OnthePageCurveandInformationRecoveryfromBlackHoles

QGSC-2019,Bariloche-Valdivia

BasedonworkwithH.Verlinde

ErikVerlinde

MotivationandOutline

• BlackHoleInformationParadox• MicroscopicDerivationofthePageCurve• DecodingtheHawkingRadiation• ConstructingtheBlackHoleInterior.

Page-curve

S = �tr�⇢ log ⇢

�

Intiial purestateofinfalling matter

Intermediateentangledstateofblackhole

withradiation

Finalpurestateofradiation

Coarsegrainedblackholeentropy

Coarsegrainedentropyofradiation

Finegrainedentanglemententropy.

Finegrainedentanglemententropy

S = �tr�⇢ log ⇢

�

S =Area

4G+ SQFT

Finegrainedgravitational+QFTentropy

Jafferis,Lewkowycz,Maldacena,SuhEngelhardt,Wall

Figurefrom:

BlackHoleEntanglementandQuantumErrorCorrection.(2012)EV&H.Verlinde

| i| 0 i

CFTcoupledtoexternalsystem.

Initialmatterstatedeterminedbyexternalsystem.

InitialCFTstate=groundstate.

U | i|0i

| i| 0 i | 0 i

UnitaryevolutionofCFTcoupledtoexternalsystem.

InitialmatterstateevolvesintomixedCFT+radiationstate

U | i|0i =X

a,n

Ca,n|ai|ni

| 0 i| i| 0 i

U | i|0i =X

a,n

Ca,n|ai|ni

EntangledstateofCFTandradiation

Unitarity impliesX

a,n

|Ca,n|2 = 1

where

a = 1, 2, . . . , drad

n = 1, 2, . . . , dcft

Page-curve

S = �tr�⇢ log ⇢

�

Intiial purestateofinfalling matter

IntermediateentangledstateoftheCFTandradiation

Finalpurestateofradiation S = log drad

S = log dcft

drad<dcft drad>dcft

BeforePagetime AfterPagetime

StatisticalderivationofthePagecurve

(Almheiri,Hartman,Maldacena,Shaghoulian,Tajdini)

(Penington,Stanford,Shenker,Yang)Analogoustoreplicawormholes

FromRenyi

tovonNeumannEntropy

Sk =1

1� klog tr

�⇢k

�

limk!1

Sk = �tr�⇢ log ⇢

�

⇢ =X

n,a,b

Ca,nC⇤n,b|aihb|

U | i|0i =X

a,n

Ca,n|ai|niThestate

givesasradiationdensitymatrix

⇢ =X

n,a,b

Ca,nC⇤n,b|aihb|

tr�⇢k

�=

X

{ni,ai}

kY

i=1

Cai,niC⇤

ni,ai+1

U | i|0i =X

a,n

Ca,n|ai|niThestate

givesasradiationdensitymatrix

Hence,wefind

Randommatrixansatz:

C⇤m,aCa,n ' 1

dcft

�n,m

Ca,n ' ei�a,n

pdcftdrad

Afterpartialsummation

Randommatrixansatz:

C⇤m,aCa,n ' 1

dcft

�n,m

Ca,n ' ei�a,n

pdcftdrad

Afterpartialsummation�1 +O

�d�1/2rad

��

Randommatrixansatz:

C⇤m,aCa,n ' 1

dcft

�n,m

Ca,n ' ei�a,n

pdcftdrad

Afterpartialsummation

Ca,nC⇤n,b '

1

drad

�ab

Andsimilarly⇣1 +O

⇣d�1/2cft

⌘⌘

�1 +O

�d�1/2rad

��

tr�⇢k

�=

X

{ni,ai}

kY

i=1

Cai,niC⇤

ni,ai+1

tr�⇢k

�=

X

{ni,ai}

kY

i=1

Cai,niC⇤

ni,ai+1a1

a2

a3

ak

ak-1

nk-1 nk

n1

n2

ni-1

ai

ni

ai+1

C

C⇤

Pagecurvefromrandommatrices:

tr�⇢k

�=

X

{ni,ai}

kY

i=1

Cai,niC⇤

ni,ai+1

tr�⇢k

�=

X

{ni,ai}

kY

i=1

Cai,niC⇤

ni,ai+1a1

a2

a3

ak

ak-1

nk-1 nk

n1

n2

ni-1

ai

ni

=X

{ni,ai}

kY

i=1

C⇤ni�1,ai

Cai,ni

ai+1

CC⇤

C⇤

Pagecurvefromrandommatrices:

