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Quantum information techniques in condensed matter: quantum equilibration, entanglement typicality, detection of topological order
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Quantum information techniques in condensed matter: quantum equilibration, entanglement typicality, detection of topological order
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QUANTUM INFORMATION TECHNIQUES IN CONDENSED MATTER: QUANTUM EQUILIBRATION, ENTANGLEMENT TYPICALITY, DETECTION OF TOPOLOGICAL ORDER by Siddhartha Santra A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS AND ASTRONOMY) May 2014 Copyright 2014 Siddhartha Santra Dedication To my Mother ii Acknowledgements This thesis is a product of the constant encouragement and inspiration I received from my advisor Paolo Zanardi at USC and mentor Alioscia Hamma at the PITP. I have learnt a lot from their different scientific personalities and expertise and owe them both an immense amount of gratitude for guiding me through my scientific infancy. I was also fortunate to be able to discuss science with Lorenzo Campos Venuti and LuigiAmicoandthankthemfortheirencouragementaswell. Thegroupmeetingsofthe Center for Quantum Information Science and Technology held in Daniel Lidar’s office wereagreatsourceofenergy-bothintermsofideasandnourishment,andIthankhim for his hospitality and his lucid explanations of many concepts. I thank Todd Brun for his excellent course on quantum information that first introduced the subject to me. I also thank Noah Tobias Jacobson and Damian Abasto who patiently welcomed me to quantum information back in 2010. ThroughoutmystayatUSC,AmitChoubeyandGregoryQuirozhavebeenscientific colleagueswhosepersonalsupporthasbeencrucialformeandIthankthemfordiscus- sions on science, religion, financial mathematics, American society, west coast rap and efficientworkoutpracticestonamejustafew. IwouldalsoliketothankDharmeshJain at CNYITP, Stony Brook who has been a companion throughout my graduate studies lending a patient ear and crucial help at many junctures. iii My thanks also go to everyone at the Department of Physics and Astronomy, USC and in particular the graduate student advisors Stephan Haas, Richard Thomson and Robin Shakeshaft for their patience and guidance. I cannot thank enough Betty Byers whoseefficiencymadeapplyingtoUSCandallthepaperworkwhileatUSCunimagin- ably easier. I am also grateful to the Dornsife College of Letters, Arts and Sciences for the award of the dissertation completion fellowship in the final year of my studies here. Finally I must thank my parents, Maya and Shyamapada, who, despite a distance of 8187 miles between my small home town of Jamshedpur in India and Los Angeles, have been my strongest source of motivation and I hope to be able to explain to them someday what I had been upto so far away. iv TableofContents Dedication ii Acknowledgements iii ListofTables vii ListofFigures viii Abstract xi Chapter1: IntroductionandPlanoftheThesis 1 Chapter2: SignaturesofEquilibrationinClosedQuantumSystems 5 2.1 Closed System Equilibration . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 A Model Closed Quantum System - Quantum XY Spin Chain . . . . . . 7 2.2.1 Choosing an Observable - The Loschmidt Echo . . . . . . . . . . 7 2.2.2 XY model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Signature of Equilibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3.1 Off-critical regime and Gaussian equilibration . . . . . . . . . . 10 2.3.2 Quasi-critical regime and universal critical equilibration . . . . . 11 2.3.3 Beyond the XY model . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Chapter3: EntanglementinthePhysicalHilbertSpace 16 3.1 Random Quantum Circuits in Quantum Information . . . . . . . . . . . . 17 3.2 Setup and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2.1 Definition of a Local Random Quantum Circuit . . . . . . . . . . 24 3.2.2 Ensemble Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2.3 Integration over the Haar measure, Algebra of Swaps . . . . . . 26 3.2.4 Entanglement Dynamics and Dynamically Closed Swap Algebras 28 3.3 Models of Local Random Quantum Circuits . . . . . . . . . . . . . . . . 30 3.3.1 Uncorrelated choices of local regions . . . . . . . . . . . . . . . . 32 3.3.2 Correlated choices of local regions . . . . . . . . . . . . . . . . . 46 v 3.3.3 Distance to the maximally mixed state . . . . . . . . . . . . . . . 58 3.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Chapter4: DetectionofTopologicalOrderinPhasesofMatter 62 4.1 Topological Order and its Detection . . . . . . . . . . . . . . . . . . . . . 63 4.2 PerturbativeResponseofR´ enyientropies-SignatureofTopologicalOrder 64 4.3 General Strategy and Mathematical Preliminaries . . . . . . . . . . . . . 66 4.3.1 General strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.3.2 R´ enyi Entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.3.3 Manifold of topologically ordered ground states . . . . . . . . . 68 4.3.4 Differential local convertibility on the ground state manifold . . 69 4.3.5 The models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.4.1 The Castelnovo-Chamon model . . . . . . . . . . . . . . . . . . . 75 4.4.2 Toric code with magnetic field along spins on rows . . . . . . . . 77 4.4.3 The Toric-Ising model . . . . . . . . . . . . . . . . . . . . . . . . 86 4.4.4 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Chapter5: ConclusionsandOutlook 91 References 95 vi ListofTables 3.1 Minimum number of iterations required to getǫ close to minimal purity. 60 vii ListofFigures 2.1 Typical behavior ofL(t). The inset shows Gaussian behavior for short times, as happens for the pure case [7]. Here L = 100, β = 6, h 0,1 = 1,γ 0 =0.5,γ 1 =0.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 P Z closetotheIsing(left)andanisotropytransition(right). AsLgrows we enter the off-critical regime and P Z becomes Gaussian. Close to the quasi-critical regime (small L) the distribution becomes a broad, generally double-peaked function. For the anisotropy transition, one can have L for which the highest amplitudes are nearly equal. This results in a collapse from two peaks to one. Parameters areβ = 40 and (left) h 0 = 0.98, h 1 = 1.02, γ 0,1 = 1.0 and L = 50 to 200 in steps of 30, (right)h 0,1 = 0.5, γ 0 = 0.01, γ 1 = 0.01 andL = 50 to 100 in steps of 10. Another way to enter the off-critical regime is to increase the temperature. SimilarplotsareobtainedreplacingLwiththetemperature T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1 An example of a random quantum circuit. Different horizontal lines indicate different qubits on which gates (possibly different) are applied at each step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2 Schematic of the RandomEdge Model. The blue circles represent local Hilbert spaces at the verticesV of the graphG= (V,E) with the edges E shown as green arrows. For the bipartionV = A∪B the change in purity for the state of the subsystemA is effected only by edges shown in red that straddle the boundary. . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 [i](Leftpanel)PurityasgivenbyEq.(4.20)withfixedtotalsizeN =10 anddifferentsubsystemsizesN A =1(blue circles),N A =2(purple squares),N A = 3(yellow rhombi),N A =4(green upright triangles),N A =5(dark blue inverted triangles). The horizontal lines are asymptotic values given by (3.42). [ii] (Right panel)Naturallogarithmofthedifferencebetweenpurityatthek’thstep and the asymptotic purity withN = 10 and same color codes forN A as in [i]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 viii 3.4 [i] (Top panel) Scaling behavior of gap (Δ) for the superoperator of the fully connected RQC. The best fit line (red) gives us the equation (in natural base): log[Δ] = −.969395log[L]+.0246366 [ii] (Bottom panel) Scaling behavior of the natural logarithm of the product of oper- ator norms for the similarity transform matrixM and it’s Inverse. The best fit line (red) giveslog[""M"" ∞ ""M −1 "" ∞ ]=1.17508+0.318L. . . . . 43 3.5 A bipartite (A,B) spin chain of length L = L A +L B with nearest- neighbor qubits interacting via 2−qubit gates (ellypses). The edge e is the one that straddles the two partitions. The gates are numbered by thesubscriptx i wherex=A,B denotesthetwohalvesofthechainand i the distance from the boundary of the two partitions. . . . . . . . . . . 47 3.6 [i] (Left panel) Purity as a function ofn c for the best sequence of uni- taries(circles) and for the worst (squares) forL A =L B =8. The smooth lineconnectingthesquaresistheanalyticexpressionforworstcase. The other curves represent the purities obtained for five random sequences. One can clearly see that these data lie between the best and the worst P nc . [ii] (Right panel) Convergence of the worst case sequence to the asymptotic formula value for different values of system/environment length. Blue circles(L = 5,L A = 2) and Yellow rhombi (L = 6,L A = 3) convergeto>99%oftheirasymptoticvaluewhereasPurplesquares(L= 8,L A =3) converges to>97%. . . . . . . . . . . . . . . . . . . . . . . . . 54 3.7 (a) (Left panel) Saturation of the sub-dominant eigenvalue forL A =L B with increasingL A for a chain of qubits. Similar behavior is observed for general qudits. (b) (Right panel) Distribution of eigenvalues for a chain of qubits with L A = L B . The orange bars represent population levels within bands of the domains for the eigenvalues withL A =L B = 200. Blue bars(superimposed on orange) are for L A = L B = 100 The gap in the spectrum can be clearly seen as the difference between the largest eigenvalue of 1 and the next at .64. . . . . . . . . . . . . . . . . . 58 4.1 The spin-1 2s (all filled circles) in the Toric Code model live on the edges of a square lattice with periodic boundary conditions. The star operator at vertex labelleds involves the product of ˆ σ x operators on the four spins (red circles) of the edges joined at the vertex. The plaquette operator for the unit cell labelledp involves the product of ˆ σ z operators on the four spins (green circles) on edges that form the cell. W 1 ,W 2 are spin flips along the two non-contractible directions of a torus (blue circles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 ix 4.2 An artist’s rendition of the lattice of spins (dark filled circles) for the Toriccodemodelwithamagneticfieldonspinsalongonlythehorizon- tal direction (shown by spins within dotted arrows). . . . . . . . . . . . . 78 4.3 Subsystem A, shown in the shaded region, of one plaquette with the spins 1,2,3,4, on the edges. The eigenvalues of the reduced density matrixρ A , involves calculating expectation values of operators on the 4 pseudo-spinsi,i+1,j,j+1, at the shown vertices [113]. . . . . . . . . . 81 4.4 R´ enyientropiesforasubsystemAofoneplaquette(showninFig.4.3), at different values of α. All entropies show monotonic behavior in both the phases: they decrease monotonically with increasing correla- tion lengthξ(λ)forλ<λ c =1 while they increase withξ(λ)forλ>1. . 82 4.5 SubsystemA,shownastheshadedregion,comprisedofatotalofseven spins which form two overlapping stars. . . . . . . . . . . . . . . . . . . . 83 4.6 The2-R´ enyientropyofasubsystemcomprisedoftwostarsA(shownin Fig. 4.5) across the phase transition atλ=1 forH =H TC +V 2 (λ). The monotonic behavior in both the phases forS 2 implies similar behavior for S α ∀α ≥ 2; whereas the general arguments presented in the text implythatforα→0theyshouldincreasetillthequantumcriticalpoint. The dotted line is the inverse of the energy gap between the ground and first excited states for the transverse field Ising model to which the perturbed gauge-fixed Toric code Hamiltonian is mapped. . . . . . . . . 85 4.7 SubsystemA, shown as the shaded region, comprised of a total of six spinswhichmakeupthespinsonaplaquetteandtwoneighboringspins to its northeast corner. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.8 BehaviorofthreerepresentativeR´ enyientropiesfortheToric-Isingmodel (V =V 3 (λ x ,λ z ))intheperturbationparameter (λ x ,λ z )-planewithinthe topologicallyorderedphaseforsubsystemAasshowninFig.(4.7). For α = .6 the entropy increases while for α = 5 it decreases monotoni- cally with increasing correlation length. The change between these two types of behavior occurs atα ≃ 1.3, the value of which was identified numerically. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 x Abstract This thesis presents insights obtained on three questions in condensed matter physics via techniques in quantum information. The first topic deals with signatures of equilibration in a closed quantum system. Using the Loschmidt echo as a representative observable, that is derived from fidelity - a popular quantity in quantum information, and by studying it’s long time statistics the roleofquantum criticalityin the equilibration dynamics ofaclosed system is analysed. Whileoff-criticalsystemsunderaquantumquenchareshowntoequilibratewell,critical systems are shown to do so poorly signified by relatively large fluctuations of the echo around it’s long time average. ThesecondtopicdealswithtypicalityofentanglementinthephysicalHilbertspace. Here,ageneralframeworkforstudyingstatisticalmomentsofphysicallyrelevantquan- tities in ensembles of quantum states generated by Local Random Quantum Circuits (LRQC) is outlined. These ensembles are constructed by finite-length random quantum circuits acting on the (hyper)edges of an underlying (hyper)graph structure. The lat- ter are designed to encode for the locality structure associated with finite-time quantum evolutions generated by physical i.e. local Hamiltonians. Physical properties of typi- calstatesintheseensembles,inparticularpurityasaproxyofquantumentanglementis studied. Theproblemisformulatedintermsofmatrixelementsofsuperoperatorswhich dependonthegraphstructure,choiceofprobabilitymeasureoverthelocalunitariesand xi circuit length. We consider different families of LRQCs and study their typical entan- glement propertiesfor finite-timeas wellas theirasymptotic behavior. In particular, for a model of LRQC that resembles closely the Trotter scheme of discretizing quantum evolutions with local Hamiltonians, we find that the area law holds in average and that the volume law is a typical property (that is, it holds in average and the fluctuations around the average are vanishing for the large system) of physical states. The area law arises when the evolution time isO(1) with respect to the sizeL of the system, while the volume law arises as typical when the evolution time scales likeO(L). The final topic deals with the perturbative response of the set of R´ enyi entropies of a subsystem when the entire system is in a state displaying some quantum order. The characteristic behavior of the entropies is shown to be able to identify topologically non-trivialandtrivialphasesinthecaseofquantumdoublemodels. Theimplicationsof theresponsetowardsthepossibilityofsimulatingtheadiabaticevolutionwithinaphase using the protocol of Local Operations and Classical Communications are discussed . xii Chapter1 IntroductionandPlanoftheThesis As Quantum Information Theory (QIT) [100] has come of age it has ceased to be just another set of tools - of mere computational relevance - in the physicist’s kit; and has instead forced a refinement of many basic concepts of it’s parent: Quantum Theory (QT). Originating as a distillation of operational elements of QT, QIT has steadily cornered a fundamental platform so that definitions of certain concepts are now best understood through their QIT versions. Examples of such operationally-defined-but- fundamental definitions and ideas are: local unitary circuit definition of topological phases of matter [27], dependence of local equilibration in closed many-body systems on the circuit complexity of unitaries that diagonalise the governing hamiltonian [92], phase transitions as singularities of the elements of the fidelity metric [143], or investi- gatingcrossovertoquantumchaoticbehaviorviaoperatorfidelitymetric[49]tonamea few. This thesis presents results obtained towards three specific questions that have rel- evanceforboth-thequantuminformationaswellasthecondensedmattercommunities. We start by considering , in chapter 2, a long standing problem in quantum statisti- cal mechanics, that of understanding equilibration in the case of closed system unitary evolution. While in the case of an open quantum system [13], one that interacts with a thermal bath, it is possible that the system relaxes towards an equlibrium state and remains close at most instants of time, unitary dynamics cannot lead to equilibration in suchastrongsense[21]-thatofthestatebeingproximaltotheequilibriumoneexceptin 1 thetrivialcasewhentheinitialstateitselfisastationarystateoftheHamiltonian. How- ever, equilibration may occur in a weaker sense - of observables taking values close to their long-time average with high probability. Concentration of the probability density fortime-dependentexpectationvaluesofgenericobservables,aroundsomemeanvalue, mightthusindicateequilibrationinaclosedsystem. Byconsideringaspecificexample, the XY quantum spin chain, and perturbing it using a quantum quench, we study the effectthatproximitytocriticalitycanhaveforclosedsystemequilibrationintheweaker sense described above. The results in this chapter thus shed light on the role of initial quantum conditions of the Hamiltonian on unitary equilibration [19]. We begin chapter (2) by explaining the problem in section (2.1) followed by the choice of our model of a closed quantum system and observable in section (2.2). The signatures of off-critical and quasi-critical equilibration are presented in section (2.3) and the chapter concludes with comments in section (2.4). Next, in chapter 3 we review something as sine qua non as the Hilbert space - the backdrop to everything kinematic or dynamical, in the canonical formulations of QT, proceedingwiththebeliefthatquantumcomputersshouldbeabletosimulaterealphys- ical systems [44, 36] and hence only that fraction of the Hilbert space that they can explore in a reasonable amount of time should be called physical. Because this fraction turns out to be exponentially small in the number of elementary units in a Quantum Many Body (QMB) system, some authors [108] have termed the full Hilbert space a “convenient illusion” and expressed a desire to describe quantum phenomenon within this physical subset accessible to us. It is thus important to ask what properties of the full Hilbert space hold in the putative physical Hilbert space. Here we use QIT tech- niques to understand the typical entanglement properties of QMB systems within the physical hilbert space and beyond. Our work is especially relevant towards the typical- ityapproachinthefoundationsofquantumstatisticalmechanicswhichseekstoexplain 2 the efficacy of the different kinds of ensembles in physics in predicting properties of individual quantum systems [52, 106]. Alargepartofthestudyofphysicsofinteractingquantummany-bodysystems,isreally doing mathematics with the metaphysical idea of causation [80]. Since causality in physics is a result of locality of interactions, the latter is one of the most crucial ingre- dients for any reasonable model of natural phenomenon. This chapter presents results on the entanglement dynamics in natural models for QMB systems whose interactions are local. A lot of the techniques used in the analysis here have overlaps with other subfields in QIT and section (3.1) is an introduction to the main idea and it’s intercon- nections. Section (3.2) introduces the necessary mathematics of group representation theory required for the analysis of the actual models in section(3.3). Section (3.4) then closes the chapter with conclusions. Finally, in chapter 4 the third question we address is detection of topological order in the ground state of a system in a some phase of matter. Topological phases are novel quantum phases [134] of matter that defy description in terms of the Landau symmetry breaking paradigm [50]. The order in such states resides in non-local quantum corre- lations and one may not use any local measurement result to detect such order. Indeed robustness against local perturbations makes topologically ordered states promising as substrates for quantum informationtasks foreg. for making quantum memories. In this chapterweshow,thatalthoughlocalmeasurementscannotsignifytopologicalorder,the perturbativeresponseoftheR´ enyientropiescalculatedforalocal(notscalingwithethe system size) region to generic Hamiltonian perturbations may provide a theoretically reliable and experimentally feasible indicator for Topological order. Section (4.1) of this chapter defines topological order and outlines known techniques for it’s detection. Theninsection(4.2)weputforwardthebasicidearelatingtheperturbativeresponseof R´ enyi entropies to the presence of topological order. This is followed by section (4.3) 3 which lays down the general strategy of our investigation along with the mathematical definitions of relevant quantities. Section (4.4) then presents results on different pertur- bations and also summarises the results. The chapter ends with concluding remarks in section (4.5). Chapter 5 then closes the thesis with conclusions and outlook for future directions of research. 