Controlled Hawking Process

by Quantum Information

Masahiro Hotta

Tohoku University

arXiv:0907.1378

ブラックホールと量子エネルギーテレポーテーション

改め

AG

SBH 4

1

Black Hole Entropy Problem

However, in 1974, S. Hawking argued that black holes emit thermal radiation by taking account of quantum effect of fields. This essentially establishes the belief that black holes are thermo-dynamical existence, which have entropy proportional to event horizon area.

In classical theory, black holes are considered to absorb matter, but emit nothing. By virtue of the no-hair theorem, a black holes was regarded as a candidate of the final state of matter.

Usually thermo-dynamical systems have statistical dynamical description using entropy concept. However, the statistical dynamical picture of black holes has not been established yet. This is called the black hole entropy problem.

A

thermal radiation

gravitational constant

In this presentation, we shed light on the problem by using a gedanken experiment of quantum energy teleportation (QET).

The argument does not require un-established physics of quantum gravity. By using only semi-classical analysis, we can make significant statements about memory of black holes.

In classical description of black holes, after the outside spacetimes are stabilized, they forget any memories except mass, charge and angular momentum (no-hair theorem). However, we can show that even after the outside spacetime become static, black holes still remember in a quantum mechanical way what they swallowed.

Thermal relaxation process of black holes is also suggested, which time scale is of order of the inverse of black hole temperature.

The QET protocols attain energy transfer

by Local Operations and Classical Communication (LOCC)

without breaking causality and local energy conservation.

Nothing!Nothing!

Let us consider two empty boxes placed separately.

energy

Let us infuse energy to one of the boxes.

Nothing!Nothing!

Something happens inside the box and outputs a numerical number 0 or 1 to be announced to the other box.

energy

Abracadabra 0 ! (or 1! )

In the other box, some operation is performed

dependent on the announced abracadabra (0 or 1).

After that, energy can be extracted from the box

which was empty at the beginning.

energy

Abracadabra 0 ! (or 1! )

teleported energy!

Scientific magic tricks can achieve this amazing magic !!

“Nothing”

POINT: WHAT is “NOTHING” in Quantum Physics ?

無

POINT: WHAT is “NOTHING” in Quantum Physics ?

NOTHING in Quantum Physics

=GROUND STATE with Quantum Fluctuation

(Quantum Fluctuation)

GROUND STATE

entanglement

○ Spin Chains

○ Trapped Ions

○ Massless Quantum Fields

○ Quantum Hall Edge Currents

○ 1 Dim. BEC (Trapped Atoms)

○ Carbon Nano Tubes

○ SQUID Chains

：

QET protocols can be considered

in a lot of quantum systems including

←Today’s talk

Today, let us focus on the protocol for quantum fields in near-horizon regions of black holes.

Let us grasp a general idea of

this QET mechanism !

M.Hotta, J. Phys. Soc. Jp. 78, (2009) 034001

M.Hotta, Phys.Lett.A372(2008)5671

M.Hotta, Phys.Rev.D78(2008)045006

Amplitude

of fluctuation

n

Quantum Fluctuation in the Ground State of

Many-Body System

Alice Bob

The ground state has many components of quantum fluctuation as superposition of states. In the above figure, red and blue lines simply describe those different components.

gspin chain

Amplitude

of fluctuation

n

gUB

0 gHUUg BB

Amount of energy increases on average !

If an unitary operation independent of Alice’s measurement result acts on Bob’s qubit, the blue-lined component may become suppressed, but the red-lined component becomes large. Thus, on average, positive amount of energy must be infused in the fluctuation.

Amplitude

of fluctuation

Alice Bob

n

Quantum Fluctuation in the Ground State

entanglement

In QET case, we use the ground-state entanglement

between fluctuation around Alice and fluctuation around Bob .

