Based on M. Hotta, Y. Nambu and K. Yamaguchi, arXiv:1706.07520 .

Soft-Hair Enhanced Entanglement Beyond Page Curves

in Black-Hole Evaporation Qubit Models Masahiro HottaTohoku University

Introduction

InfallingParticleLarge BH

Large black-hole spacetimes are conventionally described merely by classical geometry, and nothing cannot go out of the event horizon.

Black hole is black.

This picture drastically changes, because black holes can emit thermal flux due to quantum effect.

BHBHR GMk

cTπ8

3=

(Hawking, 1974)

BHBH GMr 2=

Black hole ain’t so black!

Unitarity breaking?

Information is lost!?

Thermalradiation

Thermalradiation

Only thermal radiation?

Largeblack hole

Small black hole

Ψ

thermalρ

thermalUU ρ≠ΨΨ †ˆˆ

The Information Loss Problem Hawking (1976)

Why is the information loss problem so serious?

Too small energyto leak the huge amount of information.(Aharonov, et al 1987; Preskill 1992.)

If the horizon prevents enormous amount of information from leaking until the last burst of BH, only very small amount of BH energy remains, which is not expected to excite carriers of the information and spread it out over the outer space.

Small BH

HRn

HRnHR nnp∑=ρ

HRAHR

HRHR Ann

HRnHRAunp∑=Ψ

Mixed state

Composite system in a pure state

From a modern viewpoint of quantum information,Information Loss Problem = Purification Problem of HR

Hawking radiation system

Partner systemEntanglement

What is the partnerafter the last burst?

(1) Nothing, Information Loss

(2) Exotic Remnant (Aharonov, Banks, Giddings,…)

(3) Baby Universe (Dyson,..)

(4) Radiation Itself (Page,…) Black Hole Complementarity (‘t Hooft, Susskind, …) Firewall (Braunstein, AMPS, …)

What is the purification partner of the Hawking radiation?

(4) Radiation

Black Hole Complementarity

InfallingParticle

From the viewpoint of free-fall observers, no drama happens across the horizon.

Large BH

Classical Horizon

(4) Radiation

Black Hole Complementarity

HawkingRadiation

InfallingParticle

Stretched HorizonInduced byQuantumGravity

From the viewpoint of outside observers,the stretched horizon absorbs and emits quantum information so as to maintain the unitarity.

Large BH

PlanckUnruh ET >>

(4) Radiation

Firewall

A FIREWALL at the horizon burns out free-fall observers. The inside region of BH does not exist!

Free-fallobserver

Large BH

FIREWALL

(1) Nothing, Information Loss

(2) Exotic Remnant (Aharonov, Banks, Giddings,…)

(3) Baby Universe (Dyson,..)

(4) Radiation Itself (Page,…) Black Hole Complementarity (‘t Hooft, Susskind, …) Firewall (Braunstein, AMPS, …)

(5) Zero-Point Fluctuation Flow (Wilczek, Hotta-Schützhold-Unruh )

What is the purification partner of the Hawking radiation?

Hawking Particle

Zero-Point Fluctuation of Quantum Fields

ψ(5) Zero-Point Fluctuation Flow

entangled

(Wilczek, Hotta-Schützhold-Unruh)

Entanglement Sharing

Energy Cost of the Partner≠

→ Avoiding the planck-mass remnant problem.

First-principles computation of time evolution of entanglement between an evaporating BH and Hawking particles is not able to be attained.

A popular conjecture → The Page CurveDon Page, Phys. Rev. Lett. 71, 3743 (1993).

The partner entangled with a Hawking particle remains elusive due to the lack of quantum gravity theory to date.

Page’s Strategy for Finding States of BH Evaporation:Nobody knows exact quantum gravity dynamics. So let’s gamble that the state scrambled by quantum gravity is one of TYPICAL pure states of the finite-dimensional composite system! That may not be so bad!

HR

BH

HHR

HBH

dim

,dim

=

=

HRln

EES

)0(7.0 BHPage MM ≈

BHHR

HRBH lnln ≈

<<Page Time>>

Maximum value of EE is attained at each time, and is equal to thermal entropy of smaller system.

