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EntanglementSpectrum,Topological EntanglementEntropyandaQuantum Hammersley-CliffordTheorem FernandoG.S.L.Brandão MicrosoftResearch basedonjointworkwith Kohtaro Kato UniversityofTokyo MIT2016 QuantumInformationTheory Goal:Laydownthetheoryforfuturequantum-basedtechnology (quantumcomputers,quantumcryptography,…) Quant.Comm. Entanglementtheory Q.errorcorrec.+FT Quantumcomp. Quantumcomplex.theo. QuantumInformationTheory Goal:Laydownthetheoryforfuturequantum-basedtechnology (quantumcomputers,quantumcryptography,…) Ultimatelimitsto information transmission Quant.Comm. Entanglementtheory Q.errorcorrec.+FT Quantumcomp. Quantumcomplex.theo. QuantumInformationTheory Goal:Laydownthetheoryforfuturequantum-basedtechnology (quantumcomputers,quantumcryptography,…) Ultimatelimitsto information transmission Entanglementasa resource Quant.Comm. Entanglementtheory Q.errorcorrec.+FT Quantumcomp. Quantumcomplex.theo. QuantumInformationTheory Goal:Laydownthetheoryforfuturequantum-basedtechnology (quantumcomputers,quantumcryptography,…) Ultimatelimitsto information transmission Entanglementasa resource Quantumcomputers aredigital Quant.Comm. Entanglementtheory Q.errorcorrec.+FT Quantumcomp. Quantumcomplex.theo. QuantumInformationTheory Goal:Laydownthetheoryforfuturequantum-basedtechnology (quantumcomputers,quantumcryptography,…) Ultimatelimitsto information transmission Entanglementasa resource Quantumcomputers aredigital Quant.Comm. Entanglementtheory Q.errorcorrec.+FT Quantumcomp. Quantumcomplex.theo. Quantumalgorithms withexponential speed-up QuantumInformationTheory Goal:Laydownthetheoryforfuturequantum-basedtechnology (quantumcomputers,quantumcryptography,…) Ultimatelimitsto information transmission Entanglementasa resource Quantumcomputers aredigital Quant.Comm. Entanglementtheory Q.errorcorrec.+FT Quantumcomp. Quantumcomplex.theo. Quantumalgorithms withexponential speed-up Ultimatelimitsfor efficientcomputation QITConnections QIT Quant.Comm. Entanglementtheory Q.errorcorrec.+FT Quantumcomp. Quantumcomplex.theo. QITConnections CondensedMatter Stronglycorr.systems Topologicalorder Spinglasses QIT Quant.Comm. Entanglementtheory Q.errorcorrec.+FT Quantumcomp. Quantumcomplex.theo. QITConnections CondensedMatter Stronglycorr.systems Topologicalorder Spinglasses StatMech QIT Quant.Comm. Entanglementtheory Q.errorcorrec.+FT Quantumcomp. Quantumcomplex.theo. Thermalization Thermo@nano scale Quantum-to-Classical Transition QITConnections CondensedMatter HEP/GR Stronglycorr.systems Topologicalorder Spinglasses Topolog.q.fieldtheo. Blackholephysics Holography StatMech QIT Quant.Comm. Entanglementtheory Q.errorcorrec.