tr�⇢k

�=

X

{ni,ai}

kY

i=1

Cai,niC⇤

ni,ai+1a1

a2

a3

ak

ak-1

nk-1 nk

n1

n2

ni-1

ai

ni

=X

{ni,ai}

kY

i=1

C⇤ni�1,ai

Cai,niai+1

CC⇤

Pagecurvefromrandommatrices: C⇤m,aCa,n ' 1

dcft

�n,m

=1

dk�1cft

tr�⇢k

�=

X

{ni,ai}

kY

i=1

Cai,niC⇤

ni,ai+1a1

a2

a3

ak

ak-1

nk-1 nk

n1

n2

ni-1

ai

ni

=X

{ni,ai}

kY

i=1

C⇤ni�1,ai

Cai,niai+1

CC⇤

Statisticaloriginofreplicawormholes:(Almheiri,Hartman,Maldacena,Shaghoulian,Tajdini)

(Penington,Stanford,Shenker,Yang)

C⇤C

=1

dk�1cft

DecodingtheHawkingRadiation

Kitaev,YoshidaHayden,PeningtonAnalogoustoHayden-Preskill protocol

InformationreleasedbyblackholeintoHawkingradiation

| 0 i

| i =X

i

↵i | i i

TheamountofinformationthatcanberetrievedfromtheradiationafterPagetimeisboundedby

i = 1, 2, . . . , dmat

Smat Srad � Scft =

log dmat log drad � log dcft =

(Hayden,Penington)

| i

Wemayallowstatesoftheform

providedthat

| 0 i | i =X

i

↵i | i i

U | i|0i =X

i

↵i Cia,n|ai|ni Howtorecover?| i

SCRAMBLING UNSCRAMBLE

Hayden-Preskill protocol

Yoshida,Kitaev

AfterPagetimetheblackholeismaximallyentangledwiththeradiation.

Asmallamountofinformationthrownintotheblackholecanafterascramblingtime(andemissionofafewHawkingquanta)bedecodedmerelyfromtheradiation.

SCRAMBLING UNSCRAMBLE

Hayden-Preskill protocol

Kitaev,Yoshida

TherecoveryoperationRcannotactontheCFT.andonlyusestheradiation.

ToremovetheentanglementwiththeCFT,however,oneneedstointroduceanancillaryHilbertspace.

AsancillaryHilbertspacewechoosea“copy”oftheoriginalCFT.

TherecoveryoperationRtransferstheentanglementoftheCFTtotheancillaryCFT.

AsancillaryHilbertspacewechoosea“copy”oftheoriginalCFT.

TherecoveryoperationRtransferstheentanglementoftheCFTtotheancillaryCFT.

ThiswillleadtoaThermofield DoubleState

�1/2Mat

Anotherpossible recoveryoperationisthe`Petz map’Itusesreferencedensitymatricesfortheinitialmatterandfinalradiation.

Therecoverydependsonthesubspaceofinitialstates,andisapproximate.

(Hayden&Penington)

�1 +O

�d�1/2rad

��

AmoreconvenientrecoveryoperationRmakesuseoftheapproximaterelation

Onefindsthat

C⇤in,aC

ja,m ' 1

dcft

�nm�ij

U | i i|0i = Cia,n|ai|ni

R|ai|e0i = d1/2cft

X

j,m

C⇤,jm,a| j i|mi

| i =X

i

↵i | i i

RU | i|0i|e0i ' | i|TFDi

Foragenericstateinthechosencodesubspaceoftheform

therecoveryoperationRontheevolvedentangledstategives

wheretheThermofield DoubleStatetakestheform

|TFDi = 1pdcft

X

n

|n i| en i

Takenfrom

BlackHoleEntanglementandQuantumErrorCorrection.(2012)EV&H.Verlinde

Beased on

Chapter7 ofPreskill’s bookonQuantumInformation

ReconstructingtheBlackHoleInterior

vanRaamsdonk.Maldacena,Susskind

AnalogoustoEPR=ER EVandH.Verlinde

|TFDi = 1pdcft

X

n

|n i| en i =ERbridgetotheIsland

OperatorsactingontheIslandtaketheform Orad = he0|R†

OR|e0i

CanAliceandBobhaveaRendez-Vous ontheIsland?

AliceBob

Charlie

AndcantheytellCharlieabouttheirmeeting?

Conclusion:

• UsingarandommatrixAnsatzonecanderivethePagecurve,andconstructarecoveryoperatorRthatretrievestheinformationfromtheblack,createsanERbridgetotheIslandandallowsthereconstructionoftheblackholeinterior.

THANKYOU