A Brief Review of Quantum Information Aspects of Black Hole - - PowerPoint PPT Presentation

A Brief Review of Quantum Information Aspects

• f 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!?

Thermal radiation Thermal radiation Only thermal radiation? Large black hole Small black hole

Ψ

thermal

ρ

thermal

U U ρ ≠ Ψ Ψ

†

ˆ ˆ

The Information Loss Problem

Hawking (1976)

Why is the information loss problem so serious?

Too small energy to 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,

• nly 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

HR n HR n HR

n n p

∑

= ρ

HR

A

HR

HR HR

A n n HR n HRA

u n p

∑

= 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

• f the Hawking radiation?

(4) Radiation

○ Black Hole Complementarity

Infalling Particle

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

Large BH Classical Horizon

(4) Radiation

○ Black Hole Complementarity

Hawking-like Radiation Infalling Particle Stretched Horizon Induced by Quantum Gravity

From the viewpoint of

• utside 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-fall

• bserver

Large BH FIREWALL

This scenario is criticized 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

• f 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 AGE (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 AdS BH’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 AGE ( (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 AGE ( (3)~

E

I ∝ ρ

( )

~ ) exp( V O S S

thermal typical

γ − ≤ −

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

E BH

I ∝ ρ

BH HR 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 s s s

u u p ρ

Γ

This can be proven in the following way.

      = −       = + 1 , 1

.

− − + +

− = ξ ξ ξ ξ σ

( )

1 , , = + ≥         =         − =

− + ± + − − − + +

p p p p p p p ξ ξ

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

• f emerges with probability .

±

p

Thus, the average state is given by

+

σ

[ ]

− − − + + +

+ = + + Γ ξ ξ ξ ξ p p

1

( )

=

− + ξ

ξ 1 ±

[ ] [ ]

.

1 2

ρ ξ ξ ξ ξ = + = + = + + Γ Γ

− − − + + + − − − + + +

u u p u u p U U p U U p

† †

.

± ± = u

U ξ

Next, 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.

± ± =

±

ρ q ± ± ⇒ ρ

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

+ + ⇒ − −

† x x

σ σ

Therefore 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 + +

[ ]

ρ ρ Γ = '

[ ]

' ' ρ ρ 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

A A +

+

B B

+ + ⊗

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

A

B

B A

ρ ρ ⊗

From the result proven previously, it turns out that local

• peration 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

A

Γ

B

Γ

A

B

( ) ( )

∑

∞ =

⊗ = ⇒

1

) (

µ

µ ρ µ ρ µ ρ

B A AB

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.)

 , ,

2 1 A A Γ

Γ

 , ,

2 1 B B Γ

Γ 

1 1

Local Operations and Classical Communication (LOCC)

A A +

+

B B

+ + ⊗

( )

1 ) ( ,

1

= ≥ ∑

∞ = µ

µ µ p p

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

B

A

σ

A A +

+

B B

+ + ⊗

At first, Alice measures for the system A in the initial state and

• btains the eigenvalue with probability .

A A A µ µ µ

ξ ξ µ σ

∑

± =

=

1

( )

1 , , = + ≥         =         − =

− + ± + − − − + +

p p p p p p p

A A

ξ ξ

A

σ 1 ±

±

p

Alice Bob

A

B

1 ± = µ

A

σ

A A µ µ

ξ ξ

B B

+ + ⊗

Next Alice announces the obtained eigenvalue to Bob via a classical channel.

1 ± = µ

Alice Bob

A

B

B µ

Γ

1 ± = µ

A

σ

[ ]

B B B

+ + Γ ⊗

µ

A µ

Γ

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

Alice Bob

[ ]

[ ]

B B B B A A A A

+ + Γ = Γ =

µ µ µ µ

µ ρ ξ ξ µ ρ ) ( , ) (

[ ]

A A A µ µ µ

ξ ξ Γ

) ( ) ( µ ρ µ ρ

B A

⊗

µ

A

B

B µ

Γ

1 ± = µ

A

σ

A µ

Γ

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

Alice Bob

) ( ) ( µ ρ µ ρ

µ µ B A

p ⊗

∑

± =

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.

