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Multipartite entanglement sharing and characterization of many-body systems using entanglement

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Multipartite entanglement sharing and characterization of many-body systems using entanglement

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Content MULTIPARTITEENTANGLEMENTSHARINGANDCHARACTERIZATIONOF MANY-BODYSYSTEMSUSINGENTANGLEMENT by AnirbanDas ADissertationPresentedtothe FACULTYOFTHEUSCGRADUATESCHOOL UNIVERSITYOFSOUTHERNCALIFORNIA InPartialFulfillmentofthe RequirementsfortheDegree DOCTOROFPHILOSOPHY (PHYSICS) December2013 Copyright 2013 AnirbanDas Dedication Tomyfamily, ii Acknowledgments I want to express deepest gratitude toward my advisor Professor Stephan Haas, who has been my mentor during graduate school. He guided me at every step, especially during timesthatcalledfortoughdecisions. I would like to thank my immediate supervisors Dr. Silvano Garnerone and Professor JeongSanKim. IwouldalsoliketothankDr. LetianDingforusefuldiscussionsregarding entanglemententropy. IwouldliketoextendaspecialthankstoProfessorBarryC.Sandersforinvitingmeto InstituteforQuantumInformationScience(IQIS)atUniversityofCalgarywhereIcarried out a part of my research. This visit was made possible due to financial support from Alberta’s Informatics Circle of Research Excellence (iCORE) and the kind hospitality of IQIS. I thank Professor Gene Bickers, Professor Aiichiro Nakano, Professor Jin Ma and ProfessorGeraldinePetersforservingonmyPhDdissertationcommittee. Ialsothankmyfriendsingraduateschoolformakinggraduateschoolfunandhelping me adjust in a new country and my friends from high school and IIT Kharagpur for their encouragement. iii I would like to thank Professor Jayanta Kumar Bhattacharjee of IACS Kolkata, Pro- fessor Ashoke Sen of HRI Allahabad, Professor Predhiman Krishan Kaw of IPR Gand- hinagar, Dr. Sudip Sen Gupta of IPR Gandhinagar, Professor Archan S. Majumdar of SNBNCBSKolkata,ProfessorN.D.HariDassofIMScChennai,ProfessorThomasGehrmann of University of Zurich, Professor Prasanta Kumar Datta of IIT Kharagpur, Professor Arghya Taraphder of IIT Kharagpur, Professor Sayan Kar of IIT Kharagpur and late Pro- fessor Debabrata Basu of IIT Kharagpur for supervising me on various undergraduate researchprojects. I would like to thank my high school teachers Ms. Jhuma Ray and Ms. Gowri Ramachandranforprovidinginspiration. I would like to thank Ms. Pritikona Saha, Mr. Supriyo Roy, Professor Sunanda Palit, Professor Arun Sanyal and late Professor Mithil Ranjan Gupta for teaching me advanced topics in mathematics and physical science that enabled me to crack IIT-JEE and helped mesecureadmissiontotheundergraduatecollegeofmychoice. I would like to thank my wife Paromita Ghosh for being very supportive throughout mygraduateschoolandmyparents,grandparentsandmyin-lawsfortheirsupport. iv TableofContents Dedication ii Acknowledgments iii ListofFigures vii Abstract x Chapter1: Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 MultipartiteEntanglement . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 EntanglementEntropyandQuantumCriticalPhases . . . . . . . . . . . 5 1.3.1 QuantumPhaseTransitions . . . . . . . . . . . . . . . . . . . . 5 1.3.2 EntanglementEntropyandQPT . . . . . . . . . . . . . . . . . 6 1.3.3 ExamplewithXYModel . . . . . . . . . . . . . . . . . . . . . 7 1.3.4 ExamplewithIsingModel . . . . . . . . . . . . . . . . . . . . 8 1.3.5 ImportanceofEntanglemententropyinphysicalsystems . . . . 10 1.4 ThesisOutline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Chapter2: ConcurrenceandMonogamyofEntanglement 13 2.1 MOEforqubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 ExamplewithGHZstate . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 ExamplewithW-classstate . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 ViolationofMOEforhigherdimensionalqubits(qudits) . . . . . . . . . 18 Chapter3: CRENasaMeasureofMultipartiteEntanglement 20 3.1 ConcurrenceandCREN . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.2 CRENMOE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 v 3.2.1 MonogamyInequalitiesforn-qubitsystemsintermsofCREN 26 3.2.2 CRENvsConcurrence-basedMonogamyRelations . . . . . . . 28 3.3 PartiallyCoherentSuperpositionofann-QUDITGeneralizedW-class StateandVacuumState . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Chapter4: Non-Equilibriumquantumsystems 40 4.1 IsingmodelwithDzyaloshinskii-Moriyainteraction . . . . . . . . . . . 41 4.2 Entanglemententropyderivation . . . . . . . . . . . . . . . . . . . . . 43 Chapter5: PhaseCharacterizationusingEntanglement 51 5.1 StaticEntanglementEntropyandthePhaseDiagram . . . . . . . . . . 51 Chapter6: QuenchinQuantumSystems 59 6.1 QuenchDynamicsfortheIsingModel . . . . . . . . . . . . . . . . . . 59 6.2 PhysicalInterpretationofEntanglementDynamics . . . . . . . . . . . . 60 6.3 EntanglementDynamicsfollowingaQuenchwithDMterm . . . . . . . 62 Chapter7: Conclusion 69 7.1 CRENandMOE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 7.2 Entanglemententropyandmany-bodysystem . . . . . . . . . . . . . . 71 Bibliography 73 vi ListofFigures 1.1 PartitioningthesystemofsizeN intotwoparts,ablockofsizeLand restofthesystemofsizeN−L. . . . . . . . . . . . . . . . . . . . . . 4 1.2 Phase diagram of the ID XY model in the parameter space (λ,γ). The twocriticalregionsare0≤λ≤ 1,γ = 0andλ = 1ashighlightedby the red line. The regionγ = 1 describes the ID Ising model, and the regionγ = 0 correspondstotheisotropicXXchain[1]. . . . . . . . . . 9 1.3 ConcurrenceC as a function of ∆ shows that entanglement is maxi- mumatthecriticalpoint ∆ = 1 forXXZmodel[2]. . . . . . . . . . . . 10 4.1 Spectrum of the Hamiltonian in Eq. 4.1. I show the spectrum for 4 different values of ζ, while keeping h = 0.5 fixed. See also figures in[3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Phase diagram in theh−ζ plane of the model in Eq. 4.1. The dotted line is a critical line where the model shows the same universal prop- erties of the quantum Ising model in transverse field. See also figures in[3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 5.1 S L vsLath = 1.0fordifferentζ. Thescalingbehaviorchangesfrom S L ∼ 1 6 lnL on the critical line separating the ordered and disordered phasetoS L ∼ 1 3 lnLinsidethecurrent-carryingphase. . . . . . . . . . 53 5.2 Non-universal nature of S 0 . (upper panel) S 0 vs. ζ at different h; (lowerpanel)S 0 vs.hatdifferentζ. . . . . . . . . . . . . . . . . . . . 56 vii 5.3 S L vs. h at L = 60 for different ζ. Upper panel: Static entan- glement along the transition between the ordered ferromagnetic and the disordered paramagnetic phase. The peak signals the presence of long-range correlations at the critical point, which is a signature of a second-order quantum phase transition. Lower panel: Static entan- glementalongthetransitionbetweenthedisorderedparamagneticand the current-carrying phase. The sudden change in entanglement fol- lowedbytheabsenceofapeakatthecriticalpointisasignatureofthe first-orderquantumphasetransition. . . . . . . . . . . . . . . . . . . . 57 5.4 S L vs. ζ atL = 60 for differenth. h≤ 1: Static entanglement along thetransitionbetweentheorderedferromagneticandthecurrent-carrying phase. This is a first order quantum phase transition which occurs at ζ = 1. h > 1: Static entanglement along the transition between the disordered and the current-carrying phase. This is a first order QPT whichoccursatζ =h. . . . . . . . . . . . . . . . . . . . . . . . . . . 58 6.1 S 60 (t) for the quench from varioush 0 > 1 toh = 1. The dashed lines aretheleadingasymptoticresultsforlarge[4]. . . . . . . . . . . . . . . 60 6.2 Propagation of quasi particles following the quench at t < L 2 for a blockofsizeL. Seealsofiguresin[4] . . . . . . . . . . . . . . . . . . 61 6.3 Propagation of quasi particles following the quench at t = L 2 for a blockofsizeL. Seealsofiguresin[4] . . . . . . . . . . . . . . . . . . 62 6.4 Propagation of quasi particles following the quench at t > L 2 for a blockofsizeL. Seealsofiguresin[4] . . . . . . . . . . . . . . . . . . 63 6.5 Quenches from the current carrying phase. S L (t) vs. the time steps, withL = 60, h 0 = 4.0, ζ 0 = ζ = 5.0 for differenth. Note that the extent of the initial linear regime depends on the particular evolving Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 6.6 Quenchesfromthecurrentcarrying-phasewithdifferentvaluesofthe currentdrivingfieldζ.S L (t)vs. thetimestepswithL = 60,h 0 = 3.0 toh = 1 fordifferentζ 0 =ζ. . . . . . . . . . . . . . . . . . . . . . . . 66 viii 6.7 S L (t)vs. thetimestepsinsidethecurrent-carryingphasefordifferent block sizes L. Quenching is done from h 0 = 2.0 to h = 1.0 with ζ 0 =ζ = 3.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.8 S 60 (t)vs. thetimestepsforquenchestoh = 1fromvarioush 0 inside thecurrent-carryingphaseatζ =ζ 0 = 4. . . . . . . . . . . . . . . . . . 68 ix Abstract In this thesis I study multipartite entanglement sharing, and use entanglement to examine quantumphasetransitions(QPT)andequilibriuminclosedquantumsystems. Inthefirstpart,Idiscussonreplacingconcurrencebyconvex-roofextendednegativity (CREN) for studying monogamy of entanglement (MOE). I show that all proven MOE relations using concurrence can be rephrased in terms of CREN. Furthermore, I show that higher-dimensional (qudit) extensions of MOE in terms of CREN are not disproven by any of the counterexamples used to disprove qudit extensions of MOE in terms of concurrence. I further test the CREN version of MOE for qudits by considering fully or partially coherent mixtures of a qudit W-class state with the vacuum and show that the CREN version of MOE for qudits is satisfied in this case as well. Hence I prove that the CRENversionofMOEforquditsisastrongconjecturewithnoobviouscounterexamples. In the second part, I show how entanglement entropy can be used to characterize phases and study equilibrium dynamics in closed quantum systems. As an example I studytheIsingspinchainwithaDzyaloshinskii-Moriyainteraction,focusingonthestatic and dynamic properties of the entanglement entropy. I show that the effects of the addi- tional anisotropic interaction on the phase diagram and on the dynamics of the system x are captured by the properties of the entanglement entropy. In particular, the model pro- vides a way to study the quench dynamics in a system with an energy current. I consider quenches starting from an initial excited state of the Ising spin chain, and I analyze the effectsofdifferentinitialconditions. xi Chapter1 Introduction 1.1 Motivation There has been a rising interest in the field of quantum information and quantum com- putation after Peter Shor showed that splitting a numberN into its’ prime factors can be achieved in polynomial time and provides significant speed up over the classical known algorithms that run in sub-exponential time [5]. Quantum teleportation is another inter- esting application which allows information transfer without the presence of any physical channel [6]. At the heart of all this lies quantum entanglement [7], which plays a critical role [8], [9], [10]. Quantum entanglement is a form of quantum superposition where two statesAandB arecorrelatedsuchthatameasurementonstateAwillcausethestateB to immediatelycollapsetoacorrespondingstate,andthephysicalseparationbetweenAand B is immaterial. Entanglement is non-local and may not be increased by Local Operation and Classical Communication (LOCC). A simple example of entangled state is the bell state,|ψ⟩ AB = |00⟩ AB +|11⟩ AB √ 2 . Ingeneral,ifbipartitepurestatesAandB areentangled,we cannotwrite|ψ⟩ AB ̸=|ψ⟩ A ⊗|ψ⟩ B . Entanglement in bipartite quantum systems has been intensively studied, leading to a rich understanding. The situation becomes far more difficult for the case of multipartite quantumsystems,andverylittleisknownregardingitscharacterizationandquantification. One important property to characterize multipartite entanglement is known as monogamy 1 ofentanglement(MOE)[11],whichsaysthatmultipartiteentanglementscannotbefreely sharedamongtheparties. Thus,itisanimportantandnecessarytasktocharacterizeMOE to understand the whole picture of quantum entanglement in multipartite systems, as well asitspossibleapplicationsinquantum-informationtheory. Multipartite entanglement also plays an important role in many-body system. During quantumphasetransitions(QPT),thegroundstatewave-functionchangesdrastically[12]. Insuchsituations,entanglement(whichdependsonthewave-function)showscharacteris- ticbehavior[13]thatcouldbeusedtolocatethecriticalpointandcharacterizethephases. Also due to the presence of long range correlations at the critical point, entanglement scales differently with the block size [14]. One therefore can eliminate the need of order parameter for phase characterization that depends on the system and use a more universal quantitylikeentanglement. 