
λR
ρ
+L
ρ
]
1
+
[
R
ρ
+ λL
ρ
]
1
)
N
g
( λ) d λ (4.52)
w he r e N
g
( λ) d λ i s a sin g ul a r me as ur e ( the de t ai le d definit ion of ca n be found in [ 134 , 143 ]). The impor -
t a n t po in t i s th a t , due t o the a bo v e r e pr e s e n t a t ion, the monot onic it y of the me tr ic ( 4.14 ) fo l lo w s f r om the
monot onic it y of the in t e gr a nd in E q . ( 4.52 ); thi s i s w h a t i s sho w n in the the or e m of E q . ( 4.15 ). I n fa ct ,
not ic e th a t the the or e m i s sl i gh tly mor e g e ne r al ; it a pp l ie s t o a n y ope r a t or ( i . e ., not ne c e s s a r i ly he r mit i a n
a nd tr a c e le s s ), a f r e e dom th a t w e t ak e a dv a n t a g e of in the m ain t ex t.
133
Outlo ok
We h a v e e xplo r ed thr e e d i s t inct , but in t e r c onne ct e d aspe cts of qua n tum pr oc e s s e s: g e ne r a t ion of c o -
he r e nc e f r om unit a r y a nd de p h asin g e v o lut ion s, inc omp a t i b i l it y of or tho g on al a nd g e ne r al i z e d qua n tum
me as ur e me n ts, a nd the c onne ct ion be t w e e n s y mme tr ie s a nd monot one s in M a rk o v i a n qua n tum dy n a mic s .
I n al l of the thr e e cas e s, one ca n a r g ue th a t the r e i s s ome thin g of qua n tum n a tur e g o in g on ; ei the r the r e
i s no a pp a r e n t cl as sical c oun t e r p a r t , or i f the r e i s, it i s q u a l it a t i v e l y d i ff er en t .
I n a gl imps e , a c o he r e n t s u pe r posit ion i s d i s t inct f r om a n inc o he r e n t one , a ttr i but in g ch a r a ct e r i s t ic t o
qua n tum r a ndomne s s th a t m ak e it d i s t inct f r om cl as sical r a ndomne s s . Thi s be c ome s e v ide n t in the c on-
t ex t of our w ork f r om the fa ct th a t r a ndom unit a r y pr oc e s s e s, w hich in g e ne r al c or r e spond t o dy n a mic s of
und i s tur be d , i s o l a t e d qua n tum dy n a mic s, g e ne r a t e on a v e r a g e almos t m a x im al c o he r e nc e . On the othe r
h a nd , de p h asin g e v o lut ion s, w hich c or r e spond t o a cl as sical i za t ion, g e ne r a t e almos t minim al .
C o he r e nc e i s in t im a t e ly r e l a t e d t o the w a v e ch a r a ct e r of qua n tum the or y a nd the p he nome non of in t e r -
fe r e nc e . I t i s he nc e n a tur al t o ex pe ct th a t qua n tum c o he r e nc e a nd its me as ur e s t o p l a y a pr omine n t r o le in
de t e ct in g the local i za t ion tr a n sit ion . The l a tt e r h as its r oots in de s tr uct iv e in t e r fe r e nc e r e s ult in g f r om w a v e
pr op a g a t ion w ithin a d i s or de r e d me d ium . H o w e v e r , the w a v e n a tur e of a p a r t icle i s inhe r e n tly pr e s e n t only
in its qua n tum de s cr ipt ion, he nc e the p he nome non of local i za t ion for p a r t icle s i s of qua n tum n a tur e .
I nc omp a t i b le me as ur e me n ts h a v e a r g ua b ly no a n alo g ue in the cl as sical w orld . A lthou gh in a cl as sical
pr o b a b i l i s t ic de s cr ipt ion the r e ex i s t cas e s w he n d i ffe r e n t pr ope r t ie s a r e not sim ult a ne ously kno w n w ith
134
a bs o lut e c e r t ain t y , thi s i s a ttr i but e d t o inc omp le t e kno w le d g e of the s t a t e of the p h ysical s ys t e m . On the
othe r h a nd , a pur e qua n tum s t a t e i s c on side r e d t o pr o v ide a c omp le t e de s cr ipt ion of the s ys t e m .
I n the l as t cas e of s y mme tr ie s in M a rk o v i a n dy n a mic s, the situa t ion i s mor e s ubtle . I n a qua n tum de-
s cr ipt ion, s y mme tr ie s g i v e r i s e t o a one-p a r a me t e r fa mi ly of (g e ne r al ly inde pe nde n t ) monot one s thr ou gh
our c on s tr uct ion, a fa ct th a t r e l ie s on the non- c omm ut a t iv e n a tur e of ope r a t or s . I n the cl as sical ( a be l i a n )
cas e , the r e s ult in g monot one s a r e no lon g e r d i s t inct a nd the w ho le fa mi ly c o l l a ps e s t o a sin gle monot one .
I n a br o a de r f r a me w ork , our w ork be lon gs t o the br a nch of qua n tum infor m a t ion the or y th a t i s c on-
c e r ne d w ith ex p lor in g a nd a n aly z in g the pos si b i l it ie s ( a nd impos si b i l it ie s ) th a t a r e pr e d ict e d b y the l a w s
of qua n tum me ch a nic s f r om a n ope r a t ion al v ie w po in t. U nde r thi s r oof be lon g que s t ion s w ith fa r - r e a chin g
imp l ica t ion s . F or in s t a nc e , w h a t gr a n ts qua n tum c omput e r s w ith the pot e n t i al t o pr o v ide a c omput a t ion al
a dv a n t a g e? W h y doe s qua n tum e n t a n gle me n t al lo w for unbr e ak a b le cr y pt o gr a p h y ? I n w h a t t ask s ca n qua n-
tum the r m al m a chine s out pe r for m the non- qua n tum one s? On the focal po in t of al l s uch que s t ion s l ie a
w h a t a nd a h o w :
• W h a t a r e the in gr e d ie n ts in the qua n tum r e alm th a t c on s t itut e a c e r t ain t ask a tt ain a b le th a t w ould
othe r w i s e be impos si b le?
• H o w ca n the s e in gr e d ie n ts be c omb ine d t o g e the r t o de v i s e no v e l t ask s th a t a r e a b le t o out pe r for m
w h a t i s pos si b le in the non- qua n tum r e alm?