( ) ( )

. ) (

1

=         ⊗

∑

∞ = µ

µ ρ µ ρ µ

B A AB

p ENT

( )

= + + ⊗ + +

B B A A AB

ENT

Entangled states are defined as non-seperable states:

( ) ( )

) (

1

>         ⊗ ≠ ∑

∞ = µ

µ ρ µ ρ µ ρ

B A AB AB

p ENT

This definition leads to a fact that any 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

B

Quantum Channel

Ψ

1 ,

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

A

B

Non-Local Operation

interaction

Example: entanglement of pure state of two systems

[ ] [ ]

AB AB A B AB AB B A

Ψ Ψ = Ψ Ψ = Tr , Tr ρ ρ

[ ] [ ]

B B B A A A AB AB AB

EE ρ ρ ρ ρ ln Tr ln Tr ) ( − = − = Ψ Ψ

Entanglement Entropy as one of entanglement indices

Contracted State:

[ ]

− − + + + = 2 1 Bell

− − − Ψ + + + Ψ

+ = Ψ

B A B A AB

v u p v u p

Schmidt Decomposition of Pure State:

' ' ' '

, , 1 ,

ss Vs Bs ss As As s s

v v u u p p δ δ = = = ≥ ∑

± = Ψ ± Ψ

( )

2 ln = ⇒ Bell Bell EEAB

The Bell states attain the maximum value of entanglement entropy.

( )

s s s AB AB AB

p p EE

Ψ ± = Ψ

∑

− = Ψ Ψ ⇒ ln

• II. Lubkin-Lloyd-Pagels-Page Theorem,

Page Curve Hypothesis and BH Firewall Conjecture

The BH firewall conjecture is based

• n the Page curve hypothesis,

and the hypothesis was inspired by the Lubkin-Lloyd-Pagels-Page (LLPP) theorem.

A B

A

N

B

N

AB

Ψ

AB

Ψ

Typical State of AB

B A

N N <<

[ ]

AB AB B A

Tr Ψ Ψ = ρ

[ ]

A A A EE

Tr S ρ ρ ln − =

A EE

N S ln ≈

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

A A A

I N 1 ≈ ρ

Lubkin-Lloyd-Pagels-Page Theorem:

Maximal Entanglement between A and B

A B

N N ≥

∑

=

=

A

N n B n A n A AB

v u N Max

1

~ 1

Orthogonal unit vectors

⇒ =

A A A

I N 1 ρ

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

H HR H BH dim , dim = =

HR ln

EE

S

bh page

M M 7 . ≈

BH HR

G A BH S HR BH

BH EE

4 ln 1 = ≈ ⇒ << <<

HR BH ln ln ≈

○ ○

<<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:

thermal EE

S S =

• f the smaller system.

B A HR ∪ =

C BH =

B A

Late radiation

C

Early radiation

C B A , , 1<<

A C B <<

ABC

Ψ

Page Time OLD BH

OLD BH ⇒

B A C

        ⊗         = =

C B BC BC

I C I B I BC 1 1 1 ρ

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

Harrow-Hayden

NO CORRELATION BETWEEN B AND C!

B A C

        ⊗         =

C B BC

I C I B 1 1 ρ

FIREWALL!

( )

[ ]

∞ = ∂

BC

x Tr ρ ϕ

2

) (

      =               −

2 2

1 ) ( ) ( ε ρ ε ϕ ϕ O x x Tr

BC C B C

x

B

x

→ ε

ε

• III. Canonical Typicality

for Non-Vanishing Hamiltonian, and No Firewalls

Problem for Proposition I

• f 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.

| | | | B A SEE ∂ = ∂ ∝

A B

for low excited states

AB AB

≈ Ψ

← standard area law of entanglement entropy

A

V

A 2 =

∑

=

= Ψ

| | 1

~ | | 1

A n B n A n AB

v u A

A EE

V A S ∝ = ln

Not area law, but volume law for highly 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.

. =

AB

H

In BH physics, we have to treat canonical typicality with non-vanishing H in a precise manner. Then non-maximal entanglement emerges and 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）

j j j AB

E E E H =

[ ]

{ }

E E E j E

j

, | ) ( δ − ∈ = ∆

      = ∑

∆ ∈ ) (

) (

E j j j ES

E c E V

Microcanonical Energy Shell:

[ ]

        + = Ψ Ψ =

∑

− = 1 1

2

1

A

N n n n A A AB AB B A

T T I N Tr ρ

A B

) (E VES

AB ∈

Ψ

A

N

B

N

[ ] [ ]

' '

, ,

nn A n n n n n

N T T Tr T Tr T T δ = = =

†

Bloch Representation of higher-dim quantum states

[ ]

A n n

T Tr T ρ =

.