OLD BH

Page Curve

Semi-classical regime!

time

GASS thermalEE 4

==

Page Curve Conjecture for BH Evaporation:Proposition I:After Page time, BH is maximally entangled with Hawking particles. No correlation among BH subsystems.

Proposition II:)4/( GASS horizonthermalEE == after Page time.

No correlation due to quantum monogamy

Maximal Entanglement

Hawking particles

BH

)1(EEEE NSS =

Lemma:

No correlation among BH subsystems due to quantum monogamy

N= # of BH degrees of freedom

GA

NSS horizon

thermalEE 41)1()1( ==

After Page time,( ))1(

thermalthermal NSS =

[ ] ∏=

⊗=ΨΨ=N

nnHRBH Tr

1

)1(ρρ

BH

After Page time,

This relation is criticized in this talk.

BHHR

BHBH

HR TGM

T ==π8

1 08

12 <−=

BHBH GT

Cπ

∞→BHT 0→BHMas

Note that BH has negative heat capacity!

HRBH TT =

Temperature of BH is measured by temperature of Hawking radiation.

)4/()1()1( GNASS horizonthermalEE >>>>

In this talk, we construct a model of multiple qubits which reproduces thermal property of 4-dim Schwartzschild black holes, and show that the Page curve conjecture is not satisfied due to the negative heat capacity.

Our Result

I. “Page Theorem” and BH FirewallsII. Soft Hair Evaporation at HorizonIII. Multiple Qubit Model of Black-

Hole Evaporation and Breaking of Page Curve Conjecture

Outline

I. “Page Theorem” and BH Firewalls

A BANBN

ABΨ

ABΨ

Typical State of ABBA NN <<

[ ]ABABBA Tr ΨΨ=ρ

[ ]AAAEE TrS ρρ ln−=

AEE NS ln≈

Typical states of A and B are almost maximally entangled when the systems are large.

AA

A IN1

≈ρ

Lubkin-Lloyd-Pagels-Page Theorem (“Page Theorem”)

Maximal Entanglement between A and B

AB NN ≥

∑=

=AN

nBnAn

AAB

vuN

Max1

~1

Orthogonal unit vectors

⇒= AA

A IN1ρ

BAHR ∪=CBH =

BA

Late radiation

C Early radiation

CBA ,,1<<

ACB <<

ABCΨ

Page Time

OLD BH

→OLD BH

By using the theorem, AMPS and other people proposed the BH firewall conjecture.

BAC

⊗

== CBBCBC I

CI

BI

BC111ρ

Proposition I means that A and BC are almost maximally entangled with each other.

Harrow-Hayden

NO CORRELATION BETWEEN B AND C!

BAC

⊗

= CBBC I

CI

B11ρ

FIREWALL!

( )[ ] ∞=∂ BCxTr ρϕ 2)(

=

−

2

2 1)()(ε

ρεϕϕ OxxTr BC

CB

CxBx

0→ε

ε

Flaw of Page Curve Conjecture:Area law of entanglement entropy is broken, though outside-horizon energy density in BH evaporation is much less than Planck scale.

|||| BASEE ∂=∂∝

A

Bfor low-energy-density states

ABAB0≈Ψ

← standard area law of entanglement entropy

∑=

+=Ψ

||

1

~||

1 BH

nHRnBHnHRBH

vuBH

BHEE VBHS ∝= ln ← Not area law, but volume law!

BHHR

Page curve states ⇒

This is because zero Hamiltonian (or high temperature regime) is assumed in Page curve conjecture.

.0=ABH

ABΨ [ ]ABABBA Tr ΨΨ=ρ

[ ]AAAAB TrS ρρ ˆlnˆ−=

)(, βAthermalAB SS ≈

( )AA

A HZ

βρ −≈ exp1ˆ

A BAB

Ψ

.constEE BA =+

AB NN >>

0≠+≈ BAAB HHH

EE is almost equal to thermal entropy of the smaller system for typical states.