1.2 MultipartiteEntanglement For multipartite systems, quantifying the entanglement in a general way is still an open question. To overcome the difficulty in quantifying multipartite entanglement we apply bipartite entanglement to multipartite systems. Two well known measures to quantify multipartiteentanglementareconcurrenceandblockentropy. Ingeneralevaluatingentan- glement for pure and mixed states can be quite different. In this section I will discuss evaluationofentanglementforbothofthesecases. FirstIdiscussevaluatingconcurrence. 2 For any bipartite pure state|ϕ⟩ AB in ad⊗d ′ (d ≤ d ′ ) quantum system, its concurrence, C(|ϕ⟩ AB ) isdefinedas[15]: C(|ϕ⟩ AB ) = √ 2(1−trρ 2 A ), (1.1) where ρ A = tr B (|ϕ⟩ AB ⟨ϕ|). For any mixed state ρ AB , there can be multiple pure state decompositions[16]. Inthiscaseconcurrenceisdefinedas: C(ρ AB ) = min ∑ k p k C(|ϕ k ⟩ AB ), (1.2) where the minimum is take over all possible pure state decompositions, ρ AB = ∑ k p k |ϕ k ⟩ AB ⟨ϕ| k . The measure is directly related to entanglement of formation [17]. Anotherentanglementmeasurethatcanbeconsideredasadualtoconcurrenceisthecon- currence of assistance (COA) [15], and this is related to entanglement of distillation [18]. COAandconcurrencearesameforpurestatesbutformixedstates,COAisdefinedas: C(ρ AB ) = max ∑ k p k C(|ϕ k ⟩ AB ), (1.3) where the maximum is take over all possible pure state decompositions, ρ AB = ∑ k p k |ϕ k ⟩ AB ⟨ϕ| k . Entanglement entropy is another measure of multipartite entanglement that is used extensively in many-body systems and is often referred as block entanglement. For a system of size N, block entanglement gives a measure of entanglement of a block of sizeL with rest of the system (sizeN−L) (Fig. 1.1). Ifρ is the the density matrix of the wholesystem,ρ L isthereduceddensitymatrixoftheblock(obtainedbyρ L =Tr N−L (ρ)). 3 Fig.1.1: Partitioningthe systemofsizeN intotwoparts,ablock ofsizeL andrest ofthe systemofsizeN−L. EntanglemententropyS L istheentanglementbetweenblockandrestofthesystemdefined as: S L =−Tr L [ρ L ln(ρ L )] (1.4) There are several interpretation of physical meaning of multipartite entanglement. I will discuss the physical meaning of entanglement entropy, entanglement of formation and entanglement of distillation briefly. Entanglement entropy when calculated to log 2 measuresthenumberofcopiesofthestateoneneedstoshareforsendingadesiredrandom bit of information without the presence of classical channel. Entanglement of formation 4 quantifieshowmanybellstatesareneededtoprepareacopyofentangledstateviaLOCC. Entanglement of distillation quantifies how many bell states can be obtained via LOCC fromagivenentangledstate. 1.3 EntanglementEntropyandQuantumCriticalPhases Quantum critical phases causes the wave function to change drastically. Hence entangle- mentwhichdependsonthewavefunctionisexpectedtobearthesignatureofcriticality[1]. 1.3.1 QuantumPhaseTransitions Quantumphasetransitionsareusuallydrivenbycontinuousvariationofparameterλsuch as magnetic field at zero temperature, unlike classical phase transition that is driven by temperature fluctuation. The properties of this kind of phase transition are determined solely by quantum mechanics, and the fluctuations are therefore driven by Heisenberg’s uncertainty principle at T = 0. For second order QPT,when λ approaches a specific parameter λ c , the energy gap between the ground state and the lowest excited state can becomesmallerandsmalleras: ∆∼|λ−λ c | s (1.5) wheres is the critical exponent. And the characteristic length scaleξ in the system will divergeas: 1 ξ ∼|λ−λ c | ν (1.6) 5 This characteristic length could be the correlation length depending on the system. The exponentν is known as the length scale critical exponent. One can therefore establish a relationbetweenthegap ∆andlengthscaleξ as: ∆∼ 1 ξ z (1.7) The critical exponent z = s ν is called the dynamical exponent. The gapless excitation in a system suggests the divergence of the correlation length if z is finite, and therefore highlights the appearance of a critical phase. Another important feature of quantum crit- ical phases is universality, as the above scaling properties are not sensitive to the par- ticular parameters or microscopic models. Universal scaling functions can be used to describe the critical properties of physical quantities in quantum critical phases. Exam- ples of QPT include quantum Hall systems [19], superconductor-insulator transition in two-dimensionalsystem[20]andothercasesdiscussedinthisthesis. 1.3.2 EntanglementEntropyandQPT Itisveryinterestingtostudytheentanglemententropyinthesecriticalphases. Inagapped system, we have a finite correlation length. In this case, one can expect the emergence of anarealawintheentanglemententropyforalargepartitionsincethequantumcorrelations between this partition and its exterior vanish when the distance between the boundary of partition and the outside region is far larger than ξ. In this sense, approximately only the region with volume ξS (S is the area of the partition) in it’s exterior can have the correlations within it, and therefore the entanglement entropy is expected to be bounded by an area law. However, in the gapless systems, the correlation length diverges, and 6 the above argument becomes invalid. One might ask the question: Does the area law of entanglement entropy hold in critical phases? In the 1D case, the situation is now clear: the area law does not hold, and logarithmic divergence is observed for the critical systems[21],[22],[23],[24],[25],[26],[27],[28]. In the context of conformal field theory (CFT) [29], [30], it is found that states with exponentially decaying correlationS L follow the area lawS L ∼ log 2 ξ ∼ L 0 , saturating to a finite value, whereas critical states, displaying power-law decaying correlations, a logarithmiccorrectiontotheareaisalwayspresent:S L = c+ c 6 log 2 ξ,wherecisthecentral chargefromtherelatedCFT.Inhigherdimensionalsystems,ithasbeenshownrigourously thatthearealawholdsfortheentanglemententropyofthequasi-freebosonicsystems[26]. In free fermions, it is found that the area law can be violated by a multiplication factor log 2 L if a finite fermion surface exists [31], [32]. It is quite interesting to observe that in a nodal fermionic system with only fermion points the area law still holds despite that the correlationlengthisdiverging[33],[34]. 1.3.3 ExamplewithXYModel AsimplequantumcriticalsystemmightbetheIDXYspinchaininamagneticfieldwhose hamiltonianisgivenby: H XY =− ∑ [(1+γ)S x l S x l+1 +(1−γ)S y l S y l+1 +λS z l ] (1.8) The phase diagram is shown in Fig. 1.2. Here, we have two critical regions 0 ≤ λ ≤ 1,γ = 0 andλ = 1. In the caseγ = 0, we have critical exponentsz = 2,s = 1 andv = 1 2 The central charge in this universality class is equal to 1 by conformal field theory. The 7 entanglement entropy is found to scale as 1 3 log 2 L for a partition with chain sizeL. In the caseλ = 1, we have critical exponentsz = 1,s = 1 andν = 1, and the central charge in this universality class is equal to 1 2 . The entanglement entropy is found to be log 2 L for a partitionwithchainsizeL. Thedetailsofthiscalculationcanbefoundedin[22]. One can map the ID XY spin chain model to a fermionic model through the Jordan WignertransformationtoobtaintheHamiltonian[12],[35]: H XY =− 1 2 ∑ (c † i c i+1 +γc † i c † i+1 +c † i+1 c i +γc i+1 c i −2λc † i c i ) (1.9) In higher dimension, an additional non-trivial Chern-Simons term is introduced by the Jordan-Wigner transformation, and then the one-to-one mapping is lost between the spin andfermionmodels. 1.3.4 ExamplewithIsingModel Anothersimplemodel inwhichaquantumphasetransitionoccursistheone-dimensional Isingmodel: H = Σ i (σ x i σ x i+1 +hσ z i ) (1.10) Forh = 0, the first term dominates, and the spin aligns themselves along +x or−x and the system is ferromagnetic (ordered). Forh ≫ 1, the spin aligns themselves along +z or−z and the system is paramagnetic (disordered). In betweenh = 0 andh ≫ 1 there exists a critical point (for h = 1) where the system undergoes transition from ordered to disordered. The order parameter for this transition is the magnetization along x-axis (⟨σ x ⟩), which is non-zero in ordered state and 0 is disordered state. QPT can also be studied from the point of ground state wave function, since it changes drastically at the 8 Fig.1.2: PhasediagramoftheIDXYmodelintheparameterspace(λ,γ). Thetwocritical regions are 0 ≤ λ ≤ 1, γ = 0 and λ = 1 as highlighted by the red line. The region γ = 1describestheIDIsingmodel,andtheregionγ = 0correspondstotheisotropicXX chain[1]. critical point. Entanglement which essentially depends on the ground state wave function is expected to behave differently as the system passes through critical point and has been studied for various systems [14]. An example of the behavior of entanglement upon QPT asstudiedin [2]isshowninFig.1.3. 9 0 5 10 15 20 Δ 0 0.1 0.2 0.3 0.4 C(1) 0 0.5 1 0.33 0.34 0.35 0.36 0.37 0.38 0.39 Numerical results C=A 0 -A 1 (Δ−1) 2 SU(2) point . Fig. 1.3: ConcurrenceC as a function of ∆ shows that entanglement is maximum at the criticalpoint ∆ = 1 forXXZmodel[2]. 1.3.5 ImportanceofEntanglemententropyinphysicalsystems In physical systems, the study of the entanglement entropy is not straightforward. Impor- tanceofsuchastudyisdiscussedbelow[1]: • The study of the entanglement entropy is helpful to analyze and understand numer- ical simulation in quantum many-body systems. Basically, if the entanglement entropy is small in a given system, typically one can use a small number of basis states to describe thesystem. Iftheentanglemententropyfollowsapowerlawrelationship,onecanexpect that the number of the states grow exponentially with increasing the system size L in a 2Dorhigherdimensionalsystem. Inthelattercase,theefficiencyofcalculationwilldrop 10 greatly. That’s exactly the situation of the density-matrix renormalization group method- DMRG) [36], [37], which is the most successful algorithm in 1D quantum systems. The reason is that it uses a matrix product state to approximate the ground state, and in 1D quantumsystemstheentanglemententropytypicallyiswellboundedbyaconstantand,or only shows logarithmic divergence with the increase of chain size L. The study of entan- glement entropy also sheds light on the development of efficient numerical algorithms suitableforhigherdimension. • The entanglement entropy in many-body systems has a link to the physics of black hole. The Bekenstein-Hawking entropy is known to be proportional to the area of the boundary surface [38], [39]. This kind of behavior coincides with the area law behavior of the entanglement entropy in free scalar fields [40], [41]. Recently it was shown that the entanglement entropy is exactly the Bekenstein-Hawking entropy under some condi- tions[42],[43]. • Entanglement effects are found to be associated not only with the microscopic world, but also with macroscopic scales [44], [45]. For example, in [46] it was shown that the entanglement describes the magnetic behavior of the insulating magnetic salt LiHo x Yi x F 4 . Their significant finding suggests that quantum correlations can play an importantroleforthephysicalpropertiesofsolidstates. • Entanglement entropy provides an useful tool to study non-equilibrium quantum system[4],[47]. Italsoprovidesawayforstudyingexcitedquantumstates[47]. 