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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!