A Brief Review of Quantum Information Aspects of Black Hole Evaporation

Reference: M. Hotta and A. Sugita, Prog. Theor. Exp. Phys, 123B04 (2015).M. Hotta, R. Schützhold and W. G. Unruh, Phys. Rev. D 91, 124060 (2015). M. Hotta, Y. Nambu, and K. Yamaguchi, arXiv:1706.07520.+ ….

Unitarity breaking?

Information is lost!?

Thermalradiation

Thermalradiation

Only thermal radiation?

Largeblack hole

Small black hole

Ψ

thermalρ

thermalUU ρ≠ΨΨ †ˆˆ

The Information Loss ProblemHawking (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

Purification Problem of Hawking Radiation: from a modern viewpoint of information loss

Hawking radiation system

Partner system

(1) Nothing, Information Loss

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

(3) Baby Universe (Dyson,..)

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

What is the final 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

Hawking-likeRadiation

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

(4) Radiation

○ Firewall A FIREWALL on the horizon burns out free-fall observers. The inside region of BH does not exist!This is argued from monogamy of entanglement.

Free-fallobserver

Large BH

FIREWALL

This scenario iscriticized in this talk.

(1) Nothing, Information Loss

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

(3) Baby Universe (Dyson,..)

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

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

What is the final purification partner of the Hawking radiation?

Gravitational zero energy states with supertranslation charges

Hawking (2015)

Canonical typicality for non-vanishing Hamiltonians yields non-maximal entanglement among black holes and the Hawking radiation,which makes spacetimes smooth without breaking monogamy. Thus, no reason to have BH firewalls.

~MASSAGE (1)~

Typical states must be Gibbs states for smaller quantum systems with very high precision. If we have stable Gibbs states for old Schwarzschild BH’s (and small AdSBH’s), the heat capacity must be positive. Actually, it is negative. Thus the states of BH evaporation are not typical!

Inevitable Modification of the Page Curve

~MASSAGE (2)~

Microcanonical states are far from typical for finite-temperature old BH’s,even though canonical states are typical and the large entropy O(V) is merely different from the microcanonical entropy by O(1). Entropy difference between a typical state and the canonical state must be exponentially small!

~MASSAGE (3)~

EI∝ρ

( ) 0~)exp( VOSS thermaltypical γ−≤−

You cannot use the microcanonical state in the typicality argument for evaporating black holes.

EBH I∝ρ

BHHR

E

δ−E

Plan of this talk

I. Brief Review of Quantum Entanglement II. Lubkin-Lloyd-Pagels-Page Theorem,

Page Curve Hypothesis and BH Firewall Conjecture

III. Canonical Typicality for Non-Vanishing Hamiltonians

Yields No Firewalls

IV. Summary

I. Brief Review of Quantum Entanglement

Quantum entanglement is a correlation, which cannot be generated by local operations and classical communication for many-body systems and quantum fields.

As an example, let us consider a ½ spin system in the up state of the z component of the spin for simplicity.

++

Let us assume that any quantum operation for the system is available.

Then, from this initial state, any quantum state of the system can be generated:

[ ] .∑±=

=++Γ=s

sss uupρ

Γ

This can be proven in the following way.

=−

=+

10

,01

.−−++ −= ξξξξσ

( )1,0

,

=+≥

=

−=

−+±

+

−−

−

++

ppp

pp

pp

ξξ

For the up state , we first measure an observable given by

Then, eigenvalue of emerges with probability . ±pThus, the average state is given by

+

σ

[ ] −−−+++ +=++Γ ξξξξ pp1

( )0=−+ ξξ

1±

[ ][ ].

12

ρ

ξξξξ

=+=

+=++ΓΓ

−−−+++

−−−+++

uupuup

UUpUUp ††

.±± = uU ξNext, let us perform a unitary operation which satisfies

This yields the final state which we request.

