Quantum Entanglement in Holography...In holography, von Neumann entropy is given by the area of a...
Transcript of Quantum Entanglement in Holography...In holography, von Neumann entropy is given by the area of a...
QuantumEntanglementin
HolographyXiDong
ReviewtalkStrings2018,OIST,Okinawa,Japan
June27,2018
Quantumentanglement
• Correlationbetweenpartsofthefullsystem
• Simplestexample:|Ψ⟩ = |↑⟩|↓⟩ − |↓⟩|↑⟩
Quantumentanglement
Generalcase:
Entanglementiscomplicated
• Entanglementisnotjustentanglemententropy.
• “Structure”of(many-body)quantumstate.
• Entanglemententropyisoneofmanyprobes.
• Otherprobes:correlators,Renyi entropy,modularHamiltonian/flow,complexity,quantumerrorcorrection,…
Entanglementiscomplicated
• Structureofentanglementisincreasinglyimportantinthestudyofmany-bodyphysics.• Stringtheoryandquantumfieldtheoryare(special)many-bodysystems.• Strategy:studypatternsofentanglementtoaddressinterestingquestionssuchas:
ØHowdoesquantumgravitywork?ØHowdowedescribetheblackholeinterior?ØHowdowedescribecosmology?
Plan
• EntanglementinQFT
• Entanglementinholography
• Applicationsandgeneralizations
• Holographicerrorcorrection
• VonNeumannentropy:𝑆 ≝ −Tr ρ, ln ρ,
• Renyi entropy:𝑆/ ≝0
01/lnTrρ,/
• InterestinginQFTandholography
EntanglemententropyinQFT
EntanglemententropyinQFT
• Area-lawdivergence:𝑆 = # Area6789
+⋯+ 𝑆</=> + ⋯• Even𝑑:𝑆</=> = ∑ 𝑐=𝐼== ln𝜖ØE.g.CFT4:𝑆</=> = ∫ 𝑎𝑅 + 𝑎𝐾H + 𝑐𝑊 ln𝜖• Odd𝑑:𝑆</=> isfiniteandnonlocal.ØE.g.sphericaldisk:𝑆</=> ∼ 𝐹• CangeneralizetoRenyi entropy• Cangeneralizetocaseswithcorners[Allais,Balakrishnan,Banerjee,Bhattacharya,Bianchi,Bueno,Carmi,Chapman,Czech,XD,Dowker,Dutta,Elvang,Faulkner,Fonda,Galante,Hadjiantonis,Hubeny,Klebanov,Lamprou,
Lee,Leigh,Lewkowycz,McCandish,McGough,Meineri,Mezei,Miao,Myers,Nishioka,Nozaki,Numasawa,Parrikar,Perlmutter,Prudenziai,Pufu,Rangamani,Rosenhaus,Safdi,
Seminara,Smolkin,Solodukhin, Sully,Takayanagi,Tonni,Witczak-Krempa,…]
Strongsubadditivity(SSA)
𝑆,L + 𝑆LM ≥ 𝑆,LM + 𝑆,• InLorentz-invariantQFT,leadsto:ØMonotonicC-functionin2dandF-functionin3dØA-theoremin4d• Basicideain2d:
• Aparticular2nd derivativeof𝑆(𝑙) isnon-positive.• Define𝐶 𝑙 = 3𝑙𝑆′(𝑙) ⇒ 𝐶’ 𝑙 ≤ 0
[Casini,Huerta,Teste,Torroba]
ModularHamiltonian
𝐾 ≝ −ln𝜌• Makesthestatelookthermal• Nonlocalingeneral• Exceptions:1. Thermalstate𝜌 = 𝑒1[\/𝑍2. HalfspaceinQFTvacuum:
𝐾 ∼ 2𝜋a𝑧𝑇dd
[Bisognano,Wichmann]
ModularHamiltonian
3. SphericaldiskinCFTvacuum
𝐾 ∼ 2𝜋a𝑅H − 𝑟H
𝑅𝑇dd
• Generalizestowigglycases
[Casini,Huerta,Myers,Teste,Torroba]
ModularHamiltonian
• Shapedeformationgovernedby𝑇fg,leadingto:ØAveragedNullEnergyCondition(ANEC):
a𝑑𝑥i 𝑇ii ≥ 0
• Usesmonotonicityofrelativeentropy(⇔ SSA)• Relativeentropy𝑆 𝜌 𝜎 ≝ Tr 𝜌ln𝜌 − Tr 𝜌 ln𝜎• 𝑆 𝜌 𝜎 ≥ 0;𝑆 𝜌 𝜎 = 0 iff𝜌 = 𝜎• Ameasureofdistinguishability• Monotonicity:𝑆 𝜌,L 𝜎,L ≥ 𝑆 𝜌, 𝜎,[Faulkner,Leigh,Parrikar,Wang;Hartman,Kundu,Tajdini;Bousso,Fisher,Leichenauer,Wall;
Balakrishnan,Faulkner,Khandker,Wang]
ModularHamiltonian
• Shapedeformationgovernedby𝑇fg,leadingto:
ØQuantumNullEnergyCondition(QNEC):
𝑇ii ≥12𝜋
𝑆mm
• Usescausality(inmodulartime)
[Faulkner,Leigh,Parrikar,Wang;Hartman,Kundu,Tajdini;Bousso,Fisher,Leichenauer,Wall;Balakrishnan,Faulkner,Khandker,Wang]
Plan
• EntanglementinQFT
• Entanglementinholography
• Applicationsandgeneralizations
• Holographicerrorcorrection
Inholography,vonNeumannentropyisgivenbytheareaofadualsurface:
• Practicallyusefulforunderstandingentanglementinstrongly-coupledsystems.
