エンタングルメントと ブラックホール防火壁パラドクス

Based on arXiv:1306.5057

M. Hotta, J. Matsumoto and K. Funo

Entanglement and Black Hole Firewall Paradox

Introduction

Infalling Particle Large BH

Large black-hole spacetimes are conventionally described by classical geometry.

Inside cannot causally influence outside. Nothing returns out of horizon.

Black holes emit thermal flux due to quantum effect.

bhB

bhGMk

cT

8

3

(Hawking, 1974)

bhbh GMr 2 Special & General Relativity

Quantum Theory

Statistical Mechanics

bh

bhGM

T8

1

bhbh

bh

bhbh dMGM

T

dMdS 8

G

AS bh

bh4

24 bhbh rA

First Thermodynamics Law:

Black Hole Entropy:

Black Hole Temperature:

Area of Event Horizon:

1 Bkc

bhbh GMr 2

Since the advent of Hawking radiation…

Information Loss Problem of Black Hole Evaporation

Information loss? Unitarity breaking?

Thermal radiation

Thermal radiation

Only thermal radiation?

Large black hole

Small black hole

thermal

HRn

HRnHR nnp

HRAHR

HRHR An

nHRnHRA

unp

Mixed state

Composite system in a pure state

Purification of Hawking Radiation?

Hawking radiation system

Partner system

(1) Nothing, Information Loss (2) Exotic Remnant (Aharonov, Giddings,…) (3) Baby Universe (Dyson,..)

(4) Emitted Radiation Itself ←Today’s Talk ○ Black Hole Complementarity (t’ Hooft, Susskind, …) ○ Fuzzi ball, Firewall (Mathur, AMPS, …)

What is the final entangled partner of Hawking radiation?

(5) Zero-Point Fluctuation in Local Vacuum regions (Wilczek, Hotta-Matsumoto-Funo) ←Today’s Talk

○ Black Hole Complementarity

Hawking-like Radiation

Infalling Particle

Stretched Horizon Induced by Quantum Gravity

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

Large BH

In the future, the whole radiation is not in a mixed state, but in a pure state.

○ Black Hole Complementarity

Infalling Particle

From the viewpoint of free-fall observer, the stretched horizon disappears. No drama happens across the horizon.

Large BH

Classical Horizon

○ Firewall From the viewpoint of outside observer, the stretched horizon absorbs and emits quantum information so as to maintain the unitarity. However, FIREWALL on the horizon burns out free-fall observers. The inside region of BH does not exist!

Free-fall observer

Large BH

FIREWALL

Mathur and AMPS derive this conclusion from quantum information theoretical point of view.

In this talk, we argue that the information theoretical reason of Mathur and AMPS , which derives the existence of firewalls, are wrong. They misuse the results of quantum information theory. However, another firewall paradox can be posed using quantum measurement theory. The new paradox is resolved from the viewpoint of measurement energy cost.

M.H., Jiro Matsumoto and Ken Funo, arXiv:1306.5057

Review: Mathur-AMPS Strong Subaddivity Argument

Preparation for the Argument

t

x

xtx

x

x

dxdxdxdtds

222

xy tanh

Penrose Diagram Preserving causality relations among events, a spacetime is mapped into a finite region.

Penrose diagram of Minkowski spacetime

A

B

A

BC C

222

2

11

yy

dydyds

0r

0r

0r

GMr 2

GMr 2

(Radial) Penrose diagram of black hole formation

Black hole formation via gravitational collapse

Event horizon

Maximal Entanglement

N

nBnAnAB

vuN

Max1

1

BnAn vu , : complete orthonormal basis

BABABAB

AABABBA

IN

MaxMaxTr

IN

MaxMaxTr

ˆ1ˆ

,ˆ1ˆ

NTrTrS BBAAAB lnˆlnˆˆlnˆ

A B AA NH dimBB NH dim

AB

AB

Typical State of AB

1 BA NN

ABABAB Tr

BBBAB TrS ˆlnˆ

BAB NS ln

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

B

B

B IN

ˆ1ˆ

Lubkin, Lloyd-Pagels, Page

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

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

In a typical pure state of old black hole(BH) and Hawking radiation(HR) after Page time, the internal system of BH is almost maximally entangled with a part of HR, and BH entropy is almost equal to entanglement entropy. (Page)

Then the discretized model suggests:

HRH

G

ABHH

HR

BHBH

dim

,4

expdim

HRlnNln

2

1

BHHRS ,

Nln2

1

bhpage MM 7.0

24exp

4exp

,

bf

bfGM

G

AN

fixedNHRBH

BH

HR

G

ABHBHHRSHRBH BH

4ln,1

NHRBH ln2

1lnln ○

○

<<Page Time>>

Page time

HRBHt

)(

BAHR

CBH

B

A

Late radiation

CEarly radiation

CBA ,,1

ACB

Mathur and AMPS apply this Page’s argument to late-time Hawking radiation.