P o w e r f ul m a the m a t ical t oo l s h a v e be e n de v e lope d o v e r the l as t c ou p le of de ca de s t o aid the pr o gr e s s on
the s e t w o br o a d a nd f und a me n t al que s t ion s . I t i s unde ni a b le th a t our unde r s t a nd in g of the r o le of v a r i -
ous qua n tum in gr e d ie n ts, s uch as e n t a n gle me n t , s u pe r posit ion a nd c or r e l a t ion s, h as be nefit e d f r om ne w
the or e t ical t e chnique s . On the othe r h a nd , the r e a r e n ume r ous in s t a nc e s w he r e the a n s w e r s t o s e e min gly
unc onne ct e d que s t ion s in v o lv in g d i ffe r e n t qua n tum in gr e d ie n ts tur n out t o be of simi l a r n a tur e , a nd it i s
y e t uncle a r w h y .
A n in tr i g uin g ex a mp le i s pr o v ide d b y the t ask of r e s our c e m a nipul a t ion . W h a t ca n a g iv e n r e s our c e
be tur ne d in t o , unde r the a ppr opr i a t e r e s tr ict ion s on al lo w e d m a nipul a t ion s? C on side r thr e e qua n tum
r e s our c e s: en t a n g lem en t , a r i sin g f r om the in a b i l it y of d i s t a n t l a bor a t or ie s t o pe r for m j o in t qua n tum ope r a -
135
t ion s, co h er en ce , e me r g in g f r om the pr a ct ical r e s tr ict ion t o e nh a nc e qua n tum s u pe r posit ion, a nd a t h er m a l-
it y , s t e mmin g f r om the d i ffic ult y t o pr oduc e s t a t e s out of the r m al e qui l i br ium . The thr e e a for e me n t ione d
r e s our c e s a r i s e f r om d iv e r s e p h ysical s c e n a r ios . H o w e v e r , the r e s our c e in t e r c on v e r sion th a t ca n be a chie v e d
in al l cas e s i s f ul ly de s cr i be d b y the m a the m a t ical c on s tr uct ion of m aj o r i za t io n . The m ult id ime n sion al g e n-
e r al i za t ion of the l a tt e r tur ne d out t o al s o p l a y a n impor t a n t r o le in our f r a me w ork of inc omp a t i b i l it y . Thi s
ca nnot be due t o me r e ch a nc e; a ft e r al l , w h a t i s pos si b le or impos si b le should fo l lo w f r om the l a w s of qua n-
tum me ch a nic s . H o w e v e r no c ommon p h ysical ex p l a n a t ion i s kno w n for the fa ct th a t they a dmit a a n s w e r
of simi l a r ( a t le as t , m a the m a t ical ) n a tur e .
S e v e r al b i g que s t ion s a r i s e . W h a t i s the r oot of e me r g e nc e of m a j or i za t ion the or y in the s tudy of qua n-
tum r e s our c e s? H o w the d i ffe r e n t qua n tum in gr e d ie n ts a nd pr ope r t ie s, as for in s t a nc e c o he r e nc e , inc om-
p a t i b i l it y a nd s y mme tr ie s s tud ie d in thi s the si s, c on tr i but e t o m ak in g a t ask pos si b le or impos si b le? H o w
ca n one s ys t e m a t ical ly e s t a b l i sh c onne ct ion s be t w e e n d i ffe r e n t r e s our c e s?
W e hope th a t the pr e s e n t the si s c on s t itut e s a non v a ni shin g c on tr i but ion t o de mon s tr a t in g th a t ide as a nd
t oo l s f r om qua n tum infor m a t ion the or y ca n pr o v ide us ef ul in si gh ts t o the a n alysi s of qua n tum pr oc e s s e s .
136
R e fe r e nces
[1] R o l f L a nd a ue r . I nfor m a t ion i s p h ysical . Ph y sic s T o d a y , 44(5):23–29, 1991. doi: 10 . 1063 / 1 . 881299 .
[2] R o y J . Gl a ube r . The qua n tum the or y of opt ical c o he r e nc e . Ph y s . R ev . , 130:2529–2539, J un 1963.
doi: 10 . 1103 /PhysRev. 130 . 2529 .
[3] 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 .
[4] H. P . R o be r ts on . The unc e r t ain t y pr inc ip le . Ph y s . R ev . , 34:163–164, 1929. doi: 10 . 1103 /PhysRev.
34 . 163 .
[5] L e v D . L a nd a u a nd E v g e n y M . L i f shitz . C l assic a l M e c h a n ic s . Add i s on- W e sley , 1959.
[6] H einz - P e t e r B r e ue r a nd F r a nc e s c o P e tr uc c ione . Th e t h e o r y of o p en q u a n t u m sy s t em s . Ox for d U ni -
v e r sit y P r e s s, 2002. doi: 10 . 1093 /acprof:oso/ 9780199213900 . 001 . 0001 .
[7] Be r nh a r d B a um g a r tne r a nd H eide N a r nhofe r . A n alysi s of qua n tum s e mi gr ou ps w ith GK S –
L ind b l a d g e ne r a t or s: I I. Ge ne r al . 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(39):395303,
2008. doi: 10 . 1088 / 1751 - 8113 / 41 / 39 / 395303 .
[8] 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 .
[9] Ge or g ios S t y l i a r i s, 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 . C o he r e nc e- g e ne r a t in g po w e r of
qua n tum de p h asin g pr oc e s s e s . Ph y sic a l R ev iew A , 97(3):032304, 2018. doi: 10 . 1103 /PhysRevA.
97 . 032304 .
[10] 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 . R ev . A , 95:052306, 2017. doi: 10 . 1103 /PhysRevA. 95 .
052306 .
[11] 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.
[12] V itt or io G io v a nne tt i , S e th L lo y d , a nd L or e nz o M a c c one . Qua n tum- e nh a nc e d posit ionin g a nd clock
s y nchr oni za t ion . N a t u r e , 412(6845):417–419, 2001. doi: 10 . 1038 / 35086525 .
[13] V itt or io G io v a nne tt i , S e th L lo y d , a nd L or e nz o M a c c one . Qua n tum- e nh a nc e d me as ur e me n ts: be a t -
in g the s t a nd a r d qua n tum l imit. Scien ce , 306(57 00):1330–1336, 2004. doi: 10 . 1126 /science.