• Conceptuallyimportantforunderstandingtheemergenceofspacetimefromentanglement.
HolographicEntanglementEntropy
[Ryu&Takayanagi ’06]
[VanRaamsdonk ’10;Maldacena&Susskind’13;…]
[Huijse,Sachdev&Swingle ’11][XD,Harrison,Kachru,Torroba &Wang'12;…]
𝑆 = minArea4𝐺r
ProoffromAdS/CFT
Basicidea:• UsereplicatricktocalculateRenyi entropy𝑆/.• Givenbytheactionofabulksolution𝐵/ withaconicaldefect.• VonNeumannentropy:𝑆 = 𝜕/𝐼 𝐵/ |/u0.• TheRTminimalsurfaceisarelicoftheconicaldefectasdeficitangle2𝜋 1− 0
/goestozero.
• Variationofon-shellactionisaboundaryterm,givingthearea.
[Lewkowycz,Maldacena]
• Strictlyspeaking,theRTformulaappliesatamomentoftime-reflectionsymmetry.
• Fortime-dependentstates,wehaveacovariantgeneralization(HRT):
• Powerfultoolforstudyingtime-dependentphysicssuchasquantumquenches.
• CanalsobederivedfromAdS/CFT.
HRTintime-dependentstates
𝑆 = extArea4𝐺r
[Hubeny,Rangamani&Takayanagi’07]
[XD,Lewkowycz&Rangamani’16]
CorrectionstoRTformula
TheRTformulahasbeenrefinedby:
• Higherderivativecorrections
• Quantumcorrections
• Higherderivativecorrections(𝛼′):
• “Generalizedarea”𝐴gen = 𝜕/𝐼 𝐵/ |/u0• Hastheform:𝑆Wald + 𝑆extrinsiccurvature• Example:
• Appliesto(dynamical)blackholesandshowntoobeytheSecondLaw.
HigherderivativecorrectionstoRT
𝐿 = −1
16𝜋𝐺r𝑅 + 𝜆𝑅fg𝑅fg
yields𝐴gen = a 1+ 𝜆 𝑅�� −
12 𝐾�𝐾
�
�
𝑆 = ext𝐴gen4𝐺r
[XD ’13;Camps’13;XD &Lewkowycz,1705.08453;…]
[Bhattacharjee,Sarkar&Wall’15;Wall’15]
• Theprescriptionissurprisinglysimple:
• Quantumextremalsurface.• Validtoallorders in𝐺r.• Natural:invariantunderbulkRGflow.• Matchesone-loopFLMresult.• 𝑆bulk definedinentanglementwedge(domainofdependenceofachronal surfacebetween𝐴 and𝛾,)
QuantumcorrectionstoRTThesecorrections(𝐺r ∼ 1/𝑁H)comefrommatterfieldsandgravitons.
𝑆 = ext𝐴4𝐺r
+ 𝑆bulk[Engelhardt &Wall’14;XD &Lewkowycz,
1705.08453]
[Faulkner,Lewkowycz &Maldacena’13]
• Theprescriptionissurprisinglysimple:
• CanbederivedfromAdS/CFT.• Example:2ddilaton gravitywith𝑁� matterfields.ØQuantumeffectsgeneratenonlocaleffectiveaction:
ØAppearslocalinconformalgauge.Cancheckquantumextremality.