11 1.4 ThesisOutline In this dissertation, I investigate how entanglement can be shared between multiple sys- temsandhowitcanbeusedtostudynon-equilibriumquantumsystems. Inchapter2,Idiscussmonogamyrelationforqubitsintermsofconcurrenceandshow examplesofitsviolationforqudits. In chapter 3, I show how CREN can be used to replace concurrence as entanglement measureandmonogamyrelationintermsofCRENarenotviolatedforqudits. Iworkout the monogamy relation for a partially coherent superposition of n-qudit W-class state and vacuumstate. Inchapter 4,Idiscusstheusetoentanglementtostudymany-bodysystems. Inchapter 5,IdiscusstheIsingmodelwithD-Minteractionindetails. Inchapter6,Idiscussphasecharacterizationforthemodelfromchapter5usingentan- glement. Inchapter7,IdiscussentanglementdynamicsfollowingaquenchfortheIsingsystem withD-Minteractionandhowitcanbeusedtostudyquenchesforexcitedstates. Inthelastchapter,Iendthisdissertationwithconclusionandoutlook. 12 Chapter2 ConcurrenceandMonogamyof Entanglement 2.1 MOEforqubits ConsiderthreequbitsA,B,andC whichmaybeentangledwitheachother. Itwasshown that there is a trade-off between entanglement of A with B and its entanglement with C [11]. This relation is expressed in terms of a measure of entanglement called the con- currence. It was shown that the squared concurrence betweenA andB, plus the squared concurrencebetweenAandC,cannotbegreaterthanthesquaredconcurrencebetweenA andthepairBC. Thisinequalityisverystrong,inthesensethatforanyvaluesofthecon- currences satisfying the corresponding equality, one can find a quantum state consistent with those values. Thus unlikeclassical correlations, quantum correlations can’t be freely shared. In this section I am going to discuss the main results of [11] which discuss MOE for qubits. I will first discuss concurrence for qubits. LetA andB bea pair of qubits, and letthedensitymatrixofthepairbeρ AB ,whichmaybepureormixed.Spin-flippeddensity matrixwasdefinedtobe: ˜ ρ AB = (σ y ⊗σ y )ρ ∗ AB (σ y ⊗σ y ) (2.1) 13 Since bothρ AB and ˜ ρ AB are positive operators, the productρ AB ˜ ρ AB (although not hermi- tian)hasonlyrealnon-negativeeigenvalues. Letthesquarerootsoftheseeigenvalues,in decreasing order beλ 1 ,λ 2 ,λ 3 andλ 4 . Then the concurrence of the density matrixρ AB is definedas: C AB =max{λ 1 −λ 2 −λ 3 −λ 4 ,0} (2.2) IfAB ispurestate,onecanshowthatC AB = 2 √ |ρ A |. Iwillnowdiscussforagiventhree qubit pure state ABC, how is the concurrence between A and B related to the concur- rence betweenA andC. For a pure state of three qubits, the formula for the concurrence simplifies. One can write any three qubit pure state |ϕ⟩ AB−C = ∑ 1 i=0 √ λ i |ii⟩ AB−C , where λ i ≥ 0, ∑ 1 i=0 λ i = 1, this is called Schmidt decomposition (Eq. 3.1). Hence the reduced density matrixρ AB can have at most two nonzero eigenvalues. It follows that the productρ AB ˜ ρ AB alsohasonlytwononzeroeigenvalues. Thereforewehave: C 2 AB = (λ 1 −λ 2 ) 2 =λ 2 1 +λ 2 2 −2λ 1 λ 2 =Tr(ρ AB ˜ ρ AB )−2λ 1 λ 2 ≤Tr(ρ AB ˜ ρ AB ) (2.3) UsingEqn.2.3onecaneasilyshowthat: C 2 AB +C 2 AC ≤Tr(ρ AB ˜ ρ AB )+Tr(ρ AC ˜ ρ AC ) (2.4) Now let us look into the right hand side of the inequality. Expressing the pure state three qubit system|ξ⟩ in standard basis|ijk⟩ where each index can take value 0 or 1. Hence one can writeρ AB = ∑ i,j,i ′ ,j ′ a ijk a i ′ j ′ k |ij⟩⟨i ′ j ′ |. Now usingσ y |l⟩ = iϵ lm |m⟩ and⟨l|σ y = 14 −iϵ lm ⟨m|, wherel,m = 0,1 andϵ 01 =−ϵ 10 = 1 andϵ 00 =ϵ 11 = 0 and Eq. 2.1, one can write: Tr(ρ AB ˜ ρ AB ) = Σa ijk a ∗ mnk a i ′ j ′ p a m ′ n ′ p ϵ mm ′ϵ nn ′ϵ i ′ i ϵ j ′ j (2.5) Replacingϵ nn ′ϵ j ′ j withδ nj ′δ n ′ j −δ nj δ n ′ j ′ andusingsimilaridentityforϵ mm ′ϵ i ′ i onegets: Tr(ρ AB ˜ ρ AB ) = 2|ρ A |−Tr(ρ 2 B )+Tr(ρ 2 C ) (2.6) Sincetheindividualdensitymatrixareofunittrace,Eqn.2.6canbewrittenas: Tr(ρ AB ˜ ρ AB ) = 2(|ρ A |+|ρ B |−|ρ C |) (2.7) Nowonecaneasilyobtain: Tr(ρ AC ˜ ρ AC ) = 2(|ρ A |+|ρ C |−|ρ B |) (2.8) CombiningEqn.2.7andEqn.2.8: Tr(ρ AB ˜ ρ AB )+Tr(ρ AC ˜ ρ AC ) = 4|ρ A | (2.9) CombiningEqn.2.4andEqn.2.9: C 2 AB +C 2 AC ≤ 4|ρ A | (2.10) One can interpret the right-hand side of Eq. 2.10 as follows. If one regard the pair BC as a single object, it makes sense to speak of the concurrenceC A(BC) between qubit 15 A and the pairBC, because, even though the state space ofBC is four dimensional, only two of those dimensions are necessary to express the state|ξ⟩ ofABC (using the notion of Schmidt decomposition). The two necessary dimensions are those spanned by the two eigenstates ofρ BC that have nonzero eigenvalues. One may thus treatA andBC, at least for this purpose, as a pair of qubits in a pure state. As mentioned before, the concurrence forthiscaseissimply 2 √ |ρ A |. Onecanthereforerewriteourresultas: C 2 AB +C 2 AC ≤C 2 A(BC) (2.11) ThisisalsoknownastheCWKinequality. I now work out examples with two three-qubit systems that belongs to two inequiv- alent classes (one state cannot be converted to another by LOCC) of genuine tripartite entangled states by the CKW inequality [48]. One of them is the Greenberger-Horne- Zeilinger (GHZ) class [49] and the other one is the W class [48]. These states are also maximallyentangled(reduceddensitymatrixisdiagonal). 2.2 ExamplewithGHZstate A simple three qubit GHZ state is represented by|ψ⟩ ABC = |000⟩ ABC +|111⟩ ABC √ 2 . The den- sity matrix is ρ ABC = |000⟩⟨000|+|000⟩⟨111|+|111⟩⟨000|+|111⟩⟨111| 2 . Taking partial trace I have ρ A = |0⟩⟨0|+|1⟩⟨1| 2 and ρ AB = |00⟩⟨00|+|11⟩⟨11| 2 . This gives C A(BC) = 1. Now I show the details of evaluatingC AB . Sinceρ AB is a mixed state I have to use Eqn. 1.2 for this. The pure state decomposition of ρ AB can be |X⟩ = |00⟩ √ 2 and |Y⟩ = |11⟩ √ 2 . I have unnormal- ized all possible pure state decomposition (that gives the same density matrix ρ AB ) as | ˜ ϕ i ⟩ = U i1 |X⟩ +U i2 |Y⟩, where |U ij | is the element of a general unitary matrix. Since 16 we have two pure state decompositions (|X⟩,|Y⟩), our unitary matrix is 2× 2. I get p i = ⟨ ˜ ϕ i | ˜ ϕ i ⟩ = |U i1 | 2 +|U i2 | 2 2 . This gives me the normalized wavefunction|ϕ i ⟩ = | ~ ϕ i ⟩ √ p i = U i1 |00⟩+U i2 |11⟩ √ |U i1 | 2 +|U i2 | 2 . Thus I have C(ρ AB ) = minΣ i p i C(|ϕ i ⟩ AB ). I now evaluate C(|ϕ i ⟩ AB ). I have Tr B (ρ i AB ) = |U i1 | 2 |0⟩⟨0|+|U i2 | 2 |1⟩⟨1| |U i1 | 2 +|U i2 | 2 . Which gives Tr(ρ i A 2 ) = |U i1 | 4 +|U i2 | 4 (|U i1 | 2 +|U i2 | 2 ) 2 . Thus I have √ 2(1−Tr(ρ i A 2 )) = 2|U i1 | 2 |U i2 | 2 |U i1 | 2 +|U i2 | 2 . This gives p i C(|ϕ i ⟩ AB ) = |U i1 ||U i2 |. One can easilyshowthatmin ∑ i |U i1 ||U i2 | = 0. SoIhaveC 2 AB +C 2 AC = 0. ThusEqn.2.11issatisfied. 2.3 ExamplewithW-classstate AsimplethreequbitW-classstateisrepresentedby|ψ⟩ ABC = |001⟩ ABC +|010⟩ ABC +|100⟩ ABC √ 3 . This gives me ρ ABC = (|001⟩+|010⟩+|100⟩)(⟨001|+⟨010|+⟨100|) 3 . Which gives ρ A = |1⟩⟨1|+2|0⟩⟨0| 3 and ρ AB = |00⟩⟨00|+|10⟩⟨10|+|10⟩⟨01|+|01⟩⟨10|+|01⟩⟨01| 3 . It is easy to show that C 2 A(BC) = 8 9 . I now show the details of evaluatingC AB . Pure state decomposition ofρ AB is|X⟩ = |00⟩ √ 3 and |Y⟩ = |10⟩+|01⟩ √ 3 . I have unnormalized all possible pure state decomposition as | ˜ ϕ i ⟩ = U i1 |00⟩+U i2 (|10⟩+|01⟩) √ 3 . This gives p i = ⟨ ˜ ϕ i | ˜ ϕ i ⟩ = |U i1 | 2 +2|U i2 | 2 3 . Hence, |ϕ i ⟩ = | ~ ϕ i ⟩ √ p i = U i1 |00⟩+U i2 (|10⟩+|01⟩) √ |U i1 | 2 +2|U i2 | 2 . Now I evaluate C(ρ AB ) = minΣ i p i C(|ϕ i ⟩ AB ). This gives ρ i AB = |ϕ i ⟩⟨ϕ i | and we get ρ i A = |U i1 | 2 |0⟩⟨0|+U i1 U † i2 |0⟩⟨1|+U † i1 U i2 |1⟩⟨0|+|U i2 | 2 (|1⟩⟨1|+|0⟩⟨0|) (|U i1 | 2 +2|U i2 | 2 ) . Now one can easily see that Tr(ρ i A 2 ) = (|U i1 | 2 +|U i2 | 2 ) 2 +|U i2 | 4 +2|U i1 | 2 |U i2 | 2 (|U i1 | 2 +2|U i2 | 2 ) 2 . After some algebra I get ∑ i p i C(|ϕ i ⟩ AB ) = ∑ i 2|U i2 | 2 3 = 2 3 (Since ∑ i |U ik | 2 = 1forafixedk). HenceC 2 AB = 4 9 . SoIhaveC 2 AB +C 2 AC = 8 9 . ThusEqn.2.11issaturatedbytheW-classstate. One can easily see that evaluating concurrence for mixed states is not trivial and involves significant analytical calculations. Thus evaluating concurrence for a general mixedstateisexpectedtobecomputationallyintensive. 17 2.4 Violation of MOE for higher dimensional qubits(qudits) Although MOE is a typical property of multipartite quantum entanglement, it is however about the relation of bipartite entanglements among the parties in multipartite systems. Thus, the following criteria for an entanglement measure must be satisfied to have a good descriptionofthemonogamynatureofentanglementinmultipartitequantumsystems[50]. (i)Monotonicity: Thepropertythatensuresentanglementcannotbeincreasedunderlocal operationsandclassicalcommunications(LOCC). (ii)Separability: Capabilityofdistinguishingentanglementfromseparability. (iii)Monogamy: Upperboundonasumofbipartiteentanglementmeasurestherebyshow- ingthatbipartitesharingofentanglementisbounded. Thus it is important that measure of entanglement for qudits(d-dimensional states) also observeMOE. Unfortunately it has been seen that there is violation of MOE in terms of concurrence for higher dimensional quantum systems [51], [52]. I will be discussing these two case briefly. First let us look into the case in [51] for a 3⊗ 3⊗ 3 quantum system |ψ⟩ ABC = |123⟩−|132⟩+|231⟩−|213⟩+|312⟩−|321⟩ √ 6 . For pure state ρ ABC , one can easily verify C 2 A(BC) = 4 3 . To evaluate ρ AB and ρ AC , one can use|x⟩ = |23⟩−|32⟩ √ 2 , |y⟩ = |31⟩−|13⟩ √ 2 and |z⟩ = |12⟩−|21⟩ √ 2 . Using this on can easily writeρ AB = |x⟩⟨x|+|y⟩⟨y|+|z⟩⟨z| 3 . It is now easy to seeC 2 AB =C 2 AC = 1. HenceMOEisviolated. The next example is for a 3 ⊗ 2 ⊗ 2 system showing with just one higher dimen- sional system MOE fails completely [52]. The pure state in this case is |ψ⟩ ABC = √ 2|010⟩+ √ 2|101⟩+|200⟩+|211⟩ √ 6 . It can be easily seen thatC 2 A(BC) = 12 9 andC 2 AB = 8 9 . Hence 18 MOEisviolated. HencethereisaneedtodefineanentanglementmeasurethatwillobserveMOEforhigher dimension quantum systems. In the next section I will introduce convex roof extended negativity(CREN)asonesuchpossiblemeasure. 19 Chapter3 CRENasaMeasureofMultipartite Entanglement Idiscussedthatthereexistquantumstatesinhigher-dimensionalsystems[51],[52]which violatethemonogamypropertiesintermsoftheproposedentanglementmeasures,andthis exposestheimportanceofchoosingaproperentanglementmeasure. Iproposetheconvex- roof extended negativity (CREN) [53] as a powerful candidate for the criteria above. Besidesitsmonotonicityandseparabilitycriteria,IclaimthatCRENisagoodalternative for MOE without any known example violating its monogamy property even in higher- dimensional systems. I show that any monogamy inequality of entanglement for multi- qubitsystemsusingconcurrence[15]canberephrasedbyCREN,andthisCRENMOEis alsotrueforthecounterexamplesofconcurrenceinhigher-dimensionalsystems[51],[52]. 20 3.1 ConcurrenceandCREN Apart from concurrence, another well-known quantification of bipartite entanglement is the negativity [53] [54], which is based on the positive partial transposition (PPT) crite- rion [55], [56]. For a bipartite pure state|ϕ⟩ AB in ad⊗d ′ (d≤d ′ ) quantum system with theSchmidtdecomposition: |ϕ⟩ AB = d−1 ∑ i=0 √ λ i |ii⟩, λ i ≥ 0, d−1 ∑ i=0 λ i = 1, (3.1) (without loss of generality, the Schmidt basis is taken to be the standard basis), the partial transpositionof|ϕ⟩ AB is: |ϕ⟩⟨ϕ| T B = d−1 ∑ i,j=0 √ λ i λ j |ij⟩⟨ji| = d ∑ i=1 a 2 i |ii⟩⟨ii|+ ∑ i̸=j a i a j |ij⟩⟨ji| = d−1 ∑ i=0 λ i |ii⟩⟨ii|+ ∑ i<j √ λ i λ j (|ij⟩⟨ji|+|ji⟩⟨ij|). (3.2) Thus,thenegativeeigenvaluescanbe− √ λ i λ j fori<j withcorrespondingeigenvectors |ψ ij ⟩ = 1 √ 2 (|ij⟩−|ji⟩),andthenegativityN of|ϕ⟩ AB isdefinedas[50]: N(|ϕ⟩) = |ϕ⟩⟨ϕ| T B 1 −1 = 2 ∑ i<j √ λ i λ j , (3.3) where∥·∥ 1 isthetracenorm. 21 Based on the reduced density matrix of|ϕ⟩ AB , I can have an alternative definition of negativity: N(|ϕ⟩) = 2 ∑ i<j √ λ i λ j = (tr √ ρ A ) 2 −1, (3.4) whereρ A = tr B |ϕ⟩ AB ⟨ϕ|. Note that N(|ϕ⟩) = 0 if and only if |ψ⟩ is separable, and it can attain its maximal value,d−1,forad⊗dmaximallyentangledstate: |ϕ⟩ = 1 √ d d−1 ∑ i=0 |ii⟩. (3.5) (OnecaneasilycheckthisbytheLagrangemultiplier)ThusN canbeameasureofentan- glementforbipartitepurestatesinanydimensionalquantumsystem. Foramixedstateρ AB ,itsnegativityisdefinedas: N(ρ AB ) = ρ AB T B 1 −1, (3.6) whereρ T B isthepartialtransposeofρ AB . ItisknownthatPPTgivesaseparabilitycriterionfortwo-qubitsystems,anditisalsoa necessaryandsufficientconditionfornondistillabilityin2⊗nquantumsystem[57],[58]. However, in higher-dimensional quantum systems rather than 2 ⊗ 2 and 2 ⊗ 3 quan- tum systems, there exist mixed entangled states with PPT, so-called ‘bound entangled states’ [57], [59]. For this case, negativity cannot distinguish PPT bound entangled states 22 from separable states, and thus, negativity itself is not sufficient to be a good measure of entanglementevenina 2⊗nquantumsystem. One modification of negativity to overcome its lack of separability criterion is CREN [50], which gives a perfect discrimination of PPT bound entangled states and sep- arablestatesinanybipartitequantumsystem. Forabipartitemixedstatemixedstateρ AB ,CRENisdefinedas: N c (ρ)≡ min ∑ k p k N(|ϕ⟩ k ), (3.7) where the minimum is taken over all possible pure state decompositions of ρ = ∑ k p k |ϕ k ⟩⟨ϕ k |. Whereas a normalized version of the negativity depending on the dimension of the quantum systems was used to show its monotonicity [53], it can be analogously shown withthedefinitionsinEqs.