After Page time,

Mathur-AMPS strong subadditivity paradox:

B

A

Late radiation

CEarly radiation

ACB

CBBCBC I

CI

BI

CBˆ1ˆ1ˆ1

CCBBBB

nB

nBnBBSSSbb

B AAAA

,01

mCmBm

nB

nBnCCBB AAAA

ccC

bbB

11

BC are almost maximally entangled with a part of A in a typical state after Page time.

ACAB AA ,

Harrow-Hayden

B

A

Late radiation

Early radiation C

CCBBBBSSS

AA ,0

CBBBBBBCAA

SSSS

Strong subadditivy:

0)||(0 CBISSS BCCB

0)||( CBI

No correlation between B and C!

Typical-State Condition:

Summary of Typical-State Condition:

0)||( CBI

No correlation between B and C!

B is almost maximally entangled with a part of A!

1)||( BAI

B

A

C

No-Drama Condition across Horizon:

222dxdtds

222)2exp( BB ddds

cosh)exp(1

,sinh)exp(1

B

B

x

t

cosh)exp(1

,sinh)exp(1

C

C

x

t

222)2exp( CC ddds

No-Drama Condition across Horizon:

222dxdtds

cosh)exp(1

,sinh)exp(1

B

B

x

t

cosh)exp(1

,sinh)exp(1

C

C

x

t

0B

0C

C B

t

x

BC

A

C

No-Drama Condition across Horizon:

nCBin nn

n,,exp0

..expˆ4

)(

..expˆ4

)(

),(ˆ

)(

0

)(

0

chiad

xx

chiad

xx

C

C

H

B

B

H

Rindler

CB

in aa 0ˆˆexp0)()(

††

Rindler mode function

Hxx

Unruh Relation:

B

00ˆ0ˆ)()(

Rindler

C

Rindler

Baa

ABCBBCAB SSSS

B

A

C

No-Drama Condition across Horizon:

AABCBC SSS ,0

nCBin nn

n,,exp0

0)||(0 BAISSS ABBA

0)||( BAINo correlation between A and B!

Summary of No-drama Condition:

0)||( BAI

No correlation between A and B!

B is highly entangled with C!

1)||( CBI

Strong Subadditivity Paradox:

Typical-State Condition of A and B: No-drama Condition across Horizon:

0)||( CBI

No correlation between B and C.

0BBA

S

B is almost maximally entangled with A.

0)||( BAI

0BCS

B is highly entangled with C in a pure state.

No correlation between A and B.

Monogamy conflict arises between A,B, and C!

1)||( BAI 1)||( CBI

A

B

C

Typical-State Condition of A and B:

0BBBB AA

S

Quantum monogamy

AB

A

B

C

No-Drama Condition across Horizon:

0 BCBCS

Quantum monogamy

Mathur and AMPS conjectured that no-drama condition does not hold and firewalls appear on the horizon! Free-fall

observer

FIREWALL

They argue that there does not exist the interior of BH!

THE DRAMA ON THE HORIZON

At least, one of the assumptions must be wrong!

Another bold remark by Harlow and Hayden:

Typical-State Condition of A and B: No-drama Condition across Horizon:

0)||( CBI

No correlation between B and C.

B is almost maximally entangled with A.

0)||( BAI

B is highly entangled with C in a pure state.

No correlation between A and B.

Strong Complementarity Conjecture

1)||( BAI 1)||( CBI

“Different Unitary Quantum Mechanics for Different Observers.”

For Outside Observers, For Infalling Observers,

BA

C

Typical-State Condition does not hold: High-energy entanglement structure is modified so as to yield the correct description of low-energy effective field theory.

0BCS

No-Drama condition does not imply purity of BC-system state: BC system is actually entangled with both A and zero-point fluctuation V. Main contribution comes from V.

0BBA

S

Our Argument:

V

No Strong Subadditivity Paradox!

M.H., Jiro Matsumoto and Ken Funo, arXiv:1306.5057

In order to make our argument concrete, let us consider a 1+1 dim. Moving Mirror Model.