4 Chapter2 SignaturesofEquilibrationinClosed QuantumSystems In this chapter we study the signature of equilibration in a closed quantum system from a quantum information perspective. Asiscommonlyaccepted,thestateofanisolatedquantumsystempreparedatsome initial time in a state ρ(0) will evolve, according to the laws of quantum mechanics, unitarily into ρ(t) at time t. If one imagines measuring an observable O of the sys- tem at different instants of time, the average measurement result will be given by the time average ⟨O(t)⟩ ∶= 1 T ∫ T 0 ⟨O(t)⟩dt, where T is the measurement time. Since T is much larger than the microscopic time scales of the system it is often set to infinity for mathematical clarity. Now, the postulates of quantum statistical mechanics assert thatthetime-averagedexpectationvalueshouldbeindistinguishablefromthatobtained using the statistical microcanonical ensemble. Although this postulate is confirmed by a number of numerical simulations [111], to date no explanation exists for why this is so. In other words, the mechanisms of thermalization in quantum systems are unknown though there exist possible approaches such as the eigenstate thermalization hypothe- sis [37, 119] or normal typicality [51, 132, 123]. The results presented here should be seen as a step in the direction of understanding the aforementioned quantum statistical mechanics’ postulate better via exact results, at least for some particular cases, which can serve to guide our intuition [19]. 5 2.1 ClosedSystemEquilibration Wefocusonthesimplestpossiblesetup-anidealizedquantumsystemisolatedfromits enviroment-alsoonewhichavoidsalldecoheringordissipativeeffects. Toexperimen- tallydeterminewhetherasystemhasequlibrated,oneisinterestedinthefull,long-time statisticsofagenericobservable ⟨O(t)⟩. Intuitivelyspeaking,goodequilibrationwould mean that the time-dependent expectation values tend to concentrate around an average valuewhereaspoorequilibrationwouldimplythatthevariationintheobservedvaluesis large. Asarepresentativeobservablethatcanpossiblycapturetheequilibrationdynam- icsofacontinuumofobservablesO,weconcentrateontheLoschmidtecho(whichisa quadratic functional of the wavefunction) (2.2.1) since it is derived from the overlap of the time evolved and original wavefunctions and does not depend on any specific form ofO. We study its exact, long-time distribution function and investigate the effects that proximity to critical points has on the equilibration dynamics. In the thermodynamic limit, also called the off-critical regime, i.e. when the system size is much larger than all length scales of the system, we see that a central limit theorem result applies lead- ing to universal Gaussian equilibration at least for the class of quadratic observables in free fermionic systems. In the opposite regime of quasi-criticality, where the correla- tion length is equal to or larger than the system size, we again find universal behavior, although one in which fluctuations are large and thermalization does not occur. 6 2.2 A Model Closed Quantum System - Quantum XY SpinChain 2.2.1 ChoosinganObservable-TheLoschmidtEcho As mentioned above, the choice of the Loschmidt echo as the observable we investi- gateismotivatedbyitsdefinition,freeofanyparticularobservable,onlyintermsofthe wavefunctions describing the state of the system. The scenario we consider here is that of a quantum quench, generalized to the mixed case. A closed system is initialized in the state ρ 0 commuting with the Hamiltonian H 0 . The system is then instantaneously quenchedandlefttoevolveaccordingtoHamiltonianH 1 . Thisisanimportantgeneral- ization,sinceinprinciplethereisnoreasonwhytheinitialstateofthesystemshouldbe pure. In particular, foritsexperimentalrelevanceweuseGibbsinitialstatesρ 0 ∼e −βH 0 . Suchasituationisinfactoftenrealizedinthelaboratorybyfirstthermalizingthesystem by putting it in contact with an external reservoir and then detaching the reservoir. The quantity we consider is the Loschmidt echo (LE), which generalized to the mixed case is given by L(t)=F(ρ(t),ρ 0 ), F(ρ,σ)= (Tr ! ρ 1 2 σρ 1 2 ), (2.1) whereF is the Uhlmann fidelity [127] which characterizes the degree of distinguisha- bility between two mixed states. Note that if either (or both) of ρ and σ is pure, the Uhlmann fidelity simplifies toF(ρ,σ)=Tr(ρσ). 7 2.2.2 XYmodel The model we investigate here is the quantum XY chain in a transverse magnetic field H =− L i=1 1+γ 2 σ x i σ x i+1 + 1−γ 2 σ y i σ y i+1 +hσ z i . (2.2) AJordan-WignertransformationbringsEq.(2.2)toaquadraticforminFermioper- atorsc i ,andhencecanbeexactlydiagonalized. AtzerotemperaturethemodelEq.(2.2) displays two kinds of quantum phase transition lines inthe (h,γ) plane. For h = ±1 and γ ≠ 0 the model is in the Ising universality class described by a c = 1 2 confor- mal field theory (CFT). Instead, in the segmentγ = 0, "h" ≤ 1 the underlying CFT has central charge c = 1. To specify completely the problem we one fixes the boundary conditions (BCs). As is customary [6], to avoid unnecessary complications we fix anti- periodic BCs on the Fermi operators i.e. c L+1 = −c 1 . Diagonalization brings Eq. (2.2) to free Fermion form: H =∑ k 2Λ k η † k η k . Our choice of BCs fixes quasimomenta to be quantized according tok = (2n+1)π L,n =L 2,...,L 21, whereas the single-particle dispersion isΛ k = & (cosk+h) 2 +γ 2 sin 2 k. The Loschmidt echo has been shown for the XY chain to be [109], L(t)=!k >0f k (Λ 1 k t), withf k (Λ 1 k t)= [ 1+ & c 2 k −(c 2 k −1)α k sin 2 (Λ 1 k t) 1+c k ] 2 , (2.3) where c k = cosh(βΛ 0 k ), α k = sin 2 (δθ k ), δθ k = θ 1 k −θ 0 k and θ k = arctan[γsin(k) (h+ cos(k))]. From its explicit form one can read off a number of important points which we is used extensively in the following: i) the timedependence is governed byL 2 fre- quenciesΛ 1 k ,ii)theLEisaproductofanextensivenumberofterms,andinparticulariii) the LE is a product ofL 2 functions over theL 2 allowed values ofk. The dependence 8 Figure 2.1: Typical behavior of L(t). The inset shows Gaussian behavior for short times, as happens for the pure case [7]. HereL=100, β =6, h 0,1 =1,γ 0 =0.5,γ 1 =0.8 onk isanalyticeverywhereexceptforthecriticalpoints(γ =0and "h"≤1or "h"=1and γ ≠0). No singularity other than those expected at criticality emerges. Typical behavior ofL(t) is depicted in Fig. (2.1). The LE quickly drops from unity att = 0 and then oscillates about its average value, with almost periodic revivals [68]. Our aim is to study the distribution function of the LE seen as a random variable over infinite time equipped with the uniform measure [21, 22]. The probability density of the LE can be written asP L (x)∶=δ(L(t)−x), where the bar denotes the time average (i.e. ¯ f = lim T→∞ T −1 ∫ T 0 f(t)dt). Saying that the LE spends most of the time close to a certain value corresponds to a concentration result for P L (x). The moments of the LE can be computed using cumulant expansion techniques developed in [21]. Here one has the additional complication given by the presence of the square-root in Eq. (2.3), which must first be expanded into an infinite series. The result for the first moment is ¯ L = ∏ k>0 f 1 k , with f 1 k = 1− (1−c −1 k ) α k 2 + 2c k (1+c k ) 2 [ 2 π E(b k )+b k 4−1]. Here, b k = (1−c −2 k )sin 2 (δθ k )andE isthecompleteellipticintegralofthesecondkind. Expanding f 1 k inthesmallquenchregime,thatisuptothesecondorderinδθ k ,oneisabletorelate the dynamical quantity ¯ L to a static quantity. Specifically, one obtains ¯ L ≃F(ρ 0 ,ρ 1 ) 2 , 9 whereρ 0,1 are Gibbs states with HamiltoniansH 0,1 . This result extends the pure state result ¯ L=Tr(¯ ρ 2 )≃ "⟨ψ 0 "ψ 1 ⟩" 4 which can be recovered sendingβ→∞[22]. The distribution function for the LE in the Ising model (i.e. γ = 1) at zero temper- ature was considered in [21]. Through numerical simulations it was argued that, in the off-critical regime, two different behaviors were observed. The distribution of the LE was seen as similar to an exponential one, (P L (x) ≃ Θ(x)e −x! ¯ L ¯ L) or to a bell-shaped Gaussian-looking one. In the next section both of these conjectured results are unified. 2.3 SignatureofEquilibration It is observed that quantum criticality plays a crucial role in determining the equilibra- tion dynamics of a closed system. Roughly, speaking a quench - quantum perturbation to the system - equilibrates poorly when the system is close to criticality compared to the off-critical regime. Below we focus on each of these regimes separately. 2.3.1 Off-criticalregimeandGaussianequilibration The form of the LE (2.3) suggests that the LE should be thought-of as a product of variables and it is helpful to study it using the new variableZ = lnL. It can be shown that, under a very mild hypothesis, the variable Z satisfies the standard central limit theorem (CLT). In particular, in the off-critical regime, asL→∞, the rescaled variable Y = (Z− ¯ Z) √ L tends in distribution to a Gaussian with zero mean and well-defined variance. This can be shown if all the cumulants of Z scale extensively, so that for the rescaled variableY we getκ n (Y )∝L 1−n!2 forn≥2 whileκ 1 (Y )=0 by construction. Hence only the first two cumulants of Y survive in the L→ ∞ limit, thus showing Gaussianity ofY . In turn, Gaussianity ofY implies that the LE is approximately Log- Normallydistributed. Thisexplainsthebehaviorobservedin[21],asaLog-Normalhas 10 regimes where it looks approximately exponential or Gaussian. In order to prove our assertion we need the (logarithm of the) moment generating function ofZ, M Z (λ) ∶= e λZ =L λ . Making the reasonable assumption that theL 2 frequenciesλ 1 k are rationally independent (that is, linearly independent over the field of rational numbers), one can use the theorem of averages [128] to compute the time-average ofL λ as a phase space average over anL 2-dimensional torus. Using numerical simulations it may be shown thatapossiblerationaldependenceisverymildanditwouldbequiteunluckytoproduce enough correlations to invalidate the CLT. With RI, one obtains obtain M Z (λ)=! k>0 g k (λ), g k (λ)= 1 2π 2π 0 [f k (θ)]dθ. (2.4) Thus,M Z (λ) =exp∑ k>0 lng k (λ). The last steps of the proof come from the fact that lng k (λ)asafunctionofk isRiemannintegrable,withafiniteintegral,providedweare away from critical points. Moreover, in the same region of parameters, lng k (λ) (and so its integral overk) is analytic inλ. With LargeL, one obtains ln[M Z (λ)] ≃LG(λ) with G(λ) = ∫ π 0 lng k (λ)dk (2π) analytic in λ. Differentiating with respect to λ one can verify that all cumulants of Z are extensive. This implies that, one has that the central limit theorem holds anywhere away from the critical points: no other source of singularity emerges other than those expected at criticality. In fact, by direct inspection ofthesecondcumulantκ 2 ,whereκ 2 (Z)=∑ k>0 κ 2 (k)withκ 2 (k)=m 2 (k)−[m 1 (k)] 2 , andm n (k)= 1 2π ∈ 2π 0 [ln(f k (θ))] n dθ withn=1,2,onecanshowthatκ 2 (Z)≤const.×L also at the critical points and hence it never scales super-extensively. 2.3.2 Quasi-criticalregimeanduniversalcriticalequilibration For a small quench close to a critical point when the initial state is a pure state, no observable (except for trivial constants of motion) thermalizes [22]. It can be shown 11 thatthisresultgeneralizestothemixedcaseconsideredhere. Moreover,someuniversal featuresoftheunderlyingcriticaltheoryshowupinthelong-timedistributionfunction. For the reasons explained above, the right quantity to look at is the Log of the LE. In the small quench regime, it is helpful to expand the Log of the LE up to the first non-zero order inδθ k . The constant terms add up to contribute to the average and, dropping fourth-order terms and going to the energy variableω j =2Λ 1 k j one gets lnL(t)= ¯ Z+ j a j cos(tω j ), (2.5) where the amplitudes are given bya(k)= (1−c −1 k )(δθ k ) 2 anda j =a(k j ). Note that the quantity (2.5) is in fact a sum ofL 2 independent random variables. This can be shown assumingagainRIofthefrequenciesω j . Usingtheergodictheoremonerealizesthatthe moment generating function of lnL is simply the product ofL 2 generating functions. Taking the Fourier transform, one sees that each variable is distributed according to P j (x)=π −1 Θ(a 2 j −x 2 ) ! a 2 j −x 2 , with zero mean and variancea 2 j 2. Itisinstructivetoconsiderwhatcanhappenatcriticalityandinwhichsensewecan expect violation of the CLT. As explained above, the total variance, which in the small quench regime reads κ 2 (Z) = (1 2)∑ j a 2 j , cannot grow more than extensively. But the other extreme is possible, namely the variancesa 2 j 2 can go to zero asL increases, and this can happen for most of the L 2 variables. When this is the case, Eq. (2.5) effectively represents a sum of very few independent variables, and the CLT regime cannot be reached. Here we notice that for an infinitesimal quench κ 2 is related to the fidelity susceptibility, a central object in the so-called fidelity approach to quantum phase transitions [144, 20]. Close to criticalitya j is a rapidly decreasing function ofj, so that only few ampli- tudes are appreciably different from zero. In this situation, a good approximation to the 12 distributionfunctionforZ isgivenretainingthen max largestamplitudesa j inEq.(2.5). Choosingn max =1, the distribution is the just encounteredP jmax with square-root singu- larities at ±a jmax . With n max = 2 the distribution is still a very spread double-peaked one, with logarithmic singularities at ¯ Z ± a 1 − a 2 as shown in [21]. Using the ergodic theorem it can be shown that this distribution is precisely the density of states (DOS) of a tight-binding model in two dimensions, with anisotropic couplings. In gen- eral, the distribution function obtained by keeping n max amplitudes is the density of states of a hypercubicn max dimensional tight binding model with anisotropic couplings a j #2 (j = 1,...,n max ) in each direction. Adding more and more amplitudes, eventually theCLTsetsinandthedistributionapproachesasingle-peakedGaussian. Clearly,when n max issmallthedistributionfunctionisveryspreadwithalargevariance,sothermaliza- tion does not take place. Let us now discuss the behavior ofa j close to criticality. The XY model has two different kinds of critical regimes characterized by different under- lying effective field theories and we consider separately both critical regimes. First, we note that increasing the temperature simply has the effect of multiplying a(k,T = 0) by a factor (1−cosh −1 (Λ k #T )) ≤ 1. At the Ising transition a large peak ina(k) close tok = π is observed. The reason for the peak has to be ascribed to the single-particle energy vanishing asω =v(k−π) (wherev =2 γ is a velocity). At finite size the quasi- momentak take only discrete values. Correspondingly, most of the weight is absorbed bythosekswhichfallinthepeak. Otheramplitudesa(k j )areconsiderablysmaller. As a result, a good approximation to the distribution can be given by a 2D DOS as shown in Fig. (2.2), left panel. The situation at the anisotropy transition (γ = 0 line) is very similar, with some notable difference due to the precise character of thec = 1 CFT. As can be easily seen, a(k) now has two peaks, due to the presence of two chiral (Majorana) Fermions corre- spondingtothetwobranchesofω =v k−k F . Thedouble-peakedformofa(k)hassome 13 Figure 2.2: P Z close to the Ising (left) and anisotropy transition (right). As L grows we enter the off-critical regime andP Z becomes Gaussian. Close to the quasi-critical regime (small L) the distribution becomes a broad, generally double-peaked function. Fortheanisotropytransition,onecanhaveLforwhichthehighestamplitudesarenearly equal. Thisresultsinacollapsefromtwopeakstoone. Parametersareβ =40and(left) h 0 =0.98, h 1 =1.02, γ 0,1 =1.0 andL=50 to 200 in steps of 30, (right)h 0,1 =0.5, γ 0 = 0.01, γ 1 = 0.01 andL = 50 to 100 in steps of 10. Another way to enter the off-critical regime is to increase the temperature. Similar plots are obtained replacingL with the temperatureT . detectableconsequenceonthestructureofthedistributionfunction. Namely,according to different quantization of quasimomenta (and damping factor due to temperature) the allowedvaluesofk canfallsymmetricallydisplacedamongthepeaks. Whenthisisthe caseweobserve,adistributionfunctiongiventhe2DDOSwitha 1 =a 2 . Inthiscasethe two peaks of the distribution merge into a single one, as can be seen in Fig. (2.2) right panel at L = 60, 90. 14 2.3.3 BeyondtheXYmodel One can make arguments in support of the validity in general of this scenario for small quenches. Restricting, for simplicity, to zero temperature and assuming a com- pletely generic, non-degenerate Hamiltonian H = ∑ n E n n⟩ ⟨n , the LE reads L(t) = ¯ L+ 2∑ n>m p n p m cos(t(E n −E m )), where p n = ⟨n ψ 0 ⟩ 2 for an initial state ψ 0 ⟩. If we now consider the logarithm of the LE and expand it in the small quench param- eter (that is in the perturbing potential V, which we assume to be extensive), then up to second order one obtains lnL(t) = ¯ Z +2∑ n>0 p n cos(t(E n −E 0 )), where for a small quench p n = ⟨n V 0⟩ 2 #(E n −E 0 ) 2 . Assuming additionally RI for the energy gaps, we return to the previous situation with a j = 2p j , namely CLT away from crit- icality, meaning Gaussian equilibration. Note that the total variance is at most exten- sive: κ 2 (Z) = 2∑ n>0 p 2 n ≤ 2∑ n>0 p n = 2χ, where χ is the fidelity susceptibility and is extensive by the extensivity of V and the assumption of non-criticality [20]. In the quasi-critical regime only a few terms of the sum dominate, thus breaking the CLT and leading to a universal, poorly equilibrating regime. 2.4 ConcludingRemarks In this chapter we considered the finite temperature generalization of the Loschmidt echo (LE) after a quantum quench. It was shown, that under a very mild hypothe- sis, away from critical points the LE is Log-Normally distributed, whereas for small quenchesclosetocriticalitythedistributionapproachesthatofthedensityofstatesofa D-dimensional anisotropic tight binding model, where D can be considered small (e.g. D = 1, 2). Although these results could be obtained analytically for the XY model con- sidered here, it seems a reasonable conjecture that such behavior is in fact general and not restricted to solvable models. 15 Chapter3 EntanglementinthePhysicalHilbert Space Oneofthepromisesthatapotentialquantumcomputershouldcomewithisit’sabilityto simulate physical quantum systems [36, 44]. One should be able to explore the physics in real world quantum systems by running the quantum computer appropriately. While the range of systems that may be simulated would be limited by the physical size of the computer, one would also be limited to a reasonable amount of time for the simulation. ForquantumsystemsthisamountstoexploringonlyafractionofthetotalHilbertspace. The physics thus explored would represent the possibilities within what may be called thephysicalHilbertspace. Moreprecisely,weadoptasadefinitionthephysicalHilbert spacetobethesubsetofthefullmathematicalHilbertspaceascribedtoasystemwhich may be explored under the action of the operator governing the dynamics of the system in a time polynomial in the system size [108]. As a theoretical model of a quantum computer, in this chapter, we consider circuits of random unitary quantum gates [99, 67, 66]. These circuits generate ensembles of quantum states, given a fiducial state, analyzing which we can make statistical statements about properties of the physical Hilbert space. In particular, we study here the dynamics and asymptotics of bipartite entanglement as quantified by the purity of the reduced state on a subsystem. 