Amplitude of fluctuation

n

AE

Measurement by A Information around B

via entanglement

Alice’s measurement specifies the value of α and its corresponding fluctuation component. In the figure, the blue-lined component is selected and the red-lined component vanishes . Because of the entanglement, Alice’s measurement result α includes information about fluctuation around Bob.

Amplitude

of fluctuation

n

AE

Local Unitary Operation by B

By getting information about α from Alice, Bob knows how the fluctuation in front of Bob behaves. Bob can choose an appropriate unitary operation in order to suppress the fluctuation.

Amplitude

of fluctuation

A

n

AE

BE

BExtraction of Energy from Spin Chain

BU

Local Operation of Bob

dependent of measurement results

By squeezing the fluctuation locally, Bob can obtain energy from the spin chain. The extracted energy was hidden in Bob’s region from the start !

Thus no energy carrier is hired in the QET protocol !!

BE

AE

BE

BE

244

1BHBH GMA

GS

ABHBH EGMS 8 BABHBH EEGMS 8

Measurement information

Impact to Black Hole Entropy Problem ?

1bit

A

B

QET in BH spacetime

=

Controlled Hawking Process by Measurement Information

)(AM

)(BU

t

x

AE

BE

b=0 or 1

Horizon

1x 2x

0x

t=0

t=T

unitary operationand energy extraction

measurementandenergyinjection

positiveenergy flux

negativeenergy flux

BE

AEControlled Hawking Process by

Measurement Information

0X

1X

IH

FH

positive energy flux

HX

AE

If B does not perform

the operation, ….

0X

1X

IH

FH

QETH

positive energy flux

negativeenergy flux

HX

AE

BE

If B performs the operation, the horizon recedes !

The black-hole entropy decreases !

0

0

1

AE

0101kD

AE

AE

AE

Horizon

AE

AE

1

AE

AE

（１）

For N fields, A performs the measurement. measurement results

The result-dependent wave packets are absorbed by the black hole.

The amount of energy of the wave packets does not depends on the measurement result.

0101kD

0

0

1

1

BE

BE

BE

BE

BE

BE

BE

BE

Shifted Horizon

extracted energy

（２）

B performs result-dependent operations for N fields.

absorbed information about the measurement result

The wave packets with negative energy are absorbed by the black hole and recede the horizon.

0101kD

0

1

1

1

Horizon

'BE

'BE

'BE

kD1011

BE

'BE

'BE

'BE

BE

If B uses incorrect information for the operations, the amount of energy extracted by QET decreases.

The amount of energy infused by the first measurement is independent of the measurement result. Thus, the outside spacetimes of black holes are also independent of the result from the classical theoretical point of view. This implies that the classical black holes do no remember the detailed information of falling matters.

However, the black holes still remember what they swallowed in a quantum mechanical way. Via the QET protocol, we can extract energy from the black holes by using correct measurement results. If the operations is based on incorrect information of the measurement result, the amount of extracted energy decreases.

Note that the amount of energy decreases in time. This describes thermal relaxation of the black holes. The time scale of the relaxation is of order of inverse of the black hole temperature, as naturally expected.

Brief Summary

In the later, let me explain details by use of equations !

In order to discuss classical behaviors of horizon shift of black holes explicitly, let us adopt a two-dimensional solvable model, CGHS model, for simplicity.

The obtained results are essentially the same as those of spherically symmetric black holes of Einstein gravity in higher dimensions.

Classical CGHS MODEL

N

nnfRegxdS

1

22222

2

144

Black Hole solution

21

2222 11 dreM

deM

ds rr

M

rH ln2

1

Event Horizon

M

eM

ex r2

Using the light-cone coordinates,

xx

dxdxds

22

1

Because the horizons stay at , the black hole geometries can be described by the flat metric in near-horizon regions.

0x

dxdxds2

Exact Solution with Falling Matter

x yzTdzdy

Mxx

dxdxds

)(2

1 2

2

N

nLn xfxT

1

2)()(

)()( xfxff RnLnn

Quantum fields in the classical black-hole spacetime can be analytically solved.