QuantumcorrectionstoRTThesecorrections(𝐺r ∼ 1/𝑁H)comefrommatterfieldsandgravitons.
𝑆 = ext𝐴4𝐺r
+ 𝑆bulk
[XD &Lewkowycz 1705.08453]
[CGHS;Russo,Susskind&Thorlacius ’92]
𝐿 = −12𝜋 𝑒1H� 𝑅 + 4𝜆H +
𝑁�96𝜋 𝑅
1𝛻H 𝑅
HolographicRenyi entropy
𝑆/ ≝1
1 − 𝑛lnTrρ,/
• Givenbycosmicbranesinsteadofminimalsurface
𝑛H𝜕/𝑛 − 1𝑛
𝑆/ =Area CosmicBrane/
4𝐺r
[XD]
• Cosmicbranesimilartominimalsurface;theyarebothcodimension-2andanchoredatedgeof𝐴.
• Butbraneisdifferentinhavingtension𝑇/ =/10�/��
.
• Backreacts onambientgeometrybycreatingconicaldeficitangle2𝜋 /10
/.
• Usefulwayofgettingthegeometry:findsolutiontoclassicalaction𝐼total = 𝐼bulk+𝐼brane.• As𝑛 → 1:probebranesettlesatminimalsurface.• One-parametergenerationofRT.
𝑛H𝜕/𝑛 − 1𝑛
𝑆/ =Area CosmicBrane/
4𝐺r
[XD]
• Whydoesthisarealawwork?• BecauseLHSisamorenaturalcandidateforgeneralizingvonNeumannentropy:
• Thisisstandardthermodynamicrelation
with𝐹/ = − 0/lnTrρ,/,𝑇 =
0/
Trρ,/ ispartitionfunctionw/modularHamiltonian− lnρ,• 𝑆/� ≥ 0 generally. Automaticbyarealaw!
𝑆/� ≝ 𝑛H𝜕/𝑛 − 1𝑛
𝑆/ = −𝑛H𝜕/1𝑛lnTrρ,/
𝑆/� = −𝜕𝐹/𝜕𝑇
𝑛H𝜕/𝑛 − 1𝑛
𝑆/ =Area CosmicBrane/
4𝐺r
[Beck&Schögl ’93][XD]
JLMSrelations
𝐾 =Area�4𝐺r
+ 𝐾bulk• CFTmodularflow=bulkmodularflow
𝑆(𝜌|𝜎) = 𝑆bulk(𝜌|𝜎)• TwostatesareasdistinguishableintheCFTasinthebulk.• Bothrelationsholdtoone-looporderin1/N.
[Jafferis,Lewkowycz,Maldacena,Suh]
Plan
• EntanglementinQFT
• Entanglementinholography
• Applicationsandgeneralizations
• Holographicerrorcorrection
Phasesofholographicentanglemententropy
• First-orderphasetransition• SimilartoHawking-Pagetransition• Mutualinformation𝐼 𝐴, 𝐵 ≝ 𝑆, + 𝑆L − 𝑆,Lchangesfromzerotononzero(atorder1/𝐺r).• HolographicRenyi entropyhassimilartransitions.
PhasesofholographicRenyientropy• Canhavesecond-orderphasetransitionat𝑛 =𝑛��=�• Needasufficientlylightbulkfield• Basicidea:increase𝑛 ~ decreaseT⇒ bulkfieldcondenses.• Examples:1. Sphericaldiskregion2. In𝑑 = 2:multipleintervals
[Belin,Maloney,Matsuura;XD,Maguire,Maloney,Maxfield]
• RTsatisfiesstrongsubadditivity:
• Alsosatisfiesotherinequalitiessuchasmonogamyofmutualinformation
and
• Providenontrivialconditionsforatheorytohaveagravitydual.• Holographicentropyconefortime-dependentstates?• AdS3/CFT2:sameasthestaticcase.
HolographicEntropyCone
𝑆,L + 𝑆LM + 𝑆M, ≥ 𝑆,LM + 𝑆, + 𝑆L + 𝑆M
𝑆,LM + 𝑆LM� + 𝑆M� + 𝑆� , + 𝑆 ,L≥ 𝑆,LM� + 𝑆,L + 𝑆LM + 𝑆M� + 𝑆� + 𝑆 ,
[XD &Czech,toappear]
[Headrick &Takayanagi]
[Hayden,Headrick&Maloney]
[Bao,Nezami,Ooguri,Stoica,Sully&Walter]
Timeevolutionofentanglement
• Considerthethermofield double(TFD)state:
|Ψ⟩ =1𝑍¡𝑒[ ¢/H|𝑛⟩£|𝑛⟩¤/
• DualtoeternalAdSblackhole:
[Maldacena]
Timeevolutionofentanglement
• ConsidertheunionofhalfoftheleftCFTandhalfoftherightCFTattime𝑡:
• Itsentanglemententropygrowslinearlyin𝑡.• RelatedtothegrowthoftheBHinterior.
