Download Entanglement Spectrum MIT 2016

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!