The model mimics gravitational collapse of spherical matter shell and really generates Hawking radiation. Besides, it is completely unitary!

⇒This unitary model is one of the best quantum systems for checking the reasoning of Mathur and AMPS in quantum black holes.

x

x

mirror trajectory

)(

xfx

)(ˆ xin

)(ˆ xout

)(

xfxMirror Trajectory:

))((ˆ)(ˆ,ˆ xfxtx inin

0|ˆ)( xfx

Boundary Condition:

Solution:

xtx

dxuaxuaxinin

in

0

)()()(ˆ)(ˆˆ

†

xixu

exp

4

1)(

00ˆ )(in

ina

The in-vacuum state:

Standard quantization:

)(exp4

1)(

xfixv

dxvaxvaxinin

out

0

)()()(ˆ)(ˆˆ

†

dxuaxuaxoutout

out

0

)()()(ˆ)(ˆˆ

†

xixu

exp

4

1)(

Out-field is quantized as

))((ˆ)(ˆ xfx inout

The out-field can be described by in-field operators via .

Scattered-Wave Mode Function:

x

exf

1ln

1)(

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

xxf )(

xxf

exp

1)(

The mirror does not move in the past.

The mirror accelerates and approaches the light trajectory,

.0

x

acceleration

2

120)/1(ˆ0 TxT inin

2T

acceleration

The mirror emits thermal flux in the late time!

Temperature:

1exp

10ˆˆ0

)()(

T

aa in

inin

in

†

x

x

)(

xfx

Mirror Trajectory Describing Formation of Eternal BH

2

12ˆ TT

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

b

c

cb

b

b

c

a

a a

a

x

x

Mirror Trajectory Describing Complete Evaporation of BH

)(

xfx

0ˆ T

2

12ˆ TT

in0

out

Completely Unitary!

Our Argument (1): Typical-state condition should be modified in order to reproduce a correct description of low-energy field theory.

0BBA

S

C

CC

B

nBBin

AA

nn11

0

BA

C

'B

'A

FIREWALLS (Bousso, Marolf&Polchinski )

Continuum Limit of Typical-State Condition?

'C CB ,

Rindler Singurarity

T

T

CBBC IC

IB

ˆ1ˆ1ˆ

BC

Tat the boundary of B and C.

0BBA

S

.,,exp0

nBCAin nn

n

The scenarios of Mathur , AMPS and Bousso correspond to regularization without scale and translational symmetries !

CB

nBCA

CB

nBCAin

nn

nnn

1

1

,,

,,explim0

CBBCBC

CB

m

BC IC

IB

nn ˆ1ˆ1,,ˆ

1

⇒ Singluarity on Rindler horizon = FIREWALL

← (Would-be) typical state!

This picture can be reproduced by a limit with a bad regularization!

However, for regularization with scale and translational invariance,

CBR

R

BC IC

IB

HZ

ˆ1ˆ1ˆ2exp1

ˆ

nBCAin nn

n,,exp0

aa '','

n

BCAin nnn ,','exp0

0)||( CBI 0BBA

S

Our Argument (2): No-drama condition does not imply the purity of BC system, if local independence between A and B holds.

0BCS

No-Drama Condition across Horizon:

nCBin nn

n,,exp0

x

BCt

/)(

)(exp

i

B

tx

i

C

AB

AB

The support of Rindler mode functions are not localized! Overlap of A and B cannot be neglected.

←AMPS purity of BC system

Note: Strict localization cannot be attained by superposing one-particle states. (J. Knight 1961)

nCVABVin

fp

nnn

,,exp0

In order to impose strict localization of A, B and C, local vacuum regions must be introduced.

pV

fV

CB

A

fp

fp

VBCAV

nCVABV

nnn

,,exp

Localized BC system is actually entangled with A and zero-point fluctuations:

0ˆ T

0ˆ T

.0BCS

Our Argument (3):

<<Information Loss Problem>> It may be possible that main entangled partner of Hawking radiation is zero-point fluctuations in local vacuum regions.

Hawking Radiation

Hawking Radiation

Zero-point Fluctuation

Black Hole Evaporation

Zero-point Fluctuation

x

ex

1

1

cxx

xx

CB

A

A

B

C

What is the entangled partner of Hawking radiation?

Is it the final informative shock waves emitted by BH burst?

?C

No!!