(3.6)and(3.7). Now consider the relation between CREN and concurrence. For any bipartite pure state|ϕ⟩ AB inad⊗d ′ quantumsystemwithSchmidtrank2: |ϕ⟩ = √ λ 0 |00⟩+ √ λ 1 |11⟩, (3.8) 23 wehave: N(|ϕ⟩) = |ϕ⟩⟨ϕ| T B 1 −1 = 2 √ λ 0 λ 1 = √ 2(1−trρ 2 A ) =C(|ϕ⟩), (3.9) whereρ A = tr B (|ϕ⟩⟨ϕ|). In other words, negativity is equivalent to concurrence for any pure state with Schmidt rank 2, and consequently it follows that for any 2-qubit mixed stateρ AB = ∑ i p i |ϕ i ⟩⟨ϕ i |: N c (ρ AB ) =min ∑ i p i N(|ϕ i ⟩) =min ∑ i p i C(|ϕ i ⟩) =C(ρ AB ), (3.10) wheretheminimaaretakenoverallpurestatedecompositionsofρ AB . 24 Similar to the duality between concurrence and COA, we can also define a dual to CREN, namely CRENOA, by taking the maximum value of average negativity over all possiblepurestatedecomposition. Furthermore,foratwo-qubitsystem,Ihave: N a c (ρ AB ) =max ∑ i p i N(|ϕ i ⟩) =max ∑ i p i C(|ϕ i ⟩) =C a (ρ AB ), (3.11) where maxima are taken over all pure state decompositions of ρ AB andN a c (ρ AB ) is the CRENOAofρ AB . FromtheanalysisofCRENandCRENOA,wecanseethatCRENcanbeconsideredas ageneralizedversionofconcurrencefrom2-qubitsystems. Thus,havingthemonotonicity and separability criteria of CREN, it is natural to investigate MOE in terms of CREN for multi-qubitsystemsandpossiblehigher-dimensionalquantumsystems. 3.2 CRENMOE Inthree-qubitsystems,monogamyinequalityintermsofconcurrencewasfirstintroduced inin[11],as: C 2 A(BC) ≥C 2 AB +C 2 AC , (3.12) whereC A(BC) =C(|ψ⟩ A(BC) ) is the concurrence of a 3-qubit state|ψ⟩ A(BC) for a bipartite cut of subsystems betweenA andBC andC AB = C(ρ AB ). Similarly, its dual inequality 25 intermsofCOA,whichcanbeinterpretedasthemaximumoftheaverageofconcurrence takenoverallpossiblepurestatedecompositionsofthemixedstate: C 2 A(BC) ≤ (C a AB ) 2 +(C a AC ) 2 , (3.13) has been shown in [60]. Later, the CKW inequality has been generalized into n-qubit systems[61],anditsdualinequalityforn-qubitsystemshasalsobeenintroduced[62]. However, a quantum state in a 3⊗ 3⊗ 3 quantum system was found that violates the CKW inequality [51], and recently another counterexample was found in a 3⊗2⊗2 quantum system [52]; therefore the CKW inequality only holds for multi-qubit systems, andevenatinyextensioninanyofthesubsystemsleadstoaviolation. Inthischapter,Ishowthatallthemonogamyinequalitiesforqubitsusingconcurrence can be rephrased by CREN, and this CREN monogamy inequality is still true for the counterexamplesin[51],[52]. 3.2.1 MonogamyInequalitiesforn-qubitsystemsintermsofCREN For any pure state |ψ⟩ A 1 ···An in an n-qubit system A 1 ⊗···⊗A n where A i ∼ = C 2 for i = 1,...,n,ageneralizationoftheCKWinequality: C 2 A 1 (A 2 ···An) ≥C 2 A 1 A 2 +···+C 2 A 1 An , (3.14) was conjectured [11] and proved [61]. Another inequality, which can be considered to be dualtoEq.(3.14)wasalsointroducedin[62]: C 2 A 1 (A 2 ···An) ≤ (C a A 1 A 2 ) 2 +···+(C a A 1 An ) 2 . (3.15) 26 Now, consider these inequalities in terms of CREN. First, note that anyn-qubit pure state |ψ⟩ A 1 ···An can have a Schmidt decomposition with at most two non-zero Schmidt coefficientswithrespecttothebipartitecutbetweenA 1 andtheothers. Thus,byEq.(3.9), Ihave: C A 1 (A 2 ···An) =N cA 1 (A 2 ···An) . (3.16) Furthermore,foranyreduceddensitymatrixρ A i A j of|ψ⟩ A 1 ···An ontotwo-qubitsubsystems A i ⊗A j ,itisatwo-qubitmixedstate;therefore,byEqs.(3.10)and(3.11),Ihave: C A i A j =N cA i A j , C a A i A j =N c a A i A j , (3.17) fori,j ∈{1,··· ,n}, i̸=j. Thus,Ihavethefollowingtheorem. Theorem1. Foranyn-qubitpurestate|ψ⟩ A 1 ···An : N cA 1 (A 2 ···An) 2 ≥N cA 1 A 2 2 +···+N cA 1 An 2 (3.18) and N cA 1 (A 2 ···An) 2 ≤ (N c a A 1 A 2 ) 2 +···+(N c a A 1 An ) 2 (3.19) whereN cA 1 (A 2 ···An) =N(|ψ⟩ A 1 (A 2 ···An) ),N cA 1 A i =N c (ρ A 1 A i ) andN c a A 1 A i =N c a (ρ A 1 A i ) fori = 2,...,n. Proof. ItisadirectconsequencefromtheoverlapofCRENandconcurrenceinEqs.(3.16) and(3.17),aswellasthemonogamyinequalitiesinEqs.(3.14)and(3.15)byconcurrence. 27 In[63],anothermonogamyinequalityofentanglementforthree-qubitsystemsinterms ofthe originalnegativity[54] wasproposed. Fora three-qubitstate|ψ⟩ ABC ,it wasshown that: N A(BC) 2 ≥N AB 2 +N AC 2 (3.20) whereN AB 2 =∥ρ T B AB ∥ 1 −1 andN AC 2 =∥ρ T C AC ∥ 1 −1 are the original negativities ofρ AB andρ AC respectively. Duetotheconvexityoftheoriginalnegativity,onecaneasilyseethatCRENisalways an upper bound of the original negativity. In other words, for any bipartite mixed state ρ AB : N c (ρ AB )≥N(ρ AB ). (3.21) From Theorem 1 together with Eq. (3.21), I have the following corollary which encapsu- latestheresultofEq.(3.20). Corollary1. Foranyn-qubitpurestate|ψ⟩ A 1 ···An : N A 1 (A 2 ···An) 2 ≥N cA 1 A 2 2 +···+N cA 1 An 2 (3.22) Thus, besides concurrence, CREN is another good entanglement measure in multi- qubitsystemsforMoE. 3.2.2 CRENvsConcurrence-basedMonogamyRelations Two counterexamples in [51], [52] are, in fact, all known counterexamples showing the violation of the CKW inequality in higher-dimensional quantum systems. Here I show thattheystillhaveamonogamyrelationintermsofCREN. 28 Counterexample1. ([51]) Letusconsiderapurestate|ψ⟩in 3⊗3⊗3quantumsystemssuchthat: |ψ⟩ ABC = 1 √ 6 (|123⟩−|132⟩+|231⟩ −|213⟩+|312⟩−|321⟩). (3.23) Since|ψ⟩ ABC is pure, it is easy to checkC 2 A(BC) = 4 3 . For mixed statesρ AB andρ AC , it was shown that any pure state in any pure state ensemble has the same constant value, 1,asitsconcurrence,whichimpliesC 2 AB =C 2 AC = 1. ThereforeIhave: C 2 AB +C 2 AC = 2≥ 4 3 =C 2 A(BC) , (3.24) whichisaviolationoftheCKWinequalityinhigher-dimensionalquantumsystems. Now, consider the case of using CREN as the entanglement measure for the state in Eq.(3.23). Since|ψ⟩ ABC ispure,itcanbeeasilycheckedthat: N A(BC) =N cA(BC) = (tr √ ρ A ) 2 −1 = 2. (3.25) ForN cAB ,considerρ AB whosespectraldecompositionis: ρ AB = 1 3 (|x⟩ AB ⟨x|+|y⟩ AB ⟨y|+|z⟩ AB ⟨z|), (3.26) 29 where, |x⟩ AB = 1 √ 2 (|23⟩−|32⟩), |y⟩ AB = 1 √ 2 (|31⟩−|13⟩), |z⟩ AB = 1 √ 2 (|12⟩−|21⟩). (3.27) By a straightforward calculation, it can be shown that for arbitrary pure states |ϕ⟩ AB =c 1 |x⟩ AB +c 2 |y⟩ AB +c 3 |z⟩ AB with|c 1 | 2 +|c 2 | 2 +|c 3 | 2 = 1,theirreduceddensity matrixρ A = tr B |ϕ⟩ AB ⟨ϕ|hasthesamespectrum{ 1 2 , 1 2 ,0}[64]. Thus,Ihave: N(|ϕ⟩ AB ) = (tr √ ρ A ) 2 −1 = 1, (3.28) for any|ϕ⟩ AB that is a superposition of|x⟩ AB ,|y⟩ AB and|z⟩ AB . By the Hughston-Jozsa- Wootters (HJW) theorem [65], any pure state in any pure state ensemble of ρ AB can be realizedasasuperpositionof|x⟩ AB ,|y⟩ AB and|z⟩ AB thusIhave: N c (ρ AB ) = min ∑ k p k |ϕ⟩ k ⟨ϕ k |=ρ AB ∑ k p k N(|ϕ⟩ k ) = 1 3 (N(|x⟩ AB )+N(|y⟩ AB )+N(|z⟩ AB )) =1. (3.29) 30 SinceEq.(3.23)isasymmetric,Ialsohaveasimilarresultforρ AC ,whichis: N c (ρ AC ) = min ∑ k p k |ϕ⟩ k ⟨ϕ k |=ρ AC ∑ k p k N(|ϕ⟩ k ) = 1 3 (N(|x⟩ AC )+N(|y⟩ AC )+N(|z⟩ AC )) =1. (3.30) Now,fromEq.(3.25)togetherwithEqs.(3.29)and(3.30),Ihave: N cA(BC) 2 = 4≥ 1+1 =N cAB 2 +N cAC 2 . (3.31) In other words, even though the state|ψ⟩ in Eq. (3.23) is a counterexample of the CKW inequalityinthree-qutritsystemsintermsofconcurrence,itstillshowsamonogamyprop- ertyintermsofCREN. Counterexample2. ([52]) Considerapurestate|ψ⟩in 3⊗2⊗2quantumsystemssuchthat: |ψ⟩ ABC = 1 √ 6 ( √ 2|010⟩+ √ 2|101⟩+|200⟩+|211⟩). (3.32) It can be easily seen that C 2 A(BC) = 12 9 whereas C 2 AB = C 2 AC = 8 9 , which implies the violation of the CKW inequality. However, by using a similar method to the previous example, I haveN cA(BC) 2 = 4 whereasN cAB 2 =N cAB 2 = 8 9 , which implies the example inEq.(3.32)alsoshowsamonogamypropertyintermsofCREN. Thus, CREN is a powerful alternative for MoE in multipartite higher-dimensional quantumsystemswithoutanytrivialcounterexamplesofar. 31 3.3 Partially Coherent Superposition of an n-QUDIT GeneralizedW-classStateandVacuumState As the first step toward general CREN MOE studies in higher-dimensional quantum sys- tems, I propose a class of quantum states inn-qudit systems consisting of partially coher- ent superpositions of a generalized W-class state [52] and the vacuum, |0⟩ ⊗n , and show that this class saturates CREN MOE for any arbitrary partition of the set of subsystems. I also show that the CREN value of the proposed class and its dual, CREN of Assistance (CRENOA)coincide,andtheyarenotaffectedbythedegreeofcoherencyinthesuperpo- sition. Thisisparticularlyimportantbecausethesaturationofmonogamyrelationimplies that this class of multipartite higher-dimensional entanglement can have a complete char- acterization by means of its partial entanglements, and the characterization is not even affectedbyitsdecoherency. The GHZ state and the W-class states show extreme differences in terms of the CKW and its dual inequalities: The CKW and its dual inequalities are saturated by W-class states, whereas the terms for reduced density matrices in the inequalities always vanish forGHZ-classstates. SincethesaturationoftheCKWinequalitybyW-classstatescanbe interpretedasagenuinetripartiteentanglementwithacompletecharacterizationbymeans ofitspartialentanglements,W-classstateshereareespeciallyinteresting. It was shown that there also exists a class of states in n-qudit systems which satu- rate a monogamy relation [52]. By using concurrence as the entanglement measure, the monogamyinequalitiesareshowntobesaturatedbyincoherentsuperpositionsofagener- alizedn-quditW-classstate[52]andthevacuum,|0⟩ ⊗n . 32 In this section, we propose a class of quantum states inn-qudit systems consisting of partiallycoherentsuperpositionsofageneralizedW-classstateandthevacuum,andshow that they have the saturation of the monogamyrelation in terms of CREN. This saturation isalsotrueforanarbitrarypartitionofthesetofsubsystems,anditisnotevenaffectedby thedegreeofcoherency. NowIreprisethedefinitionofann-quditgeneralizedW-classstate[52]: W d n ⟩ A 1 ···An = d−1 ∑ i=1 (a 1i |i0···0⟩+a 2i |0i···0⟩ +···+a ni |00···0i⟩), d−1 ∑ i=1 (|a 1i | 2 +|a 2i | 2 +···+|a ni | 2 ) = 1, (3.33) whichisacoherentsuperpositionofalln-quditproductstateswithHammingweightone. ApartiallycoherentsuperpositionofageneralizedW-classstateand|0⟩ ⊗n isgivenas: ρ A 1 ···An =p W d n ⟩⟨ W d n +(1−p)|0⟩ ⊗n ⟨0| ⊗n +λ √ p(1−p)(| W d n ⟩ ⟨0| ⊗n +|0⟩ ⊗n ⟨ W d n ), (3.34) whereλ is the degree of coherence with 0 ≤ λ ≤ 1. For the case thatλ = 1, Eq. (3.34) becomes a coherent superposition of a generalized W-class state and |0⟩ ⊗n , and it is an incoherent superposition, or a mixture when λ = 0. In other words, Eq. (3.34) is an n-qudit state where the product state of Hamming weight zero is in a partially coherent superpositionwithallthestatesofHammingweightone. 33 The state in Eq. (3.34) can also be interpreted by means of decoherence. In other words,Eq.(3.34)canbeconsideredastheresultingstatefromacoherentsuperpositionof ageneralizedW-classstateand|0⟩ ⊗n : |ψ⟩ A 1 ,···An = √ p W d n ⟩ + √ 1−p|0⟩ ⊗n , (3.35) after some decoherence process so-called phase damping [16], which can be represented as: ρ A 1 ···An = Λ(|ψ⟩⟨ψ|) =E 0 |ψ⟩⟨ψ|E † 0 +E 1 |ψ⟩⟨ψ|E † 1 +E 2 |ψ⟩⟨ψ|E † 2 , (3.36) withKrausoperatorsE 0 = √ λI,E 1 = √ 1−λ(I−|0⟩⟨0|)andE 2 = √ 1−λ|0⟩⟨0|. Now, I will show that the monogamy relation of the state in Eq. (3.34) in terms of CRENissaturatedwithrespecttoanyarbitrarypartitionofthesetofsubsystems. Further- more, the entanglements, measured by CREN, of the state in Eq. (3.34) and its reduced densitymatrixontoanysubsystemwithrespecttoanybipartitecutarenotaffectedbythe degreeofcoherencyλ. First,IwillconsidertheCRENofρ A 1 ···An inEq.(3.34)withrespecttothebipartitecut betweenA 1 andtheothers. ThestateinEq.