16 3.1 Random Quantum Circuits in Quantum Informa- tion We begin this chapter by briefly surveying the concept of random quantum circuits (RQCs) across the field of quantum information. Then we present a general formal- ism and explicitly solve certain models of Local Random Quantum Circuits where the graphs, on which the circuits act, encode the interaction between degrees of freedom thatarephysicallyproximal. Inthissensetheymaybeconsideredtobethemostnatural models of random quantum circuits and their analysis as we show in section 3.3 lets us make statistical statements about entanglement dynamics in natural systems such as time-dependent local Hamiltonians. Unlikequantumcircuitsforspecificalgorithmsconstructedwithatargetoutputstate inmind[117,55],RQCsareageneralclassofcircuitswhereaseriesoftwo-quditgates chosen from a universal gate set are applied to an input state where the choices of the gates and their supports are picked from probability distributions defining the circuit, Fig. (3.1) [67, 66, 142, 71, 10, 70, 17, 76]. For example, in a circuit comprised of qubits, the supports of gates from a universal gate set on 2-qubits may be chosen with uniformprobabilityoveranytwoqubitswhilethegatesmightbechosentosimulatethe Haar measure onU(4). Their main usage, in the Schr¨ odinger picture, is as a theoretical gadget to ana- lyze ensembles of states that are produced due to interactions (unitaries), which model ensembles of related physical systems, acting on a fiducial state upon which the RQC acts. Correspondingly in the Heisenberg picture RQCs generate ensembles of unitaries that model the same interactions - of relevance to the information processing task at hand. In fact whether the randomness in the model is intrinsic (or by design) or due to incomplete knowledge, for example having access to only specific moments of the 17 Figure3.1: Anexampleofarandomquantumcircuit. Differenthorizontallinesindicate different qubits on which gates (possibly different) are applied at each step. distribution of the interactions, RQCs provide a calculational framework to evaluate physical quantities of interest. Indeed random unitary operators play a fundamental role in quantum information justasrandomnumbersinclassicalcomputationandinformation[43]. Assuch,theyare a necessary ingredient of well known quantum information protocols in quantum data- hiding[39],quantumcryptography[69,75,2],Fidelityestimationforquantumchannels [35] etc. However the generation of Haar distributed random unitaries on large many- body quantum systems is inefficient as the number of required quantum gates grows exponentially with the number of qubits. To address this issue, a significant amount of work has been done towards the theory of t-designs which are ensembles of unitaries that reproduce the statistical moments of the Haar measure up to ordert, either exactly or approximately. In the following we discuss ideas relatingt−designs to RQCs and comment on specific uses of unitary designs for particulart’s. 18 ● Unitarydesigns In many algorithms that make use of random states or unitary operators, efficiently produced pseudo-random operators may be substituted for exactly sampled Haar uni- taries. Theextenttowhichthepseudo-randomoperatorsmimictheuniformdistribution is quantified via the notion oft−designs [35, 71, 10, 70, 17, 76]. Mathematically,anensembleofquantumstates {p i , ψ i ⟩},wherep i ’sareprobability weights for the states ψ i ⟩in the ensemble, is an exact statet−design if: ! i p i ( ψ i ⟩ ⟨ψ i ) ⊗t = dψ( ψ⟩ ⟨ψ ) ⊗t , (3.1) where in the integration to the right is taken over the left invariant Haar measure on the unit sphere in C d and normalized to 1, i.e., ∫ dψ = 1. Analogously an ensemble of unitary operators {p i ,U i }, where p i ’s are probability weights for the unitaries U i ∈ U(C d ) U † U = 1 1 d in the ensemble, is an exact unitary t−design if for all operators ρ∈B(C d ): ! i p i (U i ) ⊗t ρ(U † i ) ⊗t = dU(U) ⊗t ρ(U † ) ⊗t , (3.2) where the integration is taken over the unitarily invariant Haar measure onU(C d ). One gets approximate state and unitary designs by relaxing the conditions in Eq. (3.1) and Eq. (3.2) as follows: A state design is {p i , ψ i ⟩}approximate if (1−ǫ) dψ( ψ⟩ ⟨ψ ) ⊗t ≤! i p i ( ψ i ⟩ ⟨ψ i ) ⊗t ≤ (1+ǫ) dψ( ψ⟩ ⟨ψ ) ⊗t , (3.3) while a unitary design {p i ,U i }is anǫ-approximate unitary design if ! i p i (U i ) ⊗t ρ(U † i ) ⊗t − dU(U) ⊗t ρ(U † ) ⊗t ♢ ≤ǫ, (3.4) 19 where the superoperator norm () ♢ , known as the diamond norm, is defined for any superoperator T ∶ B(C d )→ B(C d ) and X ∈ B(C d ) as T ♢ = sup d T ⊗ 1 1 d ∞ = sup d sup X≠0 (T⊗1 1 d )X 1 X 1 . The operational interpretation of the diamond norm of the dif- ferencebetweentwoquantumoperationsT 1 ,T 2 isthatityieldsthelargestpossibleprob- abilityofdistinguishingthetwooperationsifweareallowedtohavethemactonpartof an arbitrary, possibly entangled, state. A unitary design is a stronger notion than a state design since applying unitaries of the formU ⊗t whereU is picked from the ensemble {p i ,U i } to any pure state in (C d ) ⊗t should lead to a sample of states indistinguishable from the uniformly random - Haar measure case. Below we describe in brief the uses of RQCs primarily in the role oft-designs for certain specifict’s. ● Quantum Data Hiding Herethegoalistohideclassicaldatainbipartitemixedstates[39,42],meaningtwopar- tiesAliceandBobcannotlearnthesecret(classicalbitencodedontotheglobalstate),if they do not share entanglement and can only perform quantum operations locally sup- plemented with unlimited both-ways classical communication (LOCC). Essentially the ideaisthatlocallyrandomizedstatescanonlyhaveresidualquantumcorrelationswhich cannot be accessed via LOCC. To hide classical data in quantum states the classical bit b = 0,1 is encoded onto two statesρ n 0 ,ρ n 1 ∈H 2 n ⊗H 2 n, i.e., bipartite states on (n+n) qubits, which should be orthogonal Tr(ρ n 0 ,ρ n 1 ) = 0, for reliable retrieval. RQCs come into the picture for quantum data hiding in the preparation of these states, since one requires 2-designs ∫ dU Haar U ⊗2 (.)(U † ) ⊗2 (also called 2-twirls or Bilateral twirls) with U ∈U(C 2 n ). These two-twirls are efficiently implementable using RQCs. ● Quantum Cryptography 20 Inmanyquantumcryptographicprotocols[75],completelypositivetrace-preserving mapsR∶B(C d )→B(C d ),knownasRandomizersarerequiredthatperformthefollow- ing function: R(φ)− 1 1 d ∞ ≤ ǫ d for all statesφ∈C d , (3.5) whichimpliesthatalleigenvaluesofR(φ)lieintheinterval [(1−ǫ)#d, (1+ǫ)#d]. Italso impliestheweakerestimatethat R(φ)−1 1#d 1 ≤ǫmeaningthatthedistinguishabilityof therandomizedstatefromthetotallymixedstateislessthanǫ#2. ThusweseethatRQCs which act as 1-designs, i.e., satisfy Eq. (3.4) witht=1, are candidate Randomizers and finddiverseusessuchasinconstructionofapproximateprivatequantumchannels[95], provingexistenceofquantumstateswithlockedclassicalcorrelations[38,9]andremote preparation of arbitrary d-dimensional pure quantum states [7]. ● Fidelity estimation Unitary2-designsarealsousefulinthecontextofexperimentallyestimatingtheaverage fidelity of a quantum channel [35]. Here one considers a general quantum channel, Λ∶ B(C d )→B(C d ),conjugatedbyrandomlychosenunitaryoperationsfromtheHaarmea- sureonU(d)resultinginaconjugatedchanneldefinedasE µ (Λ)∶= ∫ dUU † Λ(UρU † )U. Fromtheformoftheconjugatedchannelitisclearthattherelevantunitarydesigninthis protocol is a 2-design which may either be achieved by in-place circuits whose explicit construction in this context was provided in [35] or by using RQCs to form a 2-design. ● Fast Scrambling Random quantum circuits are also of interest in the decoupling approach [122, 15] to the quantum information where their potential usage lies as “scramblers”. The term scrambling, which arose in the context of the black hole information paradox, usually 21 refers to the process of mapping most initial pure product states under a unitary trans- formation to states which are macroscopically entangled. This means that the reduced states on any subsystem containing a constant fraction λN,0 < λ ≤ 1#2 of the total numberN of local degrees of freedom is close to the totally mixed state. Scrambling therefore is also related to the question of how closed quantum many-body systems reach equilibrium under the unitary dynamics governed by their own Hamiltonians. Technically, a scrambler is a Hamiltonian such that the associated unitary time evolution operator U(t) = exp(−iHt) takes any two intial pure product states ψ 0 ⟩ = ⊗ N i=1 ψ i ⟩, φ 0 ⟩ = ⊗ N i=1 φ i ⟩ to pure states ψ(t)⟩ = U(t) ψ 0 ⟩ and φ(t)⟩ = U(t) φ 0 ⟩ at time t such that the reduced density matrices ρ ψ (t) = Tr B ( ψ(t)⟩ ⟨ψ(t) ) and ρ φ (t) = Tr B ( φ(t)⟩ ⟨φ(t) ) on the subsytem A of size λN are close to each other in their trace distance for all timest≥t ∗ , i.e., D(ρ ψ (t),ρ φ (t))∶= 1 2 ρ ψ −ρ φ 1 ≤ ǫ 2 . (3.6) Iftheearliesttimet ∗ requiredfortheevolutionoperatorU(t)tosatisfyIneq.(3.6)scales logarithmically, i.e.,t ∼ log(N) as we increase the total system size A∪B =N for a fixed fraction0<λ<1#2, then the HamiltonianH may be termed a fast scrambler. It has been conjectured [116, 121] that the fastest scramblers require scrambling times that saturate the criteria, i.e., t ∗ = logN and further that black holes are the fastest scramblers in nature. Of our interest, i.e., in quantum information, an example of a time-dependent Hamiltonian that scrambles in a time O(logN) with high proba- bility, is known from the work in [35]. However an ideal fast scrambler would be a time-independent Hamiltonian that requires no time-dependent tuning and some results towards this, including two examples a) Brownian quantum circuits and b) Antiferro- magnetic Ising model on random graphs, have been presented recently in [88] which 22 satisfy a weaker notion of scrambling. In fact, certain models of random quantum cir- cuits(butnotlocalinthesenseconsideredinthischapter)havebeenshowntobescram- blers with scrambling timest ∗ = O((logN) 3 ) in [16]. Thus a natural question to ask in the context of the local random quantum circuits presented in here is whether they maybeusedasscramblersandtheircorrespondingscramblingspeed. Inthissectionwe briefly surveyed the many uses of RQCs prevalent in Quantum Information, Communi- cation and Computation (QICC). The examples illustrate the importance of generating random states and random unitary operators as resources for QICC. In the examples above, the central trick is to identify subsets of the unitary group thatadequatelysimulatespecificstatisticalmomentsoftheHaar-measurefortheopera- tional task at hand and to identify efficient local gate decompositions for these subsets. It is worth remarking at this point that often the RQCs employed for the purpose of simulating specific Haar-moments employ gates that do not respect geometric locality of the degrees of freedom. That is, while these are local gates which act on a bounded number of degrees of freedom, these degrees of freedom are not required to be local in awaythatrespectsphysicalrealityforexamplethesecircuitsneednotactonnearestor next-nearest neighbor interaction graphs [71]. 3.2 SetupandNotation We adopt the mathematical formalism developed by Zanardi in [142] which is natural yet sufficiently sophisticated to achieve our mathematical objectives. Thus we have a collection of ‘local’ Hilbert spaces {h i } i∈V on the vertices of the graph G = (V,E). The graphG represents the interaction graph of local gates in our circuit and finiteness of its cardinality V , is assumed. For each subsetS ⊂ V we have an affiliated Hilbert space H S = ⊗ i∈S H i as well as the operator algebra A(S) ∶= B(H S ). For S 1 ⊂ S 2 23 the algebras follow the inclusionA(S 1 ) ⊂ A(S 2 ) as we make no distinction between a 1 ∈ A(S) and a 2 = a 1 ⊗ 1 1 S 2 −S 1 ∈ A(S 2 ). Therefore A(S) ⊂ A(V ) ∀S ⊂ V and A(V ) ≅ ⋃ S⊂V A(S). For simplicity each local Hilbert space has the same finite dimension, i.e., h i ≅ C d ∀i ∈ V and thus the global state space H V is isomorphic to (C d ) ⊗V . 3.2.1 DefinitionofaLocalRandomQuantumCircuit A Local Random Quantum Circuit family C k [Ξ,L,q (k) ] of depth k is defined by the following data: 1. A family of probability densities Ξ ∶= {dµ S },S ⊂ V over the groups U(S)∶= {U ∈A(S) U † =U −1 }. 2. A subsetL of the power set2 V . Elements ofL are referred to as the local regions. 3. A probability law: q (k) ∶L k ↦ [0,1] S∶= (S 1 ,S 2 ,...,S k )↦q (k) (S), ∑ S∈L kq (k) (S)=1. Given this data, a LRQC of depth k is a unitary in U(V ) of the form U S ∶= U k U k−1 ...U 1 where a)∀i∈ {1,...,k} !⇒ U i ∈U(S i )and b)S∈L k . Thus the U S are unitary-valued random variables distributed according to the law q (k) (S)dU S = q (k) (S)∏ k i=1 dµ S i (U S i ). In this work we will we will always take the measure over the local unitaries to be the Haar measure. There are two layers of ran- domness involved in the process. While the stochastic process in whichk local regions ofthesetofverticesV arechosenaccordingtothejointdistributionq (k) accountforthe experimental lack of enough spatial resolution when applying unitary gates in a circuit, theHaaruniformchoiceofunitariesactingontheselocalregionstakesintoaccountthe lack of precise control over exactly which unitary is applied. 24 3.2.2 EnsembleMaps TheactionoftheLRQCscanbeunderstoodineithertheSchr¨ odingerpicturewhereone studiesitsactionontheinitialinputstateorintheHeisenbergpicturewherethedynam- ics affect the operators whose expectation values are then evaluated with respect to the same initial state frozen in time. Thus there is a one-to-one correspondence between the family of circuitsC k [Ξ,L,q (k) ] and an ensemble of statesE k (Ξ,L,q (k) , ψ i ⟩) that maybeconsideredastheoutputoftheLRQCsuponinputofthefiducialpurestate ψ i ⟩. For the case of a totally factorized pure state ψ i ⟩ = ⊗ V j=1 ψ i j ⟩ the dependence on the initial state drops out of purity calculations and the ensembles of circuits and states are equivalent. In either case the full statistical content of the family of LRQCs can be studied by analyzinganaturallyassociatedsetofunitaltracepreservingcompletelypositiveunique mapsR p ∶ A(V ) ⊗p ↦ A(V ) ⊗p . This is because for any fixed observable O ∈ A(V ), the mapping X O ∶ U ↦ ⟨ω,U † OU⟩ defines a classical random variable whose statis- tics describethefluctuations ofthe purely quantum expectationvalue ⟨ω,U † OU⟩while it’s moments are given byµ p (X O ) = X O (U) p U = ⟨ω,U † OU⟩ p U = ⟨ω p , (U † OU) p ⟩ U = ⟨ω p , (U † OU) p U ⟩∶= ⟨ω p ,R p (O ⊗p )⟩. For a given LRQC family the mapsR p are defined by: R p (O)= ! S∈L k q (k) (S)R p,S (O) (3.7) where, R p,S (O)∶= dU S (U † S ) ⊗p OU ⊗p S O ∈A(V ) ⊗p . (3.8) The expectation value, and all higher moments, of any observable O ∈ A(V ) can beexpressedastheHilbert-SchmidtinnerproductoftheinitialinputstateoftheLRQC with the image ofO under the relevant ensemble mapR p . 25 Consider for exampleφ ∶A(O)↦C O↦φ(O), the expectation value ofO given the initial stateω of the circuit, then we have that: φ(O)∶= ! S∈L k q (k) (S) dU S ⟨ω,U † S OU S ⟩ !⇒ φ(O)= ⟨ω,R 1 (O)⟩=µ 1 (X O ). (3.9) The distribution law for anyX O is determined by the Fourier transform of the char- acteristic functionχ O (t)∶=∑ ∞ p=0 (it) p p! µ p (X O )and formally, χ A (t)= ⟨ω ∞ ,R ∞ (A ∞ )⟩, x ∞ ∶=⊕ ∞ p=0 x p , x=ω,A, (3.10) whereR ∞ (t) ∶= ⊕ ∞ p=0 (it) p p! R p is a formal CP-map over the full operator Fock space A ∞ ∶=⊕ ∞ p=0 A(V ) ⊗p . 3.2.3 IntegrationovertheHaarmeasure,AlgebraofSwaps In this section we will mostly be concerned about the ensemble maps R 1 ∶ A(V )↦ A(V ) andR 2 ∶A(V ) ⊗2 ↦A(V ) ⊗2 obtained with the unitarily invariant measure over localunitaries-calledtheHaarmeasure. Intuitivelythisisanaturalmeasuretoexpress thelackofcontroloverexactlywhichlocalunitarygetsappliedintheLRQCsinceitcan be thought of as the generalization to the unitary group of the unit normalized uniform measure 1#(r 2 −r 1 ) on a real interval [r 1 ,r 2 ]. Thus ifU,V ∈U(d),UU † =VV † = 1 1 d where U(d) is the group of d×d-complex unitary matrices, then dU = d(VU) and ∫ dU = 1. Using a result from the representation theory of unitary groups [53], known as the Schur-Weyl duality we have that forU ∈U(d),O ∈A(H≅C d ): 26 dUUOU † = Tr(O) d 1 1. (3.11) ThisisbecauseintegratingwiththeHaarmeasureovertheadjointactionofunitaries U ∈U(d), is a projection into its commutant algebra. For a matrix representationU d ∶ U → U(d) ⊂ GL d u→ U of the unitary group U where GL d is the group of d×d- complexmatrices,thecommutantalgebraCom(U d ),isspannedbyallelementsofGL d which commute with each element ofU(d), i.e., Com(U d )∶= {T ∈GL d TU =UT ∀U ∈U(d)}. (3.12) Inthecaseofanirreduciblerepresentationoftheunitarygroupthecommutantalge- bra is spanned by the identity (1 1) operator on the vector space carrying the represen- tation. The normalization in Eq. (3.11) can then be obtained by noticing that the l.h.s. preserves the trace of the operatorO. For the case of the reducible representationU d 2 ∶U→U(d)⊗U(d) u→U ⊗U, the commutant algebra of the matrix representationU d 2 is spanned by the identity (1 1) and theswapT H∶H ′ operatorswhere 1 1,T H∶H ′ ∈A(H⊗H ′ ), H≅H ′ andtheHaarintegration over the adjoint action ofU ⊗U ∈U(d)×U(d)onO ∈A(H⊗H ′ ≅C d 2 )gives: dUU ⊗U OU † ⊗U † = Tr(OΠ + ) d 2 (d 2 +1)#2 Π + + Tr(OΠ − ) d 2 (d 2 −1)#2 Π − , (3.13) where,Π ± = (1 1±T H∶H ′)#2canbeseenastheprojectionsontothetotallysymmetric (TS) and totally antisymmetric (TAS) irreducible subspaces ofH⊗H ′ under the action ofU ⊗U. Note that the R.H.S. of Eq. (3.13) involves the operators Π ± with non-trivial supportonlyoverH⊗H ′ thatcoincideswiththesupportoftheunitaryU⊗U ∈A(H⊗ 27 H ′ ). As we show later, the iterative action of the LRQC keeps increasing the size of the support and asymptotically the operators have non-trivial support over the entire H V =⊗ i∈V H i . The swap operatorsT H A ∶H A ′ on the subspacesH A ,H A ′ ⊂H V ,H V ′, H A ≅H A ′ form an Abelian groupT V of order2 V with elements of degree twoT 2 H A ∶H A ′ = 1 1 H A ∶H A ′ , i.e., T V ∶= {T H A ∶H A ′ A≅A ′ ,A⊂V }. (3.14) The groupT V is isomorphic to the power set ofV endowed with the internal opera- tion of symmetric set differenceΔ [142]: T A 1 T A 2 =T A 1 ΔA 2 , A 1 ΔA 2 ∶= (A 1 −A 2 )∪ (A 2 −A 1 ) (∀A 1 ,A 2 ⊂V ). (3.15) Thus the algebra of swaps is isomorphic to the algebra of sets, i.e.,C(T A ) ≅C(A). In the next section we show how the group algebra C(T V ) of swaps helps us study the dynamics of average purity. 3.2.4 EntanglementDynamicsandDynamicallyClosedSwapAlge- bras For a system in a pure state, ψ⟩, the amount of entanglement between a subsystem,A, and its complement,B, is quantified by the entanglement entropy (EE), of the state for theA,B bipartition defined as the Von Neumann entropy of the reduced density matrix of the subsystem:S VN (ψ) ∶= −Tr A (ρ A logρ A ), ρ A = Tr B ( ψ⟩ ⟨ψ ). The evolution of the entanglement entropy with the number of iterations of the LRQC is a mathematical description of bipartite entanglement dynamics generated upon the initial pure product state ψ i ⟩=⊗ V j=1 ψ i j ⟩which has anS VN (ψ i )=0. 28 The Von Neumann entropy however is a particular limit of a range of entanglement measurescalledtheR´ enyientanglemententropiesparametrizedbyacontinuousparam- eterα that are defined as, S α (ψ)∶= 1 1−α logTr A (ρ α A )∀α≥0, (3.16) whileS VN (ψ)=lim α→1 S α (ψ). TheR´ enyientropiesobeytheorderingrelationS α (ψ)≥ S α ′(ψ) ∀α ′ ≥ α, which fact we use to lower bound the entanglement entropy by the 2-R´ enyi entropy, i.e., S VN (ψ) ≥ S 2 (ψ). This we do because the latter captures the essential, feature of an entanglement monotone, i.e., a quantity that never increases underlocalquantumoperationsandclassicalcommunications(LOCC)onasharedstate betweenAandB,iseasiertocomputeandmoreoverhasaphysicalinterpretationasthe expectation value of an observable on two copies of the Hilbert spaceH V . To see that the purity can indeed be expressed as the expectation of an observ- able we consider the replica trick for traces of products of operators: Tr V (QP ) = Tr V,V ′(Q⊗P T V∶V ′) ∀ Q,P ∈ A(V ) H V ≅ H V ′ where T V∶V ′ is the operator swap- ping the isomorphic Hilbert spaces H V ,H V ′. Thus for the purity P, we have that P = Tr A ρ 2 A = Tr A (Tr B ( ψ⟩ ⟨ψ ) 2 )= Tr V∶V ′(( ψ⟩ ⟨ψ ) ⊗2 T A∶A ′) where the operatorT A∶A ′ swaps two copies of only the subspacesH A ⊂H V ,H A ′ ⊂H V ′ with trivial action over H B ,H B ′. Using the concavity of the log function one can writeS 2 (ρ A ) = −log(ρ 2 A ) ≥ −log(ρ 2 A ) =∶ −log(P ) and thus the average entanglement dynamics of the LRQCs can thenbeunderstoodasthedynamicsoftheaverage2-R´ enyientropywhichinturncanbe lower bounded by a quantity calculated from the average purity. 