ρ++

ρ

The inverse process can be also achieved. Let us measure the z component of the spin. Then, the state collapses to one of the eigenstates.

±±=± ρq±±⇒ρ with probability

If the down state emerges, the spin flip operation is performed so as to get the up state.

++⇒−− †

xx σσTherefore the state is transformed into the up state with unit probability.

ρ

++

Therefore, any quantum state can be transformed into an arbitrary different state by quantum operations.

ρ 'ρ

++

[ ]ρρ Γ='

[ ]'' ρρ Γ=

This result might sound trivial, but by using a similar argument for many-body systems, very interesting facts are revealed. The concept of entanglement is one of them, as seen next.

Entanglement is an index which describes a sort of quantum correlation between quantum systems. In order to grasp the concept of entanglement, let us next consider two ½ spin systems A and B.

A B

AA ++ BB ++⊗

Let us assume that each state is pure and the up stateat the first stage.

A BBA ρρ ⊗

From the result proven previously, it turns out that local operation to each system transforms the initial pure state just into a product of mixed states without any correlation.

Arbitrary state of A Arbitrary state of B

AΓ BΓ

A B

( ) ( )∑∞

=

⊗=⇒1

)(µ

µρµρµρ BAAB p

If classical communication is also allowed, we can show that the composite system gets classical correlation. (The term “classical” is used in the context of communication theory.)

,, 21 AA ΓΓ ,, 21 BB ΓΓ

0 11

Local Operations and Classical Communication (LOCC)

AA ++ BB ++⊗

( ) 1)(,01

=≥ ∑∞

=µ

µµ pp Arbitrary state of A Arbitrary state of B

(called Seperable State)

Available State Change:

Local Operations Local Operations

Classical Communication

Simple Example of Generating a Separable State with Correlation:

A BAσ

AA ++ BB ++⊗

At first, Alice measures for the system A in the initial state and obtains the eigenvalue with probability .

AAA µµµ

ξξµσ ∑±=

=1 ( )1,0

,

=+≥

=

−=

−+±

+

−−

−

++

ppp

pp

pp

AA ξξ

Aσ1±±p

Alice Bob

A B1±=µAσ

AA µµ ξξ BB ++⊗

Next Alice announces the obtained eigenvalue to Bobvia a classical channel.

1±=µ

Alice Bob

A B BµΓ1±=µAσ

[ ]BBB ++Γ⊗ µ

AµΓ

Finally, Alice and Bob perform arbitrary local operations dependent on the eigenvalue for their systems.

Alice Bob

[ ] [ ]BBBBAAAA ++Γ=Γ= µµµµ µρξξµρ )(,)(

[ ]AAA µµµ ξξΓ )()( µρµρ BA ⊗

µ

A B BµΓ1±=µAσ

AµΓ

Then, the average state of the composite system becomesa separable state with correlation.

Alice Bob

)()( µρµρµ

µ BAp ⊗∑±=

Because LOCC is just a classical process in communication, entanglement between A and B is defined as a quantum effect such that the amount of entanglement never increases by LOCC. Besides, a product state of pure states has minimum entanglement and the value is zero.

( ) ( ) .0)(1

=

⊗∑

∞

=µ

µρµρµ BAAB pENT

( ) 0=++⊗++ BBAAABENT

Entangled states are defined as non-seperable states:

( ) ( ) 0)(1

>

⊗≠∑

∞

=µ

µρµρµρ BAABAB pENT

This definition leads to a fact thatany separable state has zero entanglement.

Entanglement is a purely quantum effect and generates interesting phenomena, including break of Bell’s inequality and quantum teleportation.

A BQuantum Channel

Ψ1,0

Increasing entanglement requires non-local operations directlyconnecting A to B, or quantum channels which can transport not only classical but also quantum information (quantum states).

