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!