29 The calculation of the average purity afterk-iterations of the LRQC requires calcu- lating Haar measure averages over the local unitaries. Starting with the fiducial com- pletelyfactorizedstateω = ψ i ⟩ ⟨ψ i =⊗ j∈V ψ i j ⟩ ⟨ψ i j ,theaveragepurityP,overunitaries U S that act on the local regionsS drawn by the RQC is given by P U S = dµ (U S)Tr[(U S ωU † S ) ⊗2 T A∶A ′] = dµ (U S)Tr[ω ⊗2 (U † S ) ⊗2 T A∶A ′U ⊗2 S ] =Tr[ω ⊗2 dµ (U S)(U † S ) ⊗2 T A∶A ′U ⊗2 S ] = ⟨ω ⊗2 ,R 2 (T A∶A ′)⟩. (3.17) The ensemble maps R 2 thus allow one to study the dynamics of average purity and this can be done in terms of dynamically closed subalgebras of swaps defined in sec. (3.2.3). The group algebraC(T V ) of swaps is a 2 V -dimensional Abelian subalge- bra ofA(V ) ⊗2 under matrix multiplication. We refer toC(T ) V as the swap algebra of V. Forbrevity,fromnowonwerepresentbyT A theswaponH A ,H A ′,i.e.,T A =T H A ∶H A ′ whenever the meaning is clear from the context. 3.3 ModelsofLocalRandomQuantumCircuits In this section we present and analyse a couple of different models for Local Random Quantum CircuitsC k [Ξ,L,q (k) ]. As mentioned in the previous section we fix the mea- sureofintegrationoverthelocalunitariestobetheHaarmeasure,i.e.,Ξ=dU Haar . The different models presented here then correspond to choices for the local regionsL and the probability distribution over subsets of the local regionq k (S),S∈L k . 30 ThechoicesofthelocalregionsLareoftenphysicallymotivatedwhereasdepending on the kind of dynamics that is to be modelled the choice of the probability distribution q k (S)may be categorized into two main types: • Uncorrelated LRQC: Here the choices of the subsets of local regions at each level of the circuit is not correlated with the previous choices made. Hence with {p (j) } k j=1 denotingk distribution laws overL we have that, q (k) (S)= k j=1 p (j) (S j ). (3.18) • CorrelatedLRQC:Herethechoicesofthesubsetsofthelocalregionateachsuc- cessive level of the circuit depends on the L × L stochastic matrices {M (j) } k−1 j=1 thatgovernthechoicesofsubsetsateachstepj =1,...,k. Thusthejointprobabil- itydistribution oversubsets ofsuch a timedependent Markovian process is given by, q (k) (S)=M (k−1) (S k ,S k−1 )M (k−2) (S k−1 ,S k−2 )...M (1) (S 2 ,S 1 )q (1) (S 1 ). (3.19) In the following sections we present examples of both Correlated and Uncorrelated LRQCs. Through a combination of analytical and numerical techniques we show how the purity, and hence the 2-R´ enyi entropy, evolves as a function of the depth of the circuit. The result is that starting from an initial totally factorized state, families of LRQCswithadepthwhichissmallcomparedtothesubsystemsize,generateensembles of states whose entanglement as measured by the 2-R´ enyi entropy is small. For times that scale asO( A )the states are maximally entangled. 31 3.3.1 Uncorrelatedchoicesoflocalregions ● RandomEdgeModel This is an example of an uncorrelated LRQC where the vertices of a graph G = (V,E) host local Hilbert spacesH i ≅C d ∀i∈V so that the full Hilbert space on which the circuit acts is given byH V =⊗ V i=1 H i . The local interactions can only occur between vertices (i,j)ofthegraphthatareconnectedbyanedgewithinthesetofedgesE. Thus intermsoftheformalismpresentedintheprevioussectionthelocalregionsherearethe edges, i.e.,L =E. At each step the LRQC draws an edgeX ∈E according to the flat unit normalized measureq ∶X ⊂E→q(X) ∈ [0,1], withq(X) = 1 E . Conditioned on the extraction of the edgeX, a unitary with non-trivial support onX is drawn with the Haar measuredµ Haar (U X)=∶dU X . The family of such Local Random Edge Quantum Circuits is therefore represented by: C k [Ξ=dU X ,L=E,q (k) ( ¯ X)= ( 1 E ) k ], (3.20) where, ¯ X = (X 1 ,X 2 ,...,X k )isthesequenceofedgesthatthecircuitextracts. Whilethe data above defines the circuit, its input is a fiducial totally factorized pure state ψ i ⟩ = ⊗ j∈V ψ i j ⟩ in the graph degrees of freedomj ∈V. Then the ensemble of states that such a circuit produces may be represented byE k [Ξ =dU X ,L =E,q (k) ( ¯ X) = ( 1 E ) k , ψ i ⟩]. As the LRQC applies Haar random local unitaries to the stochastically picked edges at each step (equivalent to a level in the depth) the average purity (2-R´ enyi entropy) of the reduced density matrix on any distinguished subsetA ⊂V decreases (increases) monotonically. In fact for considerations of purity the dependence on the fiducial state ψ i ⟩dropsoutcompletelyandanyinitialpuretotallyfactorizedstategeneratesidentical purity dynamics. 32 A B Figure 3.2: Schematic of the Random Edge Model. The blue circles represent local Hilbert spaces at the vertices V of the graph G = (V,E) with the edges E shown as green arrows. For the bipartion V = A∪B the change in purity for the state of the subsystemA is effected only by edges shown in red that straddle the boundary. Inoursettingthen,thesubsystemAandit’scomplementB formadisjointpartition oftheentiresetoflocalHilbertspaces,A∩B =φ,A∪B =V. Thepurityofthereduced stateonAchangesiffaninteractionoccursacrosstheboundaryseparatingAandB. For all interactions that occur exclusively in either part of the bipartition there is no change in the purity of the reduced state as these do not entangle the degrees of freedom across thebipartition. Ifwedenoteby∂A⊂E thoseedgesthatgoacrosstheboundary,that,is 33 thosethat havenon-nullintersectionwith bothAandB,theaverage purityforonestep of the circuit is given by (withω = ψ i ⟩ ⟨ψ i ): P = X∈E∂A q(X)Tr[ω ⊗2 " dU X (U † X ) ⊗2 T A (U X ) ⊗2 ] + X∈∂A q(X)Tr[ω ⊗2 " dU X (U † X ) ⊗2 T A (U X ) ⊗2 ] = X∈E∂A q(X)×1+ X∈∂A q(X){N d Tr[ω ⊗2 T A∪X ]+N d Tr[ω ⊗2 T AX ]} = (1−Q)+2QN d , (3.21) whereQ=∑ X∈∂A q(X)= ∂A E is the net probability of a boundary node ofA interacting with one inB andN d = d"(d 2 +1) is a group-theoretical factor that comes about as a resultoftakingthetraceofthetensorfactorintheoperatorT A =⊗ i∈A T i ,whosesupport overlaps withX, with respect to Π + X = (1 1+T X )"2. In going to the third line above we have also used the fact that Tr[ω ⊗2 T X ] = 1∀X ⊂V. Since local unitaries completely internaltoAorB,i.e.,withnosupportacrossthebipartitioncannotaffectthepuritywe geta1fortheintegralinthefirsttermontheR.H.S.. Notethatthisaveragingprocedure over the boundary unitary executed once leads to an algebra of theT X operators where with probabilityQ= ∂A E the LRQC generates an equal superposition ofT A∪X andT AX (X beingtheedgeonwhichtheunitaryacts)andwiththecomplementoftheprobability (1−Q)itleavestheoperatorT A invariant. Thisamountstosayingthatthenextiteration of the LRQC sees two new effective subsystems A∪X and A'X with with relative weightQ"2 each and the original subsystem with weight (1−Q). The LRQC now iterates this procedure k-times by drawing edges and unitaries according to the respective distributions q(X) = 1 E and dU X . Under the assump- tion that the degree of each vertex in the graph is o( ∂A ) which implies that the 34 boundary length changes negligibly due to the algebra of Eq. (3.21), i.e., we may take ∂A∪e ≈ ∂A'e ≈ ∂A then one can show that the purity fork iterations goes as P k ≥ (1−Q(1−2N d )) k . (3.22) One can understand the physical content of these calculations by considering the ther- modynamic limit of large E >> ∂A ⇒ Q small. In this limit the average 2-R´ enyi EntropyS 2 canbelowerbounded(usingconcavityofthelogfunction)bythelogarithm (base 2) of the average purity, i.e., S 2 (k)∶=−logP k ≥−logP k ⇒S 2 (k)≥−k log(1−Q(1−2N d ))∼Qk(1−2N d )= ∂A E k(1−2N d ). (3.23) The numberk of iterations corresponds to the time for any dynamics that the Random EdgeLRQCmaymodel,soEq.(3.23)implieslinearincreaseoftheentropywithtime. It alsoimpliesthatthe2-R´ enyientropyisproportionaltotheboundaryoftheregion ∂A . Indeed for this model, all α−R´ enyi entropies can be shown to be proportional to the areaoftheboundaryandincreasinglinearlywithtime,i.e.,S α (k)≥ 1 1−α log{Tr[ρ α A ]}∝ ∂A k. Notethatthequantitye p ∶=1−2N d = (d−1) 2 "(d 2 +1)istheHaaraverageofthe entangling power for unitaries acting on ad×d-dimensional system introduced in [? ]. Eventually in thek→∞ limit, i.e., fork ≫ A , the asymptotic purity can be shown to reach the value ¯ P ∞ = d A +d B d A∪B+1 . ● UniformlyRandomEdgesin1-D This is an example of an uncorrelated Random Edge Model in 1-D where the graph G = (V,E) is a linear chain of local degrees of freedom length V = L. The vertices of the graph are labelled by the indicesi = 1,2,..,L, while the local regions on which the LRQC acts is the set of nearest neighbor edges given by: E = {(1,2), (2,3),...(L− 35 1,L)}. WechooseasoursubsytemA,acontiguoussetofspinsoflength1<l<Lwith the aim of finding the purity of the reduced state onA with iterations of the circuit. The analysis of this case demonstrates the power of the ensemble map formalism as the purity dynamics is restricted to the (L+ 1)-dimensional subspace of the alge- bra of swaps CT V spanned by all the swap operators on the subsets of the vertices: T S ,S = 1,..,i,i = 1,..,L and T φ = 1 1. We denote this basis by kets in the Hilbert- schmidt space through the correspondence 0⟩ = T φ , 1⟩ = T {1} , 2⟩ = T {1,2} ,..., L⟩ = T {1,2,..,L} and this subspace can be shown to be invariant under the edge mapsR e (.)∶= ∫ dU e U ⊗2 e (.)U †⊗2 e ,e∈E. TheEnsemblemapRinthiscaseisauniformlinearcombinationofedgemaps,i.e., R= 1 L−1 ∑ L−1 i=1 R i,i+1 and in the basis defined just above, its action is given by: R 0⟩= 0⟩, R L⟩= L⟩, R i⟩=a i⟩+b( i−1⟩+ i+1⟩), i=1,2,..,L, (3.24) where a = (L−2)"(L−1) and b = (N d )"(L−1). Thus the superoperatorR admits a non-Hermitian matrix representation R in our chosen basis given by R = R 1 +R 2 whereR 1 = 0⟩ ⟨0 +b 0⟩ ⟨1 + L⟩ ⟨L +b L⟩ ⟨L−1 andR 2 =a1 1 L−1 +bA L−1 . Here 1 1 L−1 denotes the identity matrix in the (L−1)-dimensional subspace spanned by { i⟩} L−1 i=1 andA L−1 istheadjacencymatrixofthepathgraphoflength (L−1)whoseverticesare labelled by the same basis vectors. The matrixR can be diagonalized by a non-unitary transformationV asR =VDV −1 withD = diag(λ µ ) L µ =0 with the eigenvalues given by {λ µ } µ = {1,a+2bcos(πh"L),(h = 1,2,3,..,L−1),1}. Writing theL−1 eigenvalues different from 1 as 1−Δ h where Δ h = 1 L−1 (1−2N d cos(πh"L)) ∈ (0,1) one can write the spectral gap asΔ∶=1−min{Δ h } L−1 h=1 which is attained ath=1 and is given by, Δ= 1 L−1 (1−2N d cos(π"L))≥ 1 L (1−2N d )=e p "L. (3.25) 36 The spectral gap of the matrix R controls the rate of relaxation to the asymp- totic state which is exponentially close (in the size of the complement B ) to a max- imally entangled pure state across the A,B bipartition. Starting with a subsystem of size A = l, i.e., in the ensemble map formalism a state l⟩ = T l , the average purity of the subsystem after any number of steps k is given by: ¯ P k =∑ L j=0 (R k ) j,l = ∑ L j=0 (VDV −1 ) k j,l = ∑ L j=0 (VD k V −1 ) j,l = ∑ L µ =0 λ k µ (∑ L j=0 V j,µ )(V −1 ) µ,l . Clearly in the asymptotic limitk→ ∞, only the contributions from the unit eigenvalues survive, i.e., ¯ P ∞ = (∑ L j=0 V j,0 )(V −1 ) 0,l +(∑ L j=0 V j,L )(V −1 ) L,l = (d L−l +d l )"(d L +1)whileforanyfinite k the difference to this value is given by (with the sum over odd values ofh between 1 andL−1): ¯ P k − ¯ P ∞ = 2 L h odd (1−Δ h ) k sin(πlh"L){ 2N d sin(πh"L) 2N d cos(πh"L)−1 +cot(πh"2L)}. (3.26) One can use this result for the average purity after a finite number of steps to bound its difference from ¯ P ∞ as ¯ P k − ¯ P ∞ ≤le −kΔ C with ad-dependent constantC =O(1). Further using the lower bound on the spectral gap Eq. (3.25) of the matrix R implies that, k ≥ L e p (log(C"ǫ)+logl) ⇒ ¯ P k − ¯ P ∞ ≤ǫ. (3.27) ● RandomEdgeModelonthecompleteGraph We now consider a variation of the Random Edge Model where we relax the con- dition of local interactions by taking the graph G = (V,E) to be the complete graph K V ,thatis,agraphinwhicheverytwoverticesareconnectedbyanedgebelongingto the setE = V ×V. The number of edges is of course E = + N 2 ,. As before, the RQC draws an edgeX according to flat unit normalized measureq ∶X ∈E→q(X)∈ [0,1], withq(X)= 1 E . ConditionedtotheextractionoftheedgeX,aunitarywithsupporton 37 X is drawn with the Haar measuredµ Haar (U X). The analysis of this case highlights the utility of the superoperator formalism. We introduce a bipartition in the system by V =A∪B withA∩B =∅,withcardinalities A =N A and B =N B andN =N A +N B . The average purity of the reduced state on subsystem A after the k-th iteration of the circuit is given by: P k =Tr[ω ⊗2 R k (T A )]= ⟨ω ⊗2 ,R k (T A )⟩. (3.28) ToseewhatformRtakeswehavetoseethataftereachextraction,iftheextractededge is not straddling the bipartition then the superoperator has trivial action. Therefore, the probabilityq(A) thatR(T A ) has a trivial action is given by the probability of drawing anedgecompletelyinsideAorcompletelyinsideB =V *A,thatis,q(A)= ! N A 2 "+! N−N A 2 " ! N 2 " . Otherwise,R(T A )has a non trivial action: R(T A )=q(A)T A + N d E {N B i∈A T Ai +N A j∈B T A∪j }. (3.29) NoticethatthesuperoperatorRactsonthespaceofswapoperatorsT A . Thatis,forany subsetA ⊂ V,R(T A ) is an operator in the Hilbert-Schmidt space onH ⊗2 . Therefore, the matrix elements ofR in the space of the shift operators are given by R B,A = ⟨T B R T A ⟩. (3.30) Notice now that thek-th iteration of the superoperator corresponds to matrix multipli- cation, that is, (R k ) B,A = ⟨T B (R) k T A ⟩. (3.31) 38 In order to find a convenient expression forR for this model, we show how to map thesuperoperatorR(T A )inEq.(3.29)actingontheswapoperatorstoaspinoperatorin an abstract 2 N -dim Hilbert space. First of all, consider the mapping between subsets of V to a pure vector which is a member of the computational basis in an abstract 2 N -dim Hilbert space: K ∶A⊂V→⊗ i∈V χ A (i)⟩ = ψ A ⟩ withχ A (i) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 i∈A 1 i∉A , (3.32) Here,χ A (i) is the indicator function for the nodei ∈V. By considering all the subsets X in the power setP(V ) it is obvious that the states ψ X ⟩, X ⊂ V form a complete orthonormalbasisforthe2 N dimensionalHilbertspaceofN spin1/2particles. Wewill callthisspacetheabstractqubitspaceandoperatorsinthisspacearedenotedwithahat on top. On this space let us introduce the total spin operators ˆ S α = 1 2 ∑ i∈V σ α i whereσ α i are the Pauli matrices acting at the i’th spin. Note that then: ˆ S Z ψ A ⟩= 1 2 (N A −N B ) ψ A ⟩, (3.33) whichallowsus topromotethenumbersN A ,N B inEq.(3.29)tooperatorsN A → ˆ N A = ( ˆ S Z +N"2),N B → ˆ N B = (N"2− ˆ S Z ),inthesensethatN A = ⟨ψ A ˆ N A ψ A ⟩andsimilarly for ˆ N B . With these definitions, we have q A = 1− N A N B N(N−1) = 1− 1 E [(N"2) 2 − ( ˆ S Z ) 2 ]. 39 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææææææææææææææææææææææææææææææææææææææææææææ à à à à à à à à à à à à à à à à à à à ààààààààààààààààààààààààààààààààààààààààà ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôô 10 20 30 40 50 60 k 0.2 0.4 0.6 0.8 P k æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à à ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô 10 20 30 40 50 60 k 1 2 3 4 5 6 Log@P k P D Figure 3.3: [i] (Left panel) Purity as given by Eq. (4.20) with fixed total size N = 10 and different subsystem sizes N A = 1 (blue circles),N A = 2 (purple squares),N A = 3 (yellow rhombi),N A = 4 (green upright triangles),N A = 5 (dark blue inverted triangles). The horizontal lines are asymptotic values given by (3.42). [ii] (Right panel) Natural logarithm of the difference between purity at thek’th step and the asymptotic purity withN =10 and same color codes forN A as in [i]. Similarly we now define the total raising ˆ S + = ˆ S x +i ˆ S y and lowering operators ˆ S − = ˆ S x −i ˆ S y . Then we obtain ˆ S − ψ A ⟩= j∈B ψ A∪j ⟩ (3.34) ˆ S + ψ A ⟩= i∈A ψ Ai ⟩. (3.35) Now we can define the (hatted) operator ˆ R acting on the abstract qubit space as: ˆ R∶= (1− 1 E [(N"2) 2 −( ˆ S Z ) 2 ]) + N d E [ ˆ S + (N"2− ˆ S Z )+ ˆ S − (N"2+ ˆ S Z )]. (3.36) ˆ Risanon-Hermitianoperatorintheabstractqubitspacespannedbythevectors ψ A ⟩. In this space, its matrix elements read ( ˆ R) B,A = ⟨ψ B ˆ R ψ A ⟩. We now show that iterations oftheRQCprotocolintheoriginalspacecorrespondtoiterationsoftheoperator(3.36) in the abstract space. To this aim, an important observation is that in the original space 40 thetraceofthepowersofthesuperoperatorRistakenwithrespecttoω ⊗2 whichbelongs tothetotallysymmetricsubspaceofH ⊗2 . Wecanfinallyrelatetheaveragepurityofthe subsystemA to a sum over matrix elements of the matrix ˆ R k : P k (A)=Tr[ω ⊗2 R k (T A )]= ⟨ω ⊗2 , B⊂V (R k ) B,A T B ⟩ = B⊂V (R k ) B,A 1 /0000000000000000100000000000000002 ⟨ω ⊗2 ,T B ⟩= B⊂V ( ˆ R k ) B,A = B⊂V ⟨ψ B ! ˆ R k !ψ A ⟩, (3.37) and thus from Eq. (3.37) one can see that in the abstract qubit space P k = (⟨0! +⟨1!) ⊗N ( ˆ R) k !ψ A ⟩, (3.38) because∑ B⊂V !ψ B ⟩= (!0⟩ +!1⟩) ⊗N ≡ !ψ symm ⟩. Notice that the state !ψ symm ⟩ is invariant under the action of the projector Π S = 1 N! ∑ σ∈S N σ onto the symmetric subspace of the abstractqubitspace,whereσ ∈S N aretheelementsofthesymmetricgrouponN labels. With this in mind and the fact that Total spin operators commute with Π S we can write Eq. (3.38) as: P k = ⟨ψ symm !Π S ( ˆ R) k Π S !ψ A ⟩. (3.39) Note that Π S !ψ A ⟩ ∈ H J=Jmax , i.e., Π S is the projector onto the highest total spin sub- space. In our case then it projects onto theJ =N 2 subspace. Since [( ˆ R) k ,Π S ]=0 we focusonjusttheH J=Jmax subspace. HoweverΠ S !ψ A ⟩isnotnormalized. Normalization involves some algebra to relate states in the tensor product basis for the abstract qubit space to the total spin basis: suppose then that the z-component of spin for some state in the tensor product basis of L spins is m. ThenN up +N down =N, 1 2 (N up −N down ) = 41 m !⇒ N up = (N 2 +m) thus the number of distinct states in the tensor product basis with m as their z-component is % N Nup & = % N N2+m &. The state in the total spin basis in the H J=Jmax subspace that has the same z-component is a symmetric combination of these distinct states with appropriate normalization. It turns out that: Π S !m⟩= 1 $ % N N2+m & !N 2,m⟩. (3.40) With this in mind we see that Π S !ψ A ⟩ = 1 # $ N N A % !N 2, 1 2 (2N A −N)⟩. Inserting this and the identity in theH J=Jmax subspace into Eq. (3.39) we obtain: P k = 1 $ % N N A & N2 m=−N2 ⟨ψ symm !Π S !L 2,m⟩ ×⟨N 2,m!( ˆ R) k !N 2, 1 2 (2N A −N)⟩ =C(N,N A ) N2 m=−N2 , - N m +N 2 .( ˆ R) k m, 1 2 (2N A −N) , (3.41) whereC(N,N A )= $ N A !(N−N A )! N! andthematrixelementofthek iteratedsuperoperator are ( ˆ R) k m, 1 2 (2N A −N) = ⟨ N 2 ,m!( ˆ R) k ! N 2 , 1 2 (2N A −N)⟩. NotethatEq.(4.20)expressespuritydynamicsofad N -dimensionalsysteminterms of dynamics in an exponentially smaller (N +1)-dimensional space. From Eq. (3.36) one can see that for N A = 0, ˆ R!ψ A ⟩ = 1 as well as for N A = N, ˆ R!ψ A ⟩ = 1, all other eigenvaluesoftheoperatorbeinglessthan1. Ifoneassumesthatthesearetheonlytwo fixed points of the operator then asymptotically the non-zero eigenspace is spanned by 42 2.5 3.0 3.5 4.0 4.5 5.0 5.5 ln@LD 5.0 4.5 4.0 3.5 3.0 2.5 ln@ D 10 20 30 40 50 L 5 10 15 ln@ÈÈMÈÈ ! ÈÈM 1 ÈÈ ! D Figure 3.4: [i] (Top panel) Scaling behavior of gap (Δ) for the superoperator of the fully connected RQC. The best fit line (red) gives us the equation (in natural base): log[Δ]= −.969395log[L]+.0246366[ii](Bottompanel)Scalingbehaviorofthenatural logarithm of the product of operator norms for the similarity transform matrixM and it’s Inverse. The best fit line (red) giveslog[!!M!! ∞ !!M −1 !! ∞ ]=1.17508 +0.318L. the symmetric and antisymmetric combinations ψ 0 ⟩+ ψ N ⟩ √ 2 , ψ 0 ⟩− ψ N ⟩ √ 2 with only the sym- metric combination contributing to the purity which reaches the value P k→∞ = d 2N−N A +d N+N A d N (d N +1) . (3.42) The typical behavior of purity is shown in Fig. (3.3). An obviously important question is: How fast in k does the protocol take the purity to within ǫ > 0 of the asymptotic value? The answer to this question can be related to the gap of the matrix ˆ R in the H J=Jmax subspace. FromEq.