A BNon-Local Operation

interaction

Example: entanglement of pure state of two systems

[ ] [ ]ABABABABABBA ΨΨ=ΨΨ= Tr,Tr ρρ

[ ] [ ]BBBAAAABABABEE ρρρρ lnTrlnTr)( −=−=ΨΨ

Entanglement Entropyas one of entanglement indices

Contracted State:

[ ]−−+++=2

1Bell

−−−Ψ+++Ψ +=Ψ BABAAB vupvup

Schmidt Decomposition of Pure State:

'''' ,,1,0 ssVsBsssAsAss

s vvuupp δδ ===≥ ∑±=

Ψ±Ψ

( ) 2ln=⇒ BellBellEEAB

The Bell states attain the maximum value of entanglement entropy.

( ) ss

sABABAB ppEE Ψ±=

Ψ∑−=ΨΨ⇒ ln

II. Lubkin-Lloyd-Pagels-Page Theorem, Page Curve Hypothesis and BH Firewall Conjecture

The BH firewall conjecture is based on the Page curve hypothesis, and the hypothesis was inspired bythe Lubkin-Lloyd-Pagels-Page(LLPP) theorem.

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:

Maximal Entanglement between A and B

AB NN ≥

∑=

=AN

nBnAn

AAB

vuN

Max1

~1

Orthogonal unit vectors

⇒= AA

A IN1ρ

Let us assume that Hilbert-space dimensions of black holes and Hawking radiation become finite due to quantum gravity effect.

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

bhpage MM 7.0≈

BHHR

GABHSHRBH BH

EE 4ln1 =≈⇒<<<<

HRBH lnln ≈○

○

<<Page Time>>

Maximum value of EE is attained at each time.

OLD BH

Simplified Page Curve

Page Curve Hypothesis for BH Evaporation:

Proposition I:When the dimension of the BH Hilbert space is much larger or less than that of Hawking radiation, BH and HR in a typical pure state of quantum gravity share almost maximal entanglement. In other words, quantum states of the smaller system is almost proportional to the unit matrix.

Proposition II:

thermalEE SS = of the smaller system.

BAHR ∪=CBH =

BA

Late radiation

C Early radiation

CBA ,,1<<

ACB <<

ABCΨ

Page Time

OLD BH

OLD BH ⇒

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

III. Canonical Typicality for Non-Vanishing Hamiltonian,and No Firewalls

Problem for Proposition I of Page Curve Hypothesis:

The area law of entanglement entropy is broken in a sense of ordinary many body physics, though outside-horizon energy density in BH evaporation is much less than the Planck scale.

|||| BASEE ∂=∂∝

A

Bfor low excited states

ABAB0≈Ψ

← standard area law of entanglement entropy

AVA 2=

∑=

=Ψ||

1

~||

1 A

nBnAnAB

vuA

AEE VAS ∝= ln

Not area law, but volume law forhighly excited states!

A

B

LLPP Typicality ⇒

Qubit network model

This is because zero Hamiltonian (complete degeneracy) is assumed in the LLPP theorem. This is also an implicit premise of the Page curve hypothesis.

.0=ABH

In BH physics, we have to treat canonical typicality with non-vanishing H in a precise manner.Then non-maximal entanglement emergesand makes near-horizon regions smooth. Thus no firewalls appear.

M. Hotta and A. Sugita, Prog. Theor. Exp. Phys, 123B04 (2015).

Microcanonical Energy Shell （not a tensor product of the sub-Hilbert spaces）

jjjAB EEEH =

[ ]{ }EEEjE j ,|)( δ−∈=∆

= ∑∆∈ )(

)(Ej

jjES EcEVMicrocanonicalEnergy Shell:

[ ]

+=

ΨΨ=

∑−

=

1

1

2

1 AN

nnnA

A

ABABBA

TTIN

Trρ

A B )(EVESAB∈ΨAN BN

[ ] [ ] '',0, nnAnnnnn NTTTrTTrTT δ===†

Bloch Representation of higher-dim quantum states

[ ]Ann TTrT ρ=

.)(

∑∆∈

=ΨEj

jjABEc

== ∑∆∈ )(

dim)(dimEj

jjES EcEVD

1))(exp( >>∝ BB NVD γ

BA NN <<

for ordinary systems.