) (

∑

∆ ∈

= Ψ

E j j j AB

E c

      = =

∑

∆ ∈ ) (

dim ) ( dim

E j j j ES

E c E V D

1 )) ( exp( >> ∝

B B N

V D γ

B A

N N <<

for ordinary systems.

Volume of B Hilbert space dimension of B

n

T

Evaluate for

∫

= p d c p c f f

D

) ( ) (

( )

k j jk kk jj k j k j jj j j

D D c c c c D c c

' ' ' ' ' ' ' '

) 1 ( 1 , 1 δ δ δ δ δ + + = =

∗ ∗ ∗

        − ∝

∑

∆ ∈

1 ) (

) ( 2 E j j

c c p δ

1 ) ( =

∫

p d c p

D

Uniform Ensemble on Mircrocanonical Energy Shell:

( )

1 1 1

1 1 2 2

2

+         ≤ −

∑

− =

D T N Tr

A

N n n A A A A

ρ ρ

( ) ( )

1 10 exp ) exp(

23

>> = ∝ O V D

B

γ

Max eigenvalue

)) (exp(

B A A

V O γ ρ ρ − ≤ −

( )

1

2 2

+ ≤ − D T T T

n n n

B

N 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 .

 + + =

B A

H H H

Negligibly small

. const E E H

B A

= + =

Harlow argued a canonical typicality in a weak interaction limit.

AB

Ψ

[ ]

AB AB B A

Tr Ψ Ψ = ρ

[ ]

A A A AB

Tr S ρ ρ ln − =

) (

,

β

A thermal AB

S S ≈

( )

A A A

H Z β ρ − ≈ exp 1

A B

AB

Ψ

. const E E

B A

= +

A B

N N >>

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

B A C

FIREWALL?

( )

[ ]

? ) (

2

∞ = ∂

BC

x Tr ρ ϕ

Harlow pointed out a possibility that BH firewalls may exist even after canonical typicality with non-zero Hamiltonian. , B A HR ∪ =

C BH = C B A , , 1<< A C B <<

( ) ( ) ( ) ( )

C B C B BC

H H H H β β β ρ − ⊗ − = + − ∝ exp exp exp

No Correlation, just like

C B

I I ⊗

arXiv:1409.1231 Harlow,

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

( ) ( )

BC C B BC

V H H + + − ∝ β ρ exp

A C B <<

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

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

( ) ( )

BC C B BC

V H H + + − ∝ β ρ exp

BC

V

B

H

C

H

[ ]

? ! lim ∞ =

BC BCV

Tr ρ

→

BC

V

Harlow’s worry: Actually, a correlation exists between B and C for small interactions.

( ) ( )

' ' ' exp '

' ' ' ' ' ' C B C B C B BC

V H H + + − ∝ = β ρ ρ

[ ] [ ]

∞ < =

BC C B BC BC

V Tr V Tr '

' '

ρ ρ

BC

V '

' B

H

'

' C

H

'

' 'C B

V

' ' '

' ' ' ' C B C B BC C B

V H H V H H + + = + +

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 B

AB

Ψ

E E E

B A

= +

[ ]

AB AB B A

Tr Ψ Ψ = ρ )) ( ( / ) ) ( exp( E Z H E

A A A

β β ρ − =

[ ]

[ ]

thermal A A A A EE

S Tr Tr S = − ≤ − = ρ ρ ρ ρ ln ln

Arbitrary state: Gibbs state:

B

H

A

H

[ ] [ ]

( )

[ ]

( )

1 ln

2 1

− − − − − =

A A A A A A A A A

Tr E H Tr Tr I ρ λ ρ λ ρ ρ

If a stable Gibbs state exists, it attains the maximum

• f the von Neumann entropy with average energy

fixed.