(3.34)hasapurestatedecompositionas: ρ A 1 ···An =( √ p W d n ⟩ +λ √ 1−p|0⟩ ⊗n )( √ p ⟨ W d n +λ √ 1−p⟨0| ⊗n ) +( √ (1−p)(1−λ 2 )⟨0| ⊗n )( √ (1−p)(1−λ 2 )|0⟩ ⊗n ). (3.37) 34 Now,let: | ˜ ψ 1 ⟩ = √ p W d n ⟩ +λ √ 1−p|0⟩ ⊗n , | ˜ ψ 2 ⟩ = √ (1−p)(1−λ 2 )|0⟩ ⊗n (3.38) be two unnnormalized states in ann-qudit system. Then, by the HJW theorem [65], any otherpurestatedecompositionofρ A 1 (A 2 ···An) = ∑ r i=1 | ˜ ϕ i ⟩⟨ ˜ ϕ i |ofsizer canberealizedby the choice of anr×r unitary matrix (u ij ) such that| ˜ ϕ i ⟩ = u i1 | ˜ ψ 1 ⟩ +u i2 | ˜ ψ 2 ⟩. In other words, with the normalization of| ˜ ϕ i ⟩ = √ p i |ϕ i ⟩, we can consider an arbitrary pure state decompositionofρ A 1 (A 2 ···An) = ∑ r i=1 p i |ϕ i ⟩⟨ϕ i |witharbitrarysizer. Byusingthemethodintroducedin[52],Icandirectlyevaluatetheaveragenegativityof the pure states|ϕ i ⟩ for an arbitrary pure state decomposition ofρ A 1 (A 2 ···An) . After tedious butstraightforwardcalculations,itcanbeshownthattheaveragenegativityisindependent fromthechoiceofaunitarymatrix (u ij ),whichis: ∑ i p i N(|ϕ i ⟩) = 2p √ A(1−A), (3.39) whereA = 1− ∑ d−1 j=1 |a 1j | 2 . Thus,bythedefinitionofCREN,Ihave: N c (ρ A 1 (A 2 ···An) ) =min ∑ i p i N(|ϕ i ⟩) = 2p √ A(1−A) =max ∑ i p i N(|ϕ i ⟩) =N a c (ρ A 1 (A 2 ···An) ), (3.40) 35 where the minimum is taken over all possible pure state decompositions ofρ A 1 (A 2 ···An) = ∑ i p i |ϕ i ⟩⟨ϕ i |. Furthermore, it can be seen from Eq. (3.39) that this average value is also invariant underthedegreeofcoherencyλ. Inotherwords,nomatterhowmuchamountofdecoher- enceinEq.(3.36)happenstothestateinEq.(3.35),itsentanglementispreserved. Now,forN cA 1 A i withi = 2,...,n,Iwillfirstconsiderthecasewheni = 2,whereasall the other cases are analogously following. By tracing over all subsystems exceptA 1 and A 2 fromρ A 1 ···An ,Iget: ρ A 1 A 2 =p d−1 ∑ i,j=1 (a 1i a ∗ 1j |i0⟩⟨j0|+a 1i a ∗ 2j |i0⟩⟨0j|+a 2i a ∗ 1j |0i⟩)⟨j0|+a 2i a ∗ 2j |0i⟩⟨0j| +(A 2 +1−p)|00⟩⟨00|+λ √ p(1−p) d−1 ∑ k=1 [(a 1k |k0⟩+a 2k |0k⟩)⟨00| +a ∗ 1k |00⟩(⟨k0|+a ∗ 2k ⟨0k|)], (3.41) withA 2 = 1− ∑ d−1 j=1 (|a 1j | 2 +|a 2j | 2 ). Nowconsidertwounnormalizedstates: | ˜ ψ 1 ⟩ = √ p d−1 ∑ i=1 (a 1i |i0⟩+a 2i |0i⟩)+λ √ 1−p|00⟩, | ˜ ψ 2 ⟩ = √ A 2 +(1−p)(1−λ 2 )|00⟩, (3.42) thenIhave ρ A 1 A 2 =| ˜ ψ 1 ⟩⟨ ˜ ψ 1 |+| ˜ ψ 2 ⟩⟨ ˜ ψ 2 |. (3.43) 36 Thus all possible pure states in an arbitrary pure state decomposition ofρ A 1 A 2 of size r can be realized as a linear combination of| ˜ ψ 1 ⟩ and| ˜ ψ 2 ⟩ by choosing anr×r unitary matrix. Again, by using a similar method to the case ofρ A 1 ···An , it can been shown that the average negativity ofρ A 1 A 2 is invariant under the choice of pure state decomposition, whichis: N cA 1 A 2 = 2p √ (1−A)(A−A 2 ) =N c a A 1 A 2 . (3.44) Furthermore, rather surprisingly, this average value is also invariant under the degree of coherency. In other words, no matter how much amount of decoherence in Eq. (3.36) happens,itdoesnotevenaffecttheentanglementbetweenthesubsystemsA 1 andA 2 . Similarly,Icanhave: N cA 1 A i = 2p √ (1−A)(A−A i ) =N c a A 1 A 2 , i = 3,...,n (3.45) withA i = 1− ∑ d−1 j=1 (|a 1j | 2 +|a ij | 2 ),andthus, n ∑ i=2 N c 2 A 1 A i = n ∑ i=2 (N c a A 1 A i ) 2 (3.46) In other words, I have obtained a saturation of the CREN monogamy relation for an n-quditstateinEq.(3.34),andthissaturationdoesnotdependonthechoiceofcoherency λ. 37 For any arbitrary partitionP ={P 1 ,··· ,P m } of the set of subsystems, it was shown that ann-qudit generalized W-class state can be also considered as anm-partite general- izedW-classstate[52],thatis: |W d n ⟩ A 1 ···An = d−1 ∑ i=1 (a 1i |i···0⟩+···+a ni |0···i⟩) = d−1 ∑ i=1 |˜ x 1i ⟩ P 1 ⊗···⊗| ⃗ 0⟩ Pm +···+| ⃗ 0⟩ P 1 ⊗···⊗|˜ x mi ⟩ Pm = d−1 ∑ i=1 √ q 1i |i⟩ P 1 ⊗···⊗|0⟩ Pm +···+ √ q mi |0⟩ P 1 ⊗···⊗|i⟩ Pm =|W d m ⟩ P 1 ···Pm , (3.47) where, |˜ x si ⟩ Ps =a (n 1 +···+n s−1 +1)i |i···0⟩ Ps + ···+a (n 1 +···+ns)i |0···i⟩ Ps (3.48) and √ q si |x si ⟩ Ps =|˜ x si ⟩ Ps , | ⃗ 0⟩ Ps =|0···0⟩ Ps (3.49) withrenaming|x si ⟩ Ps =|i⟩ Ps and| ⃗ 0⟩ Ps =|0⟩ Ps fors∈{1,...,m}. ThereforeEq.(3.34)canalsobeconsideredtobeapartiallycoherentsuperpositionof an m-partite generalized W-class state and the vacuum, |0⟩ P 1 ···Pm , and thus the result in Eq.(3.46)isalsotrueforanyarbitrarypartitionofthesetofsubsystems. Not only for the case of multi-qubit systems and the counterexamples in Sec. 3.2, CREN also shows a strong monogamy relation of entanglement for a class of n-qudit 38 statesinapartiallycoherentmixtureofageneralizedW-classstateandthevacuum. Thus, the CREN version of MoE is a strong conjecture for qudit systems with no obvious coun- terexamples. In the next few sections I will now explore the application of entanglement in under- standingphysicalsystems. 39 Chapter4 Non-Equilibriumquantumsystems Recentexperimentshaveshownthatitispossibletostudyunitarynonequilibriumdynam- ics in quantum systems on long time scales [66], [67], [68], [69]. One important result of these investigations has been the observed lack of thermalization. The reason for this non-thermal behavior is attributed to the near integrability of the system. This observa- tion has motivated recent studies of the non-equilibrium properties of integrable quantum modelHamiltonians[70],[71],[72],[73],[74]. Theinterestinthistopicisalsomotivated byitsrelevancetoavarietyofexperimentalsituations,includingcoldatoms[75],Penning traps[76]andJosephson-junctionarrays[77]. Therearemanywaysinwhichitispossibletodriveasystemoutofequilibrium. Quan- tum quenches, and coupling to baths with different temperatures (or potentials) are the mostcommon protocols. While in the quenchscenario the focus ison the unitary dynam- ics of the full system, in the second case one usually deals with an effectiv description of the dynamics of the subsystem only. Different out-of-equilibrium dynamics have never- theless similar characteristics, for example the presence of currents (of particles, energy, orheat). In this section I will discuss our system which is based on the work done in [3] on Ising model with the presence of energy current. The paper discusses on how to obtain a non-equilibrium quantum systems and uses traditional approach to study such a system. 40 An overviewof this will help in making the connection while I use entanglement to study thesamesystem. 4.1 IsingmodelwithDzyaloshinskii-Moriyainteraction I consider an Ising spin chain in transverse field H I , with an additional Dzyaloshinskii- Moriya(DM)interactionH DM . ThetotalHamiltonianH I +H DM isdefinedasfollows: H =− ∑ j [ 1 2 σ x j σ x j+1 + h 2 σ z j + ζ 8 ( σ x j σ y j+1 −σ y j σ x j+1 ) ] , (4.1) where h is the external magnetic field, and ζ is the coupling parameter determining the strength of the DM interaction. Such an anisotropic interaction is present in many low- dimensional materials with the necessary crystal symmetry, and it originates from spin- orbit coupling [78], [79], [80], [81], [82], [83], [84]. Furthermore, the DM interaction is ofrelevanceinquantuminformationtheory,sinceitplaysanimportantroleinthephysics ofquantumdots[85],andinfault-tolerantquantumcomputation[86]. Adding the DM term to H I does not affect the solvability of the model [87], and interestingly enough it provides the system with a richer phase diagram. These features have beeen used in [3] to study the effective out-of-equilibrium quantum dynamics of the model. H DM can be viewed as a current term. The reason for this is the following. The equationofmotionforthelocalenergydensityofH I ,definedby: ϵ j =− 1 4 σ x j (σ x j+1 +σ x j−1 )− h 2 σ z j , (4.2) 41 SoIhave: ˙ ϵ j = i ~ [H I ,ϵ j ]. (4.3) One can write the time derivative of the energy current as the divergence of the energy current: ˙ ϵ j =C j −C j+1 , (4.4) with, C j ∝σ y j ( σ x j−1 −σ x j+1 ) . (4.5) It thus follows thatH DM is precisely the sum over all sites of the local currents ∑ j C j . Therefore,theground-stateexpectationvalueofH DM : J ≡⟨ ∑ j ζ 8 ( σ x j σ y j+1 −σ y j σ x j+1 ) ⟩, (4.6) becomes an order parameter indicating the presence of an energy current. Once the total Hamiltonianhas beendiagonalized, which can bedone with theusual Jordan-Wignerand Bogoliubovtransformations: H = ∑ q Λ q b † q b q , (4.7) theeffectoftheDMinteractionisclearlyobservedatthesingle-particlelevel. Thesingle- particlespectrumisgivenby: Λ q = √ 1+h 2 +2hcosq +ζsinq, (4.8) withq∈ [−π,π)themomentumofthequasi-particle. Ascanbeseenintheaboveexpres- sion and in Fig. 4.1, the DM interaction makes the spectrum non-symmetric with respect 42 to q = 0. Note that the ground state of the Hamiltonian including the DM interaction is the same for all values of ζ in the interval [0,1]. In particular this means that, within this range of values, the Ising model in a transverse fieldH I and the system described by Eq. 4.1 have the same ground state. Beyond ζ = 1 (with h ≤ ζ) the Fermi sea starts to be populated by the modes in between the zeros of the single particle spectrum (i.e. betweenq + andq − in Fig. 4.1). This implies that the ground state is not anymore that of H I . Furthermore, since the DM term commutes with the rest of the Hamiltonian, at the many-bodylevelwhenζ = 1wemusthavealevelcrossingbetweenthegroundstateofH I and some previously excited Hamiltonian eigenstates. Fig. 4.2 shows the phase diagram of the model [3]. There are three regions: ferromagnetic, polarized paramagnetic, and the so-called current phase, characterized by J ̸= 0. The current phase is gapless, and the two-pointcorrelationfunctionsshowapower-lawbehaviorwithanoscillatoryamplitude, ⟨σ x l σ x l+n ⟩ gs ∼ Q(h,ζ) √ n cos(kn)whereQisanon-universalfunctionandk≡ arccos 1 ζ [3]. Iwillnowdiscussthedetailedmathematicalstepsforevaluatingentanglemententropy. 4.2 Entanglemententropyderivation In this section I will discuss a detailed description of the steps involved in first evalu- ating the entanglement entropy of the Hamiltonian in Eq. 4.1, and then calculating its time evolution. After the standard sequence of Jordan-Wigner and Bogoliubov trans- formations the Hamiltonian is in the diagonal form H = ∑ π k=−π Λ k b † k b k , with Λ k = 1 2 ( √ 1+h 2 +2hcosk +ζsink). 43 −3 −2 −1 0 1 2 3 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 q Λ q ζ=0 ζ=0.5 ζ=1 ζ=2 q − q + Fig. 4.1: Spectrum of the Hamiltonian in Eq. 4.1. I show the spectrum for 4 different valuesofζ,whilekeepingh = 0.5fixed. Seealsofiguresin[3]. The density matrix of the subsystem of size L, embedded in a system of size N, can be obtainedtracingouttherestofthesystem[88],[89],[90]: ρ L =Tr N−L (ρ) =A 0 e −H , (4.9) 44 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 ζ h ordered phase disordered phase current phase Fig. 4.2: Phase diagram in the h−ζ plane of the model in Eq. 4.1. The dotted line is a critical line where the model shows the same universal properties of the quantum Ising modelintransversefield. Seealsofiguresin[3]. whereA 0 isanormalizationconstantandH isthequadratichermitianoperator: H = L ∑ i,j=1 c † i V i,j c j + 1 2 (c † i W i,j c † j −c i W i,j c j ). (4.10) 45 H canbediagonalizedviaageneralizedBogoliubovtransformation. Thereduceddensity matrixhastheform: ρ L =A 0 exp[− L ∑ q=1 ε q d † q d q ] (4.11) UsingTr(ρ L ) = 1, I getA 0 = Π L q=1 1 1+exp(−εq) . This gives the final form of the density matrixas: ρ L = Π L q=1 exp(−ε q d † q d q ) 1+exp(−ε q ) . (4.12) Definingν q ≡ 1−exp(−εq) 1+exp(−εq) ,Icanwrite: ρ q ≡   1+νq 2 0 0 1−νq 2   , (4.13) andalso: ρ L = Π L q=1 1+ν q 2 exp[−ln( 1+ν q 1−ν q )d † q d q ] = Π L q=1 ( 1+ν q 2 −ν q d † q d q ) = L ⊗ q=1 ρ q . (4.14) 46 ItisimportanttonotethattheJordan-Wignertransformationmapsthe2 L ×2 L dimensional densitymatrixto2L×2Ldimension. Thiseasesourcomputationsignificantly. Usingthe factthat exp(d † q d q lnλ) = 1+(1−λ)d † q d q ,onehasfortheentanglemententropy: S L =−Tr(ρ L ln(ρ L )) = L ∑ q=1 [ln(1+exp(−ε q ))+ ε q 1+exp(ε q ) ] =− L ∑ q=1 ( 1+ν q 2 ln 1+ν q 2 + 1−ν q 2 ln 1−ν q 2 ). (4.15) Now I will calculate ν q , from which I can obtain the block entropy. ν q is given by the expectationvalueofd † q d q andd q d † q : ⟨d † q d q ⟩ =Tr(ρ L d † q d q ) = exp(−ε q ) 1+exp(−ε q ) = 1−ν q 2 ⟨d q d † q ⟩ =Tr(ρ L d q d † q ) = 1 1+exp(−ε q ) = 1+ν q 2 . (4.16) I define four 2L× 1 column vector: D ≡   d d †   , C ≡   c c †   , ¯ D ≡   d † d   and ¯ C ≡   c † c   , where d = (d 1 ,...,d L ) t , and similarly for c. The previous Bogoliubov transformationscanbeexpressedinacompactmatrixnotationas: D =   g h h g   C, (4.17) 47 and, ¯ D t = ¯ C t   g t h t h t g t   , (4.18) whereg andhareL×Lmatrices. IntermsofexpectationvaluesIhave: ⟨D ¯ D t ⟩ =   g h h g   ⟨C ¯ C t ⟩   g t h t h t g t   . (4.19) Now consider a quantum quench protocol. Initially the system is prepared in the ground state of an HamiltonianH ′ , and suddenly one of the parameters is changed, and