Volume of B

Hilbert space dimension of B

nTEvaluate for

∫= pdcpcff D)()(

( )kjjkkkjjkjkj

jjjj

DDcccc

Dcc

''''''

''

)1(1

,1

δδδδ

δ

++

=

=

∗∗

∗

−∝ ∑

∆∈

1)()(

2

Ejjccp δ 1)( =∫ pdcp D

Uniform Ensemble on Mircrocanonical Energy Shell:

( )1

11 1

1

222

+

≤− ∑

−

= DT

NTr

AN

nn

AAAA ρρ

( )( ) 110exp)exp( 23 >>=∝ OVD Bγ

Max eigenvalue

))(exp( BAA VO γρρ −≤−

( )1

22

+≤−

D

TTT

nnn

BN independent!

Sugita Theorem (2006)

Hotta-Sugita (2015) as a response to a BH firewall debate with Daniel Harlow

Private Communication with D. Harlow about “Jerusalem Lectures on Black Holes and Quantum Information”, arXiv:1409.1231 .

++= BA HHHNegligibly small

.constEEH BA =+=

Harlow argued a canonical typicality in a weak interaction limit.

ABΨ [ ]ABABBA Tr ΨΨ=ρ

[ ]AAAAB TrS ρρ ln−=

)(, βAthermalAB SS ≈

( )AA

A HZ

βρ −≈ exp1

A BAB

Ψ

.constEE BA =+

AB NN >>

Without any proof, Harlow argued these only in the weak interaction limit.

BAC

FIREWALL?

( )[ ] ?)( 2 ∞=∂ BCxTr ρϕ

Harlow pointed out a possibility that BH firewalls may exist even after canonical typicality with non-zero Hamiltonian.

,BAHR ∪= CBH =

CBA ,,1<< ACB <<

( )( )( ) ( )CB

CBBC

HHHHββ

βρ−⊗−=

+−∝expexp

exp

No Correlation, just like CB II ⊗

arXiv:1409.1231Harlow,

However, the worry is useless.We can prove nonexistence of firewalls for general systems by using the general theory of canonical typicality.

( )( )BCCBBC VHH ++−∝ βρ expACB <<

Irrespective of the strength of the interaction between B and C,

M. Hotta and A. Sugita, Prog. Theor. Exp. Phys, 123B04 (2015).

( )( )BCCBBC VHH ++−∝ βρ exp

BCVBH CH

[ ] ?!lim ∞=BCBCVTr ρ0→BCV

Harlow’s worry:

Actually, a correlation exists between B and C for small interactions.

( )( )'''exp' '''''' CBCBCBBC VHH ++−∝= βρρ

[ ] [ ] ∞<= BCCBBCBC VTrVTr '''ρρ

BCV''BH ''CH

'''CBV

''' '''' CBCBBCCB VHHVHH ++=++

Merely an ordinary local operator of C’

Border shift does not change physics at all.

No firewall!

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

A BAB

ΨEEE BA =+

[ ]ABABBA Tr ΨΨ=ρ

))((/))(exp( EZHE AAA ββρ −=

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

Arbitrary state:

Gibbs state:

BHAH

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

If a stable Gibbs state exists, it attains the maximum of the von Neumann entropy with average energy fixed.

0=Iδ

)(/)exp( ββρ AAA ZH−=

[ ] [ ]AAAA TrTr ρρρρ lnln −≤−

Conventional “proof”:

Unfortunately, the typicality argument cannot be applied to Schwarzschild BH evaporation!