1104149 .
137
[14] R a fal D e mk o w icz - D o br za ń sk i a nd L or e nz o M a c c one . U sin g e n t a n gle me n t a g ain s t no i s e in qua n-
tum me tr o lo g y . Ph y s . R ev . Lett . , 113(25):250801, 2014. doi: 10 . 1103 /PhysRevLett. 113 . 250801 .
[15] 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.
[16] D a nie l A L id a r a nd T odd A B r un . Q u a n t u m er r o r co r r e c t io n . C a mbr id g e U niv e r sit y P r e s s, 2013.
[17] F e r n a ndo G . S . L . B r a nd ã o , M ich ałH or ode ck i , J on a th a n O ppe nheim, J os e p h M . R e ne s, a nd
R o be r t W . S pe k k e n s . R e s our c e the or y of qua n tum s t a t e s out of the r m al e qui l i br ium . Ph y s . R ev .
Lett . , 111:250404, 2013. doi: 10 . 1103 /PhysRevLett. 111 . 250404 .
[18] G i l a d Gour , M a rk us P . M ül le r , V a r un N a r asimh a ch a r , R o be r t W . S pe k k e n s, a nd N ic o le Y un g e r
H alpe r n . The r e s our c e the or y of infor m a t ion al none qui l i br ium in the r mody n a mic s . Ph y sic s R e-
p o r ts , 583:1 – 58, 2015. doi:https://doi.org/ 10 . 1016/j.physrep. 2015 . 04 . 003 .
[19] M a tt e o L os t a gl io , K a mi l Kor z e k w a , D a v id J e nnin gs, a nd T e r r y R udo lp h . Qua n tum c o he r e nc e ,
t ime-tr a n sl a t ion s y mme tr y , a nd the r mody n a mic s . Ph y s . R ev . X , 5:021001, 2015. doi: 10 . 1103 /
PhysRevX. 5 . 021001 .
[20] Gr e g or y S E n g e l , T e s s a R C al houn, El i za be th L R e a d , T a e- K y u A hn, T om áš M a nčal , Y ua n- C h un g
C he n g , R o be r t E B l a nk e n ship , a nd Gr ah a m R Fle min g. E v ide nc e for w a v e l i k e e ne r g y tr a n s -
fe r thr ou gh qua n tum c o he r e nc e in p hot os y n the t ic s ys t e m s . N a t u r e , 446(7137):782–786, 2007.
doi: 10 . 1038 /nature 05678 .
[21] M as oud M o h s e ni , P a tr ick R e be n tr os t , S e th L lo y d , a nd A l a n A spur u- G uz i k . E n v ir onme n t -as si s t e d
qua n tum w al k s in p hot os y n the t ic e ne r g y tr a n sfe r . Th e J o u r n a l of C h em ic a l Ph y sic s , 129(17):1 1B603,
2008. doi: 10 . 1063 / 1 . 3002335 .
[22] S us a nn a F H ue l g a a nd M a r t in B P le nio . V i br a t ion s, qua n t a a nd b io lo g y . Co n t em p o r a r y Ph y sic s ,
54(4):181–207, 2013. doi: 10 . 1080 / 00405000 . 2013 . 829687 .
[23] H o w a r d M W i s e m a n a nd Ge r a r d J M i l bur n . Q u a n t u m m e as u r em en t a n d co n tr o l . C a mbr id g e U ni -
v e r sit y P r e s s, 2009.
[24] 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.
[25] 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 .
[26] 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 . M o d. Ph y s . , 91:025001, 2019.
doi: 10 . 1103 /RevModPhys. 91 . 025001 .
[27] 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.
[28] 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 . , 89:041003, 2017. doi: 10 . 1103 /RevModPhys. 89 . 041003 .
138
[29] 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 . , 113:140401,
2014. doi: 10 . 1103 /PhysRevLett. 113 . 140401 .
[30] J o hn P r e sk i l l . L e ctur e not e s for p h ysic s 229: Qua n tum infor m a t ion a nd c omput a t ion, 1998.
[31] 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 , 94:052336, 2016. doi: 10 . 1103 /PhysRevA. 94 . 052336 .
[32] 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 s . R ev . A , 94:052324, 2016. doi: 10 . 1103 /PhysRevA. 94 . 052324 .
[33] 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 . ,
116:120404, 2016. doi: 10 . 1103 /PhysRevLett. 116 . 120404 .
[34] J ul io I de V ic e n t e a nd A lex a nde r S tr e lts o v . Ge n uine qua n tum c o he r e nc e . 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 , 50(4):045301, 2016. doi: 10 . 1088 / 1751 - 8121 / 50 / 4 / 045301 .
[35] U tt a m S in gh, L in Z h a n g , a nd A r un K um a r P a t i . A v e r a g e c o he r e nc e a nd its t y p ical it y for r a ndom
pur e s t a t e s . Ph y sic a l R ev iew A , 93(3):032125, 2016. doi: 10 . 1103 /PhysRevA. 93 . 032125 .
[36] L in Z h a n g , U tt a m S in gh, a nd A r un K P a t i . A v e r a g e s ube n tr op y , c o he r e nc e a nd e n t a n gle me n t of
r a ndom mi xe d qua n tum s t a t e s . A n n a ls of Ph y sic s , 377:125–146, 2017. doi: 10 . 1016/j.aop. 2016 .
12 . 024 .
[37] D a v id P e r ez - G a r c i a , M ich a e l M W o l f , D e ne s P e tz , a nd M a r y Be th R usk ai . C on tr a ct iv it y of posit iv e
a nd tr a c e-pr e s e r v in g m a ps unde r l p nor m s . J o u r n a l of M a t h em a t ic a l Ph y sic s , 47(8):083506, 2006.
doi: 10 . 1063 / 1 . 2218675 .
[38] 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 , 95:052307, 2017. doi: 10 . 1103 /PhysRevA.
95 . 052307 .
[39] 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,
2018. doi: 10 . 1063 / 1 . 4997146 .
[40] L in Z h a n g , Z hi h a o M a , Z hi h ua C he n, a nd S h a o - M in g F ei . C o he r e nc e g e ne r a t in g po w e r of unit a r y
tr a n sfor m a t ion s v i a pr o b a b i l i s t ic a v e r a g e . Q u a n t u m I nf o r m a t io n P r o ce ssi n g , 17(7):186, 2018. doi:
10 . 1007 /s 11128 - 018 - 1928 - 4 .