+FT Quantumcomp. Quantumcomplex.theo. Thermalization Thermo@nano scale Quantum-to-Classical Transition QITConnections CondensedMatter HEP/GR Stronglycorr.systems Topologicalorder Spinglasses Topolog.q.fieldtheo. Blackholephysics Holography StatMech QIT Quant.Comm. Entanglementtheory Q.errorcorrec.+FT Quantumcomp. Quantumcomplex.theo. Thermalization Thermo@nano scale Quantum-to-Classical Transition Exper.Phys. Iontraps,linearoptics, opticallattices,cQED, superconduc.devices, manymore ThisTalk Goal:giveoneexampleoftheseemergingconnections: Studyscalingofentanglement insomephysicalstates QIT RelativeEntropy StrongSubadditivity Fawzi-RennerBound CondensedMatter Groundstates Arealaw EntanglementSpectrum ThermalStates AreaLaw Arealawassumption:ForeveryregionX, X Xc correlationlength Topological entanglement entropy (Kitaev,Preskill ‘05,Levin,Wen‘05) Expectedtoholdinmodelswithacorrelationlength. Butnotalwaystrue. Inthistalkweconsiderthisformofarealawasanassumption andanalyse whatareitsconsequences. QuantumInformation1.01: Fidelity …it’sameasureofdistinguishabilitybetweentwo quantum states. Giventwoquantumstatestheirfidelityisgivenby Ittellshowdistinguishabletheyarebyanyquantum measurement QuantumInformation1.01: RelativeEntropy …it’sanothermeasureofdistinguishability betweentwoquantum states. Def: Givesoptimalexponentfordistinguishingthetwostates (inasymmetrichypothesistesting;Stein’sLemma) Pinsker’s inequality: ConditionalMutualInformation Given, Strongsub-additivity: ConditionalMutualInformation Given, Strongsub-additivity: Fawzi-Renner‘14: ConditionalMutualInformation Given, Strongsub-additivity: (Fawzi-Renner‘14)If,thereisachannel s.t. CanreconstructthestateABCfromreductiononABby actingonBonly ConsequenceofAreaLaw: StateReconstruction Arealawassumption:ForeveryregionX, A B C l Topological entanglement entropy A B C correlationlength ForeveryABCwithtrivialtopology: (Kitaev ‘12)impliesthestatecanbecreatedbyshort-depthcircuit (Kim‘14)Impliesthestatecanbeconstructedfromlocalparts TopologicalEntanglementEntropy (Kitaev,Preskill ‘05,Levin,Wen‘05) Arealawassumption:ForeveryregionX, A B B l C ConditionalMutualInformation: Assumingarealawholds: Topological entanglement entropy correlationlength EntanglementSpectrum X Xc :eigenvaluesofreduceddensity matrixonX AlsoknownasSchmidteigenvalues ofthestate EntanglementSpectrum X Xc :eigenvaluesofreduceddensity matrixonX AlsoknownasSchmidteigenvalues ofthestate (Haldane, Li’08,….) ForFQHE,entanglementspectrum matchesthelowenergiesofaCFTacting ontheboundary of X EntanglementSpectrum X :eigenvaluesofreduceddensity matrixonX Xc AlsoknownasSchmidteigenvalues ofthestate (Haldane, Li’08,….) ForFQHE,entanglementspectrum matchesthelowenergiesofaCFTacting ontheboundary of X (Cirac,Poiblanc, Schuch,Verstraete ’11,….) NumericalstudieswithPEPS. Fortopologically trivialsystems(AKLT,Heisenberg models): entanglement