Actually, from our result, the typical state must be a Gibbs state, but…

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

GTME BH π8

1== 0

81

2 <−=GTdT

Edπ

( )[ ]BHBH HTrZ ββ −= exp)(

( )02

2

>−

=T

EE

dTEd

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

Thus, a system of a black hole and Hawking radiation is not in typical states, at least in the sense of the Page curve hypothesis, during BH evaporation. Because we have no stable Gibbs state,“ thermal entropy” of Schwarzschild BH ( ) is not needed to be a upper bound of entanglement entropy.

)4/( GA

)4/( GASS thermalEE ≈≤

ABΨ

ABU

MicrocanonicalEnergy Shell

.constEtotal =

ABΨ is a typical state with almost certaintyafter a relaxation time.

In ordinary quantum systems,

HRBH +Ψ

HRBH IU ⊗

)(emissionU

Sub-Hilbert space of non-typical states

The state of BH evaporation can be non-typical until the last burst.

Fast scrambling of BHdoes not contribute to entanglement between BH and HR.

Non-chaotic HR emission generated by smooth space time curvature outside horizon

If so, how is the Page curve modified?

The moving mirror model is totally unitary. So we are able to learn how the information can be retrieved.

The model is a tool to explore the Page curve hypothesis and its modification by using various mirror trajectories.

+x−x

mirrortrajectory

)( −+ = xfx

)(ˆ +xinϕ

)(ˆ −xoutϕ

)( −+ = xfxMirror Trajectory:

))((ˆ)(ˆ −− = xfx inout ϕϕ

( ) ))((ˆ)(ˆ,ˆ −+ −= xfxtx inin ϕϕϕ

0|ˆ)( =−+= xfxϕBoundary

Condition:

Solution:

Scattering Relation:

( )xtx ±=±

∂∂

−∂∂

−= −−

−−

−−

−−−

−−

223

)()(

23

)()(

2410)(ˆ0

xfxf

xfxfxT inin π

:ˆˆ::ˆˆ:ˆoutoutT ϕϕϕϕ −−−−−− ∂∂=∂∂=

))((ˆ)(ˆ −− = xfx inout ϕϕ

Out-going energy flux:

derived from

(Dynamical Casimir Effect)

( )−−−+ +−== xexfx κ

κ1ln1)(

Moving Mirror Model in 1+1 dim. mimics 3+1 dim. spherical gravitational collapse.

−− ≈−∞≈ xxf )(

( )−− −−≈∞≈ xxf κκ

exp1)(

The mirror does not move in the past.

The mirror accelerates and approaches the light trajectory,

.0=+xacceleration

2

120)/1(ˆ0 TxT inin

πκ =>>−−−

πκ2

=Tacceleration

The mirror emits thermal flux in the late time.

Temperature:

12exp

10ˆˆ0 )()(

−

∝ω

κπ

ωω inoutout

in aa †

Hawking Radiation!

+x−x

Hawking Radiation

Mirror Trajectory:

( )[ ]−+ −+−= xx κexp1ln

1+1 dim Moving Mirror Modelas analogue of Hawking radiation emission

πκ2

=T

At rest → Gradually accelerated→ Uniform acceleration

+x−x

mirrortrajectory

)( −+ = xfx

−1x

−2x A

B

Bε−−

2x

ε+−1x

[ ]ABABBA Tr ΨΨ=ρ

[ ]AAAB TrS ρρ ln−=

Entanglement Entropy of Emitted Radiation

( )

∂∂−

= −−

−−

−−

)()()()(ln

121

122

2

12

xfxfxfxfSAB ε

Entanglement entropy:

Lattice spacing (UV cutoff)

( )( )

−∂∂

−=

−=

−−−−

−−

−−

2

1212

2

12

)(

)()(

)()(ln121

xxxfxf

xfxf

SSS ABvac

ABren

Renormalized entanglement entropy:

Holzhey-Larsen-Wilczek

+x−x

Hawking Radiation

Mirror Trajectory:

( )( )( )