[41] S h unlon g L uo a nd Y ua n S un . P a r t i al c o he r e nc e w ith a pp l ica t ion t o the monot onic it y pr o b le m of
c o he r e nc e in v o lv in g sk e w infor m a t ion . Ph y s . R ev . A , 96:022136, 2017. doi: 10 . 1103 /PhysRevA.
96 . 022136 .
[42] S h unlon g L uo a nd Y ua n S un . Qua n tum c o he r e nc e v e r s us qua n tum unc e r t ain t y . Ph y s . R ev . A ,
96:022130, 2017. doi: 10 . 1103 /PhysRevA. 96 . 022130 .
[43] M ich a e l W o l f . Qua n tum ch a nne l s & ope r a t ion s: G uide d t our , 2012. UR L : https://www- m 5 .
ma.tum.de/foswiki/pub/M 5 /Allgemeines/MichaelWolf/QChannelLecture.pdf .
139
[44] R ich a r d J o z s a , D a nie l R o b b , a nd W i l l i a m K . W oott e r s . L o w e r bound for a c c e s si b le infor m a t ion in
qua n tum me ch a nic s . Ph y s . R ev . A , 49:668–6 77, 1994. doi: 10 . 1103 /PhysRevA. 49 . 668 .
[45] B r uc e C Be r ndt , R on ald J E v a n s, a nd Ke nne th S W i l l i a m s . G a uss a n d J a co bi s u m s . W i ley N e w Y ork ,
1998.
[46] R oe Goodm a n a nd N o l a n R W al l a ch . R epr e se n t a t io n s a n d i n v a r i a n ts of t h e c l assic a l g r o u p s , v o lume 68.
C a mbr id g e U niv e r sit y P r e s s, 2000.
[47] P a o lo Za n a r d i . L ocal r a ndom qua n tum c ir c uits: E n s e mb le c omp le t e ly posit iv e m a ps a nd s w a p
al g e br as . J o u r n a l of M a t h em a t ic a l Ph y sic s , 55(8):082204, 2014. doi: 10 . 1063 / 1 . 4891604 .
[48] 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.
[49] I n g e m a r Be n g ts s on, Å s a E r ic s s on, M a r e k K uś, W oj c ie ch T a dej , a nd K a r o l Ż y cz k o w sk i . B irk hoff ’ s
po ly t ope a nd uni s t och as t ic m a tr ic e s, n= 3 a nd n= 4. 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 ,
259(2):307–324, 2005. doi: 10 . 1007 /s 00220 - 005 - 1392 - 8 .
[50] A a r on C a rl S mith . U ni s t och as t ic m a tr ic e s a nd r e l a t e d pr o b le m s . I n M a t h em a t ic s a n d Co m pu t i n g ,
p a g e s 239–250. S pr in g e r , 2015. doi: 10 . 1007 / 978 - 81 - 322 - 2452 - 5 _ 16 .
[51] 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 , 100:224204, 2019. doi: 10 . 1103 /PhysRevB. 100 .
224204 .
[52] 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 .
[53] 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 .
[54] 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 .
[55] 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, 2015. doi: 10 . 1146 /
annurev- conmatphys- 031214 - 014726 .
[56] 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 .
[57] 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 , 6:0 41028, 2016. doi: 10 . 1103 /PhysRevX. 6 . 041028.
[58] 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 .
140
[59] 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 .
[60] 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 1:052120, 2015. doi: 10 . 1103 /PhysRevA. 91 . 052120 .
[61] 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.
[62] 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 . , 101:190403, 2008. doi: 10 . 1103 /PhysRevLett. 101 . 190403 .
[63] 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 , 79:061103, 2009. doi: 10 . 1103 /PhysRevE. 79 .
061103 .
[64] 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 , 81:022113, 2010. doi: 10 . 1103 /PhysRevA. 81 . 022113 .
[65] 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 . , 107:010403, 2011.
doi: 10 . 1103 /PhysRevLett. 107 . 010403 .
[66] 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 .
[67] 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.
[68] 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 .
[69] 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 .
[70] 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 . , 91:021001, 2019. doi: 10 . 1103 /
RevModPhys. 91 . 021001 .
[71] 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 .
[72] 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 .
141
[73] 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 .
[74] 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.
[75] 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 .
[76] 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 .
[77] 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 .
[78] 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 .
[79] 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 .
[80] 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 . , 99:100603, 2007. doi: 10 . 1103 /PhysRevLett.99 .
100603 .
[81] 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 sic s R ep o r ts , 697:1 – 87, 2017. doi:
https://doi.org/ 10 . 1016 /j.physrep. 2017 . 07. 001 .
[82] 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 .
[83] 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 .
[84] 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 .
[85] 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 .
[86] 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, 2008. doi: 10 . 1103 /PhysRevLett. 101 . 120603 .
[87] 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.
142
[88] 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 inc omp a t i b i l it y of qua n tum me as ur e me n ts