Remark: for ordinary weakly interacting quantum systems, entanglement entropy is upper bounded by thermal entropy, as long as stable Gibbs states exist.

A BAB

Ψ EEE BA =+[ ]ABABBA Tr ΨΨ=ρ

))((/))(exp( EZHE AAA ββρ −=

[ ] [ ] thermalAAAAEE STrTrS =−≤−= ρρρρ lnln

Gibbs state:

BHAH [ ] AAAA EHTr =ρ

[ ] [ ]( ) [ ]( )1ln 21 −−−−−= AAAAAAAAA TrEHTrTrI ρλρλρρ

If a stable Gibbs state exists, thermal entropy of the smaller system is the maximum value of EE for any state with average energy fixed.

( ) 0=AI ρδ

( ) ( ))(/)exp( AAAAA EZHE ββρ −=

[ ] [ ] thermalAAAA STrTr =−≤− ρρρρ lnln

Conventional “proof”:

[ ] ( )AAAAA EEHTr ββρ =⇒=

No stable Gibbs state for Schwarzschild BH due to negative heat capacity! (Hawking –Page, 1983)

( )[ ]BHBH HTrZ ββ −= exp)(

( )02

2

>−

=T

EE

dTEd

If there exists a stable Gibbs state, the heat capacity must be positive.

( )T/1=β

Thus, a system of a black hole and Hawking radiation is not in a typical state, at least in the sense of the LLPP theorem. Because we have no stable Gibbs state, “ thermal entropy” of Schwarzschild BH is not needed to be a upper bound of entanglement entropy.

)4/( GASEE ≤

II. Soft Hair Evaporation at Horizon

(1) Nothing, Information Loss

(2) Exotic Remnant (Aharonov, Banks, Giddings,…)

(3) Baby Universe (Dyson,..)

(4) Radiation Itself (Page,…) Black Hole Complementarity (‘t Hooft, Susskind, …) Firewall (Braunstein, AMPS, …)

(5) Zero-Point Fluctuation Flow (Wilczek, Hotta-Schützhold-Unruh )

(5)’ Soft Hair without Energy(Hotta-Sasaki-Sasaki (2001), Hawking-Perry -Strominger (2015))

What is the purification partner of the Hawking radiation?

Gravitational vacuum degeneracy with zero energy plays a crucial role. Hawking, Perry and Strominger call it soft hair.

Bondi-Metzner-Sachs soft hair

Horizon soft hairInformation

S. W. Hawking, M. J. Perry and A. Strominger, PRL 116, 231301 (2016).M. Hotta, J. Trevison and K. Yamaguchi, Phys. Rev. D94, 083001 (2016).S. W. Hawking, M. J. Perry and A. Strominger, JHEP 05, 161 (2017).

M. Hotta, K. Sasaki and T. Sasaki, Class. Quantum Grav. 18, 1823 (2001).M. Hotta, Phys.Rev. D66 124021 (2002) .A. Strominger, JHEP07,152 (2014).

0≈ω

0≈EHawkingParticles

(Feb 3rd, 2017 at Cambridge)

0>T

Horizon soft hair of black holes comes from “would-be” gauge freedom of general covariance, diffeomorphisms (Hotta et al, 2001). This is a similar mechanism of emergence of momentum eigenstates orthogonal to each other.

Black Hole at Rest

'x

'y

0=px

y

ωωωω

coshsinh',sinhcosh'

txttxx

+=+=

Running Black Hole

'x

'y

ωsinh' mp =x

y

We must have Lorentz covariance in the theory.

'x

'y

x

yRunning Black Hole

'x

'y

ωsinh' mp =x

y

0sinh0 === ωmpp

Lorentz transformation, one of general coordinate transformations, generates an infinite number of physical states with different values of momentum.

ωsinhmp =

Horizon soft hair of black holes comes from “would-be” gauge freedom of general covariance, diffeomorphisms (Hotta et al, 2001). This is a similar mechanism of emergence of momentum eigenstates orthogonal to each other.

Soft hair of black holes comes from “would-be” gauge freedom of general covariance.