= I δ

) ( / ) exp( β β ρ

A A A

Z H − =

[ ]

[ ]

A A A A

Tr Tr ρ ρ ρ ρ ln ln − ≤ −

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)

GT M E

BH

π 8 1 = =

8 1

2 <

− = GT dT E d π

( )

[ ]

BH BH

H Tr Z β β − = exp ) (

( )

2 2

> − = T E E dT E d

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 /( G A

) 4 /( G A S S

thermal EE

≈ ≤

AB

Ψ AB

U

Microcanonical Energy Shell

. const Etotal =

AB

Ψ

is a typical state with almost certainty after a relaxation time.

In ordinary quantum systems,

HR BH +

Ψ

HR BH

I U ⊗

) (emission

U

Sub-Hilbert space of non-typical states

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

Fast scrambling of BH does 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

mirror trajectory

) (

− + =

x f x

) ( ˆ

+

x

in

ϕ

) ( ˆ

−

x

• ut

ϕ

) (

− + =

x f x

Mirror Trajectory:

)) ( ( ˆ ) ( ˆ

− − =

x f x

in

• ut

ϕ ϕ

( )

)) ( ( ˆ ) ( ˆ , ˆ

− + −

= x f x t x

in in

ϕ ϕ ϕ | ˆ

) (

=

− + =

x f x

ϕ

Boundary Condition:

Solution: Scattering Relation:

( )

x t x ± =

±

                ∂ ∂ − ∂ ∂ − =

− − − − − − − − − − − 2 2 3

) ( ) ( 2 3 ) ( ) ( 24 1 ) ( ˆ x f x f x f x f x T

in in

π

: ˆ ˆ : : ˆ ˆ : ˆ

• ut
• ut

T ϕ ϕ ϕ ϕ

− − − − − −

∂ ∂ = ∂ ∂ =

)) ( ( ˆ ) ( ˆ

− − =

x f x

in

• ut

ϕ ϕ

Out-going energy flux:

derived from

(Dynamical Casimir Effect)

( )

−

− − +

+ − = =

x

e x f x

κ

κ 1 ln 1 ) ( Moving Mirror Model in 1+1 dim. mimics 3+1 dim. spherical gravitational collapse.

− −

≈ −∞ ≈ x x f ) (

( )

− −

− − ≈ ∞ ≈ x x f κ κ exp 1 ) (

The mirror does not move in the past. The mirror accelerates and approaches the light trajectory,

. =

+

x

acceleration

2

12 ) / 1 ( ˆ T x T

in in

π κ = >>

− − −

π κ 2 = T

acceleration

The mirror emits thermal flux in the late time. Temperature:

1 2 exp 1 ˆ ˆ

) ( ) (

−       ∝ ω κ π

ω ω in

• ut
• ut

in

a a

†

Hawking Radiation!

+

x

−

x

Hawking Radiation

Mirror Trajectory:

( )

[ ]

− +

− + − = x x κ exp 1 ln 1+1 dim Moving Mirror Model as analogue of Hawking radiation emission π κ 2 = T

At rest → Gradually accelerated → Uniform acceleration

+

x

−

x

mirror trajectory

) (

− + =

x f x

− 1

x

− 2

x A B B

ε −

− 2

x ε +

− 1

x

[ ]

AB AB B A

Tr Ψ Ψ = ρ

[ ]

A A AB

Tr S ρ ρ ln − =

Entanglement Entropy

• f Emitted Radiation

( )

        ∂ ∂ − =

− − − − − −

) ( ) ( ) ( ) ( ln 12 1

1 2 2 2 1 2

x f x f x f x f S AB ε

Entanglement entropy:

Lattice spacing (UV cutoff)

( ) ( ) 

       − ∂ ∂ − = − =

− − − − − − − − 2 1 2 1 2 2 1 2 ) (

) ( ) ( ) ( ) ( ln 12 1 x x x f x f x f x f S S S

AB vac AB ren

Renormalized entanglement entropy:

Holzhey-Larsen-Wilczek

+

x

−

x

Hawking Radiation

Mirror Trajectory:

( ) ( ) ( )

     − + − + − =

− − +

h x x x κ κ exp 1 exp 1 ln BH Evaporation Trajectory

At rest → Uniform Acceleration → At rest

100 200 300 400 500 600 700 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010

x y

) (

− − −

x T

−

x

500 , 1 = = h κ

100 200 300 400 500 600 700 5 10 15 20

x y

EE

S ∆

− 2

x

2 , 500 , 1

1

− = = =

−

x h κ

Thanks to Daniel Harlow Page time

Page Curve for BH Evaporation Trajectory

BH

M

BH

M

HR BH

⊗ Ψ

HR BH +

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

( ) ( ) ( )

     − + − + − =

− − +

h x x x λ κ exp 1 exp 1 ln

Planck-energy last burst with a tiny amount of information

Possible modification

• f the Page curve,

assuming local dynamics.