r e l a t iv e t o a b asi s . Ph y s . R ev . L ett . , 123:070401, 2019. doi: 10 . 1103 /PhysRevLett. 123 . 070401 .
[89] E S chr öd in g e r . Z um hei s e nbe r gs che n un s ch ä r fe pr inz ip . Sit z u n g s b er ic h t e der P r eussisc h en Ak a dem ie
der W issen sc h aft en , Ph y si k a l isc h -m a t h em a t isc h e K l asse , 14:296–303, 1930.
[90] L or e nz o M a c c one a nd A r un K . P a t i . S tr on g e r unc e r t ain t y r e l a t ion s for al l inc omp a t i b le o bs e r v a b le s .
Ph y s . R ev . Lett . , 113:260401, 2014. doi: 10 . 1103 /PhysRevLett. 113 . 260401 .
[91] D a v id D e uts ch . U nc e r t ain t y in qua n tum me as ur e me n ts . Ph y s . R ev . Lett . , 50:631–633, 1983. doi:
10 . 1103 /PhysRevLett. 50 . 631 .
[92] H a n s M a as s e n a nd J . B . M . U ffink . Ge ne r al i z e d e n tr op ic unc e r t ain t y r e l a t ion s . Ph y s . R ev . Lett . ,
60:1103–1106, 1988. doi: 10 . 1103 /PhysRevLett. 60 . 1103 .
[93] M a tthi as C hr i s t a nd l a nd A ndr e as W in t e r . U nc e r t ain t y , mono g a m y a nd lock in g of qua n tum c or -
r e l a t ion s . I n IS IT 2005: P r o ce e d i n g s of t h e I n t er n a t io n a l S y m p o si u m o n I nf o r m a t io n Th e o r y , p a g e s
879–883. I EEE, 2005.
[94] P a tr ick J . C o le s, R o g e r C o l be ck , L i Y u , a nd M ich a e l Z w o l ak . U nc e r t ain t y r e l a t ion s f r om simp le
e n tr op ic pr ope r t ie s . Ph y s . R ev . Lett . , 108:210405, 2012. doi: 10 . 1103 /PhysRevLett. 108 . 210405 .
[95] K a mi l Kor z e k w a , M a tt e o L os t a gl io , D a v id J e nnin gs, a nd T e r r y R udo lp h . Qua n tum a nd cl as sical e n-
tr op ic unc e r t ain t y r e l a t ion s . Ph y s . R ev . A , 89:042122, 2014. doi: 10 . 1103 /PhysRevA. 89 . 042122 .
[96] 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 . M o d. Ph y s . , 89:015002, 2017. doi: 10 . 1103 /RevModPhys.
89 . 015002 .
[97] 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):1–12, 2019. doi: 10 . 1038 /
s 42005 - 019 - 0179 - 8 .
[98] S h unlon g L uo . H ei s e nbe r g unc e r t ain t y r e l a t ion for mi xe d s t a t e s . Ph y s . R ev . A , 72:042110, 2005.
doi: 10 . 1103 /PhysRevA. 72 . 042110 .
[99] S h unlon g L uo . Qua n tum unc e r t ain t y of mi xe d s t a t e s b as e d on sk e w infor m a t ion . Ph y s . R ev . A ,
73:022324, 2006. doi: 10 . 1103 /PhysRevA. 73 . 022324 .
[100] D . L i , X . L i , F . W a n g , H. H ua n g , X . L i , a nd L . C . K w e k . U nc e r t ain t y r e l a t ion of mi xe d s t a t e s b y me a n s
of w i gne r - ya n as e- dys on infor m a t ion . Ph y s . R ev . A , 79:052106, 2009. doi: 10 . 1103 /PhysRevA. 79 .
052106 .
[101] P a o lo G i b i l i s c o , D a nie le I mp a r a t o , a nd T omm as o I s o l a . U nc e r t ain t y pr inc ip le a nd qua n tum fi she r
infor m a t ion . i i . J o u r n a l of m a t h em a t ic a l p h y sic s , 48(7):072109, 2007. doi: 10 . 1063 / 1 . 2748210 .
[102] P a o lo G i b i l i s c o , F umio H i ai , a nd D é ne s P e tz . Qua n tum c o v a r i a nc e , qua n tum fi she r infor m a t ion,
a nd the unc e r t ain t y r e l a t ion s . I E E E T r a n sa c t io n s o n I nf o r m a t io n Th e o r y , 55( 1):439–443, 20 09. doi:
10 . 1109 /TIT. 2008 . 2008142 .
143
[103] N a n L i , S h unlon g L uo , a nd Y ua n y ua n M a o . Qua n t i f y in g the qua n tumne s s of e n s e mb le s . Ph y s . R ev .
A , 96:022132, 2017. doi: 10 . 1103 /PhysRevA. 96 . 022132 .
[104] T ei k o H einos a a r i , J uk k a Kiuk as, a nd D a nie l R eitzne r . N o i s e r o bus tne s s of the inc omp a t i b i l it y of
qua n tum me as ur e me n ts . Ph y s . R ev . A , 92:022115, 2015. doi: 10 . 1103 /PhysRevA. 92 . 022115 .
[105] R oope U o l a , C os t a n t ino B udr oni , Otf r ie d G ühne , a nd J uh a - P e k k a P e l lonp ä ä . One-t o - one m a pp in g
be t w e e n s t e e r in g a nd j o in t me as ur a b i l it y pr o b le m s . Ph y s . R ev . Lett . , 115:230402, 2015. doi: 10 .
1103 /PhysRevLett. 115 . 230402 .
[106] T ei k o H einos a a r i , J uk k a Kiuk as, D a nie l R eitzne r , a nd J us si S ch ultz . I nc omp a t i b i l it y br e ak in g qua n-
tum ch a nne l 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 , 48(43):435301, 2015. doi:
10 . 1088 / 1751 - 8113 / 48 / 43 / 435301 .
[107] E rk k a H a a p as alo . R o bus tne s s of inc omp a t i b i l it y for qua n tum de v ic e s . J o u r n a l of Ph y sic s A : M a t h e-
m a t ic a l a n d Th e o r et ic a l , 48( 25):255303, 2015. doi: 10 . 1088 / 1751 - 8113 / 48 / 25 / 255303 .
[108] D . C a v alca n t i a nd P . S kr z y pcz y k . Qua n t it a t iv e r e l a t ion s be t w e e n me as ur e me n t inc omp a t i b i l it y ,
qua n tum s t e e r in g , a nd nonlocal it y . Ph y s . R ev . A , 93:052112, 2016. doi: 10 . 1103 /PhysRevA. 93 .
052112 .
[109] C l a ud io C a r me l i , T ei k o H einos a a r i , a nd A le s s a ndr o T o i g o . S t a t e d i s cr imin a t ion w ith pos tme as ur e-
me n t infor m a t ion a nd inc omp a t i b i l it y of qua n tum me as ur e me n ts . Ph y s . R ev . A , 98:012126, 2018.