[Hartman,Maldacena]
Holographiccomplexity• SuchgrowthoftheBHinteriorisconjecturedtocorrespondtothelineargrowthofcomplexityofthestate.• Complexity:minimum#ofgatestoprepareastateØComplexity= VolumeØComplexity= Action
[Stanford,Susskind; Brown,Roberts,Susskind,Swingle,Zhao]
Traversablewormhole• TFDisahighlyentangledstate,butthetwoCFTsdonotinteractwitheachother.• Dualtoanon-traversablewormhole.
• Canbemadetraversablebyaddinginteractions.• Quantumteleportationprotocol.
[Gao,Jafferis,Wall;Maldacena,Stanford,Yang;Maldacena,Qi]
Einsteinequationsfromentanglement• TheproofofRTreliedonEinsteinequations.• Cangointheotherdirection:derivingEinsteinequationsfromRT.• Basicidea:firstlawofentanglemententropy
𝛿𝑆 = 𝛿 𝐾§ under𝜌 → 𝜌 + 𝛿𝜌• ApplytoasphericaldiskintheCFTvacuum,sothat𝐾§islocallydeterminedfromtheCFTstresstensor.• Bothsidesarecontrolledbythebulkmetric:viaRTontheleft,andviaextrapolatedictionaryontheright.• Givesthe(linearized)Einsteinequations.
[XD,Faulkner,Guica,Haehl,Hartman,Hijano,Lashkari,Lewkowycz,McDermott,Myers,Parrikar,Rabideau,Swingle,VanRaamsdonk,...]
Bitthreads
• ReformulateRTusingmaximal flowsinsteadofminimalsurfaces.
• Flow:𝛻 ¨ 𝑣 = 0, 𝑣 ≤ 𝐶• Maxflow-mincuttheorem:
max>a𝑣,
= 𝐶minª~,
Area 𝑚
[Freedman,Headrick;Headrick,Hubeny]
Bitthreads
• Oneadvantageisthatmaxflowschangecontinuouslyacrossphasetransitions,unlikemincuts:
• ThisbitthreadpicturehasbeengeneralizedtoHRT.[Freedman,Headrick;Headrick,Hubeny]
Entanglementofpurification
• Givenamixedstate𝜌,L ,itmaybepurifiedas|𝜓⟩,,LL.• Entanglementofpurification:min
®𝑆 𝜌,,m
• ConjecturedtobeholographicallygivenbytheentanglementwedgecrosssectionΣ,L°±² :
[Takayanagi,Umemoto,...]
Modularflowasadisentangler
• “Modularminimalentropy”isgivenbytheareaofaconstrainedextremalsurface.• Giventworegions𝑅,𝐴,definethemodularevolvedstate: 𝜓 𝑠 ⟩ = 𝜌¤=´ 𝜓⟩• Andthemodularminimalentropy:
𝑆¤ 𝐴 = min´𝑆, |𝜓(𝑠)⟩
• Holographicallygivenbyaconstrainedextremalsurface(i.e.extremalexceptatintersectionswiththeHRTsurface𝛾¤ of𝑅).
[Chen,XD,Lewkowycz&Qi,appearingtoday]
Modularflowasadisentangler
• “Modularminimalentropy”isgivenbytheareaofaconstrainedextremalsurface.• Obviouswhenthemodularflowislocal:
• Seemstobegenerallytrue(canbeprovenind=2).• Ausefuldiagnosticofwhether𝛾, and𝛾¤ intersect.