Horizon

0=horizonQ

),(' φθττ T+=

qQ horizon ='

Horizon

We must have horizontal asymptotic symmetry as a part of general covariance in the theory.

),('),,('

φθφφθθ

Φ=Θ=

Supertranslation:

Superrotation:

Soft hair of black holes comes from “would-be” gauge freedom of general covariance.

Horizon

qQ horizon ='

00 === qQQ horizonhorizon

Supertranslation and superrotation generate an enormous number of physical states with different values of holographic charges.

Horizon

qQhorizon =

),(' φθττ T+=

),('),,('

φθφφθθ

Φ=Θ=

BMS Supertranslation:

Horizon Superrotation:

Near Horizon

⊗

Null Future Infinity Poincare Symmetry:

cxx +Λ='Horizon Supertranslation:

),(' φθCuu +=

including time translation generated by Hamiltonian H

[ ] [ ] 0,, =⊗⊗=⊗⊗ horizonSRhorizonST QIIHQIIHNear-horizon symmetry provides degeneracy of Hamiltonian.

BH Symmetries

(BMS Superrotation?)

Collapsing Matter

Collapsing Matter

Horizon

An infinite number of supertranslationcharges at the horizon might store whole quantum information of absorbed matter.

0)( =∂ µµ iJ

≈

GAOW hairsoft 4

ln

( ))4/( GAreaO⊗ψ

Unitarity?

Soft hair of BH

Hawking radiation

(Hotta-Sasaki-Sasaki, 2001) (Hawking –Perry-Strominger, 2017)

III. Multiple Qubit Model of Black-Hole Evaporation and Breaking of Page Curve Conjecture

BHGMT

π81

=

ΨΨ=

HGT

π81

Hawking Temperature Relation:

Key Idea: Construct many-body systems which reproduces this relation.

This condensed matter model yields negative heat capacity. Bekenstein-Hawking entropy can be defined even if the systems do not have actual horizons.

( )G

AGr

GHG

THd

S horizonBHBH 44

44

24 22

==ΨΨ

=ΨΨ

= ∫ππ

Implicitly assumed that partition function of the total system is well defined. Thus, the heat capacity is always positive.

EES

)(tn2/N N

Page time

Decaying qubits of BH into qubits of HR in Page Model

[ ]ABABBA Tr ΨΨ=ρ

[ ] )(ˆ)(ln)()( tUdttTrtS AAAEE ∫−= ρρ

Hawking Radiation Black Hole

)(ASS thermalEE =BH entropy

In order to incorporate negative heat capacity, we consider not only emission of Hawking particles but also decreasing of BH degrees of freedom (soft hair) ! (Hotta-Nambu-Yamaguchi 2017)

Hawking ParticleEmission

Zero-Energy Soft Hair Evaporation

Cf. Negative heat capacity in D0 particle model,E. Berkowitz, M. Hanada, and J. Maltz, Phys. Rev. D94, 126009, (2016).

0≈E

0≈E

0≈E

0≈E

0≈E

0≈E

2T1T21 TT <

Negative heat capacity induced by soft particle decays

otot EtE =)( 1 otot EtE ≈)( 2

HighLowLarge Number of Particles Small Number of Particles

If radiation coupling is switched on, the temperature of emitted radiation rises!

1 2 3 4 N-1 N( )

NNN Ut +−+−+==Ψ

−1321)( ˆ0

N qubits of A at Initial Time

Energy conserved random unitary operation(Fast Scrambling)

Our Model:0ˆ †

+=+ a0ˆ †

−=− a

One of typical pure states of N qubits

Our Model 0ˆ †

+=+ a0ˆ †

−=− a

0

Hawking Particle

Evaporating Soft Hair

0000)1( +−−+++=ωoH

Fast-Scrambled Pure-State Black Hole at Initial Time

Soft Hair Emission

SHHRo HHHH ++=

( )∑=

+=N

nzo nH

11)(

2σω

( )