thenewHamiltonianisdenotedbyH. Thequasi-particleoperatorvectorB ′ k ≡ (b ′ k , b ′ † −k ) t is associated with H ′ , and the vector B k ≡ (b k , b † −k ) t is associated with H. Simi- larly for C k , which is associated with bare vacuum fermions. Define also the matrix R µ (α) ≡ cos( α 2 )I + iσ µ sin( α 2 ), where σ µ are the Pauli matrices, and µ = x,y,z. It can be easily seen thatC k = R x (θ k )B k , andC k = R x (θ ′ k )B ′ k , withθ k a parameter of the Bogoliubovtransformation[91]. FromwhichwecanwriteB k =R x (θ ′ k −θ k )B ′ k . When a quench takes place, the time evolution of the fermion operators is given by B k (t) =e −iHt B k e iHt . IcanwriteB k (t) =S z (−2Λ k t)B k ,where: S z (−2Λ k t) =   e −i k t 0 0 e −i −k t   . Notice that the energy spectrum of the Hamiltonian in Eq.4.1 is not symmetric, which meansthatingeneralΛ −k ̸= Λ k . Inordertoevaluatethetwo-pointcorrelationfunctionsI 48 considerdifferentcases. Whentheinitialstateofthesystemisinthenon-current-carrying regionIhave: ⟨B ′ k B ′ k † ⟩ =   1 0 0 0   . (4.20) Iftheinitialstateofthesystemisinthecurrent-carryingphaseandk∈ (k 1 ,k 2 ),wherek 1 andk 2 arethezerosofthespectrumthen: ⟨B ′ k B ′ k † ⟩ =   0 0 0 0   . (4.21) Fork lyingbetween−k 1 and−k 2 Ihave: ⟨B ′ k B ′ k † ⟩ =   1 0 0 1   . (4.22) AcompactwayofexpressingEq4.20,Eq.4.21,andEq.4.22isgivenby: ⟨B ′ k B ′ k † ⟩ =   1 2 (1+ | k | k ) 0 0 1 2 (1− | −k | −k )   . (4.23) FinallyIcanwrite: ⟨C k (t)C † k (t)⟩ =R x (θ k )⟨B k (t)B † k (t)⟩R † x (θ k ) =R x (θ k )S z (−2Λ k t)⟨B k B † k ⟩S † z (−2Λ k t)R † x (θ k ) =R x (θ k )S z (−2Λ k t)R x (θ ′ k −θ k )⟨B ′ k B ′ k † ⟩ ×R † x (θ ′ k −θ k )S † z (−2Λ k t)R † x (θ k ). (4.24) 49 Notice that the above expression is the same if I consider S z (−2Λ k t), with Λ k the sin- gle particle spectrum of Eq. 4.1, or if I consider S z (−2Λ k t), with Λ k the single particle spectrum of the Ising Hamiltonian without the DM term. This can be seen with a direct calculation. Forexample,oneentryoftheabovecorrelationmatrixisgivenby: 2⟨c † −k (t)c −k (t)⟩ = E 1 +E 2 +(E 2 −E 1 )cosθ k cos(θ ′ k −θ k ) +(E 1 −E 2 )cos[t(Λ k +Λ −k )]sinθ k sin(θ ′ k −θ k ), (4.25) whereE 1 ≡ 1 2 (1+ | k | k ) andE 2 ≡ 1 2 (1− | −k | −k ). The argument ofS z appears only in the argumentofthetrigonometricfunctioninsuchawaythattheDMcontributionisirrelevant (seeEq.4.8). ThisprovesthatthetimeevolutionofthecorrelationmatrixinEq.4.24,with respect to the ground-state of Eq. 4.1, is the same as the time evolution of the correlation matrix with respect to the Ising Hamiltonian in transverse field, with respect to an excited stateoftheIsingHamiltonian. 50 Chapter5 PhaseCharacterizationusing Entanglement To characterize the system’s behavior I focus my attention on the Entanglement Entropy (EE),asmeasuredbytheVonNeumannentropy,whichisacentralquantityinthecharac- terizationofnonequilibriumquantumdynamics[29]. 5.1 StaticEntanglementEntropyandthePhaseDiagram In this section I show how entanglement can be used to characterize the different phases of the model. To measure the EE, I consider a bipartition of the spin chain into two subsystems A and B. For this setup a good measure of EE between the two partitions is given by the von Neumann entropyS A ≡ −Trρ L lnρ L , where ρ L is the ground-state reduceddensitymatrixofthesubsystemAwithLspins. It is known that for critical one-dimensional systems the EE scales logarithmicaly in the subsystem size, with a prefactor given by the central charge of the associated Confor- malFieldTheory(CFT): S L = c 3 lnL+S 0 , (5.1) 51 wherec is the central charge andS 0 is a non-universal constant [14], [21]. On the other hand, in the non-critical region of the phase diagram, the entanglement entropy saturates toavaluewhichdependsonthecorrelationlengthξ: S L ∝ c 3 lnξ. (5.2) Both Eq. 5.1 and Eq. 5.2 characterize the ground-state properties of EE for one- dimensionalsystems. Apartfromthegroundstateitisalsoofinteresttoinvestigateentanglementproperties of excited states. Recently, two works have appeared on this topic. In [92] it has been shown that there are excited states for which the logarithmic scaling of EE can have pref- actors different from the ground state, and that for some excited states the scaling can be extensive in the subsystem size, instead of logarithmic. In [93] the authors have studied the connection between EE for excited states and properties of the associated CFT not contained in the central charge. The EE of excited states is of interest also in the context of quantum quenches since in this setting the system is unitarily driven from the initial groundstatetoanexcitedstate. The Hamiltonian in Eq. 4.1 naturally fits in this set of problems. Following the dis- cussion from the previous section the model I am considering allows me to study the EE of some excited states ofH I , simply by tuning the coupling constant associated with the DM term. The excitations I can consider in this way are characterized by the modes in betweenthezerosofthesingleparticlespectrumthatgetpopulatedwhenζ > 1andh<ζ (Eq. 4.8). For these states the EE can be evaluated analogously to the ground state ofH I (seealso[94],[95],[96]forrelatedanalyticalstudy). 52 0 1 2 3 4 4.5 0.4 0.9 1.4 1.9 2.4 ln L S L ζ=0.5 ζ=1.5 ζ=4.5 Fig. 5.1: S L vs L at h = 1.0 for different ζ. The scaling behavior changes from S L ∼ 1 6 lnLonthecriticallineseparatingtheorderedanddisorderedphasetoS L ∼ 1 3 lnLinside thecurrent-carryingphase. Now I consider in detail the entanglement properties of the different phases shown in Fig. 4.2. First, I compare the scaling of the entanglement in the non-current-carrying critical regions and in the region where an energy current is present. Fig. 5.1 shows the result of the simulations for the scaling behavior of the ground-state EE withζ < 1 and h = 1. For all values of ζ ∈ [0,1] one observes the same scaling result. In the non- current-carrying region critical states are present only on the h = 1 line of the phase diagram. On this line, separating the ferromagnetic and polarized paramagnetic phases, the ground state of the system is the same as in H I . This implies that the EE scaling is logarithmic with a prefactor of c/3, and c = 1/2. Note that also the entire current- carrying phase (ζ > 1 andh < ζ) is gapless, and in this sense critical. At any point in this phase one can observe logarithmic scaling of the EE in the subsystem size. This is 53 consistent with the discussion in the previous section on the algebraic decay of the two point correlation function. Interestingly, the prefactor of the logarithmic scaling of EE in thecurrentphaseistwiceaslargeastheprefactorinthenon-currentphase. Thisdoubling reflects the increased number of zeros in the single-particle spectrum, consistent with the results of [94], [95], [96]. In fact, when ζ > 1 the ground state of Eq. 4.1 is the filled Fermi sea of modes inbetween q − and q + (see Fig. 4.1). Since the ground-state in the current phase is effectively an excited eigenstate ofH I , one could expect a scaling of EE that is extensive in the system size, as shown in [92] for excited states. The reason why this is not the case is due to the nature of the single-particle spectrum, which at most can have two zeros (see Fig. 4.1), and thus does not satisfy the requirements found in [92] for anextensivescalingofEE. TheDMinteractionintheHamiltonianaffectsalsothesub-leadingterminthescaling of EES L = 1 3 lnL+S 0 (h,ζ). Deriving the analytical form of the sub-leading order term S 0 (h,ζ)iscomplicated. Nonetheless,onecaninvestigatethistermnumerically. InFig.5.2 one can see thatS 0 is constant on the critical lineh = 1 withζ ≤ 1. As soon asζ > 1 andh < ζ,S 0 increases, but becomes almost constant for largeζ. AlsoS 0 is maximum at h = 0, and S 0 is minimum at the critical point h = ζ. From the behavior of S 0 one canconcludethatagivenblockhasthehighestentanglementwhenallthenegativemodes (q∈ [−π,0))intheFermiseaarefilled. ConsequentlyEEincreaseswithhighervaluesof theenergycurrent. I now consider the differences between the critical lines shown in the phase diagram (Fig. 4.2), separating different phases. The only second-order quantum phase transition is foundalongattheh = 1line(withζ ≤ 1),whichcorrespondstotheIsingquantumphase transition (see Fig. 5.3). On the hand the boundaries of the current-carrying phase with 54 both the paramagnetic and the ferromagnetic phases are characterized by a levelcrossing. This translates into a sudden jump in EE (see Fig. 5.3 and Fig. 5.4). The value of EE is always higher in the current carrying phase because of the presence of long-range corre- lations that decay algebraically. The plots in Fig. 5.3 and Fig. 5.4 show that controlling the DM term can be used as an entanglement switch. The amount of entanglement can be driven by the DM coupling term or the magnetic field, which are controllable parameters inopticallattices[97]. 55 0 1 2 3 4 5 0.4 0.6 0.8 1 ζ S 0 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 h S 0 h=0.2 h=0.5 h=1 h=2 ζ=4 ζ=5 ζ=6 Fig. 5.2: Non-universal nature ofS 0 . (upper panel)S 0 vs. ζ at differenth; (lower panel) S 0 vs.hatdifferentζ. 56 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 h S L ζ<1.0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 h S L ζ=2.0 ζ=3.0 ζ=4.0 Fig. 5.3: S L vs. h at L = 60 for different ζ. Upper panel: Static entanglement along the transition between the ordered ferromagnetic and the disordered paramagnetic phase. The peak signals the presence of long-range correlations at the critical point, which is a signature of a second-order quantum phase transition. Lower panel: Static entanglement along the transition between the disordered paramagnetic and the current-carrying phase. Thesuddenchangeinentanglementfollowedbytheabsenceofapeakatthecriticalpoint isasignatureofthefirst-orderquantumphasetransition. 57 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 ζ S L h=0.5 h=1 h=2 h=3 Fig. 5.4: S L vs. ζ at L = 60 for different h. h ≤ 1: Static entanglement along the transition between the ordered ferromagnetic and the current-carrying phase. This is a first order quantum phase transition which occurs atζ = 1. h > 1: Static entanglement along the transition between the disordered and the current-carrying phase. This is a first orderQPTwhichoccursatζ =h. 58 Chapter6 QuenchinQuantumSystems In this chapter I will mainly discuss the Ising system driven out of equilibrium with a quantum quench in the presence of an energy current. In the usual quench protocol the systemispreparedintheground-stateintheabsenceofanycurrent,andsubsequentlythe dynamicsdrivesittosomeexcitedstate. ThemodelHamiltonianIconsiderhereallowsfor the study of two different new situations: a quench from the ground-state in the presence ofanenergycurrent,andaquenchfromaninitialexcitedstateintheabsenceofanenergy current. The two scenarios are associated with different Hamiltonians which, in a sense, aredualtoeachother. 6.1 QuenchDynamicsfortheIsingModel Inthis section, I will focus onthe quench dynamics of the EE. Quenching providesa way to excite a system, initially prepared in the ground state, and to subsequently study the non-equilibrium dynamics of the model (in the following, I will denote with a subscript 0 the value of the parameters describing the initial Hamiltonian). Quench dynamics for the ising model have been studied in details in [29], [4]. Here I discuss some of the previous workdonewhichwillbenecessarytounderstandmyresults. Fig.6.1[4]showsevolution of entanglement for the ising model from various h 0 > 1 (non-critical point) to h = 1 (critical point) for a block of size 60. One can see there is a linear evolution till t ∼ L 2 59 0 10 20 30 40 50 60 t 0 3 6 9 12 15 18 21 24 S 60 h 0 =1.1 h 0 =1.5 h 0 =2 h 0 =5 h 0 =∞ Fig. 6.1: S 60 (t) for the quench from varioush 0 > 1 toh = 1. The dashed lines are the leadingasymptoticresultsforlarge[4]. followedbysaturation. Alsoonecanseethattheasymptoticvaluereacheddependsonthe quench size. Now I will discuss the phenomenological picture described in [4] to explain thisphenomena. 