doi: 10 . 1103 /PhysRevA. 98 . 012126.
[110] C l a ud io C a r me l i , T ei k o H einos a a r i , a nd A le s s a ndr o T o i g o . Qua n tum inc omp a t i b i l it y w itne s s e s .
Ph y s . R ev . Lett . , 122:130402, 2019. doi: 10 . 1103 /PhysRevLett. 122 . 130402 .
[111] R oope U o l a , T r i s t a n Kr a ft , J i a n g w ei S h a n g , X i a o - D on g Y u , a nd Otf r ie d G ühne . Qua n t i f y in g
qua n tum r e s our c e s w ith c onic pr o gr a mmin g. Ph y s . R ev . Lett . , 122:130404, 2019. doi: 10 . 1103 /
PhysRevLett. 122 . 130404 .
[112] P a ul S kr z y pcz y k , I v a n Š u p ić, a nd D a nie l C a v alca n t i . A l l s e ts of inc omp a t i b le me as ur e me n ts g iv e
a n a dv a n t a g e in qua n tum s t a t e d i s cr imin a t ion . Ph y s . R ev . Lett . , 122:130403, 2019. doi: 10 . 1103 /
PhysRevLett. 122 . 130403 .
[113] God f r ey H a r o ld H a r dy , J o hn E de n s or L ittle w ood , a nd Ge or g e P ó lya . I n e q u a l i t ie s . C a mbr id g e U ni -
v e r sit y P r e s s, 1988.
[114] 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 . , 81:865–942, 2 009. doi: 10 . 1103 /RevModPhys. 81 . 865 .
[115] Ge or g ios S t y l i a r i s, 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 . C o he r e nc e- g e ne r a t in g po w e r of
qua n tum de p h asin g pr oc e s s e s . Ph y s . R ev . A , 97:032304, 2018. doi: 10 . 1103 /PhysRevA. 97 .
032304 .
[116] S a m ue l K a rl in a nd Y os ef R inott. C omp a r i s on of me as ur e s, m ult iv a r i a t e m a j or i za t ion, a nd a pp l ica -
t ion s t o s t a t i s t ic s . I n S t u d ie s i n Eco n o m etr ic s, T i m e Ser ie s, a n d M u l t i v a r i a t e S t a t is t ic s , p a g e s 465–489.
El s e v ie r , 1983.
144
[117] 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 .
[118] 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 sic a l R ev iew A , 95(5), 2017. doi: 10 . 1103 /PhysRevA. 95 .
052306 .
[119] Thom as Dur t , Be r tho ld - Ge or g E n gle r t , 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 . On m utual ly
unb i as e d b as e s . I n t er n a t io n a l j o u r n a l of q u a n t u m i nf o r m a t io n , 8(04):535–640, 2010. doi: 10 . 1142 /
S 0219749910006502 .
[120] A l f r e d H or n . D oub ly s t och as t ic m a tr ic e s a nd the d i a g on al of a r ot a t ion m a tr i x . A m er ic a n J o u r n a l of
M a t h em a t ic s , 76(3):620–630, 1954.
[121] P M A l be r t i . W a ch s e nde übe r g a n gs w ahr s cheinl ichk eit e n und v o l l s t ä nd i g posit iv e s t och as t i s che
tr a n sfor m a t ione n . W iss . Z . K M U Leip zi g , M a t h.- N a t u r w iss . R , 31(1):3–10, 1982.
[122] P e t e r M A l be r t i , Be r nd C r e l l , A r min U hlm a nn, a nd C hr i s t i a n Z y l k a . Or de r s tr uctur e ( m a j or i za -
t ion ) a nd ir r e v e r si b le pr oc e s s e s . I n E r n s t - C hr i s t op h H as s a nd P e t e r J or g P l a th, e d it or s, V er n et z t e
W issen sc h aft en : C r o ss l i n k s i n N a t u r a l a n d So ci a l Scien ce s , p a g e s 281–290. L o g os V e rl a g Be rl in, 2008.
[123] T ei k o H einos a a r i , T ak a y uk i M iya de r a , a nd M á r io Z im a n . A n in v it a t ion t o qua n tum inc omp a t i b i l it y .
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(12):123001, 2016. doi: 10 . 1088 / 1751 - 8113 /
49 / 12 / 123001 .
[124] 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 . S y mme tr ie s a nd monot one s in m a rk o v i a n qua n tum dy n a m-
ic s . Q u a n t u m , 4:261, 2020. doi: 10 . 22331 /q- 2020 - 04 - 30 - 261 .
[125] J un J o hn S ak ur ai a nd J im N a po l it a no . M o der n q u a n t u m m e c h a n ic s . C a mbr id g e U niv e r sit y P r e s s,
2017. doi: 10 . 1017 / 9781108499996.
[126] V ict or V . A l be r t a nd L i a n g J i a n g. S y mme tr ie s a nd c on s e r v e d qua n t it ie s in L ind b l a d m as t e r e qua -
t ion s . Ph y s . R ev . A , 89:022118, 2014. doi: 10 . 1103 /PhysRevA. 89 . 022118 .
[127] V ict or V . A l be r t. A s y mpt ot ic s of qua n tum ch a nne l s: c on s e r v e d qua n t it ie s, a n a d i a b a t ic l imit , a nd
m a tr i x pr oduct s t a t e s . Q u a n t u m , 3:151, 2019. doi: 10 . 22331 /q- 2019- 06 - 06 - 151 .
[128] S t e p he n D . B a r tle tt , T e r r y R udo lp h, a nd R o be r t W . S pe k k e n s . R efe r e nc e f r a me s, s u pe r s e le ct ion
r ule s, a nd qua n tum infor m a t ion . R ev . M o d. Ph y s . , 79:555–609, 2007. doi: 10 . 1103 /RevModPhys.
79 . 555 .
[129] 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 . The the or y of m a nipul a t ion s of pur e s t a t e as y mme-
tr y: I. B asic t oo l s, e quiv ale nc e cl as s e s a nd sin gle c op y tr a n sfor m a t ion s . N ew J o u r n a l of Ph y sic s ,
15(3):033001, 2013. doi: 10 . 1088 / 1367 - 2630 / 15 / 3 / 033001 .
[130] 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 . E x t e nd in g N oe the r ’ s the or e m b y qua n t i f y in g the as y mme-
tr y of qua n tum s t a t e s . N a t u r e co m m u n ic a t io n s , 5:3821, 2014. doi: 10 . 1038 /ncomms 4821 .
[131] M a tt e o L os t a gl io , K a mi l Kor z e k w a , a nd A n t on y M i lne . M a rk o v i a n e v o lut ion of qua n tum c o he r e nc e