(3.36)itiseasytoseethatthematrixrepresentationRhas elements: R p,q ≡ ⟨ N 2 ,p! ˆ R! N 2 ,q⟩ =δ p,q f(p) + N d !E! [( N 2 +q) , N 2 ( N 2 +1) −p(p −1)δ p,q+1 +( N 2 −q) , N 2 ( N 2 +1) −p(p +1)δ p,q−1 ], (3.43) 43 which implies that the above matrix R is tri-diagonal and satisfies the condition R k,k+1 ⋅R k+1,k ≥ 0. Such a matrix is similar to a Hermitian matrix. One can then diag- onalize the Hermitian matrix and asymptotics can be calculated by the power method of eigenvalues for the derived Hermitian Matrix. We then have thatSRS −1 =H !⇒ ∃ U s.t. U † HU = D !⇒ U † SRS −1 U = D where D,U,S are diagonal, Unitary and Invertible matrices respectively. Using the spectral resolution ofD one can write, USRS −1 U † =∑ i λ i !λ i ⟩⟨λ i ! !⇒ R k =S −1 U † ∑ i λ k i !λ i ⟩⟨λ i !US. Then withM =U † S we obtain from Eq. (3.39) for the case !A!=N 2: P k =C(N,N A ) N+1 α=1 λ k α ⟨ψ + !M −1 !λ α ⟩⟨λ α !M !ψ 0 ⟩ = 2 N2 $ % N N2 & N+1 α=1 λ k α a α , (3.44) where !ψ + ⟩ is the normalized !ψ symm ⟩ and !ψ 0 ⟩ = !J =N 2,m=0⟩ are unit normalized vectors anda α = !⟨ψ + !M !λ α ⟩⟨λ α !M −1 !ψ 0 ⟩!. Assuming that the eigenvalues ofR are soarrangedthatλ 1 =λ 2 =1 and therestarearrangedin anon-increasing orderwefind: !P k −P ∞ !=C(N,N A )! L+1 α=3 λ k α a α ! ≤C(N,N A ) L+1 α=3 !λ α ! k !⟨ψ + !M !λ α ⟩⟨λ α !M −1 !ψ 0 ⟩! ≤C(N,N A )!λ 3 ! k L+1 α=3 !⟨ψ + !M !λ α ⟩!!⟨λ α !M −1 !ψ 0 ⟩! ≤C(N,N A )!λ 3 ! k × ' * * + L+1 α=3 !⟨ψ + !M !λ α ⟩! 2 ' * * + L+1 α=3 !⟨λ α !M −1 !ψ 0 ⟩! 2 ≤C(N,N A )λ k 3 !!M!! ∞ !!M −1 !! ∞ . (3.45) 44 WerequiretheR.H.Softheaboveexpressiontobelessthanǫ. Therefore,takingthe logarithmoftheinequalityC(N,N A )λ k 3 ++M++ ∞ ++M −1 ++ ∞ ≤ǫweobtain(usingλ 3 =1−Δ): k ≥ logC(N,N A )+log++M++ ∞ ++M −1 ++ ∞ +log11ǫ log 1 1−Δ . (3.46) Sincelog 1 1−Δ ≥Δ, requiring k ≥ logC(N,N A )+log++M++ ∞ ++M −1 ++ ∞ +log11ǫ Δ , (3.47) makes sure that inequality Eq. (3.46) is also fulfilled. Indeed the bound in Eq. (3.47) is a rather weak lower bound as for all graph sizesN that we studied numerically the asymptotic values for any size of the subsystemN A <N were reached much before the above bound. The main difficulty in estimating k min (ǫ,N,N A ), i.e., the minimum number k of iterations required to reach within ǫ accuracy of the asymptotic value resides in the calculation of the operator norms of the similarity transform matrices M, its inverse M −1 and the gap Δ. This amounts to diagonalizing the non-Hermitian matrix R in the maximal total spin subspace. However this is an exponentially reduced problem of diagonalization in a (N +1)-Dim space compared to ad N dimensional one. Numerical study shows that the gap Δ of R to has an algebraic dependence on N as shown in Fig. (3.4). From the least-squares best fit line we can evaluate Δ=e .025 N .97 ≈ 1.025 N . (3.48) 45 Moreover, from the lower panel of the same figure we find a linear dependence of the logarithm of the product of operator norms of the matricesM,M −1 , i.e., log[++M++ ∞ ++M −1 ++ ∞ ]=1.17508+0.318N. (3.49) Thus logC(L) ≈ L→∞ α 1 logL, log[++M++ ∞ ++M −1 ++ ∞ ] ≈ α 2 L + β, 1 Δ ≈ α 3 L where α 1 ,α 2 ,α 3 ,β =O(1)we finally obtain the scaling k ≥(α 1 logN +α 2 N +β)α 3 N +log(11ǫ)α 3 N =O(N 2 ). (3.50) 3.3.2 Correlatedchoicesoflocalregions ● ContiguousEdgeModel In this section we consider an example of a LRQC where the choices of the local regionsinsuccessiveiterationsofthecircuitarecorrelated. Aswewillarguethismodel intends to mimic evolution of a multi-partite system under a local (time-dependent) Hamiltonian. Here, all the edgesE of graphG = (V,E) are acted on by the RQC with 2-local unitaries in some particular order ‘σ’ - denoting an ordered sequence of edges. A LRQC of widthO(N) picks this ordering of nodes such that all nodes in the graph are acted on by nearest neighbour unitaries. One pass through such a circuit is called a cycle. This procedure is then iterated throughn c cycles. The composite unitary of a cycleistheσ−orderedproductofunitarieswithsupportonnearestneighboursaccording to the graphG. While our general formulation of this model extends to graphs with any geometry in any number of dimensions we present a detailed analysis of 1-D graphs and the 2-D square lattice. 46 Figure 3.5: A bipartite (A,B) spin chain of lengthL =L A +L B with nearest-neighbor qubits interacting via 2−qubit gates (ellypses). The edgee is the one that straddles the two partitions. The gates are numbered by the subscriptx i wherex =A,B denotes the two halves of the chain andi the distance from the boundary of the two partitions. ● ContiguousEdgeModelontheLinearChainLetusstartwiththe1-Dgraph ofFig.(3.5),withN =L,andintroduceabipartitionintosubsystemsAandB oflengths L A ,L B . The graphG = (V,E) is thus a linear graph ofL vertices and (L−1) nearet neighboredges. ThesitesinAontheleftoftheboundaryarelabeledbyi A =1 A ,...,L A increasingtowardstheleftwhilethesitesinB arelabeledbyi B =1 B ,...,L B increasing towards the right. The edges on the chain are labeled by a i =< (i+1) A ,i A > and the edgesinB arelabeledbyb i =<i B ,(i+1) B >. TheRQCchoosesanorderingamongall theedgesinthechain,andthenoneachedgeactswithaHaardistributed2-quditunitary operator in the given order. The ensemble therefore contains all possible permutations of the list of edges.Thus the possible number of ordered sequences is +E+! = (L−1)!, i.e., the size of the permutation group on+E+ labels. Forn c =O(1), the average purity depends strongly on the order in which the RQC chooses the edges, while whenn c exceeds the subsystem size, ordering does not really count, as we shall see in the following. In any case, we can consider the orderings 47 that give the two extreme situations, that is, the minimum and maximum decrease of purity, i.e., maximum entangling power which we call the best sequence and the one that corresponds to the minimal decrease of purity or minimum entangling power is termed the worst sequence. LetU σ denote the ordered product of 2-qudit unitaries over all the edges inE with the order given by the permutationσ, i.e.,U σ =U σ(e 1 ) ...U σ(e E ) . The ensemble and its measure are then given by E k {Ξ=dU σ ,L=E,q (k) ( ¯ X)=( 1 +E+! ) k ,+ψ i ⟩} dU σ =δ(U −U σ ) ! X∈E dµ Haar (U X ). (3.51) The physical meaning of the expressions above is as follows: The local region for theLRQCistheentiresetofedgesE ofthegraphGtakeninsomeorderofpermutation. Since each permutation is equally likely the probability of choosing one out of these is equal to the inverse of the total number of possible permutations. The measure of inte- gration over the local unitaries is a product of measures over each of the local unitaries acting on the nearest neighbor edges inE, in the order specified by the permutationσ. We now discuss why sequences of unitaries corresponding to different permu- tations yield different purities. We start by showing which sequence produces the greatest decrease of purity, that we dub best case. The process of generating the sequence of local unitaries according to some permutation is called generating a cycle. This amounts to choosing edges x i ∈ E according to some probability distribution p (i) (x i+1 +x i ),i = 1,2,...(L− 2) such that the ordered sequence x 1 ,x 2 ,...,x L−1 corre- sponds to the edges chosen according to the permutation σ. Let us consider then the sequence U best = U e U A U B where U e is the unitary straddling the edge, i.e., acting on 1 A and 1 B andU A = U a 1 U a 2 ...U a L A −1 is the internal structure ofU A whereU a 1 means 48 a 2-qudit unitary with support on the qudits the nearest of which is at a distance of 1 lattice spacing from the boundary. Note that the internal structure of the unitary is non- deformable since [U a i ,U a i+1 ] ≠ 0 as they share a node. With the same convention the internal structure of U B = U b 1 U b 2 ...U b L B −1 . Physically this corresponds to the LRQC choosing and applying all possible unitaries on the outermost nodes in A and B fol- lowed by two nodes 1 lattice spacing closer to the boundary and so on till the nodes 1 A ,2 A and 1 B ,2 B are acted on. Finally there is a boundary interaction through U e . Thus the probability distribution corresponding to choosing the edges that create this particular permutation of edges, the best sequence, is a cycle of length (circuit depth) k =L A +L B −1 =L−1 where the (L A −1) levels of the circuit choose edges inA as follows: The first set of nodes to be acted upon by the LRQCx 1 is chosen as p (1) (x 1 )= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 iff+x 1 +=2 ∧ D(x 1 ,1 A )=L A −2 0 otherwise , (3.52) while the next L A − 2 levels choose edges depending on the choice of nodes in the previous step p (i) (x i +x i−1 )= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 iff+x i−1 ∩x i +=1 ∧ D(x i ,1 A )=L A −(i+1) 0 otherwise , wherei=2,3,...,(L A −1)andthedistancebetweentwosetsD(x i ,x j )=min y (+y i −y j +) istheminimumdifferencebetweenanytwoelementsbelongingtothetwodifferentsets. Inourcasethisdifferenceisthedifferenceofthepositionlabels(1,2,...,L A ). Similarly 49 of the next (L B −1) levels of the circuit the first set of nodes x L A in B is picked as follows: p (L A ) (x L A +x L A −1 )= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 iff+x L A +=2 ∧ D(x L A ,1 B )=L B −2 0 otherwise , while the next(L B −2)sets are chosen as p (j) (x j +x j−1 )= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 iff+x j ∩x j−1 +=1 ∧ D(x j ,1 B )=L B −(j+1) 0 otherwise , wherej =2,3,(L B −1). Finally the RQC chooses the boundary edge as follows: p (L A +L B −1) (x L A +L B −1 +x L A +L B −2 )= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 iff+x L A +L B −1 +=2 ∧ +x L A +L B −1 ∩A,B+=1 0 otherwise . One can similarly devise probability distributions for the RQC that generates any desiredorderingoftheedges. However,fornow,withthemeasuredefinedinEq.(3.51) and iterating the algebra of Eq. (3.13), one finds that the purity is given by nested inte- grals: 50 P=Tr$ω ⊗2 d[U B ]d[U A ]dU e (U † B U † A U † e ) ⊗2 T A (U e U A U B ) ⊗2 % =Tr$ω ⊗2 d[U B ]((U † B ) ⊗2 d[U A ]((U † A ) ⊗2 dU e (U † e ) ⊗2 T A U ⊗2 e )U ⊗2 A )U ⊗2 B !% =Tr[ω ⊗2 (N 2 d T A +N 3 d T A−1 +N 4 d T A−2 ..+N L A d T A−L A +2 +N L A d T A−L A +N 2 d T A +N 3 d T A+1 +N 4 d T A+2 ...+N L B d T A+L B −2 +N L B d T A+L B )] =(N 2 d +N 3 d +N 4 d +....+N L A d +N L A d )+(N 2 d +N 3 d +N 4 d +....+N L B d +N L B d ) = N 2 d (1−N L A −1 d ) (1−N d ) +N (L A −1) d + N 2 d (1−N L B −1 d ) (1−N d ) +N (L B −1) d . (3.53) The notationT A+r means the swap acting onS = A∪1 B ,2 B ,...,r B and similarlyS = A−r =A&1 A ,2 A ,...,r A . ForL A ,L B reasonablylargewefindfromEq.(3.53)thatatthe conclusion of the first cycle, i.e., iteration 1, P nc=1 ≈ 2N 2 d (1−N d ) , (3.54) while forn c ≪L A ,L B P nc ≈' 2N 2 d (1−N d ) * nc , (3.55) andagainwerecallthatthisisforthebestcasesequence. AlsonotethatfromEq.(3.13) we see that for each non-trivial action of the averaging procedure over the unitaries we getadecreaseofpuritybythesameamount,(1−2N d ),thusasequencewhichmaximizes the number of non-trivial actions will result in the maximum decrease of purity. Now we want to compare the above case with the sequence that produces the least decreaseinpurity(ortheleast2-R´ enyientropy),ortheworstcase. Weseethatbykeep- ing the same internal structure ofU A ,U B and comparing the purities corresponding to 51 the four possible cases: (1)U =U e U A U B , (2)U =U B U e U A , (3)U =U A U e U B , (4)U = U B U A U e wecaneasilyseethatforn c =1thesequence(4)isindeedtheworstcasesce- nario. Numerically, we find that the the sequence 4 performs the worst also for generic large n c . As n c > 1, the decrease of purity also depends on what is the ordering of the unitaries in the products U A and U B . We find numerically that the least decreas- ing sequence of unitaries (worst case) is given by choosing U worst = U B U A U e with U A =U a L A −1 ....U a 2 U a 1 andU B =U b L B −1 ....U b 2 U b 1 . Let us quantify the purity using the worst sequenceU worst as a function ofn c . Forn c =1, we have: P nc=1 =Tr[ω ⊗2 (N d T A−1 +N d T A−1 )]=2N d . (3.56) We can also obtain the exact expression for the purity after any number of iterations n c ≤L A where the size of the environment isL B ≥L A . We find: ¯ P nc = m=nc−1 m=0 Tr[ω ⊗2 , C(n c ,m) 2 N (nc+m) d (T A−(m+1) +T A+(m+1) )] =2s nc {1− ((1−N d )N d ) nc 2n c −1 n c ! 2 F 1 (1,2n c ;1+n c ;N)} = 2N nc d (1−N d ) nc {1−f(n c )}, (3.57) whereC(n c ,m)=2" nc+m−1 m #, s= N d (1−N d ) ,and 2 F 1 (1,2n c ;1+n c ;N)istheGaussHyper- geometricfunction. Therefore,ourformalismallowsustoobtaintypicalityofthepurity for arbitrary depth of the RQC. Whenn c =O(1), the subsystem is very far from being the Haar case, and the RQC is not at−design for anyt, indeed, the system features an area law for the entanglement. As n c increases, entanglement propagates in the bulk 52 at distance ∼ n c from the boundary. For large values of n c , the expression simplifies becausef(n c →Large)→0, and we see that P nc→Large ≈ 2N nc d (1−N d ) nc . (3.58) Comparing this expression to the one for the purity in the best case sequence Eq. (3.55), we see that, as long as n c is smaller than the system size, the best case sequence is better by a factor of 1 2 (2N d ) nc , see Fig. (3.6,top panel). Nevertheless, the same figure shows that after system size is reached, the two cases converge to a similar value. In the following, we show that the asymptotic value is independent of the order- ing. This meansthat thereis an c abovewhichthereis anonsetforthe independenceof the ordering. The numerical results shown in Fig. (3.6,top panel) suggest that the onset happens atn c ∼L A . We can give a justification of why the above given sequences are indeed the ones that decrease the purity the most (best) or the least (worst). Atn c = 1 of course all the unitaries acting afterU e are not entangling at all. Withn c growing, all the unitaries that are acting beforeU e allow some entanglement to be generated among the qubits. The sequences that go towards the boundary bring entanglement towards it, while the ones which start with the boundary and go outwards bring entanglement away. In this case, indeed, the site 1 A would get very entangled with the bulk in A (the same occurs to theB side) and by monogamy this does not allow to effectively transfer entanglement information across the partition. This scenario also shows how in the worst case the RQC would entangle nodes at the same length as the iteration number. At this point, we want to look for a result about the maximum purity of the n c −iterated ensembleE nc . The purity is a monotonically decreasing function because it is obtained by iterated application of CP maps, for alln c . As we pointed out above, the 53 æ æ æ æ æ æ æ æ æ à à à à à à à à à ì ì ì ì ì ì ì ì ì ò ò ò ò ò ò ò ò ò 1 2 3 4 5 6 7 8 n c 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 P n c æ æ æ æ æ æ æ æ æ æ à à à à à à à à à à ì ì ì ì ì ì ì ì ì ì 2 4 6 8 10 n c 0.2 0.4 0.6 0.8 P n c Figure 3.6: [i] (Left panel) Purity as a function of n c for the best sequence of uni- taries(circles) and for the worst (squares) forL A =L B =8. The smooth line connecting thesquaresistheanalyticexpressionforworstcase. Theothercurvesrepresentthepuri- tiesobtainedforfiverandomsequences. Onecanclearlyseethatthesedataliebetween the best and the worstP nc . [ii] (Right panel) Convergence of the worst case sequence totheasymptoticformulavaluefordifferentvaluesofsystem/environmentlength. Blue circles(L = 5,L A = 2) and Yellow rhombi (L = 6,L A = 3) converge to > 99% of their asymptotic value whereas Purple squares(L =8,L A =3) converges to >97%. different results above hold in the region of validity for the scaling ofn c . Of course the absolute minimum of the purity of the reduced state cannot be less than ( 1 d ) L A corre- spondingtothetotallymixedstate. Toanswerquestionsabouttheaveragedistanceofa state inE nc asymptoticallyto the totally mixed state onthe subsytemA weresort tothe superoperatorformulationalittlelater. Aswenoticedabove,plotinFig.(3.6,Toppanel) also shows that for largen c the ordering does not count. This means that our scheme does indeed mimic the Trotter scheme as far as the statistics of the reduced system is concerned. In other words, the average decrease of purity in this model approximates the average decrease of purity obtained by evolving with local time dependent Hamil- tonians and using the Trotter scheme. The irrelevance of ordering for largen c can be understoodmathematicallyinthesuperoperatorformalism,whatwenowgoondescrib- ing for this model. The chain superoperator is aσ−ordered product of non commuting projections,i.e.,R chain =R σ(E) .....R σ(1) ,becauseofthenon-commutavityofthelocal 54 unitariesthatmakeuptheproduct. Infacttheproductsinthesuperoperatorisinreverse order of unitaries. From the first line of Eq. (3.53), one can see that U =U n U n−1 ....U 1 !⇒ R chain =R 1 ....R n−1 R n , (3.59) wherethesubscriptsonU denotesupportsforthesame. Notethat,forafixedsequence, R † chain ≠ R chain and therefore R chain is not a hermitian operator. Nevertheless, the averaged sum over all possible sequences,R = 1 n! ∑ σ R σ , is Hermitian. Since our defi- nitionofthebestsequenceisU =U e U A U B withaspecificinternalstructureofunitaries withinU A andU B and that for the worst sequence isU = U † A U † B U e Eq. (3.59) implies thatR best =R † worst . One can understand the action of the superoperator by studying it’s action on the non-Orthnormal basis of the swapT X , X ⊂ V operators. We regard the swapoperatorsasketsinthe ( V +1)−dimensionalsubspaceS =span{ i⟩}, 0≤i≤ V of the Hilbert-schmidt space on (H V ) ⊗2 . This subspace is the space of swap operators acting on alli qubits from one end of the chain where 0 ≤i ≤ V . The correspondence i⟩ =T i = 1 1 V !i ⊗T i then implies that the Hilbert-schmidt inner product ⟨i,j⟩ ≠δ i,j . In this basis the matrix representations of these superoperators are real and so it turns out that the eigenvalues ofR best and that ofR worst are identical. This means that for suffi- ciently large iteration numbern c which sequence we choose does not matter while the rate of approach to the asymptotic value of purity is dictated by just the gap (1−λ 2 ) in either case. The difference in initial decays of the purity for the sequences lies in the fact that the eigenvectors correspoding to identical eigenvalues are different. Let us now explain the action ofR chain in our chosen basis. Consider the action of the superoperatorR chain corresponding to the best sequence for the linear chain which 55 has A = L A , B = L B when it takes as an argument some T X , X ⊂ V which is the swap on (H X ) ⊗2 subspace. Then the possibilities are: R chain ( 0⟩)=1, R chain ( V ⟩)=1 R chain ( i⟩)= i−1 p=0 N 1+p d i+1−p⟩+N i d 0⟩, 1<i<L A R chain ( L A ⟩)= L A −1 p=1 N 1+p d L A +1−p⟩+N L A d 0⟩+ V −L A −1 p=1 N 1+p d L A −1+p⟩+N L A d 2L A ⟩ R chain ( i⟩)= V −i−1 p=0 N 1+p d i−1+p⟩+N V −i d V ⟩,L A <i< V . (3.60) In this basis the matrix representation R of R chain cannot be guaranteed to be even Normal thus we may only attempt a Jordan decomposition of R but this is enough to understandtheiterativebehavioroftheRQC.Thepurityofthegeneratedensembleafter any number of iterationsk is given by: P nc = i⟩∈S ⟨i R nc L A ⟩. (3.61) FromEq.(3.59)weseethat R chain ≤∏ e∈E R e ≤1whichmeansthatalleigenvalues λ, are less than equal to 1 in modulus and hence asymptotically only the contribution from fixed points survive. Note that each of the projectionsR e has two eigenvectors 1 1 and T V (which are swaps on no nodes and all nodes respectively), with eigenvalue 1. 56 If one assumes then that the common eigenspace ofR chain is spanned by 1 1 andT V we find that the asymptotic value of the purity is: P nc→∞ = ⟨ω ⊗2 ,R ∞ chain (T A )⟩ = ⟨ω ⊗2 , 1 1+T V ! d L (d L +1) ⟩⟨ 1 1+T V ! d L (d L +1) ,T A ⟩+ 0 '********************************************************+********************************************************, ⟨ω ⊗2 , 1 1−T V ! d L (d L +1) ⟩⟨ 1 1−T V ! d L (d L +1) ,T A ⟩ = 1 ! d L (d L +1) ⟨ 1 1+T V ! d L (d L +1) ,T A ⟩ = d 2L−L A +d L+L A d L (d L +1) , (3.62) wherethenormalizationforthesymmetricandantisymmetriccombinationsofthebasis vectors for the common eigenspace is obtained by setting ⟨ 1 1+T V 2C , 1 1+T V 2C ⟩ = Tr[ 1 1+T V 2C 2 ] = 1 !⇒ C = ! d L (d L +1). We have verified this result numerically using small sys- tem lengths and the convergence withn c to the value given by Eq. (3.62) is shown by Fig. (3.6). ForthespecificcaseofL A =L B wefoundnumericallythatthesub-dominanteigen- valueλ 2 saturates at a value of (2N d ) 2 , Fig. (3.7) with increasing system size. Indeed numericalevidenceindicatesthatthespectrumofR chain atleastforthiscaseisentirely positive 0 ≤λ∶λ∈Spec(R chain )≤ 1. All that happens upon increasing the system size isthatthepopulationofeigenvaluesinthesubregionsofthedomain0≤λ≤λ 2 increases proportionally, except for the largest eigenvalue of 1 whose number remains equal to 2, see Fig. (3.7). This corroborates our assumption that there exist only two fixed points. 