−+

−+−= −

−+

hxxx

κκ

exp1exp1ln

BH Evaporation Trajectory

At rest → Uniform Acceleration → At rest

0 100 200 300 400 500 600 7000.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.010

x

y

)( −−− xT

−x

500,1 == hκ

0 100 200 300 400 500 600 7000

5

10

15

20

x

y

EES∆

−2x

2,500,1 1 −=== −xhκ

Thanks to Daniel Harlow

Page time

Page Curve for BH Evaporation Trajectory

BHMBHM

HRBH0⊗Ψ

HRBH +Φ

Young black hole Old black hole

In order to reproduce the Page curve, very strange time evolution induced by nonlocality is required for the mirror trajectories!

Quite different time schedules of information leakage for black holes with the same mass.

+x−x

Quantum Gravity

Mirror Trajectory:

Hawking Radiation

( )( )( )

−+

−+−= −

−+

hxxx

λκ

exp1exp1ln

Planck-energy last burst with a tiny amount of information

Possible modificationof the Page curve,assuming local dynamics.

κλ >>Planck energy scale

BH scale

0 100 200 300 400 500 600 7000

10

20

30

40

50

x

y

EES∆

−2x

2,500,100,1 1 −==== −xhλκ

Modified Page Curve for BH Evaporation

“Page time”

Contributionof zero-point fluctuationwithout energy cost for information storage

All of the information comes out at the end by zero-point fluctuation flow.

+x

−x

MirrorTrajectory

HawkingParticle

Zero-PointFluctuation

Entangled Pairin Vacuum State

Quantum Gravity at the End

Information Retrieval without Energy at the End

The entangled partner of the Hawking particle is zero-point fluctuation with zero energy. (Wilczek, Hotta-Schützhold-Unruh)

Particle A Particle B

HawkingParticle Zero-Point Fluctuation Flow

with Zero Energy

Entanglement

Entangled Partner

(Wilczek, Hotta-Schützhold-Unruh, cf. Hawking-Perry-Strominger)

Therefore, the information loss problem may not be so serious, because small (or zero) amount of energy is enough to carry huge amount of quantum information in principle.

0=r

Zero-point fluctuation

Hawking Radiation

in0

Apparenthorizon

Collapsing Shell

Black Hole EvaporationZero-point fluctuation

BA

Late radiation

Early radiationC

AABCBC SSS == ,0

BCABCBAB SSSS −+≥

Strong subadditivy:

Strong Subadditivity “Paradox”

ABAB SSS +≥Page Curve Hypothesis ABA SS >

No Drama:

BA

Late radiation

Early radiationC

Strong Subadditivity “Paradox”

ABABA SSSS +≥>

Page Curve Hypothesis ABA SS >

BS>0

BA

Late radiation

Early radiationC

Strong Subadditivity “Paradox”

Remnant& Zero-Point Fluctuation Flow

ABA SS <

until the last burst.

Thus, no strong subadditivity paradox!

0 100 200 300 400 500 600 7000

10

20

30

40

50

x

y

EES∆

−2x

We don’t care the no drama condition breaks at the last burst, because the horizon is affected by quantum gravity.

0=> + flcptzrABA SS

The last burst

Summary ○ Adopting canonical typicality for nondegenerate systems with nonvanishing Hamiltonians, the entanglement becomes non-maximal, and BH firewalls do not emerge.

○ Typical states must be Gibbs states for smaller quantum systems. If we have stable Gibbs states for old Schwarzschild BH’s (and small AdS BH’s), the heat capacity must be positive. Because it is actually negative, the states of BH evaporation are not typical.

⇒ Inevitable Modification of the Page Curve

Note: for a large AdS BH and Hawking radiation in a thermal equilibrium, the entanglement entropy equals the thermal entropy of the smaller system.

IV. Quantum Measurement of BH Firewalls

“Another Firewall Paradox”

M. Hotta, J. Matsumoto and K. Funo, Phys. Rev. D89, 124023 (2014)

Free-fall observers do not encounter firewalls when come across event horizonin an average meaning.