1212

2

12

)()(

)()(ln

12

1),(

RR

RREE

xfxf

xfxfRRS

22 Rxx

R

x

11 Rxx

(Holzhey-Larsen-Wilczek)

Entanglement Entropy between R and its compliment:

This formula shows that shock waves, which are confined in a very narrow space, are not entangled with anything!

R

0

)()(

)()(ln

12

1),(

1212

2

12

RR

RREE

xfxf

xfxfRRS

Gosh! What is actually the entangled partner of Hawking radiation?

⇒ Zero-point fluctuations in local vacuum regions (Wilczek 1992, Hotta-Matsumoto-Funo 2013)

Entanglement without energy cost!

x

ex

1

1

cxx

xx

Non-entangled highly energetic pulse

Non-entangled highly energetic pulse

Hawking radiation with constant flux

Zero-point fluctuation entangled with HR

Zero-point fluctuation entangled with HR

2

12ˆ

HRTT

0ˆ T

0ˆ T hxT

ˆ

xT ˆABC

D0t

Tt

Hawking radiation with constant flux

2

12ˆ

HRTT

fV

pV

0r

Zero-point fluctuation entangled with HR

Hawking Radiation

Non-entangled highly energetic pulse in0

Apparent horizon

Collapsing Shell

Zero-point fluctuation entangled with HR

Non-entangled highly energetic pulse

Free-fall observers do not encounter firewalls when come across event horizon!

Infalling Particle Large BH

Classical Horizon

Firewall paradox of AMPS is resolved in this model!

In fact, NO FIREWALLS in an average meaning in moving mirror models.

)(ˆ xT

in0

However, we have another firewall paradox in the moving mirror model. The point is Reeh-Schlieder theorem in quantum field theory.

in0

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 of L′ can be arbitrarily closely reproduced by acting a polynomial of local operators of E′ on .

'L

'E

L

Ein0

in0

x

EOT

EL

EO

'ˆ

EO

The Reeh-Schlieder property ensures much entanglement of the system in the final state of the moving-mirror model.

EO

.0 outin

This property is maintained in the time evolution of

L

E

iEM

Li

?FW)( fwh xgx

i

LiE i

measured post-measurement state of L

Imagine that, besides the background Hawking radiation, a wave packet with positive energy of the order of the radiation temperature appears at in the post-measurement state of L. Then the above firewall (FW) appears at .

fwxx

Li

fwxx

)( fwh xgx

Firewall Measurement Paradox:

If measurement operator of E is constructed from Reeh-Schlieder operation, an arbitrary post-measurement state of L can emerge.

'ˆ

iEM

'Li

'L

'E

Li

L

)( fwh xgx

Resolution of the Paradox from a viewpoint of Quantum Measurement Energy Cost

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

in0

'ˆˆ

iEiE MM

IMM iE

i

iE ˆˆ† IMM iE

i

iE ''ˆˆ†

iniEiELininiELiEini

L MMxTMxTMxT 0ˆˆ)(00ˆ)(ˆ0)( '''''''

††

If is not singular, no outstanding peak of energy flux appears.

''ˆˆ

iEiE MM†

E

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

)(

0ˆˆ0

0ˆˆ)(0

''

''

fwhfw

iniEiEin

iniEiEinxgxE

MM

MMxT

†

†

planckfw Erl

E 12

1

E L

x

T

Lil

For any non-singular measurement,

)1(

0ˆˆ0

0ˆˆ0

''

''

OMMI

MMr

iniEiEin

iniEiEin

†

†

⇒No Firewall appears!

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

)( fwh xgx

in0

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

0110101011

Singular measurements, which yield firewalls, generally require preparation of a divergent amount of energy in the measurement region before the measurement is performed and this energy is expected to provide a large back reaction to the spacetime. The effect may cause formation of a new black hole in the measurement region and enclose the measurement device within the event horizon before it outputs results.

iEM

⇒ Firewall Information Censorship (Hotta-Matsumoto-Funo,2013)

Summary ○ Strong subadditivity paradox is a superficial one. If the models which allow correct continuum limit to low-energy field theory, typical-state condition does not hold. If strict localization of subsystems (A,B and C), no-drama condition does not imply purity of BC system. Acutually, both A and zero-point fluctuation are entangled with the BC system. No firewall is required by the entanglement monogamy argument.

○ Reeh-Schlieder theorem rises a measurement-based firewall problem. However, the amount of measurement energy of firewalls becomes divergent. The effect may cause formation of a new black hole in the measurement region and enclose the measurement device within the event horizon before it outputs results. ⇒ Firewall Information Censorship