[Chen,XD,Lewkowycz&Qi,appearingtoday]
EntanglementindeSitterholography• dS/dScorrespondence:
𝑑𝑠µ¶7·¸H = 𝑑𝑤H + sinH 𝑤 𝑑𝑠µ¶7
H
• Suggestsadualdescriptionastwomattersectorscoupledtoeachotherandto𝑑-dimensionalgravity.• Reminiscentof(butdifferentfrom)theTFDexamplediscussedpreviously.• Inparticular,thetwosidesareinamaximallyentangledstate (toleadingorder),asshownbyentanglementandRenyi entropies.• MatchesandgivesaninterpretationoftheGibbons-Hawkingentropy.[XD,Silverstein,Torroba]
Plan
• EntanglementinQFT
• Entanglementinholography
• Applicationsandgeneralizations
• Holographicerrorcorrection
Whyquantumerrorcorrection?• 𝜙(𝑥) canberepresentedon𝐴 ∪ 𝐵,𝐵 ∪ 𝐶,or𝐴 ∪ 𝐶 .• ObviouslytheycannotbethesameCFToperator.• Definingfeatureforquantumerrorcorrection.• Holographyisaquantumerrorcorrectingcode.• Reconstructionworksinacodesubspace ofstates.
[Almheiri,XD,Harlow’14]
AdS/CFTasaQuantumErrorCorrectingCode
BulkGravity
• Low-energybulkstates
• DifferentCFTrepresentationsofabulkoperator
• Algebraofbulkoperators
• Radialdistance
QuantumErrorCorrection
• Statesinthecodesubspace
• Redundantimplementationofthesamelogicaloperation
• Algebraofprotectedoperatorsactingonthecodesubspace
• Levelofprotection[Almheiri,XD,Harlow]
Tensornetworktoymodels
[Pastawski,Yoshida,Harlow&Preskill ’15;Hayden,Nezami,Qi,Thomas,Walter&Yang’16;Donnelly,Michel,Marolf &Wien’16;…]
Inotherwords:∀ 𝑂� intheentanglementwedgeof𝐴,∃ 𝑂, on
𝐴,s.t. 𝑂,|𝜙⟩ = 𝑂�|𝜙⟩ and𝑂,½ 𝜙 = 𝑂�
½ 𝜙 holdfor∀ |𝜙⟩ ∈ 𝐻�ÀµÁ .
Anybulkoperatorintheentanglementwedgeofregion𝐴mayberepresentedasaCFToperatoron𝐴.
Entanglementwedgereconstruction 𝑂�
Thisnewformofsubregion dualitygoesbeyondtheold“causalwedgereconstruction”:entanglementwedgecanreachbehindblackholehorizons.
Validtoallordersin𝐺r.
Anybulkoperatorintheentanglementwedgeofregion𝐴mayberepresentedasaCFToperatoron𝐴.
Entanglementwedgereconstruction 𝑂�
• Goalistoprove:
• Thisisnecessaryandsufficientfor∃𝑂,,s.t. 𝑂, 𝜙 = 𝑂� 𝜙 and𝑂,
½ 𝜙 = 𝑂�½ 𝜙
• WLGassume𝑂� isHermitian.• Considertwostates|𝜙⟩,𝑒=ÄÅÆ|𝜙⟩ incodesubspace𝐻� :
Reconstructiontheorem𝜙 [𝑂�, 𝑋,̅] 𝜙 = 0
𝑆 𝜌,̅ 𝜎,̅ = 𝑆 𝜌�Ë 𝜎� = 0
[XD,Harlow&Wall1601.05416]
[Almheiri,XD &Harlow’14]
𝜙 𝑒1=ÄÅÆ𝑋,̅𝑒=ÄÅÆ 𝜙= 𝜙 𝑋,̅ 𝜙
𝜙 [𝑂�, 𝑋,̅] 𝜙 = 0
𝑂�
RTfromQEC
• WesawthatentanglementwedgereconstructionarisesfromJLMSwhichisinturnaresultofthequantum-correctedRTformula.• Cangointheotherdirection.• Inanyquantumerror-correctingcodewithcomplementaryrecovery,wehaveaversionoftheRTformulawithone-loopquantumcorrections:
𝑆, = 𝐴 + 𝑆�
[Harlow]
ConclusionandQuestions• PatternsofentanglementprovideaunconventionalwindowtowardsunderstandingQFTandgravity.
ØCanwegeneralizetheentanglement-basedproofofc-,F-,anda-theoremstohigherdimensions?• Wediscussed𝛼′ correctionstotheRTformula.ØIsthereastringyderivationthatworksforfinite𝛼′?• WesawthatRTwithone-loopquantumcorrectionsmatchesQECwithcomplementaryrecovery.
ØDohigher-ordercorrectionsmeansomethinginQEC?ØWhatconditionsforacodetohaveagravitydual?ØHowdowefullyenjoyallofthisinthecontextofblackholeinformationproblemandcosmology?
Thankyou!