spectrummatchestheenergiesofalocalHamiltonian onboundary Fortopological systems(Toric code):needsnon-local Hamiltonian EntanglementSpectrum X :eigenvaluesofreduceddensity matrixonX Xc AlsoknownasSchmidteigenvalues ofthestate (Haldane, Li’08,….) ForFQHE,entanglementspectrum matchesthelowenergiesofaCFTacting ontheboundary of X (Cirac,Poiblanc, Schuch,Verstraete ’11,….) NumericalstudieswithPEPS. Fortopologically trivialsystems(AKLT,Heisenberg models): entanglement spectrummatchestheenergiesofalocalHamiltonian onboundary Fortopological systems(Toric code):needsnon-local Hamiltonian Howgeneralarethesefindings?Canwemakethemmoreprecise? Result1:BoundaryState thm 1 Supposesatisfiesthearealawassumption.Then A B B C Result1:BoundaryState thm 1 Supposesatisfiesthearealawassumption.Then Suppose.Thenthereisalocals.t. B2 B3 … Bk-2 Bk-1 B1 B2k … Bk+2 Bk Bk+1 Result1:BoundaryState thm 1 Supposesatisfiesthearealawassumption.Then Suppose.Thenthereisalocals.t. Local”boundaryHamiltonian” Non-local”boundaryHamiltonian” Result1:BoundaryState thm 1 Supposesatisfiesthearealawassumption.Then Suppose.Thenthereisalocals.t. Obs:Correlationlengthofthestatedeterminestemperature ofthethermalstate() Result2:EntanglementSpectrum thm 2 Supposesatisfiesthearealawassumption with.Then … X B1 B2 B3 Bl-1 Bl X’ Result2:EntanglementSpectrum thm 2 Supposesatisfiesthearealawassumption with.Then If,thenforeveryk thereisno localHamilatonian H s.t. Fromthm 1tothm 2 X B X’ Fromthm 1tothm 2 X B X’ sinceisapurestate Fromthm 1tothm 2 X B X’ Fromthm 1tothm 2 X B If, X’ Howtoprovethm 1? We’llstartwiththesecondpart.Recap: Suppose.Thenthereisalocals.t. B2 B3 … Bk-2 Bk-1 B1 B2k … Bk+2 Bk Bk+1 Byarealaw: Theideaistoshowthisimpliesthestateisapproximatelythermal MarkovNetworks x7 x3 x9 x1 x6 x4 x2 x8 x5 x10 Wesayr.v.x1,…,xn onagraphG formaMarkovNetworkif xi isindendentofallotherx’sconditionedonitsneighbors I.e.LetNi besetofneighborsofvertexi.Thenforeveryi, Hammersley-CliffordTheorem x7 x3 x9 x1 x6 x4 x2 x8 x5 x10 (Hammersley-Clifford ‘71) LetG=(V,E)beagraphandP(V)beapositive probabilitydistributionoverr.v.locatedattheverticesofG.Thepair (P(V),G)isaMarkovNetworkif,andonlyif,theprobabilityPcanbe expressedasP(V)=eH(V)/Zwhere isasumofrealfunctionshQ(Q)ofther.v.incliquesQ. QuantumHammersley-Clifford Theorem q7 q3 q9 q1 q6 q4 q2 q8 q5 q10 (Leifer, Poulin ‘08, Brown,Poulin ‘12) Ananalogousresultholdsreplacing classicalHamiltoniansbycommuting quantumHamiltonians (obs:quantumversionmorefragile;onlyworksforgraphswithno3cliques) Canwegetasimilarcharacterizationforgeneralquantumthermalstates? ApproximateQuantum Hammersley-CliffordTheorem? A l B C Def:Wesayaquantumstateisa(l,eps)approximateMarkovnetworkifforevery regionsABCs.t. BshieldsAfromC andBhaswidthl, Conjecture:ApproximateMarkovNetworksareequivalenttoGibbs