57 (a) (b) Figure3.7: (a)(Leftpanel)Saturationofthesub-dominanteigenvalueforL A =L B with increasing L A for a chain of qubits. Similar behavior is observed for general qudits. (b) (Right panel) Distribution of eigenvalues for a chain of qubits withL A = L B . The orangebarsrepresentpopulationlevelswithinbandsofthedomainsfortheeigenvalues withL A =L B = 200. Blue bars(superimposed on orange) are forL A =L B = 100 The gapinthespectrumcanbeclearlyseenasthedifferencebetweenthelargesteigenvalue of 1 and the next at .64. 3.3.3 Distancetothemaximallymixedstate With this value of the asymptotic purity Eq. (3.62) we can bound the average dis- tance of states in our ensemble reduced to the subsystem A, from the totally mixed state on it, σ = 1 1 A d L A . Indeed we see that the trace distance, D(ρ,σ) = 1 2 ρ−σ 1 ≤ 1 2 ! rank(ρ−σ) ρ−σ 2 while 1 2 √ ρ−σ ρ−σ 2 = 1 2 √ ρ−σ ! Tr[ρ 2 −ρσ−σρ+σ 2 ] = 1 2 √ d A # Tr(ρ 2 )+Tr(σ 2 )− 2 d A = 1 2 √ d A ! P (ρ)−P (σ)≤ 1 2 √ d A $ % % & ǫ '*****************************+******************************, P (ρ)−P (σ) = 1 2 √ d A √ ¯ ǫ= 1 2 # d L A d L B = 1 2 d L A −L B 2 . (3.63) 58 Thus for d B ≫ d A the LRQC of the contiguous edge model takes any initial pure totally factorized state indistinguishably close to a maximally entangled pure state with respect to theA,B bipartition such that on the small subsystemA the reduced state is the totally mixed. The exact analytical calculation of the variance for such models is difficult. How- ever, we can resort to a Markov type inequality for positive valued random variable to assert that the output states of the LRQCs in the various ensembles typically have very small probabilities of being significantly different from their average: P(X >µ ) ≤ E(X) µ !⇒ P(X > ¯ P ∞ )≤ ¯ P ∞ µ = d −L A +d −L B µ . (3.64) In this section we have analyzed in detail two main types of Local random quantum circuits with a) uncorrelated choices of local regions and b) correlated choices of local regions. Of the former type we have analyzed three specific instances i) Random edge modelonageneralgraphinanydimensions3.3.1ii)choosingnearestneighboredgeson a graph in 1-D 3.3.1 and iii) random choices of edges on a completely connected graph 3.3.1. For the correlated case we studied the contiguous edge model 3.3.2 where the unitaries acting on local regions mimic a discrete time quantum evolution with a local Hamiltonian. In all these cases the asymptotic purity, after a large number of iterations (much greater than the subsystem size), reaches a value that can be seen as the sum of minimal purities for the subsytemA, and its complementB, i.e., ¯ P ∞ ≅ 1 d A + 1 d B . How fast does the purity approach this asymptotic state under the action of the LRQCs depends in all cases on the gap of the ensemble mapR 2 considered as an oper- ator in the space of swap operators acting on relevant subsets of the total degrees of freedom. In Table 3.1 we summarize the dependence of the minimum required value of iterationsk min for the condition ¯ P ∞ − ¯ P k <ǫ to hold for the different models. 59 Table 3.1: Minimum number of iterations required to getǫ close to minimal purity. Model k min Terminology REM on a general graph #E#"#∂A# #E#: No. of Edges, #∂A#: Boundary length Nearest neighbor edges in 1-D Llogl L=Total length, l= subsystem length REM on the complete graph L 2 L=Total size of the graph Contiguous edge model L L=Total size of the graph It is clear from the table that the fastest approach to asymptotic purity is attained by the contiguous edge model which is to be expected since an extensive number (∼L) of parallel gates act on the degrees of freedom where L is the total size of the graph. We defined physical ensembles of states acting on initial totally factorized states by a circuit of lengthk of random and independent unitaries with local support. We studied thetypicalityofentanglementbymeansofthepurityofthereducedstateandfoundthat for a timek =O(1), the typical purity obeys the area law. Thus, the upper bounds for area law are actually saturated, on average. Similarly, we proved that by means of local evolution a subsystem of linear dimensionsL is typically entangled with a volume law when the time scales with the size of the system. Moreover, we showed that for large values ofk the reduced state becomes very close to the completely mixed state for all these models. 3.4 ConcludingRemarks In this chapter we have summarized a general formalism to analyze Local Random QuantumCircuitsandpresentedseveralexamplesofthelatter. Thebasicphysicalques- tionweansweredwastheevolutionofentanglementgeneratedbytheLRQCdynamics. Since the proxy for entanglement was the purity, and through it the 2-R´ enyi entropy, we showed that the relevant ensemble map was the second moment operatorR 2 for the 60 circuits. The iterative action of the circuits was shown to lead to an algebra of swap operators on subsets of the total degrees of freedom. The fixed point of this map cor- responds to a normalized symmetric linear combination of the identity and the swap actingonalllocaldegreesoffreedom,whiletherateofapproachtothisfixedpointwas shown to depend on the gap of the operator representing R 2 in the space of relevant swaps. Using techniues from group theory, operator algebras and matrix analysis we showed the typicality of area law for short times and volume law for times scaling with the system size in a particular model of LRQC - the contiguous edge model. 61 Chapter4 DetectionofTopologicalOrderin PhasesofMatter In recent years a central thrust of research in quantum many-body theory and quantum information science has been the identification and characterization of novel phases of matterwhichcannotbeadequatelydescribedbytheLandausymmetrybreakingmecha- nism [50]. These phases are generically exhibited by ground states of strongly interact- ing systems in two spatial dimensions. Quantum spin liquids [140], topological insula- tors [72], and anyonic systems [134], are examples that are of immediate interest to the condensed matter community and important for quantum information processing tasks as well [45, 134, 14, 48, 96]. Because the low energy states of these gapped systems do not break any symmetry of the Hamiltonian there exists no local observable whose expectationvaluesmaybetakenasanorderparameterdenotingthephase[50];however despitesharingthesamesymmetriestheremayexistphasesthatexhibitdifferentphysi- cal properties [120]. The non-symmetry-breaking quantum order [134] in such systems thus needs careful definition and characterization. To this end, methods of varying reli- ability and feasibility have been proposed [134, 62, 64, 85, 89, 57, 136]. In this chapter we focus on a certain class of spin liquids providing topologically ordered phases of matter and outline a scheme for detecting the phase in which the systemresidesbasedontheperturbativeresponseofthesetofR´ enyientropiestogeneric perturbations. 62 4.1 TopologicalOrderanditsDetection According to the most common definition, topological phases of quantum matter are those that have a ground state degeneracy (with respect to all local observables) protected by the topology of the lattice on which the spin Hamiltonian is defined [134,84,135]. Usinginsightsfromquantuminformationtheoryanalternativedefinition of a topologically non-trivial state is one which cannot be adiabatically connected to a productstate(inthelatticedegreesoffreedom)usingunitarycircuits(withfiniterange) of constant depth (not scaling with lattice size). This means that local unitary quantum circuits cannot completely remove the entanglement and the residue can be thought of aslongrangeentanglement[27]. Theorderinsuchstatescanthenbeidentifiedthrough the values (zero for topologically trivial states) of carefully constructed quantities, such as the topological entanglement entropy [134, 62, 64, 85, 89, 57, 56], which charac- terize the correlation between different subregions of the many-body system, or as the Wilson loop[46, 54]. It must be pointed out that such figures of merit for Topological Order (TO) are efficient, provided that length scales of the system much larger than the correlation length ξ are inspected. This makes the detection of the topological order experimentally challenging, because it involves a state tomography of a macroscopic portionofthesystem. Moreover,thereevenexiststateswhicharetopologicallyordered according to one definition but not the other for e.g. a Chern insulator is topologically ordered according to the circuit definition but has zero topological entropy (TE) [60] andconverselystateswithnon-zerovalueofthelatterbutadiabaticallyconnectibletoa product state using local unitary circuits [73]. Indeed, even certain classical states may have a non-zero TE [23, 73]. Thus our understanding of indicators for TO is far from complete. Calculation of the Topological Entanglement Entropy as an indicator of TO was based on the idea that TO is a property of the wavefunction [89], whereas previously 63 such order was supposed to be manifest only in the dynamical properties for example quasiparticlestatisticsandedgeexcitations[3,137]. Themethodoutlinedinthischapter is a step towards showing that TO also manifests itself in the characteristic perturbative response of the entire set of R´ enyi entropies (a bulk property) calculated for generic subsystems. This may be seen to supplement TEE which relies on constraints on the boundary degrees of freedom for sufficiently large subsystems. There the large size of subsystems is required to cancel the contribution from local correlations - bulk contri- butions are rejected by design. In the following we show that, at least for the class of quantumdoublemodels[134],theperturbativeresponseoftheR´ enyientropiesdepends on how many and how much the degrees of freedom within the bulk of the subsystems contribute to the entanglement spectrum. 4.2 Perturbative Response of R´ enyi entropies - Signa- tureofTopologicalOrder Motivated by ideas proposed in [61] and studied further in [32] we show that the detec- tionoftopologicalquantumphasesispossiblethroughthestudyofitslocalconvertibil- ityasimpliedbythenotionofpurestatemanipulationusingLocalOperationsandClas- sical Communications (LOCC) [99, 105, 112]. More precisely, we consider whether the adiabatic transformation of a topologically ordered state, in the ground state mani- fold of a tunable Hamiltonian, can be simulated via LOCC, restricted to the two parts of the bipartition. The picture behind, is that the global character of the correlations encoded in topologically ordered phases, is expected to pose specific constraints on the possibility of simulating adiabatic evolution using LOCC as the latter protocol is lim- ited in its capacity to generate quantum coherences. The figure of merit we employ, is the differential local convertibility of the many-body state introduced by Cui et al. in 64 [33]. Quantitatively, this amounts to studying the response of the R´ enyi entropies, or equivalently,theresponseoftheentanglementspectrum[30,126,87,1,91,4,5,40,8]. Wedemonstratethat,forgenericbipartitionsandsystemswithnon-constantcorrelation length, while certain R´ enyi entropies (with R´ enyi’s parameterα ≥ α c ) decrease as the Hamiltonian is tuned towards the quantum critical point within a topologically ordered phase, others (0 ≤ α ≤ α c ) show an increase. In the topologically trivial phases, like paramagnetic and symmetry breaking phases [33, 47], however, all entropies increase monotonically as the critical point is approached. Despite the fact that the set of R´ enyi entropies by itself does not provide any extra universalinformation,comparedtotheTE,atanyfixedvalueoftheHamiltonianparam- eter[45],theperturbativeresponseoftheR´ enyientropiesprovidesafaithfulindicatorof TO,evenfortheentropiesofverysmall(sub)systems. Inotherwords,ourapproachhas anaddedvalue,inthatitinvolvestheanalysisonsubsystemswhosesizesneednotscale with the correlation length of the physical system. This implies an obvious reduction of the complexity involved in the operation to trace the topological order in the system, opening the way to much simpler experimental protocols. In section 4.3 of this chapter we explain our basic strategy, lay down the notation and quickly review the basic theory of majorization of probability vectors along with criteria for LOCC convertibility of ground states. In section 4.3.5 we present the differ- entmodels,acoupleofwhichareamenabletoexactanalyticaltreatmentwhilethemost generalcase is dealtwith numerically using 2D DMRG. In section 4.4.4 we summarize our results and conclude with comments and discussion in section 4.5 about the scope of this line of inquiry. 65 4.3 GeneralStrategyandMathematicalPreliminaries 4.3.1 Generalstrategy In this thesis, as a concrete example of a spin Hamiltonian with TO in the ground state, wechooseKitaev’sToriccode[84]withaperturbationV (λ),thatmaybetunedthrough to the topologically trivial phase. Here all perturbed HamiltoniansH TC +V (λ) have a unique quantum critical point. We choose the perturbation V so that it can drive a quantum phase transition to either a disordered paramagnetic phase, or a ferromagnet. Phase transitions of this kind have been studied in [124, 65, 79, 41, 115]. Because we wantstatementsaboutlocalconvertibilitywithinaphasetobegeneric,weaimtoobtain the reduced density matrix (specifically its eigenvalues or trace of arbitrary powers) in full generality. We then analyze the behaviour of the R´ enyi entropies with respect to λ. These entropies are functions of the eigenvalues of the reduced density matrix and the monotonicity of the entire set of entropies depends on their relative majorization, whichisapartialorderonthesetofprobabilityvectors(thevectorofeigenvalues)[98]. Finally we check if all R´ enyi entropies show monotonic behaviour within a phase or does a subset of them show opposing behaviour from the rest. In order to achieve this, weneedtosolveforthegroundstate #ψ(λ)⟩andthenobtainthereduceddensitymatrix as a function of the parametersλ. Inthischapter,theresultsareobtainedusingbothanalyticalandnumericalmethods to find the ground state and compute the R´ enyi entropies of the models. Analytically, we consider two models. One, the Castelnovo-Chamon model, possesses an exact form for the ground state. We are able to compute exactly all the R´ enyi entropies by using grouptheoreticmethods[25]. Wealsostudythetoriccodeinanexternalmagneticfield, where the field is only acting on a subset of spins. This model maps into free fermions [134, 58, 141], and is thus exactly solvable. In [58, 59], an expression was derived for 66 the 2−R´ enyi entropy for a particular subsystem in terms of correlation functions. Here, we achieve a general expression for the2−R´ enyi entropy of a generic subsystem of this model. These results are actually more general and can be applied to any lattice gauge theory. Finally,westudythetoriccodeinpresenceofIsingcouplingsinboththexandz direction. This model is not exactly solvable. We attack the problem numerically using aversionofinfiniteDMRGintwodimensions[138,94,31],basedonaMatrixProduct State(MPS) representation oftheground statemanifoldforacylinderofinfinitelength and finite width. This method has proven very useful to study topological phases [29]. 4.3.2 R´ enyiEntropies Consider a multipartite pure quantum state #ψ⟩ ∈ ⊗ N i H i . The entanglement spectrum ¯ ν = {ν 1 ,ν 2 ,...,ν d A } of the state, is defined as the set of eigenvalues of the reduced density matrix ρ A = Tr ¯ A (#ψ⟩⟨ψ#), where A is a subset of local Hilbert space indices, A ⊂ [N], with the associated Hilbert space given by H A = ⊗ j∈A H j . We call A the subsystem. The complement of the subsystemA then is ¯ A = [N'A] with its associated Hilbert spaceH ¯ A =⊗ j∈ ¯ A H j . The entanglement spectrum of a state is the crucial ingredient in the definition of R´ enyi entropies for the reduced density matrixρ A defined as: S α (ρ A )∶= 1 1−α logTr(ρ α A ) = 1 1−α log(! j ν α j ) ∀α ≥0 (4.1) KnowledgeabouttheentiresetofR´ enyientropiesS α (ρ A ) ∀α ∈ [0,∞)isequivalent to complete knowledge about the spectrum of the state itself. At specific values of the continuous parameterα, the R´ enyi entropies provide operationally important informa- tion about the state: S α=0 = logR -R being the Schmidt rank is a measure of bipartite entanglement for the state that serves as a criteria for efficient classical representation 67 of the state [129] while lim α→1 S α =S VN is the entanglement entropy of the pure state #ψ⟩, that is a measure of its distillable entanglement, entanglement cost and that of for- mation,relativeentropyofentanglementandsquashedentanglement[99]. Alsoalinear combination of 2-R´ enyi entropiesS 2 calculated for suitably chosen bipartitions, can be used as a probe of topological order [58, 59]. For product states #ψ⟩ = #ψ⟩ A ⊗ #ψ⟩ ¯ A , theentanglementspectrumcollapsestounityforoneeigenvalueandzeroforallothers: ρ 2 A =ρ A , which means that all R´ enyi entropies are zero as well. 4.3.3 Manifoldoftopologicallyorderedgroundstates We define the ground state manifold,M, of a HamiltonianH(λ) as the continuous set of ground states #ψ(λ)⟩ (in a particular topological sector) for all possible values of the control parameters λ. So M = {#ψ(λ)⟩ s.t. #ψ(λ)⟩ is the ground state ofH(λ) ∀λ = (λ 1 ,...,λ n ) ∈ R n }. As the Hilbert space is endowed with a definite tensor product structureH =H A ⊗H B , which defines a bipartition of the system, we can consider the set of reduced density matricesρ A (λ) to the subsystemA as a function ofλ, and study the behaviour of the set of R´ enyi entropiesS α (λ)withλ andα: S α (λ)∶=S α (ρ A (λ)) =S α (Tr ¯ A (#ψ(λ)⟩⟨ψ(λ)#)) ∀α ≥0, (4.2) In the next section, we show, on the back of specific examples, that the monotonicity of theentiresetS α (λ)∀αisacharacteristicofthephaseunlesstheperturbationand/orthe choice of bipartition is fine tuned. The collective behaviour can be captured succinctly by the sign of the derivative Sign[∂ λ S α (λ)] ∀α, which remains constant in the topo- logically disordered phase - negative as the perturbation is tuned away from the critical 68 point; whereas in the ordered phase ∂ λ S α (λ) < 0 for α < α c , while it is positive for α ≥α c , as we move away from the quantum critical point. 4.3.4 Differentiallocalconvertibilityonthegroundstatemanifold The class of Local Operations and Classical Communications [28] - LOCC operations - are general quantum operations augmented with classical communication. The opera- tionsallowedarelocalinthesenseofbeingrestrictedseparatelytothetwopartsofsome bipartition of the system while potentially unlimited two-way classical communication (CC)isallowedbetweenobserversofthetworegionssothatoperationsconditionedon outcomesoftheotherregionmaybeimplemented. Thisclassofoperationsismotivated by current technological capabilities as generating quantum coherences becomes expo- nentially more difficult with increasing system size as well as the difficulty in quantum data communication. Differential local convertibillity (DLOCC) is a property of a submanifoldM i ⊂M ofthegroundstatemanifoldMthatdetermineswhetherLOCCoperationsmaybeused to transform from #ψ(λ)⟩ ∈M i to another #ψ(λ+δλ)⟩ ∈M i . Mathematically we say that M i is DLOCC iff, Sign[∂ λ S α (λ)] =constant∀α ≥0∀ #ψ(λ)⟩ ∈M i . (4.3) A negative sign in the R.H.S of the condition above implies DLOCC property of M i in the direction of increasing λ. In this work we focus on submanifoldsM i that areregionsofthegroundstatemanifoldpertainingtothedifferentphases,labelledbyi, for the different Hamiltonian models we consider. Thus we frequently refer to a phase being DLOCC as well. 69 The quantity: Sign[∂ λ S α (λ)] ∀α, has operational significance with respect to traversing M i using LOCC. The results of [98, 87, 105, 91, 4, 5, 40], imply that one can use LOCC operations to transform a ground state #ψ(λ)⟩ ∈ M i to another #ψ(λ+δλ)⟩ ∈ M i , which may require access to a shared entangled state #φ⟩ (entan- glement catalyst) betweenA, ¯ A (bipartition), with probability 1, at proximal values of λ,λ+δλ, within a phase, iff the vector of Schmidt coefficients of the product state #ψ(λ+δλ)⟩#φ⟩ at the target parameter value λ+δλ, majorizes the vector of Schmidt coefficients of the state #ψ(λ)⟩#φ⟩at the initial point. Majorization is a partial order on the set of positive vectors ¯ ν λ , ¯ ν λ+δλ which, for our purposes here, are the vectors of Schmidt coefficients of the states #ψ(λ)⟩#φ⟩ and #ψ(λ+δλ)⟩#φ⟩ respectively with respect to the A, ¯ A bipartition. It compares the dis- order in one vector with respect to another. Arranging the entries of the vectors ¯ ν λ+δλ ,¯ ν λ in a non-increasing manner: (ν λ+δλ ) 1 ≥ (ν λ+δλ ) 2 ≥ (ν λ+δλ ) 3 .... ≥ (ν λ+δλ ) d and (ν λ ) 1 ≥ (ν λ ) 2 ≥... ≥ (ν λ ) d , we say ¯ ν λ+δλ majorises ¯ ν λ , i.e., ¯ ν λ ≺ ¯ ν λ+δλ iff: k ! j=1 (ν λ ) j ≤ k ! j=1 (ν λ+δλ ) j ∀k =1,2,...,d A , (4.4) Which may be called the catalytic majorization relation since the vectors represent the Schmidt coefficients of states that are a tensor product with the catalyst state #φ⟩. It should be clear that not all pairs of states #ψ(λ+δλ)⟩ and #ψ(λ)⟩ will require a catalyst for DLOCC conversion. For such states their respective vectors of Schmidt Coefficientsγ λ+δλ ,γ λ follow a majorization relationγ λ ≺γ λ+δλ without the need for the ancilliaryentanglementcatalyst #φ⟩. ThenecessaryandsufficientconditionforDLOCC 70 conversion,withorwithouttheneedforacatalystissuccintlycapturedbythecondition [87] S α (¯ γ λ ) ≥S α (¯ γ λ+δλ ) ∀α, (4.5) whichimpliesEq.