6.2 PhysicalInterpretationofEntanglementDynamics The explanation for this has been discussed in details in [4]. The initial state |ψ 0 ⟩ has higher energy relative to the ground state of the hamiltonianH which governs the subse- quent time evolution, and therefore acts as a source of quasiparticle excitations. Particles 60 Fig. 6.2: Propagation of quasi particles following the quench att < L 2 for a block of size L. Seealsofiguresin[4] emitted from different points (further apart than the correlation length in the initial state) are incoherent, but pairs of particles moving to the left or right from a given point are highlyentangled. IfthequasiparticledispersionrelationisE =E(k),theclassicalveloc- ity isv(k) = dE dk . There is an assumption that a maximum allowed speed which is taken to be 1, that is |v(k)| ≤ 1. A quasiparticle of momentum k produced at x is therefore at x +v(k)t at time t, ignoring scattering effects. Now consider these quasiparticles as they reach either A or B at time t. The field at some point x 1 ∈ A will be entangled with that at a pointx 2 ∈ B if a pair of entangled particles emitted from a pointx arrive simultaneously at x 1 and x 2 (Fig. 6.2). At t < L 2 , there is a net flux of particles inside the block causing the block and rest of the system to become more entangled with time. Theamountofquasiparticles(emittedfromthesamepoint)sharedincreaseslinearlywith time,henceentanglementincreaseslinearlywithtime(Fig.6.2). Attimet = L 2 marksthe end of linear evolution (Fig. 6.3). For timet > L 2 , the amount of quasiparticles (emitted fromthesamepoint)sharedbetweentheblockandthesystemremainsconstantwithtime eventhoughthereispropagationofparticlesthroughoutthesystem(Fig.6.4). 61 Fig. 6.3: Propagation of quasi particles following the quench att = L 2 for a block of size L. Seealsofiguresin[4] 6.3 Entanglement Dynamics following a Quench with DMterm I now focus on the case when the DM term is present with the Ising hamiltonian. As stated previously, the model in Eq. 4.1 is of interest because it combines two different mechanisms typically used to drive a system out of equilibrium: quantum quenching, and the coupling to a field originating a current in the system. Furthermore the inclusion of the DM term allows us to study a model Hamiltonian where the energy current can be controlledandusedinthequenchprotocol. 62 Fig. 6.4: Propagation of quasi particles following the quench att > L 2 for a block of size L. Seealsofiguresin[4] In my setup, the quench can either involve the magnetic fieldh, the DM couplingζ or a combination of the two. Since the DM term commutes with the Hamiltonian, a quench in ζ leaves the system in one of its eigenstates, providing a trivial evolution of the EE. On the other hand, quenches in the magnetic field give more interesting behaviors. If the quench is done with the initial state prepared in a region with no current the results are similar to those found in [29], [4], where quenches for the H I were considered. This is due to the fact that, in the absence of a current, the ground-state wave function initially is identical to that ofH I , and the time evolution is not affected by the presence of the DM term (see the derivation of Eq. 4.24 in chapter 4 for a proof of this). More interestingly, if thequenchinvolvesaninitialstateinsidethecurrentphase,newnon-trivialbehaviorscan 63 0 50 100 150 200 2 3 4 5 6 7 8 9 10 11 12 time S L h=0.5 h=0.7 h=1 h=2 Fig.6.5: Quenchesfromthecurrentcarryingphase.S L (t)vs. thetimesteps,withL = 60, h 0 = 4.0, ζ 0 = ζ = 5.0 for differenth. Note that the extent of the initial linear regime dependsontheparticularevolvingHamiltonian. beexpected,sincethegroundstatenowisradicallydifferent. Itisimportanttonoticethat the DM coupling enters only in the specification of the initial state, whereas the evolution can be effectively described by the Hamiltonian without the DM term. The calculations showingthatthisisinfactthecasecanbefoundinchapter4. I first compare the evolution of the EE for different quenches inside the current- carrying phase. Fixing the coupling constant of the DM term, and quenching only the 64 external magnetic field we obtain the results shown in Fig. 6.5. One always has an initial ballistic evolution of the EE, which grows linearly in time and saturates at some point. Quite interestengly, the saturation time (hence also the rate at which entanglement is ini- tiallybuildingup)dependsontheparticularevolvingHamiltonian. Thiswayonecancon- trol the time needed to generate the maximal asymptotic amount of entanglement. This propertyisrelevantalsofromacomputationalpointofview. Infact,DMRG-likeschemes, used for the simulation of the time evolution of quantum systems, can take advantage of the lower rate at which entanglement is generated. Knowing the regions in the phase dia- gram where such rates are lower can provide more efficient time simulations. As far as I know this is a new feature that is not present in other quench protocols considered so far in the literature. The other aspect that is important to notice in Fig. 6.5 is the special role played by the lineh = 1 in the phase diagram, which turns out to provide the maxi- mumasymptoticEEfordifferentquenchparameters. Thiscanbeunderstoodbymapping the quench forH I +H DM to a quench protocol forH I only. As stated in chapter 4, the entanglementevolutionwithrespecttoH(h,ζ)isidenticaltotheevolutionwithrespectto H(h,0). Furthermore,thegroundstateofH(h 0 ,ζ 0 ),theinitialHamiltonianinthequench protocolisalsoanexcitedeigenstateofH(h 0 ,0),becauseofthecommutativityoftheDM term with the total Hamiltonian. From this dual perspective the effect of the current is that of effectively quenching an excited eigenstate without the current term. For the Ising model, a quench from h 0 ̸= 1 yields the maximum value of S L (∞) when quenched to h = 1, because the energy gap closes ath = 1, and hence a large number of zero energy excitations can be produced. While the asymptotic value of EE depends on the particular excitedstateatthebeginningofthequench. 65 0 20 40 60 80 100 1 2 3 4 5 6 7 8 9 10 time S L ζ=4 ζ=5 ζ=10 ζ=100 Fig. 6.6: Quenches from the current carrying-phase with different values of the current driving field ζ. S L (t) vs. the time steps with L = 60, h 0 = 3.0 to h = 1 for different ζ 0 =ζ. Fig. 6.6 shows results of simulations for quenches with increasing values of the DM fieldinthecurrent-carryingphase. TheasymptoticvalueoftheEEdecreaseswithincreas- ingζ. Thisisconsistentwiththephenomenologicalpictureprovidedin[29],[4],andwith the fact that if the system starts in an excited state, the available number of unoccupied modes that can be occupied after the quench is smaller than in the case of having the ground state as an initial state. Furthermore, Fig. 6.6 shows that the time at which the EE 66 0 20 40 60 80 100 2 4 6 8 10 12 14 time S L 0 100 200 0 10 20 30 L asymptotic value L=30 L=60 L=90 L=120 Fig.6.7:S L (t)vs. thetimestepsinsidethecurrent-carryingphasefordifferentblocksizes L. Quenchingisdonefromh 0 = 2.0toh = 1.0withζ 0 =ζ = 3.0. saturatesdoesnotdependonζ,andconsequentlydoesnotdependontheparticularinitial Hamiltonian eigenstates (as long as it is not an eigenstate of the evolving Hamiltonian). The line withζ = 100 in Fig. 6.6 shows that very deep into the current phase quenching doesnotcreateentanglement. Infact,whenζ ≫h 0 quenchingthemagneticfieldisjusta smallperturbationtotheHamiltonian,whichthenapproximatelystaysinthegroundstate. 67 0 50 100 150 200 2 4 6 8 10 12 14 time S 60 h 0 =1.3 h 0 =1.5 h 0 =2.5 h 0 =3.5 0 1 2 3 0 5 10 15 20 h 0 −h asymptotic S 60 Fig. 6.8: S 60 (t) vs. the time steps for quenches to h = 1 from various h 0 inside the current-carryingphaseatζ =ζ 0 = 4. Finally, I verify that the presence of an energy current does not affect the extensivity of the asymptotic value of EE (Fig. 6.7), and its proportionality with the quench size (Fig.6.8). 68 Chapter7 Conclusion SinceIhaveinvestigatedtwodifferentaspectsofentanglement,myconclusionaredivided intotwosections. 7.1 CRENandMOE The study of higher-dimensional quantum systems is, undoubtedly, important and even necessary to quantum-information science for various kind of reasons. First, qudits for d > 2 are preferred in some physical systems such as in quantum key distribution where the use of qutrits increases coding density and provide stronger security compared to qubits[98]. Infault-tolerantquantumcomputationaswellasonquantumerror-correcting codes (QECCs), many studies are concentrated on the case of binary QECCs in a two- dimensional Hilbert space, whereas generalizations of proofs are often nontrivial when d> 2. However,asbothqubitandquditsystemsoccurinthenaturalworld,thereisnoreason to assume that a theoretical result should hold solely for two-dimensional systems. If an importantresult(e.g.,monogamyofentanglement)isshowntobetrueforthecased = 2, then this would suggest that a lot of effort should be directed towards qudit systems, as the case ford > 2 could be fundamentally different from the cased = 2. For example, a recent result [99] shows that for subsystem stabilizer codes in d-dimensional Hilbert 69 space,auniversalsetoftransversalgatescannotexistforevenoneencodedqudit,forany dimensiond,whichisknownasno-gotheoremfortheuniversalsetoftransversalgatesin QECC.Theextensionofthemultipartiteentanglementanalysis,especiallythemonogamy relation from qubit-to-qudit case is far more than trivial. The entanglement properties in higher-dimensionalsystemsarehardlyknownsofar,andthusanyfundamentalstepofthe challenges to the richness of entanglement studies for system of higher-dimensions and multipartitesystemswouldbefruitfulandevennecessarytounderstandthewholepicture ofquantumentanglement. In this thesis, I have proposed CREN as a powerful alternative for MOE in higher- dimensional quantum systems. I have shown that any monogamy inequality of entangle- ment for multiqubit systems can be rephrased in terms of CREN. Furthermore, we have pointed out the possibility of CREN MOE in higher-dimensional quantum systems by showingthatallthecounterexamplesfortheCKWinequalitysofarinhigher-dimensional quantum systems still have a monogamy inequality in terms of CREN, as well as no triv- ial counterexamples for CREN MOE so far. This task is one of the key challenges in finding a bipartite entanglement measure that meets the three criteria for qubits and for higher-dimensionalsystems. For the studies of CREN MOE in higher-dimensional quantum systems, I have pro- posedaclassofquantumstatesinn-quditsystemsthatareinapartiallycoherentsuperpo- sitions of a generalized W-class state and the vacuum. The CREN monogamy relation for theproposedclasshasbeenshowntobetrueanditalsoholdswithrespecttoanyarbitrary partitionofthesubsystems. Thus CREN is a good candidate for the general monogamy relation of multipartite entanglement, and it shows a strong evidence of its possibility even for the case of mixed 70 states in higher-dimensional systems. I believe that the analysis of CREN MOE derived here will give a full and rich reference for the study of MOE in higher-dimensional quan- tumsystems,whichisoneofthemostimportantandnecessarytasksinthestudyofquan- tumentanglement. 7.2 Entanglemententropyandmany-bodysystem I have studied the static and dynamic properties of the entanglement entropy in the Ising spin chain with a transverse field and a Dzyaloshinskii-Moriya interaction. The model is characterizedbythepresenceofanenergycurrentforcertainregionsofthephasediagram. ConcerningthestaticpropertiesIhaveanalyzedthetransitionsbetweenphaseswithno energycurrentandthephasewhereanenergycurrentispresent. Thetransitioniscaptured by a discontinuity of EE as a function of the parameters, and by a distinguishable scaling behavior in the current-carrying and non-current-carrying regions. In particular, the lead- inglogarithmictermoftheEEscalingwithrespecttothesystemsizehasaprefactorinthe current-carryingregionwhichistwiceaslargecomparedtothesecondorderIsingcritical line. Concerningthebehavioroftheentanglementevolutionfollowingaquench,themodel inEq.4.1allowsonetostudynewquenchprotocols. Theusualschemesconsiderquenches from an initial ground state. This scenario, for the model in Eq. 4.1, effectively corre- spondstoaquenchfromaninitiallyexcitedstateoftheIsingspinchainintransversefield (without DM interaction). The main result of this analysis shows that the ballistic pic- ture presented in [29], [4] is still valid, although with a significantly different aspect. In particular the entanglement saturation time in the current-carrying phase depends on the 71 details of the evolving Hamiltonian. 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Abstract (if available)