InfallingParticleLarge BH

Classical Horizon

The strong subadditivity paradox has been resolved.

However, we have another possibility of firewall emergence.

The point is Reeh-Schlieder theorem in quantum field theory.

Reeh-Schlieder theorem:The set of states generated from by the polynomial algebra of local operators in any bounded spacetime region is dense in the total Hilbert space of the field. Thus, in principle, any state can be arbitrarily closely reproduced by acting a polynomial of local operators of E’ on .

in0

in0

inn E

nnnnn xdxOxOxxa 0)(ˆ)(ˆ),,(

'111∑∫≈Ψ∀

E’ L’

Ψ⇒in0

∑∫n E

nnnnn xdxOxOxxa

'111 )(ˆ)(ˆ),,(

in0

f

'L

'E

L

E

−x

fOEˆ−−T

EL

EO

'ˆ

EO

EO

.0 fin →

Note that the Reeh-Schlieder property is maintained in the time evolution:

xdxOxOxxaO n

n EnnnnE ∑∫= )(ˆ)(ˆ),,(ˆ

111

Even in the future infinity, we may remotely generate any excitation with some probability smaller than 1.

L

E

Li

?FW

)( fwh xgx =+

∑=i

LEi if ψ

Measured Firewall!

Imagine that, besides the background Hawking radiation, a wave packet with positive energy of the order of the radiation temperature appears at . Then the firewall (FW) appears at .

fwxx =−

If measurement operator is constructed from Reeh-Schlieder operation, an arbitrary post-measurement state including firewalls can emerge.

fwxx =−

)( fwh xgx =+

Firewall Measurement Paradox:

Eiψ

EiE OM ˆˆ ∝

V. Informational Cosmic Censorship Conjecture

“Resolution of the Paradoxfrom a viewpoint of

Quantum Measurement Energy Cost”

M. Hotta, J. Matsumoto and K. Funo, Phys. Rev. D89, 124023 (2014)

'ˆ

iEM

'Li 'L

'E

Li

LBecause the mirror merely stretches the modes of the field, the future measurement is equivalent to a past measurement for the in-vacuum state.

'ˆˆ

iEiE MM ⇔

E

We can analyze the problem using past infinity.

The local measurements generally inject energy on average to the system in owing to its passivity property (Pusz and Woronowicz).Thus the measurements always require an energy cost.

Though the Reeh-Schlieder theorem is mathematically correct, it does not guarantee that the measurement energy to create is finite.

Li

in0

The two-point correlation functions for non-singular measurements simply obey a power-law decay as a function of the distance. ⇒ No Firewalls!

If is regular, no outstanding peak of energy flux appears.

'' ˆˆiEiE MM†

Hotta, Matsumoto and Funo,Phys. Rev. D89, 124023 (2014).

( ))( fwhfw xgxE −+δ

planckfw El

OE <<

<

π121

'E 'L

+x

++T

Lil

⇒No Firewall appears!

)( fwh xgx =+

If we assume FW appears at with less-than-1 probability in a finite-energymeasurement, then

Hotta, Matsumoto and Funo,Phys. Rev. D89, 124023 (2014).

)( fwh xgx =+

'E 'L

+x

++T

l+E

fw

fw

rlErE

Eπ121−

≥+

∞→+ErlE fw π12

1→

Energy cost of measurement:

Energy cost of FW measurement diverges!

fwE

Hotta, Matsumoto and Funo,Phys. Rev. D89, 124023 (2014).

)1(Or =

Huge amount of energy for firewall measurement

Black hole is formed in the measurement region during the preparation of huge energy for the firewall measurement

Event Horizon

0110101011

0110101011

0110101011

0110101011iEM

⇒ Firewall Information Censorship

Informational extension of Cosmic Censorship

M. Hotta, J. Matsumoto and K. Funo, Phys. Rev. D89, 124023 (2014)