(4.3). Inwords,onecanuseLOCCtransformations,possiblyassisted by entanglement catalysis, to transform from #ψ(λ)⟩ to #ψ(λ+δλ)⟩ provided all R´ enyi entropies show monotonically decreasing behavior in going from the initial parameter value to the final one. Thus catalytic majorization and monotonic behaviour (in α) of the whole set of R´ enyi entropies are mutual implications. For α = 1 for e.g. Ineq. (4.5) implies that a necessary condition for LOCC operations to be used to transform to the new state #ψ(λ+δλ)⟩ is for it to have a lower value of the entanglement entropy with respect to the underlying bipartition [99]. 4.3.5 Themodels We consider three different perturbations V (λ), to Kitaev’s Toric code (TC) model H TC [84]. The TC Hamiltonian H TC is defined on a 2-D system of spin-1/2 particles living on the edges of a square lattice with periodic boundary conditions in both direc- tions, Fig. (4.1). The Hilbert space size of the system defined on a square lattice of size L×L isN = 2 2L 2 . There are two different kinds of mutually commuting operators that appear in the Hamiltonian: starsA s =∏ i∈s ˆ σ x i defined at the vertices of the lattice that 71 are the products of Pauli matrices ˆ σ x i acting on the 4 edges shared by a vertex and pla- quettesB p =∏ j∈p ˆ σ z j that are products of ˆ σ z j on the 4 edges of a unit cell. The operators A s ,B p have eigenvalues±1. All our Hamiltonians then have the form H =H TC +V (λ)∶=−! s A s −! p B p +V (λ). (4.6) Note that, because∏ s A s = ∏ p B p = 1 1, there are only L 2 −1 independent operators of each kind. They constitute a complete set of commuting operators with H TC , and thereforeallexcitationsoftheunperturbedHamiltonianH TC maybelabelledbythe±1 eigenvalues of the2×(L 2 −1)operators. This means that there are 2 2L 2 −2 excited states corresponding to each of the 2 2L 2 "2 2L 2 −2 = 4 degenerate ground states which is consistent with the fact that the ground state degeneracy for a topologically ordered Hamiltonian of spin-1/2s defined on a torus is4 g withg =1 being the genus of the surface. For our purposes though, one can work in a gauge fixed sector with allB p =+1, that corresponds to an effective low energy theory withZ 2 -gauge symmetry since [A s ,∏ i ˆ σ z i ] = 0 ∀s, and the only excita- tions are those of stars, so that in this sector the Hilbert space dimension is 2 L 2 −1 again with 4 degenerate ground states. In this gauge fixed sector all eigenstates ofH TC are superpositions of loop operatorsg =∏ i∈s ˆ σ x i that are products of spin-flips on spins that are crossed by contractible closed loops in the dual lattice. The loop operators are ele- ments of the groupG that is generated by the stars. The four degenerate ground states, #ψ⟩,W 1 #ψ⟩,W 1 #ψ⟩,W 1 W 2 #ψ⟩, each define a particular topological sector within the gauge fixed sector and are related to each other by spin flips on non-contractible loops W 1 ,W 2 , along the two non-contractible directions of the Torus. 72 Figure4.1: Thespin-1"2s(allfilledcircles)intheToricCodemodelliveontheedgesof asquarelatticewithperiodicboundaryconditions. Thestaroperatoratvertexlabelleds involvestheproductof ˆ σ x operatorsonthefourspins(redcircles)oftheedgesjoinedat the vertex. The plaquette operator for the unit cell labelledp involves the product of ˆ σ z operators on the four spins (green circles) on edges that form the cell. W 1 ,W 2 are spin flips along the two non-contractible directions of a torus (blue circles). Herewefocusonthesimplestgroundstate #ψ⟩,i.e.,afixedtopologicalsectorwithin the gauge. Restricting our attention to this sector, which we callTS 1 , essentially cap- tures all the phenomenology we want to highlight as well as simplifies the calculations. Thus our analytical results pertain to this sector where in subsections 4.4.1, 4.4.2 we consider gauge invariant perturbations toH TC that take drive the system across a quan- tum critical point between a topologically ordered and disordered phase. For a discus- sion of the critical point see [130, 131, 139]. The more general perturbation 4.4.3 is studied numerically. The tool used here is a two dimensional density matrix renormal- ization group extended to infinite cylinders [29]. The ability to study a Hamiltonian on aninfinitecylinderallowsustoobtaintheentiresetofquasi-degeneratedgroundstates. From that set we chose a ground state in a given topological sector and make sure that the same choice was made for every value ofλ x andλ z in Eq. (4.9). This can be done 73 by looking at the expectation value of certain loop operators around the cylinder. For small perturbations studied here, they are close to±1, which allows one to identify the topological sector. All DMRG results presented here are converged in bond dimension, which is a refinement parameter in this calculation. The perturbations presented in this thesis are as follows: a. The Castelnovo-Chamon model This perturbation has an exponential form, V 1 (λ) =! s e −λ∑ i∈s ˆ σ z i , (4.7) that commutes with all the plaquette operators [B p ,V 2 (λ)] = 0 ∀p, i.e., it is a gauge invariant perturbation. This system shows a phase transition from a topologically ordered phase to a paramagnetic phase at the critical value ofλ ≈0.44. b. Toric code Hamiltonian with magnetic field along spins on rows. The perturba- tion here is a ˆ σ z magnetic field applied only to the spins along the rows of the square lattice (we call this direction the horizontal direction), V 2 (λ) =−λ ! h∈ horiz ˆ σ z h . (4.8) Since [B p ,V 3 (λ)] = 0 ∀p this is a gauge invariant perturbation as well that drives the TC model from a topologically ordered phase across the critical point at λ = 1 to a paramagnetic one. c. The Toric-Ising Model Here the perturbation, V 3 (λ x ,λ z ) =− ! i,µ =ˆ x,ˆ y (λ x ˆ σ x i ˆ σ x i+µ +λ z ˆ σ z i ˆ σ z i+µ ), (4.9) 74 describes the interplay between topological and antiferromagnetic orders. For generic λ x andλ z , the perturbation breaks theZ 2 gauge symmetry. The latter is preserved for eitherλ x = 0 orλ z = 0. Whenλ x (λ z ) = 0, the topological and antiferromagnetic orders are separated by a continuous quantum phase transition occuring at the critical value of λ z (λ x ) =λ c ∼1"6 [83]. 4.4 Results In this section we present analytical and numerical results that exhibit the relationship between differential local convertibility and correlation length for Hamiltonians H = H TC +V (λ), whereV (λ) =V 1 ,V 2 ,V 3 described in the previous section. 4.4.1 TheCastelnovo-Chamonmodel WestartbyobservingherethattheperturbationV 1 issuchthatthespin-spincorrelation function ⟨ˆ σ x i ˆ σ x j ⟩ λ inagroundstatewithinthetopologicalsectorTS 1 oftheHamiltonian H =H TC +∑ s e −λ∑ i∈s ˆ σ z i is zero for all values ofλ. In the sectorTS 1 , we pick a ground state #ξ⟩given by [25]: #ξ⟩ = 1 √ Z ! g∈G e (λ2)∑ i∈Λ σ z i (g) #g⟩, (4.10) where g#0⟩, is the state obtained by acting with g =∏ i A s i ,g ∈ G, that is the product of star operators, on the totally polarized all spins-up (in the z-basis) reference state #0⟩ and the termσ z i (g) = ⟨g#ˆ σ z i #g⟩ in the exponent takes the value of−1 if the spin at edge i has been flipped and +1 otherwise. Z = Z(λ) = ∑ g∈G e λ∑ i σ z i (g) is a normalization constant. NotethatwithΛdenotingthesetofallspins,∑ i∈Λ σ z i (g) =N−L(g),i.e.,the 75 sumcountsthetotalnumberofspinsinastatelessthenumberthathavebeenflippedby theoperatorg ∈Gwhichareclosedloopsorproductsofclosedloopsintheduallattice. InordertoanalyzetheDLOCCpropertiesofthismodelweneedthereduceddensity matrix for a subset of spinsA, on the whole lattice Λ =A∪B, when the whole system is in state (4.10): ρ A (λ) = 1 Z ! g∈G g ′ ∈G A e λ 2 (N−L(g)) e λ 2 (N−L(gg ′ )) x g A #0⟩ A A ⟨0#x g A g ′ A , (4.11) where the group G A = {g ∈ G#g = g A ⊗ 1 1 B } is the subgroup of G generated by stars operators acting non-trivially only on the spins in A and x g A is the restriction of the operatorsg ∈G to just the subsystemA (for details see [62, 64]). We will also need the subgroupG B = {g ∈G#g = 1 1 A ⊗g B } which includes all products of star operators that act non-trivially only on the spins inB. Then theα-R´ enyi entropy is given by: S α (ρ A ) = 1 (1−α) log 1 Z α ! g∈G e −λEg ( ! h∈G A ,g∈G B e −λE hgk ) α−1 = 1 (1−α) log 1 Z α (λ) ! g∈G e −λEg w α−1 (λ,g), (4.12) where E g = L(g)−N and w(λ,g) ∶= ∑ h∈G A ,h∈G B e −λE hgk with all the λ dependence made explicit. After a straighforward but tedious calculation one can obtain the derivative of Eq. (4.12) with respect to the parameterλ and it is given by the expression: ∂ λ S α (λ) = $⟨E g ⟩ w(λ,g) % ˜ Z(λ,α) + α (1−α) ⟨E g ⟩ Z(λ) − 1 (1−α) ⟨E g ⟩ ˜ Z(λ,α) , (4.13) 76 Here ˜ Z(λ,α) ∶= ∑ g∈G e −λEg w α−1 (λ,g) and we use averages with respect to the functions f(g) = w(λ,g), ˜ Z(λ,α),Z(λ) defined as usual: ⟨E(g)⟩ f(g) = ∑ g (f(g)E(g))"∑ g (f(g)). One can now evaluate the R.H.S. of Eq. (4.13) in the limit λ→ 0 which corresponds to small perturbations of the TC model and find that ∂ λ S α (λ) ≤0∀α. This implies that all R´ enyi entropies decrease as we move away from the point in the phase diagram with a flat entanglement spectrum. Further, it can be shown [113] that the sign of the slopes of R´ enyi entropies for fixed α do not change withinthephasefor thismodel and thuswe find thatthis modelhas DLOCC withinthe topologically ordered phase. Similarly if one considers theλ→ ∞ limit one finds that all the slopes are negative as well implying that the particular form of the perturbation V 1 leads to DLOCC in both, the topologically ordered and the paramagnetic, phases of the model. 4.4.2 Toriccodewithmagneticfieldalongspinsonrows The gauge invariant perturbation V 2 (λ) lets us analyse a model with a non-constant correlationlengthξ(λ). TheGaugefixed(B p =1∀p)Hamiltonian(4.6)uptoaconstant offset is thus: H =−! s A s −λ ! h∈ horiz ˆ σ z h , (4.14) where by h ∈ horiz, we mean that the external field is applied only to spins on edges along the rows that we take to be the horizontal direction, Fig. (4.2). To solve Eq. (4.14) we map it to an exactly solvable model that preserves the local algebraoftheterms. First,observethatthestaroperatorshaveeigenvalues±1;thennote that each ˆ σ z h operator on a horizontal link has two neighboring star operators acting on the vertices connected by the edge. Because the action of ˆ σ h z is to flip the sign of both 77 Figure 4.2: An artist’s rendition of the lattice of spins (dark filled circles) for the Toric code model with a magnetic field on spins along only the horizontal direction (shown by spins within dotted arrows). the star operators that share the spin ‘h’, {A s ,ˆ σ z h } = 0 for these neighboring stars and onecanmovetoanalternatepicturewherethestaroperatorsatavertexarereplacedby pseudo-spinoperators, ˆ τ z s ,atthesamevertexwitheigenvalues±1. Theactionof ˆ σ z h then correspondstotheactionof ˆ τ x i ˆ τ x i+1 whentheverticess,s+1sharetheedgelabelled‘h’, i.e., it flips both neighboring pseudo-spins. We callA s ,ˆ σ z h operators in the ‘σ-picture’ in contrast to the ‘τ-picture’ for operators in terms of the pseudo-spin operators ˆ τ . The map is thus given by: A s → ˆ τ z s ˆ σ z h → ˆ τ x i ˆ τ x i+1 , (4.15) 78 which maps the Hamiltonian (4.14) to: ˜ H =− ! s∈all vertices ˆ τ z s −λ ! all rows ! s∈row ˆ τ x s ˆ τ x s+1 =− ! all rows ! s∈row ˆ τ z s −λ ! all rows ! s∈row ˆ τ x s ˆ τ x s+1 = ! all rows (− ! s∈row ˆ τ z s −λ ! s∈row ˆ τ x s ˆ τ x s+1 ) =⊕ all rows H row , H row =− L ! s=1 ˆ τ z s −λ ! s∈row ˆ τ x s ˆ τ x s+1 . (4.16) Eq. (4.16) implies that the new Hamiltonian is a direct sum of 1-D quantum Ising Hamiltonians on theL rows. The ground state of ˜ H is thus given by the tensor product ofthegroundstatesofeachindividualrow,i.e., #ψ⟩ =⊗ j∈all rows #ψ j ⟩. EachrowHamilto- nianH row in the expression above is solved by mapping the Pauli spins via the Jordan- Wigner transformation to Fermions and then a Bogoliubov transformation diagonalizes the Hamiltonian to a free Fermionic form [104]. In the present paper, we consider the symmetric ground state enjoying the global spin flip symmetry of the Hamiltonian and thus < ˆ τ x i >=0 in the ground state. This model exhibits two phases as well: a topologically ordered one for weak mag- netic field and a disordered one beyond the critical valueλ = 1 [58, 141]. The results ofthismodel,whichfollowinthenextsubsections,demonstratethatforfine-tunedper- turbations one might indeed obtain differential local convertibility for specially chosen bipartitions. We remark that although we considered the symmetric ground state of the systemthisdoesnotresultinalossofgeneralityandatthesametimeeasestheanalytical presentation. For special choices of subsystems (we call these ‘thin’ subsystems for reasons that become clear in the following) we can determine the exact eigenvalues of the reduced density matrix for all values of the perturbing fieldλ and hence all the R´ enyi entropies 79 S α which show monotonic perturbative behaviour for all α. On the other hand, for systems with a ‘bulk’ some R´ enyi entropies have a different behaviour with increasing λ than others. ‘Thin’ subsystems A drastic simplification in the exact calculation of the R´ enyi entropies for the ground state of gauge theories (of which the toric code is the simplest example, theZ 2 gauge theory) can be obtained by choosing some particular partitions [58, 59]. A ‘thin’ sub- sytemA, in the lattice for the Toric code model is one where there are no star operators that can act on spins which exclusively belong to A. For example, the bipartition of spins on the lattice where subsytem A is comprised only of rows (columns) with the columns (rows) forming the complementB. Mathematically this means that the group G A only contains the identity, 1 1, which in turn implies that the reduced density matrix, ρ A , is diagonal in thez-basis of theσ-spins [64]. All loops on the real lattice are other examples, the shortest such loop being a plaquette, Fig. (4.3). Intuitively, ‘thin’ subsys- tems are those wherein all the degrees of freedom are maximally entangled, even in the unperturbed toric code model, while respecting the gauge constraints. Thus increasing correlation length cannot lead to newer non-zero values appearing in the entanglement spectrum. Such is the subsystem A that we now investigate. The reduced density matrix for a plaquette with 4 spins is a matrix of size 2 4 × 2 4 . However because of the gauge constraint,B p =1,onlythreespinsareindependentwhichmeansthatthemaximalrank of the reduced density matrix is 2 3 = 8. The diagonal entries of this matrix, (ρ A ) ¯ s¯ s , correspond to expectation values of the projector onto the different spin configurations, 80 Figure 4.3: SubsystemA, shown in the shaded region, of one plaquette with the spins 1,2,3,4, on the edges. The eigenvalues of the reduced density matrix ρ A , involves calculating expectation values of operators on the 4 pseudo-spinsi,i+1,j,j+1, at the shown vertices [113]. ¯ s = (s 1 ,s 2 ,s 3 ) ∈ {−1,1} 3 , in the ground state of the Hamiltonian (4.14) of the three independent spins, i.e., (ρ A ) ¯ s,¯ s = 1 2 3 ⟨ψ#(1+s 1 ˆ σ z 1 )(1+s 2 ˆ σ z 2 )(1+s 3 ˆ σ z 3 )#ψ⟩ = 1 2 3 (1+s 1 ⟨ˆ σ z 1 ⟩+s 2 ⟨ˆ σ z 2 ⟩+s 3 ⟨ˆ σ z 3 ⟩+s 1 s 2 ⟨ˆ σ z 1 ˆ σ z 2 ⟩+s 2 s 3 ⟨ˆ σ z 2 ˆ σ z 3 ⟩+s 3 s 1 ⟨ˆ σ z 3 ˆ σ z 1 ⟩ +s 1 s 2 s 3 ⟨ˆ σ z 1 ˆ σ z 2 ˆ σ z 3 ⟩) = 1 2 3 (1+s 1 ⟨ˆ τ x i ˆ τ x i+1 ⟩+s 2 ⟨ˆ τ x i ⟩ 2 +s 3 ⟨ˆ τ x j ˆ τ x j+1 ⟩+s 1 s 2 ⟨ˆ τ x i ⟩ 2 +s 2 s 3 ⟨ˆ τ x i ⟩ 2 +s 3 s 1 ⟨ˆ τ x i ˆ τ x i+1 ⟩ 2 +s 1 s 2 s 3 ⟨ˆ τ x i ⟩ 2 ), (4.17) where in the last line above we have used the mapping (4.15) to express the diagonal entries in terms of theτ-spins [113]. Notice that the only non-trivial expectation values 81 Figure 4.4: R´ enyi entropies for a subsystemA of one plaquette (shown in Fig. 4.3), at different values ofα. All entropies show monotonic behavior in both the phases: they decrease monotonically with increasing correlation length ξ(λ) for λ < λ c = 1 while they increase withξ(λ)forλ>1. of theτ-spins are those of two point functions since ⟨ˆ τ x i ⟩ = 0 in the symmetric ground state. The thermodynamic limit expressions [104, 6] in the entire domain ofλ is ⟨ˆ τ x i ˆ τ x i+1 ⟩= 1 π π 0 cos(φ)[cos(φ)−1"λ]+sin 2 (φ) [(1"λ−cos(φ)) 2 +sin 2 (φ)] 12 dφ 0<λ. (4.18) Thus we can calculate the trace of arbitary powers of the reduced density matrix, Tr(ρ α A ) = ∑ s 1 ,s 2 ,s 3 =−1,1 (ρ A ) α ¯ s¯ s , using which the R´ enyi entropies are given by (with T(λ)= ⟨ˆ τ x i ˆ τ x i+1 ⟩): S α (λ)= 1 1−α log[ 1 2 3α {2(1+T(λ)) 2α +2(1−T(λ)) 2α +4(1−T(λ) 2 ) α }]. (4.19) From the plot of Eq. (4.19) in Fig. (4.4) we observe that for all values of α = .01,.1,.5,1.01,2, the entropies show monotonic behaviour with λ in both the phases. 82 Figure 4.5: Subsystem A, shown as the shaded region, comprised of a total of seven spins which form two overlapping stars. While in the topologically ordered phase,λ< 1, the entropies decrease as we approach thequantumcriticalpoint,forthedisorderedregionitdecreasesaswemoveawayfrom it. Generaltreatment On the lattice, we call systems with a ‘bulk’ those that have at least one or more star operators that act on spins exclusively belonging toA. This means that the groupG A is non-trivial and the reduced density matrix for the subsystem is not diagonal anymore [64]. Consequently, the analysis of this case is considerably more involved. We refer to [62] for an introduction to the technique used to treat a gauge theory. Since the perturbation we consider is gauge invariant, indeed, we can represent the state as the sum over element of a group, and this makes the calculation possible in the formalism. We can compute exactly the reduced density matrix [113]. Moreover, we can find an exact expression for the purity: P (λ)= G B G g∈G A ,z∈Z A ⟨ψ(λ) gz ψ(λ)⟩ 2 , (4.20) 83 where, ψ(λ)⟩ is the ground state of the Hamiltonian (4.14) and G B is the cardinality ofthegroupofstaroperatorsactingexclusivelyinthecomplementofA,i.e.,G B = {g ∈ G g = 1 1 A ⊗g B }. As before, G A is the group of spin flips generated by star operators exclusively inA whileZ A is the group generated by products of ˆ σ z ’s acting on spins in A.Asanaside,wenotethatthisexpressioncanbegeneralizedtogeneralgaugetheories and quantum double models and to a general R´ enyi entropy of indexα [113]. Although in principle we can calculate the entropiesS α (λ) for each integerα, here wefocusonthe2−R´ enyientropyonly. Inparticular,wedemonstratethatithasamono- tonic behaviour in both the phases. The monotonicity of S 2 (λ) is sufficient to show that all higher entropies obey the same monotonicity because of the continuity of the entropies inα and because of their ordering relation: S α ′ ≤ S α ∀α ′ ≥ α. On the other hand in the Toric code limit atλ=0, the eigenspectrum is flat with there being2 5 equal eigenvaluessummingto1withtheremaining2 7 −2 5 =96eigenvalues,allzero. Turning on the perturbation has the effect of making some of these zero eigenvalues non-zero which shows up as an increase of lim α→0 S α and other R´ enyi entropies with α close to zero. Alternatively put, the Schmidt rank of the state ψ⟩ (λ) increases withλ with respect to bipartitions with a bulk. To analyze this case while keeping the presentation simple, we choose a subsystem A which includes the 7 spins of two neighboring stars, Fig. (4.5). For the calculations, here we use the symmetric ground state in theTS 1 sector. The evaluation of the R.H.S of Eq. (4.20) again relies on theσ−τ correspondence (4.15) and we get for the purity: 84 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 S 2 0 2 4 6 8 10 12 Figure 4.6: The 2-R´ enyi entropy of a subsystem comprised of two stars A (shown in Fig. 4.5) across the phase transition at λ = 1 for H = H TC +V 2 (λ). The monotonic behavior in both the phases forS 2 implies similar behavior forS α ∀α ≥ 2; whereas the general arguments presented in the text imply that forα→ 0 they should increase till the quantum critical point. The dotted line is the inverse of the energy gap between the groundandfirstexcitedstatesforthetransversefieldIsingmodeltowhichtheperturbed gauge-fixed Toric code Hamiltonian is mapped. P = 1 2 7 {(1+< ˆ τ x 1 ˆ τ x 2 > 2 ) 2 × (1+3< ˆ τ x 1 ˆ τ x 2 > 2 +2< ˆ τ x 1 ˆ τ x 3 > 2 +< ˆ τ x 1 ˆ τ x 4 > 2 +< ˆ τ x 1 ˆ τ x 2 ˆ τ x 3 ˆ τ x 4 > 2 +< ˆ τ z i > 2 +< ˆ τ x 1 ˆ τ z 2 ˆ τ x 3 > 2 +< ˆ τ z 2 ˆ τ x 3 ˆ τ x 4 > 2 +< ˆ τ x 1 ˆ τ z 2 ˆ τ x 4 > 2 +< ˆ τ z 2 ˆ τ z 3 > 2 +< ˆ τ z 2 ˆ τ z 3 ˆ τ x 2 ˆ τ x 3 > 2 +< ˆ τ z 2 ˆ τ z 3 ˆ τ x 1 ˆ τ x 4 > 2 +< ˆ τ z 2 ˆ τ z 3 ˆ τ x 1 ˆ τ x 2 ˆ τ x 3 ˆ τ x 4 >)}. (4.21) The2-R´ enyientropyS 2 (λ)=−log(P (λ))isshowninFig.