Download Controlled Hawking Process by Quantum Information

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
Controlled Hawking Process
by Quantum Information
Masahiro Hotta
Tohoku University
arXiv:0907.1378
Black Hole Entropy Problem
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.
A
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.
S BH
thermal
radiation
1

A
4G
gravitational
constant
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.
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.
Let us consider two empty boxes placed separately.
Nothing!
Nothing!
Let us infuse energy to one of the boxes.
energy
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! )
Nothing!
Nothing!
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.
teleported energy!
energy
Abracadabra 0 ! (or 1! )
Scientific magic tricks can achieve this amazing magic !!
POINT: WHAT is “NOTHING” in Quantum Physics ?
無
“Nothing”
POINT: WHAT is “NOTHING” in Quantum Physics ?
GROUND STATE
(Quantum Fluctuation)
NOTHING in Quantum Physics
=GROUND STATE with Quantum Fluctuation
entanglement
QET protocols can be considered
in a lot of quantum systems including
○ Spin Chains
○ Trapped Ions
○ Massless Quantum Fields ←Today’s talk
○ Quantum Hall Edge Currents
○ 1 Dim. BEC (Trapped Atoms)
○ Carbon Nano Tubes
○ SQUID Chains
：
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
Quantum Fluctuation in the Ground State of
Many-Body System
n
Alice
spin chain
Bob
g
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.
Amplitude
of fluctuation

g U B HU B g  0
Amount of energy increases on average !
n
UB g
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
Quantum Fluctuation in the Ground State
n
Alice
Bob
entanglement
In QET case, we use the ground-state entanglement
between fluctuation around Alice and fluctuation around Bob .
Amplitude of fluctuation
Measurement by A
Information around B
via entanglement
n
EA

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
Local Unitary Operation by B
 EA
n

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
Local Operation of Bob
of fluctuation
U B  
dependent of measurement results
 EA
 EB
n
B
Extraction of Energy from Spin Chain
A
 EB
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 !!
Impact to Black Hole Entropy Problem ?
S BH
1
2

A  4GM BH
4G
EA BHEEBA
SBH S8BHGM
 8BHGM
QET in BH spacetime
=
 EB Controlled Hawking Process
 EA
by Measurement Information
B
1bit
A
Measurement
information
 EB
U B ( )

M A ( )
t
negative
energy flux
 EB
 EB
unitary
operation
and
energy
extraction
t=T
positive
energy flux
EA
b=0 or 1
Horizon

x 0
measurement
and
energy
injection
t=0
Controlled Hawking Process by
Measurement Information
x1
x2
EA
x
X
0
If B does not perform
the operation, ….
HF
X1
HI
X H 
positive
energy flux
EA
X
0
If B performs the
operation, the horizon
recedes !
H QET
HF
The black-hole
entropy decreases !
X1
negative
energy flux
HI
 EB
X H 
positive
energy flux
EA
Dk  1010
（１）
For N fields, A performs the
measurement.
measurement results
1
 EA
 EA
0
 EA
1
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.
 EA
Horizon
 EA
 EA
 EA

0

 EA
 EB
（２）
B performs resultdependent operations
for N fields.
1
 EB
0
 EB
1
 EB


 EB
Dk  1010
absorbed information
about the measurement
result
 EB
The wave packets
with negative energy
are absorbed by the
black hole and
recede the horizon.
 EB
0
 EB
extracted
energy
Shifted Horizon
 EB '
If B uses incorrect
information for the
operations, the amount
of energy extracted by
QET decreases.
0
 EB '
1
 EB '
1
 EB


 EB '
 EB
Dk  1010
 EB '
1
Horizon
 EB '
0111  Dk
Brief Summary
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.
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 twodimensional 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
1

2
2
2
 2
2
S   d x  g e R  4   4   f n  
2 n 1


Black Hole solution
1
 M  2 r  2  M  2 r 
2
ds  1  e
d  1  e
 dr






2
Event Horizon
1 M 
rH 
ln  
2   
 
e
M

2 r
x 
e 

M
Using the light-cone coordinates,


dx dx
ds  
2  
1  x x
2

Because the horizons stay at x  0 , the black hole
geometries can be described by the flat metric in nearhorizon regions.

ds  dx dx
2

Quantum fields in the classical black-hole spacetime can be
analytically solved.


n
Ln
Rn
f  f (x )  f (x )
Falling-Matter Mode
Out-Going Mode
Exact Solution with Falling Matter


dx dx
ds  


x
y

2  



1  x x 
dy  dz T  ( z )


2M 
2
N

T  ( x )     f Ln ( x )

n 1


2
large-black-hole and near-horizon limit
M  ,
1
x 




dx dx
ds  


x
y

2  



1  x x 
dy  dz T  ( z )


2M 


 dx dx
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
M 
X 
x


Solution in the New Coordinates:


dX dX
ds  
X
Y
M
1
 2 X  X    dX   dZ T 


2 
2
HF
X H 
HI
Horizon-Shift
ds  
2
The outside spacetime
becomes static soon after
the matter is absorbed !


dX
'
dX
'
ds 2  
M'
 2 X ' X '

dX  dX 
M
 2 X  X 

T 
energy density
1
1
 2
 2
 t , x     ( x )    ( x )   o
4
4
    t f   x f
t
   x f 
Hamiltonian

H    t , x dx

x
vacuum state
Horizon
H 0 0
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 0 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.
In the near-horizon region, we get 1-bit information about
quantum fluctuation of a field by interacting with a twolevel spin and measuring the spin.


H m  g (t ) y     ( x)  ( x)dx 
4

g (t )   (t )
This indirect measument is described by the following
measurement operators:

 M  sin     ( x) ( x)dx 
M 0  cos    ( x)  ( x)dx 
1


4


4

NOTE:

V  0
V  T exp  i  H dt 
M b 0 0 M b  TrP I   1

b
 1
b
m
0    V 

Post-Measurement States of Quantum Fields:
b



i
i

i
b
4
4



 2M b 0 
e



1
e


2 
   exp  i   ( x)  ( x)dx 0
b


 




The amount of energy for the state is independent of the
measurement result and given by
E A    ( x)  dx
2
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.
0
 EA
Horizon
b  0, 1
 EA
Classical treatment of spacetime is valid
for the large N limit.
1
 EA
 EA
 EA
0
 EA
1
 EA
 EA

 EA
Horizon
Dk  1010
0

 EA
After time T passes, perform an operation dependent on the
measurement result for the quantum fluctuation.

U b  exp i  1
b
 p ( x )

( x)dx

real parameter fixed later
Post-Measurement State
F 
U e
b 0,1
b
 iTH
 iTH
M b 0 0 M b e Ub

Average Energy after the Operation
Tr F H   E A     
2
4
1

    p( x)
 ( y )dxdy
3

x  y  T 
   p ( x)  dx
2

By setting  
2
Tr F H   EA  EB

EB 
0
4
2
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.
b  0,1
 EB
Horizon
 EB
 EB
1
 EB
0
 EB
1
 EB


 EB
Dk  1010
 EB
 EB
0
 EB
extracted
energy
Horizon
If wrong information about the measurement result is
adopted, the operation does not extract but gives energy
to the flctuation.
3
EB ' 
4
2
When the errors happen for ¼ of N fields, the amount of
energy extracted from the black hole becomes zero on
average.
 EB '
0
 EB '
1
 EB '
1
 EB


 EB '
Dk  1010
 EB
 EB '
1
Horizon
 EB '
0111  Dk
Come back to Summary
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.