Falling-Matter Mode Out-Going Mode

1

, xM

large-black-hole and near-horizon limit

dxdx

zTdzdyM

xx

dxdxds

x y)(

21 2

2

Even if the incoming energy flux exists, the geometry is described by the flat metric near the horizon in these coordinates.

It is possible to observe horizon-shift explicitly using rescaled coordinates such that

x

MX

X Y

TdZdXXXM

dXdXds

212

2

Solution in the New Coordinates:

The outside spacetime becomes static soon after the matter is absorbed !

Horizon-Shift

IH

FH

HX

T

XX

MdXdX

ds2

2

''

'''

2

2

XXM

dXdXds

oxxxt

22 )(4

1)(

4

1,

dxxtH ,

00 H

Hamiltonian

energy density

vacuum state

fff xxt

x

t

Horizon

Quantum Energy Teleportation

in Black-Hole Spacetime

The thermal equilibrium states of quantum fields in the black-hole spacetime are described by the entangled Hartle-Hawking state. Near the horizon, this state is reduced into the Minkowski vacuum state in the large-mass limit.

Let us consider a QET protocol in the near-horizon region in order to investigate quantum properties of black holes.

0

dxxxtgH ym )()(4

)( )()( ttg

VVITrMM bbPbb 001100

dtHiTV mexp

dxxxM )()(4

cos0

dxxxM )()(4

sin1

In the near-horizon region, we get 1-bit information about quantum fluctuation of a field by interacting with a two-level spin and measuring the spin.

This indirect measument is described by the following measurement operators:

NOTE:

44 1

202

ibib

bb eei

M

0)()(exp dxxxi

Post-Measurement States of Quantum Fields:

The amount of energy for the state is independent of the measurement result and given by

dxxEA2)(

This energy is infused by the measurement device. The wave packets with this energy fall into the black hole and increase the mass and entropy of the black hole. Because the amount of energy is independent of the measurement result, the outside spacetime is the same, independent of the result.

AE

Horizon

1,0bAE

0

0

0

1

AE

0101kD

AE

AE

AE

Horizon

AE

AE

1

AE

AE

Classical treatment of spacetime is valid for the large N limit.

After time T passes, perform an operation dependent on the measurement result for the quantum fluctuation.

dxxxpiU bb )()(1exp

real parameter fixed later

Post-Measurement State

1,0

00b

biTH

bbiTH

bF UeMMeU

2 AF EHTr

Average Energy after the Operation

dxdyy

Tyxxp )(

1)(

43

dxxp 2)(

BAF EEHTr

04

2

BE

2

By setting

This implies that positive energy is transferred from the quantum fluctuation to the outside. Simultaneously, wave packets with negative energy are generated during the operation and fall into the black hole. This decreases the mass and entropy of the black hole.

BE

BE

Horizon

1,0b

0101kD

0

0

1

1

BE

BE

BE

BE

BE

BE

BE

BE

Horizon

extracted energy

If wrong information about the measurement result is adopted, the operation does not extract but gives energy to the flctuation.

When the errors happen for ¼ of N fields, the amount of energy extracted from the black hole becomes zero on average.

4

3'

2

BE

0101kD

0

1

1

1

Horizon

'BE

'BE

'BE

kD1011

BE

'BE

'BE

'BE

BE

The amount of energy infused by the first measurement is independent of the measurement result. Thus, the outside spacetimes of black holes are also independent of the result from the classical theoretical point of view. This implies that the classical black holes do no remember the detailed information of falling matters.

However, the black holes remember what they swallow in a quantum mechanical way. Via the QET protocol, we can extract energy from the black holes by using correct measurement results. If the operations is based on incorrect information of the measurement result, the amount of extracted energy decreases.

Note that the amount of energy decreases in time. This describes thermal relaxation of the black holes. The time scale of the relaxation is of order of inverse of the black hole temperature, as naturally expected.

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