κ λ >>

Planck energy scale BH scale

100 200 300 400 500 600 700 10 20 30 40 50

x y

EE

S ∆

− 2

x

2 , 500 , 100 , 1

1

− = = = =

−

x h λ κ Modified Page Curve for BH Evaporation

“Page time”

Contribution

• f zero-point fluctuation

without energy cost for information storage All of the information comes

• ut at the end by zero-point

fluctuation flow.

+

x

−

x

Mirror Trajectory

Hawking Particle Zero-Point Fluctuation Entangled Pair in 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 Hawking Particle 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.

= r

Zero-point fluctuation

Hawking Radiation

in

Apparent horizon

Collapsing Shell

Black Hole Evaporation

Zero-point fluctuation

B A

Late radiation Early radiation

C

A ABC BC

S S S = = ,

BC ABC B AB

S S S S − + ≥

Strong subadditivy:

Strong Subadditivity “Paradox”

A B AB

S S S + ≥

Page Curve Hypothesis

AB A

S S >

No Drama:

B A

Late radiation Early radiation

C

Strong Subadditivity “Paradox”

A B AB A

S S S S + ≥ >

Page Curve Hypothesis

AB A

S S >

B

S >

B A

Late radiation Early radiation

C

Strong Subadditivity “Paradox”

Remnant& Zero-Point Fluctuation Flow

AB A

S S <

until the last burst.

Thus, no strong subadditivity paradox!

100 200 300 400 500 600 700 10 20 30 40 50

x y

EE

S ∆

− 2

x

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

= >

+ flc pt zr AB A

S S

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 horizon in an average meaning.

Infalling Particle Large BH Classical Horizon

The strong subadditivity paradox has been resolved.

However, we have another possibility

• f 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 .

in

in

in n E n n n n n

x d x O x O x x a ) ( ˆ ) ( ˆ ) , , (

' 1 1 1

∑∫

≈ Ψ

∀

 

E’ L’

Ψ ⇒

in

∑∫

n E n n n n n

x d x O x O x x a

' 1 1 1

) ( ˆ ) ( ˆ ) , , (  

in

f

' L ' E L E

−

x

f OE ˆ

− −

T E L

O ˆ

ˆ

O E

O ˆ

. f

in →

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

x d x O x O x x a O

n n E n n n n E ∑∫

= ) ( ˆ ) ( ˆ ) , , ( ˆ

1 1 1

 

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

L

E

L

i

? FW

) (

fw h x

g x =

+

∑

=

i L E i

i f 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 .

fw

x x =

−

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

fw

x x =

−

) (

fw h x

g x =

+

Firewall Measurement Paradox:

E i

ψ

E iE

O M ˆ ˆ ∝

• V. Informational Cosmic

Censorship Conjecture

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

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

'

ˆ

iE

M

' L

i

' L ' E

L

i

L

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

'

ˆ ˆ

iE iE

M M ⇔

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.

L

i

in

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.

' ' ˆ

ˆ

iE iE M

M†

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

( )

) (

fw h fw

x g x E −

+

δ

planck fw

E l O E <<       < π 12 1

' E ' L

+

x

+ +

T

L

i

l

⇒No Firewall appears!

) (

fw h x

g x =

+

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

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

) (

fw h x

g x =

+

' E ' L

+

x

+ +

T

l

+

E

fw fw

rlE rE E π 12 1− ≥

+

∞ →

+

E

rl E fw π 12 1 →

Energy cost

• f measurement:

Energy cost of FW measurement diverges!

fw

E

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

) 1 ( O r =

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

iE

M ˆ

⇒ Firewall Information Censorship

Informational extension of Cosmic Censorship

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