statesofgeneralquantumlocalHamiltonians (atleastonregularlattices) ApproximateQuantumHammersleyCliffordTheoremfor1DSystems A B C thm 1. LetH bealocalHamiltonianonn qubits.Then Gibbsstate@temperatureT: ApproximateQuantumHammersleyCliffordTheoremfor1DSystems A B C thm 1. LetH bealocalHamiltonianonn qubits.Then 2. Letbeastateonn qubitss.t. foreverysplit ABCwith|B|>m,.Then ProofPart2 X1 X2 X3 m Letbethemaximumentropystates.t. Fact1(Jaynes’Principle): Fact2 Let’sshowit’ssmall ProofPart2 X1 m SSA X2 X3 ProofPart2 X1 m X2 X3 ProofPart2 X1 m X2 X3 ProofPart2 X1 m Since X2 X3 ProofPart2 X1 m Since X2 X3 ProofPart1 Recap: LetH bealocalHamiltonianonn qubits.Then WeshowthereisarecoverychannelfromBtoBC reconstructingthestateonABCfromitsreductiononAB. StructureofRecoveryMap Thereexistsanoperator𝑋" suchthat $ $ 𝜌$%&' ≈ id+ ⊗ 𝜅"→/0 𝜌+"%&' = 𝑋" tr"4 𝑋"56 𝜌+"%&' 𝑋"56 7 ⊗ 𝜌$&4 ' 𝑋"7 𝐴 𝐵; 𝐵< 𝐴 𝐵; 𝐵< 𝐴 𝐵; 𝐵< 𝐶 𝐴 𝐵; 𝐵< 𝐶 StructureofRecoveryMap Thereexistsanoperator𝑋" suchthat $ $ 𝜌$%&' ≈ id+ ⊗ 𝜅"→/0 𝜌+"%&' = 𝑋" tr"4 𝑋"56 𝜌+"%&' 𝑋"56 7 ⊗ 𝜌$&4 ' 𝑋"7 𝐴 𝐵; 𝐵< 𝐴 𝐵< 𝐵; Difficulty:𝜅"→"> isatrace-increasingmap 𝐴 𝐵; 𝐵< 𝐶 𝐴 𝐵; 𝐵< 𝐶 Repeat-until-successMethod V "→"> . Wenormalize 𝜅"→"> anddefineaCPTD-mapΛ → Succeedtorecoverwithaconstantprobability𝑝. 𝐴 ApplyΛP "Q→"Q > Success 𝐵? Fail 𝐵@?56 𝐵?56 @6 𝐵6 𝐶 & Traceout𝐵 applyΛP "R→"R"@Q "Q> 𝐵@A Fail 𝐵A ⋯ 𝐵6 𝑙 2𝑙 Fail 𝐶 Traceout @ 𝐵?56 .. 𝐶 &apply ΛP "S→"S…> Fail Success Success Obtainastate≈ 𝜌 $ %&' 𝐵@6 ⋯ q Choose𝑁 ∼ 𝑙 𝐵 = 𝒪 𝑙 A . →Totalerror=Failprobability 1 − 𝑝 J +approx.error𝒪 𝑒 5𝒪(J) = 𝒪 𝑒 5𝒪(J) . LocalityofPerturbations Thekeypointintheproof: Forashort-rangedHamiltonian𝐻,thelocalperturbationto𝐻 onlyperturb theGibbsstatelocally. 𝐼 𝑉 𝑙 AusefullemmabyAraki(Araki,‘69) For1DHamiltonian withshort-rangeinteraction𝐻, 𝑒 $YZ 𝑒 5$ − 𝑒 $[ YZ 𝑒 5$[ ≤ 𝒪 𝑒 5𝒪(J) 𝑒 5_$ → 𝑒 5_($YZ) ≈ 𝑋` 𝑒 5_$𝑋`7 𝑋` = _ _ 5 A ($[ YZ) A $[ 𝑒 𝑒 Local Proofthm 1part2 We’llstartwiththesecondpart.Recap: Suppose.Thenthereisalocals.t. Apply1DapproximatequantumHammersley-Cliffordthm toget Withl=n/m.Choosem=O(log(n))tomakeerrorsmall Proofthm 1part1 thm 1 Supposesatisfiesthearealawassumption.Then A B B C Proofthm 1part1 Wefollowthestrategyof(Katoetal‘15)forthezero-correlationlengthcase AreaLawimplies A B1 B2 B1 B2 C ByFawzi-RennerBound,therearechannelss.t. Proofthm 1part1 Define: Wehave ItfollowsthatCcanbereconstructedfromB.Therefore Proofthm 1part1 Define: Wehave ItfollowsthatCcanbereconstructedfromB.Therefore Since with So Proofthm 1part1 Since LetR2 bethesetofGibbsstatesofHamiltoniansH=HAB +HBC.Then OpenProblems • Whathappensindimbiggerthan2? • CanweprovetheapproximateMarkovpropertyforgeneral quantumstates? • Canweprovetheconverse,i.e.thatapproximatequantum MarkovNetworksareapproximatelythermal? • Aretwocopiesoftheentanglementspectrumnecessaryto getalocalboundarymodel? Thanks!