unde r s y mme tr ic dy n a mic s . Ph y s . R ev . A , 96:0 32109, 2017. doi: 10 . 1103 /PhysRevA. 96 . 032109 .
145
[132] 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 spor 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, 2012. doi:
10 . 1088 / 1367 - 2630/ 14 / 7 / 073007 .
[133] E dw a r d W itt e n . A mini -in tr oduct ion t o infor m a t ion the or y . a r X i v:1805.11965 , 2018.
[134] Kr i s t a n T e mme , M ich a e l J a me s K as t or ya no , M a r y Be th R usk ai , M ich a e l M a r c W o l f , a nd F r a nk V e r -
s tr a e t e . The χ
2
- d iv e r g e nc e a nd mi x in g t ime s of qua n tum M a rk o v pr oc e s s e s . J o u r n a l of M a t h em a t ic a l
Ph y sic s , 51(12):122201, 20 10. doi: 10 . 1063 / 1 . 3511335 .
[135] M ich a e l J . K as t or ya no , D a v id R e e b , a nd M ich a e l M . W o l f . A c ut off p he nome non for qua n tum
M a rk o v ch ain 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 , 45(7):075307, 2012. doi:
10 . 1088 / 1751 - 8113 / 45 / 7 / 075307 .
[136] M ich a e l J . K as t or ya no a nd Kr i s t a n T e mme . Qua n tum lo g a r ithmic S o bo le v ine qual it ie s a nd r a p id
mi x in g. J o u r n a l of M a t h em a t ic a l Ph y sic s , 54(5):052202, 2013. doi: 10 . 1063 / 1 . 4804995 .
[137] F r a nk H a n s e n . M e tr ic a d jus t e d sk e w infor m a t ion . P r o ce e d i n g s of t h e N a t io n a l Ac a dem y of Scien ce s ,
105(29):9909–9916, 2008. doi: 10 . 1073 /pnas. 0803323105 .
[138] E . A . M or o z o v a a nd N . N . C he n ts o v . M a rk o v in v a r i a n t g e ome tr y on m a ni fo ld s of s t a t e s . J o u r n a l of
So v iet M a t h em a t ic s , 56(5):2648–2669, 1991. doi: 10 . 1007 /BF 01095975 .
[139] D é ne s P e tz . M onot one me tr ic s on m a tr i x sp a c e s . L i n e a r Al ge br a a n d its App l ic a t io n s , 244:81 – 96,
1996. doi: 10 . 1016 / 0024 - 3795 ( 94 ) 00211 - 8 .
[140] D é ne s P e tz a nd C s a b a S ud á r . Ge ome tr ie s of qua n tum s t a t e s . J o u r n a l of M a t h em a t ic a l Ph y sic s ,
37(6):2662–2673, 1996. doi: 10 . 1063 / 1 . 531535 .
[141] D é ne s P e tz a nd M a r y Be th R usk ai . C on tr a ct ion of g e ne r al i z e d r e l a t iv e e n tr op y unde r s t och as -
t ic m a pp in gs on m a tr ic e s . I nfi n it e Di m en sio n a l A n a l y sis, Q u a n t u m P r o b a bi l it y a n d R e l a t e d T o pic s ,
1(01):83–89, 1998. doi: 10 . 1142 /S 0219025798000077 .
[142] D é ne s P e tz . Quasi - e n tr op ie s for finit e qua n tum s ys t e m s . R ep o r ts o n m a t h em a t ic a l p h y sic s , 23(1):57–
65, 1986. doi: 10 . 1016 / 0034 - 4877 ( 86 ) 90067 - 4 .
[143] A ndr e w L e s nie w sk i a nd M a r y Be th R usk ai . M onot one R ie m a nni a n me tr ic s a nd r e l a t iv e e n tr op y
on nonc omm ut a t iv e pr o b a b i l it y sp a c e s . J o u r n a l of M a t h em a t ic a l Ph y sic s , 40(11):5702–5724, 1999.
doi: 10 . 1063 / 1 . 533053 .
[144] M a r t in I de l . On the s tr uctur e of posit iv e m a ps . M as t e r ’ s the si s, T e chnical U niv e r sit y of M unich,
2013.
[145] J a r osl a v N o v otn ỳ , J iř í M a r yšk a , a nd I g or J ex . Qua n tum M a rk o v pr oc e s s e s: F r om a ttr a ct or s tr uctur e
t o ex p l ic it for m s of as y mpt ot ic s t a t e s . Th e E u r o p e a n Ph y sic a l J o u r n a l Pl us , 133(8):310, 2018. doi:
10 . 1140 /epjp/i 2018 - 12109 - 8 .
[146] E m a n ue l Kni l l , R a y mond L a fl a mme , a nd L or e nza V io l a . The or y of qua n tum e r r or c or r e ct ion for
g e ne r al no i s e . Ph y s . R ev . Lett . , 84:2525–2528, 2000. doi: 10 . 1103 /PhysRevLett. 84 . 2525 .
146
[147] P a o lo Za n a r d i . S t a b i l i z in g qua n tum infor m a t ion . Ph y s . R ev . A , 63:012301, 2000. doi: 10 . 1103 /
PhysRevA. 63 . 012301 .
[148] D a v id W . Kr i bs . Qua n tum ch a nne l s, w a v e le ts, d i l a t ion s a nd r e pr e s e n t a t ion s of O
n
. P r o ce e d i n g s of
t h e Ed i n bu r g h M a t h em a t ic a l So ciet y , 46(2):421–433, 2003. doi: 10 . 1017 /S 0013091501000980 .
[149] E . B r i a n D a v ie s . M a rk o v i a n m as t e r e qua 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 , 39(2):91–
110, 1974. doi: 10 . 1007 /BF 01608389 .
[150] R o be r t A l ick i a nd K a rl L e nd i . Q u a n t u m d y n a m ic a l sem i g r o u p s a n d a pp l ic a t io n s , v o lume 717.
S pr in g e r - V e rl a g , 2007. doi: 10 . 1007 / 3 - 540 - 70861 - 8 .
[151] W oj c ie ch R o g a , M a rk F a nne s, a nd K a r o l Ż y cz k o w sk i . D a v ie s m a ps for qub its a nd qutr its . R ep o r ts
o n M a t h em a t ic a l Ph y sic s , 66(3):311–329, 2010. doi: 10 . 1016/S 0034 - 4877 ( 11 ) 00003 - 6 .
[152] 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 sy s t em s . S pr in g e r - V e rl a g , 2012. doi: 10 . 1007 /
978 - 3 - 642 - 23354 - 8 .
[153] 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 . D y n a mical r e spon s e the or y for dr iv e n- d i s sip a t iv e
qua n tum s ys t e m s . Ph y s . R ev . A , 93:032101, 2016. doi: 10 . 1103 /PhysRevA. 93 . 032101 .
147