∫∫∫

Ψ−++Ψ+−+

Ψ∂−Ψ=

dxxxgdxxxg

dxxixH

RR

RxRHR

)(ˆ)()(ˆ)(

)(ˆ)(ˆ

†

†

( )

∫∫∫

Ψ−+Ψ−+

Ψ∂−Ψ=

dxxxfdxxxf

dxxixH

SS

SxSSH

)(ˆ)(0)(ˆ)(0

)(ˆ)(ˆ

†

†

Initial State:SRQbits

vacvact Ψ==Ω 0(

0)(ˆ)(ˆ =Ψ=ΨSSRR vacxvacx

~ BH internal dynamics

~ Hawking radiation emission

~ Soft hair emission

[ ]( ) )).(()(00)(1

)()(,,,3,2)1(

tTtptp

ttTr SRN

ρ

ρ

+−=

ΩΩ=

T

T

e

eT ω

ω

ρ−

−

+

+++−−=

1)(

Qubit Gibbs state at temperature T

One-site reduced state is described by

Hawking particle is emitted by flipping the up state to the down state. So the radiation temperature is equal to T! This model actually reproduces the Hawking temperature relation:

( ) ( )tMGtT

BHπ81

=( )( )

=

=−1

)1(

)0(4

,

TGN

HNtM oBH

ωπ

Qbit Survival Probability

[ ])1()1()1( ln ρρTrSEE −=

+

+

−+==

TTT

ppSS ththermal ωωω

exp1

1exp1ln)1(

NS

GA

NTdM

NS BHhorizonBH

BH === ∫ 411)1(

Entanglement entropy between a single site and other subsystems:

Thermal entropy per site:

Bekenstein-Hawking Entropy per site:

Page Curve Conjecture ⇒ after Page time)1()1()1(BHthermalEE SSS ==

Before Page Time

After Page Time

Last Burst of BH

)1()1()1(BHthermalEE SSS >>>>

( )1

( )1

Page time1.0)0(=

Tω

Actually, (Hotta-Nambu-Yamaguchi 2017)

)0(ln)0()1(

TT

TTSEE ≈

High temperature regime:

2ln)0()1(

TTSthermal ≈

2)0()0(4

≈

TT

TNSBH ω

>>

>> ←due to Soft Hair Contribution

←due to Time-Dependence of Survival Probability p(t)

)0(T : Initial Temperature

( ) dpT

NET

dENpT

NpEddS thththBH

)1()1()1(

+==

dpT

ESNdSdS thththermalBH

−−=

)1()1(

TdEdS th

th

)1()1( =

( ) ( )ββ /1'2ln )1()'(/1

00 th

pTp

thermalBH EddpNNpSS ∫∫+−=

Large discrepancy is generated by this term

One-body Gibbs State Relation:

First Law:

Averaged Thermal Entropy: )1(

ththermal NpSS =

( ) dpT

NET

dENpT

NpEddS thththBH

)1()1()1(

+==

dpT

ESNdSdS thththermalBH

−−=

)1()1(

TdEdS th

th

)1()1( =

=

+= 2

2

2

2 )0()0()0(0T

TOT

TOT

TSBH

Large discrepancy is generated by this term

One-body Gibbs State Relation:

First Law:

Averaged Thermal Entropy: )1(

ththermal NpSS =

BAC

Feedback to Firewall ParadoxLate radiation

Early radiationBH

B

AC

Late radiation

Early radiationBH

S Soft hair

Early Hawking radiation is entangled with zero-energy BMS soft hair, and that late Hawking radiation can be highly entangled with the degrees of freedom of the BH, avoiding the emergence of a firewall at the horizon.

SummaryWe construct a model of multiple qubits that reproduces thermal properties of 4-dim Schwartzschild BH’s. After Page time, entanglement entropy, thermal entropy and Bekenstein-Hawking entropy are not equal to each other. Entanglement entropy exceeds thermal entropy and BH entropy.

Our result suggest that early Hawking radiation is entangled with zero-energy soft hair, and that late Hawking radiation can be highly entangled with the degrees of BH freedom, avoiding the emergence of firewalls at horizons.