(4.6). Justasforthethin subsystem case, we find similar monotonicity in the approach and departure from the quantum critical point. 85 Figure 4.7: SubsystemA, shown as the shaded region, comprised of a total of six spins whichmakeupthespinsonaplaquetteandtwoneighboringspinstoitsnortheastcorner. 4.4.3 TheToric-Isingmodel Finally,weconsiderthesubsystemAconsistingofaplaquettewithtwoadjoiningspins pictured in Fig. (4.7) and numerically show that for the perturbationV =V 3 (λ x ,λ z ) = −∑ i,µ =ˆ x,ˆ y (λ x ˆ σ x i ˆ σ x i+µ +λ z ˆ σ z i ˆ σ z i+µ ), which takes the Toric code Hamiltonian from a topo- logicallyorderedphasetoaferromagneticphase,thesetofR´ enyientropiesinthetopo- logically ordered phase shows the splitting behavior. Note that neither the perturbation here nor the choice of the subsystem is fine-tuned. In other words, the lack of differ- ential local convertibility is a robust property of the topologically ordered phase and is universal. Here by universal we mean that this property should hold for all quantum systems that show similar behavior in their entanglement spectrum landscape and cor- relation length behavior. However, the value ofα for the R´ enyi index such that the sign of the derivative∂ λ S α (λ) changes, is non universal and is numerically found here to be α ≃ 1.3, see Fig. (4.8). The space of the parameters spanned is deep in the topological phase, with λ x,z ≤ 0.05. For highλ values, i.e., in the ferromagnetic phase, the sign is found to be the same (not shown in the plot) for every value of the R´ enyi index α. Thus even in this model where a phase transition occurs from a topologically ordered 86 Figure 4.8: Behavior of three representative R´ enyi entropies for the Toric-Ising model (V =V 3 (λ x ,λ z )) in the perturbation parameter (λ x ,λ z )-plane within the topologically ordered phase for subsystemA as shown in Fig. (4.7). Forα =.6 the entropy increases while for α = 5 it decreases monotonically with increasing correlation length. The change between these two types of behavior occurs atα ≃ 1.3, the value of which was identified numerically. phase to a ferromagnetic one the latter exhibits differential local convertibility whereas the former does not. 4.4.4 Summaryofresults Here we collect the main results of this section that will help formulate, in the conclu- sions, the conjecture about the splitting phenomenon of the R´ enyi entropies. • For perturbations (4.4.1) with constant correlation length and any bipartition the behaviour of the R´ enyi entropies is monotonic and there is no splitting phe- nomenon. • For perturbations (4.4.2) with non-constant correlation length and thin biparti- tioningthebehaviouroftheR´ enyientropiesismonotonicandthereisnosplitting phenomenon. 87 • For perturbations (4.4.2) with non-constant correlation length and bulk biparti- tioning the Reny’s entropies split. • For general perturbations (4.4.3) the splitting behaviour of the entropies is robust and happens without reference to the size of the subsystem as long as the subsys- tem has some bulk. 4.5 ConcludingRemarks In this chapter, we considered a paradigmatic class of topological phases, as those ones arising from the toric code with a perturbation driven by a set of control parameters λ = (λ 1 ,...,λ n ). We focused on the case where the energy gap can vanish, giving rise to a quantum phase transition to a topologically trivial phase (paramagnet). The perturbations studied affected the correlation length of the system ξ while vanishing exactly for the toric code,λ=0. Weshowedthatthetwophasescanbedistinguishedthroughaspecificnotionrelated to LOCC. This notion is known as differential local convertibility: Bipartitioning the system in A and B, the result is that two adjacent states in the topological quantum phase,genericallycannotbeconnectedbyLOCCinAandB (eveninthepresenceofa catalyst);intheparamagneticphases,incontrast,thestatesarelocallyconvertible. This is consistent with the fact that in the topogically trivial phases it is always possible to transformthegroundstatetoatotallyfactorizedstateinthephysicaldegreesoffreedom by using a local unitaryquantum circuit of fixed depth. From the quantum computation perspective, the locally convertible character of a phase implies its limited adiabatic computational power since the physical transformation may be simulated using LOCC operations which do not generate quantum coherences between the two parts of the bipartition [34, 47]. 88 The figure of merit for the nature of the response to the perturbation is expressed in terms of the R´ enyi entropies associated to a subsystemA of contiguous spins: The non-local convertible phase features a splitting behaviour of the entropies, with their partial derivative along the control parameterλ i changing sign for a particular value of the R´ enyi indexα. The splitting phenomenon is observed within the whole topological phaseirrespectiveoftheparticularformoftheperturbationorofthesubsystemA,unless it is very fine tuned - such as the ones without any bulk. The value ofα at which the splitting occurs is instead dependent on the details of the model. The understanding of this phenomenon relies on the structure of the entanglement spectrumaroundaspecialpointinthephase. Indeed,inthetopologicallyorderedphase of this model there exists an extremal point with a flat entanglement spectrum and zero correlationlength,ξ =0. Asweperturbawayfromthispoint,ifthecorrelationlengthξ alsoincreasesthennewerdegreesoffreedomgetinvolvedintheentanglementspectrum as a result of which the lower (α→ 0) entropies increase, on the other hand the higher α entropies decrease because of the algebraic suppression of the contributions from the new small but non-zero values in the spectrum and loss of contributions from the pre- viously non-zero larger eigenvalues. We comment that since similar phenomenology in the entanglement spectrum is known to be displayed in cluster states [81, 118], or more generallyinallgraphstates[97],similarfindingsintheR´ enyientropiesresponseshould apply to those as well. Our work here should be seen as supporting a growing body of evidence [32, 47] that this characteristic perturbative response should hold for a wider class of states such as quantum double models, cluster states and other quantum spin liquids. In the toric code case knowledge about the ground state degeneracy can addi- tionally distinguish its topologically ordered ground states from the latter. Compared to this, ground states of all symmetry broken phases exhibit monotonic behaviour of 89 their R´ enyi entropies with an increase in correlation length, and are thus always locally convertible [47]. In order to compute the R´ enyi entropies for the perturbed toric code, we resorted to two methods. For general perturbations that break gauge invariance, and also make the system non integrable, we used a 2D DMRG method, which can treat infinite cylinders [29]. Ontheotherhand,forthegaugeinvariantperturbation,wefoundageneralexpres- sion for the R´ enyi entropies, that can be generalized to every gauge theory. Moreover, for a particular form of the perturbation, the system is integrable, and we can find an exact analytical formula for the R´ enyi entropy. This result is technically relevant, and wouldallowto treatseveralproblems,including stabilityissues atzero[86,11, 12]and finitetemperature[24,74,93,102],theconfinementproblem[54],andtheidentification of relevant correlations [133, 101] and in the dynamical problem [77, 82, 110], i.e., the resilience of the splitting property or of topological entropies after a quantum quench [125, 59]. 90 Chapter5 ConclusionsandOutlook In this thesis we have presented results towards three specific questions in condensed matterphysicsusingideasandtechniquesfromquantuminformationtheory. Theresults as well as the methods used are instructive in that they display fully the synergistic relationship between the two broad branches. Fidelity, Random Quantum Circuits and R´ enyientropiesareallconceptsthatattractedattentionfirstinthequantuminformation community and have subsequently been found very useful also in condensed matter. Inthe problemsstudiedhere theseconceptshave beenthecentraltoolsisourinvestiga- tions. CondensedmatterphysicsinturnservedtoguideourintuitionabouthowFidelity, Ensembles of states and Entropies should behave. In the second chapter we showed how the equilibration dynamics of a closed quan- tum system is encoded in the long-time distribution function of generic observables. Using the Loschmidt echo generalized to finite temperature, it was shown that one can obtainanexactexpressionforitslong-timedistributionforaclosedsystemdescribedby a quantum XY chain following a sudden quench. In the thermodynamic limit the loga- rithm of the Loschmidt echo became normally distributed, whereas for small quenches in the opposite, quasi-critical regime, the distribution function acquired a universal double-peakedformindicatingpoorequilibration. Thesefindings,obtainedbyacentral limit theorem-type result, were argued to hold also for completely general models in the small-quench regime. It would be interesting to see in the future, answers to similar questions for non-integrable systems. 91 ThethirdchaptersummarizedageneralformalismtoanalyzeLocalRandomQuan- tum Circuits and presented several examples of the latter. The basic physical ques- tionweansweredwastheevolutionofentanglementgeneratedbytheLRQCdynamics. Since the proxy for entanglement was the purity, and through it the 2-R´ enyi entropy, we showed that the relevant ensemble map was the second moment operatorR 2 for the circuits. The iterative action of the circuits was shown to lead to an algebra of swap operators on subsets of the total degrees of freedom. The fixed point of this map cor- responds to a normalized symmetric linear combination of the identity and the swap actingonalllocaldegreesoffreedom,whiletherateofapproachtothisfixedpointwas shown to depend on the gap of the operator representing R 2 in the space of relevant swaps. Using techniues from group theory, operator algebras and matrix analysis we showed the typicality of area law for short times and volume law for times scaling with the system size in a particular model of LRQC - the contiguous edge model. The typicality results for entanglement in the physical Hilbert space are important for the foundations of quantum statistical mechanics. While generic properties of Haar distributedrandomquantumstatesareusedtomaketypicalityarguments,foreg. insub- system equilibration problems etc., the unphysical nature of Haar distributed random states (in light of the exponential difficulty in their exact sampling) raises conceptual concerns. Our work with the different kinds of LRQCs may thus be mathematically considered simply as proposals for natural probability distribution laws over the full quantum state space. From the quantum control perspective the lack of precise control overlocalunitarygatesandtheirsupportsmaybemodelledwhereboththesequantities are promoted to be random variables with associated distribution laws. The analysis of LRQCs thus allow us to make statistical statements about QIP systems with limited resources. Finally, as we saw with the contiguous edge model, suitably defined LRQC 92 circuitfamiliesmay“simulate”thedynamicsoftime-dependentlocalrandomHamilto- nians with a discrete circuit. It will be interesting to study the spectral properties of the ensemble maps for rele- vant LRQCs on different kinds of graphs. In particular the map’s fixed points and the dependence of the map spectral gap on the graph structure and size. For all the models considered here the asymptotic state was close to a maximally mixed one in trace dis- tance. It would be interesting to study the conditions under which families of LRQCs may be considered to be approximate unitaryt-designs. As long as topological order is absent, ground states of local gapped Hamiltonians can be obtained by a circuit of fixed depth from a completely factorizable state. It would be interesting to study the statistics of the entanglement in ensembles where the fiducial state is topologically ordered. For instance, we would like to know if in such ensemblesthereisanonvanishingtopologicalentropy[63,89]onaverage,andwhatare the fluctuations. The techniques presented here may prove useful to study the problem of the stability of topological phases under local unitary noise models. Finally, LRQCs, being non-relativistic quantum spin systems, must have a bounded speed(speedofsound)forthepropagationofcorrelationsinthesystem[90,107]. That is, the commutator of two local observables with disjoint supports should be bounded by a time-dependent (i.e. number of iterations) function which should depend on the parameters defining the LRQC family. While it is known that for disordered systems there is an exponential slowdown of the speed of propagation of correlations [18], it would be interesting to identify the speed of sound and correlation length in LRQCs. Finally, in the fourth chapter we showed that topologically ordered ground states can be distinguished through a specific property related to the notion of LOCC, known 93 as differential local convertibility - the response of the Renyi entropies to an exter- nal perturbation. We considered the toric code Hamiltonian with several perturba- tions and using analytical and numerical methods we showed that while a subset of these entropies increased with the correlation length, others decreased in the topologi- cally ordered phase; compared to the response in the trivial phase where all entropies increased with increasing correlation length. We conjectured that this characteristic perturbative response should hold for a wider class of states such as quantum double models, cluster states and other quantum spin liquids whereas symmetry broken states should always show monotonic behaviour of their entropies. These findings here hold experimental promise because the subsystem size can be independent of the correlation length, in addition we discussed their implications for quantum information process- ing tasks using topologically ordered states. It would also be interesting to see if the local convertibility properties -or failure thereof - hold for more general topologically ordered states without flat entanglement spectra such as fractional quantum Hall states [78, 114, 26] and chiral spin liquids [103]. Althoughthepreviouschaptersdealtwithspecificquestionsinaself-containedman- ner, a combination of the techniques outlined in them can be fruitful towards other importantquestions. Onecanforexample,perhapsstudythedynamicalproblemi.e. the resilience of topological order to quantum quenches using methods in chapter (2) and chapter (4). Similarly the techniques in chapter (3) and chapter (4) can help us under- stand the robustness of quantum memories based on topologically ordered substrates to random local unitary noise. To conclude, we hope that some of the techniques outlined in this thesis can be extended to address the questions of further interest posed here. 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Abstract (if available)
Abstract
This thesis presents insights obtained on three questions in condensed matter physics via techniques in quantum information. ❧ The first topic deals with signatures of equilibration in a closed quantum system. Using the Loschmidt echo as a representative observable, that is derived from fidelity—a popular quantity in quantum information, and by studying it's long time statistics the role of quantum criticality in the equilibration dynamics of a closed system is analysed. While off‐critical systems under a quantum quench are shown to equilibrate well, critical systems are shown to do so poorly signified by relatively large fluctuations of the echo around it’s long time average. ❧ The second topic deals with typicality of entanglement in the physical Hilbert space. Here, a general framework for studying statistical moments of physically relevant quantities in ensembles of quantum states generated by Local Random Quantum Circuits (LRQC) is outlined. These ensembles are constructed by finite‐length random quantum circuits acting on the (hyper)edges of an underlying (hyper)graph structure. The latter are designed to encode for the locality structure associated with finite‐time quantum evolutions generated by physical i.e. local Hamiltonians. Physical properties of typical states in these ensembles, in particular purity as a proxy of quantum entanglement is studied. The problem is formulated in terms of matrix elements of superoperators which depend on the graph structure, choice of probability measure over the local unitaries and circuit length. We consider different families of LRQCs and study their typical entanglement properties for finite‐time as well as their asymptotic behavior. In particular, for a model of LRQC that resembles closely the Trotter scheme of discretizing quantum evolutions with local Hamiltonians, we find that the area law holds in average and that the volume law is a typical property (that is, it holds in average and the fluctuations around the average are vanishing for the large system) of physical states. The area law arises when the evolution time is O(1) with respect to the size L of the system, while the volume law arises as typical when the evolution time scales like O(L). ❧ The final topic deals with the perturbative response of the set of Rényi entropies of a subsystem when the entire system is in a state displaying some quantum order. The characteristic behavior of the entropies is shown to be able to identify topologically non‐trivial and trivial phases in the case of quantum double models. The implications of the response towards the possibility of simulating the adiabatic evolution within a phase using the protocol of Local Operations and Classical Communications are discussed.
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Creator
Santra, Siddhartha
(author)
Core Title
Quantum information techniques in condensed matter: quantum equilibration, entanglement typicality, detection of topological order
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Physics
Publication Date
04/21/2014
Defense Date
03/25/2014
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University of Southern California
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entanglement,equilibration,OAI-PMH Harvest,quantum information,topological order
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Zanardi, Paolo (
committee chair
), Brun, Todd A. (
committee member
), Campos Venuti, Lorenzo (
committee member
), Jonckheere, Edmond A. (
committee member
), Lidar, Daniel A. (
committee member
)
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santra@usc.edu
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Tags
entanglement
equilibration
quantum information
topological order