Abstract In this thesis I study multipartite entanglement sharing, and use entanglement to examine quantum phase transitions (QPT) and equilibrium in closed quantum systems. ❧ In the first part, I discuss on replacing concurrence by convex-roof extended negativity (CREN) for studying monogamy of entanglement (MOE). I show that all proven MOE relations using concurrence can be rephrased in terms of CREN. Furthermore, I show that higher-dimensional (qudit) extensions of MOE in terms of CREN are not disproven by any of the counterexamples used to disprove qudit extensions of MOE in terms of concurrence. I further test the CREN version of MOE for qudits by considering fully or partially coherent mixtures of a qudit W-class state with the vacuum and show that the CREN version of MOE for qudits is satisfied in this case as well. Hence I prove that the CREN version of MOE for qudits is a strong conjecture with no obvious counterexamples. ❧ In the second part, I show how entanglement entropy can be used to characterize phases and study equilibrium dynamics in closed quantum systems. As an example I study the Ising spin chain with a Dzyaloshinskii-Moriya interaction, focusing on the static and dynamic properties of the entanglement entropy. I show that the effects of the additional anisotropic interaction on the phase diagram and on the dynamics of the system are captured by the properties of the entanglement entropy. In particular, the model provides a way to study the quench dynamics in a system with an energy current. I consider quenches starting from an initial excited state of the Ising spin chain, and I analyze the effects of different initial conditions.

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Creator Das, Anirban (author)

Core Title Multipartite entanglement sharing and characterization of many-body systems using entanglement

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Physics

Publication Date 10/07/2013

Defense Date 05/08/2013

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

Tag convex-roof extended negativity (CREN),entanglement,monogamy of entanglement (MOE),multipartite entanglement,OAI-PMH Harvest,quantum phase transition,quenches

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Haas, Stephan W. (committee chair)

Creator Email anirban.das1981@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-335012

Unique identifier UC11296413

Identifier etd-DasAnirban-2082.pdf (filename),usctheses-c3-335012 (legacy record id)

Legacy Identifier etd-DasAnirban-2082.pdf

Dmrecord 335012

Document Type Dissertation

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Rights Das, Anirban

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

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

Repository Name University of Southern California Digital Library

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