Abstract (if available)

Abstract We explore from an information-theoretic perspective three aspects of quantum processes with no apparent classical counterpart: (i) generation of coherence in quantum evolutions, (ii)incompatibility of quantum measurements, and (iii) implications of symmetries in Markovian quantum dynamics. We develop a family of measures that quantify the average effectiveness of a quantum evolution to generate quantum coherence out of incoherent states. We examine its typical behavior and employ it to study localization in quantum mechanics. Measurement incompatibility is famously expressed through uncertainty relations. Here we lay down a framework to capture incompatibility by means of an ordering. The main idea behind the definition relies on measurement emulation

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University of Southern California Dissertations and Theses

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Asset Metadata

Creator Styliaris, Georgios (author)

Core Title Coherence generation, incompatibility, and symmetry in quantum processes

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 07/16/2020

Defense Date 02/20/2020

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag coherence,incompatibility,Markovian dynamics,OAI-PMH Harvest,quantum mechanics,quantum processes

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Zanardi, Paolo (committee chair), Brun, Todd (committee member), Campos Venuti, Lorenzo (committee member), Lidar, Daniel (committee member), Mitra, Urbashi (committee member)

Creator Email stiliaris@gmail.com,styliari@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-330374

Unique identifier UC11663804

Identifier etd-StyliarisG-8684.pdf (filename),usctheses-c89-330374 (legacy record id)

Legacy Identifier etd-StyliarisG-8684.pdf

Dmrecord 330374

Document Type Dissertation

Rights Styliaris, Georgios

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA