Soft-Hair Enhanced Entanglement Beyond Page Curves in Black-Hole - - PowerPoint PPT Presentation
Soft-Hair Enhanced Entanglement Beyond Page Curves in Black-Hole Evaporation Qubit Models Masahiro Hotta Tohoku University Based on M. Hotta, Y. Nambu and K. Yamaguchi, arXiv:1706.07520 . Introduction Large black-hole spacetimes are
Based on M. Hotta, Y. Nambu and K. Yamaguchi, arXiv:1706.07520 .
Masahiro Hotta
Tohoku University
Infalling Particle Large BH
Large black-hole spacetimes are conventionally described merely by classical geometry, and nothing cannot go out of the event horizon.
Black hole is black.
This picture drastically changes, because black holes can emit thermal flux due to quantum effect.
BH B HR
GM k c T π 8
3
=
(Hawking, 1974)
BH BH
GM r 2 =
Black hole ain’t so black!
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, 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
HR n HR n HR
n n p
∑
= ρ
HR
HR HR
A n n HR n HRA
u n p
= Ψ
Mixed state
Composite system in a pure state
From a modern viewpoint of quantum information, Information Loss Problem = Purification Problem of HR
Hawking radiation system
Partner system
Entanglement
What is the partner after the last burst?
(1) Nothing, Information Loss (2) Exotic Remnant (Aharonov, Banks, Giddings,…) (3) Baby Universe (Dyson,..) (4) Radiation Itself (Page,…) ○ Black Hole Complementarity (‘t Hooft, Susskind, …) ○ Firewall (Braunstein, AMPS, …)
What is the purification partner of 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 Radiation Infalling Particle Stretched Horizon Induced by Quantum Gravity
From the viewpoint of
the stretched horizon absorbs and emits quantum information so as to maintain the unitarity.
Large BH
Planck Unruh
E T >>
(4) Radiation
○ Firewall
A FIREWALL at the horizon burns out free-fall observers. The inside region
Free-fall
Large BH FIREWALL
(1) Nothing, Information Loss
(2) Exotic Remnant (Aharonov, Banks, Giddings,…) (3) Baby Universe (Dyson,..) (4) Radiation Itself (Page,…) ○ Black Hole Complementarity (‘t Hooft, Susskind, …) ○ Firewall (Braunstein, AMPS, …) (5) Zero-Point Fluctuation Flow (Wilczek, Hotta-Schützhold-Unruh )
What is the purification partner of the Hawking radiation?
Hawking Particle Zero-Point Fluctuation
(5) Zero-Point Fluctuation Flow
entangled
(Wilczek, Hotta-Schützhold-Unruh)
Entanglement Sharing Energy Cost of the Partner
→ Avoiding the planck-mass remnant problem.
First-principles computation of time evolution of entanglement between an evaporating BH and Hawking particles is not able to be attained. A popular conjecture → The Page Curve
Don Page, Phys. Rev. Lett. 71, 3743 (1993).
The partner entangled with a Hawking particle remains elusive due to the lack of quantum gravity theory to date.
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
) ( 7 .
BH Page
M M ≈
BH HR
HR BH ln ln ≈
○ ○
<<Page Time>>
Maximum value of EE is attained at each time, and is equal to thermal entropy of smaller system. OLD BH
Page Curve
Semi-classical regime! time
G A S S
thermal EE
4 = =
Page Curve Conjecture for BH Evaporation: Proposition I: After Page time, BH is maximally entangled with Hawking particles. No correlation among BH subsystems. Proposition II:
horizon thermal EE
after Page time.
No correlation due to quantum monogamy
Maximal Entanglement
Hawking particles
BH
) 1 ( EE EE
Lemma:
No correlation among BH subsystems due to quantum monogamy N= # of BH degrees of freedom
horizon thermal EE
) 1 ( ) 1 (
After Page time,
) 1 ( thermal thermal
NS S =
[ ] ∏
=
⊗ = Ψ Ψ =
N n n HR BH
Tr
1 ) 1 (
ρ ρ
BH
After Page time,
This relation is criticized in this talk.
BH HR
BH BH HR
T GM T = = π 8 1 8 1
2 <
− =
BH BH
GT C π ∞ →
BH
T →
BH
M
as Note that BH has negative heat capacity!
HR BH
T T =
Temperature of BH is measured by temperature of Hawking radiation.
) 1 ( ) 1 (
horizon thermal EE
In this talk, we construct a model of multiple qubits which reproduces thermal property of 4-dim Schwartzschild black holes, and show that the Page curve conjecture is not satisfied due to the negative heat capacity. Our Result
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
Lubk ubkin-Lloyd-Pagel els-Page T e Theo eorem em ( (“Page e Theo eorem em”)
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
C BH =
B A
Late radiation
C
Early radiation
C B A , , 1<<
ABC
Ψ
Page Time OLD BH
→OLD BH
By using the theorem, AMPS and other people proposed the BH firewall conjecture.
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
Area law of entanglement entropy is broken, though outside-horizon energy density in BH evaporation is much less than Planck scale.
for low-energy-density states
AB AB
≈ Ψ
← standard area law of entanglement entropy
∑
= +
= Ψ
| | 1
~ | | 1
BH n HR n BH n HR BH
v u BH
BH EE
V BH S ∝ = ln
← Not area law, but volume law!
Page curve states ⇒
This is because zero Hamiltonian (or high temperature regime) is assumed in Page curve conjecture.
AB
AB
Ψ
[ ]
AB AB B A
Tr Ψ Ψ = ρ ˆ
[ ]
A A A AB
Tr S ρ ρ ˆ ln ˆ − =
) (
,
β
A thermal AB
S S ≈
( )
A A A
H Z β ρ − ≈ exp 1 ˆ
A B
AB
Ψ
. const E E
B A
= +
A B
N N >>
B A AB
EE is almost equal to thermal entropy of the smaller system for typical states.
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
Gibbs state:
B
H
A
H
[ ]
A A A A
E H Tr = ρ
[ ] [ ]
( )
[ ]
( )
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, thermal entropy of the smaller system is the maximum value of EE for any state with average energy fixed.
( )
=
A
I ρ δ
( ) ( ))
( / ) exp(
A A A A A
E Z H E β β ρ − =
thermal A A A A
S Tr Tr = − ≤ − ρ ρ ρ ρ ln ln
Conventional “proof”:
[ ]
( )
A A A A A
E E H Tr β β ρ = ⇒ =
No stable Gibbs state for Schwarzschild BH due to negative heat capacity! (Hawking –Page, 1983)
( )
[ ]
BH BH
H Tr Z β β − = exp ) (
( )
2 2
> − = T E E dT E d
If there exists a stable Gibbs state, the heat capacity must be positive.
( )
T / 1 = β
Thus, a system of a black hole and Hawking radiation is not in a typical state, at least in the sense of the LLPP
“ thermal entropy” of Schwarzschild BH is not needed to be a upper bound of entanglement entropy.
(1) Nothing, Information Loss (2) Exotic Remnant (Aharonov, Banks, Giddings,…) (3) Baby Universe (Dyson,..) (4) Radiation Itself (Page,…) ○ Black Hole Complementarity (‘t Hooft, Susskind, …) ○ Firewall (Braunstein, AMPS, …)
(5) Zero-Point Fluctuation Flow (Wilczek, Hotta-Schützhold-Unruh ) (5)’ Soft Hair without Energy (Hotta-Sasaki-Sasaki (2001), Hawking-Perry -Strominger (2015))
What is the purification partner of the Hawking radiation?
Gravitational vacuum degeneracy with zero energy plays a crucial role. Hawking, Perry and Strominger call it soft hair.
Bondi-Metzner-Sachs soft hair
Horizon soft hair
Information
≈ E
Hawking Particles
(Feb 3rd, 2017 at Cambridge)
> T
Horizon soft hair of black holes comes from “would-be” gauge freedom of general covariance, diffeomorphisms (Hotta et al, 2001). This is a similar mechanism of emergence of momentum eigenstates
Black Hole at Rest
' x ' y = p x y
ω ω ω ω cosh sinh ' , sinh cosh ' t x t t x x + = + =
Running Black Hole
' x ' y ω sinh ' m p = x y
We must have Lorentz covariance in the theory.
' x ' y x y
Running Black Hole
' x ' y ω sinh ' m p = x y sinh = = = ω m p p
Lorentz transformation, one of general coordinate transformations, generates an infinite number of physical states with different values of momentum.
ω sinh m p =
Horizon soft hair of black holes comes from “would-be” gauge freedom of general covariance, diffeomorphisms (Hotta et al, 2001). This is a similar mechanism of emergence of momentum eigenstates
Soft hair of black holes comes from “would-be” gauge freedom of general covariance.
Horizon
=
horizon
Q
) , ( ' φ θ τ τ T + =
q Q horizon = '
Horizon
We must have horizontal asymptotic symmetry as a part of general covariance in the theory.
) , ( ' ), , ( ' φ θ φ φ θ θ Φ = Θ =
Supertranslation: Superrotation:
Soft hair of black holes comes from “would-be” gauge freedom of general covariance.
Horizon
q Q horizon = '
= = = q Q Q
horizon horizon
Supertranslation and superrotation generate an enormous number of physical states with different values of holographic charges.
Horizon
q Qhorizon =
) , ( ' φ θ τ τ T + = ) , ( ' ), , ( ' φ θ φ φ θ θ Φ = Θ =
BMS Supertranslation: Horizon Superrotation: Near Horizon
Null Future Infinity Poincare Symmetry:
c x x + Λ = '
Horizon Supertranslation:
) , ( ' φ θ C u u + =
including time translation generated by Hamiltonian H
, , = ⊗ ⊗ = ⊗ ⊗
horizonSR horizonST
Q I I H Q I I H
Near-horizon symmetry provides degeneracy of Hamiltonian.
BH Symmetries
(BMS Superrotation?)
Collapsing Matter Collapsing Matter
Horizon
An infinite number of supertranslation charges at the horizon might store whole quantum information of absorbed matter.
) (
= ∂
µ µ i
J
≈ G A O W
hair soft
4 ln
( )
) 4 /( G Area O ⊗
ψ
Unitarity?
Soft hair of BH
Hawking radiation (Hotta-Sasaki-Sasaki, 2001) (Hawking –Perry-Strominger, 2017)
BH
GM T π 8 1 =
Ψ Ψ = H G T π 8 1
Hawking Temperature Relation: Key Idea: Construct many-body systems which reproduces this relation. This condensed matter model yields negative heat capacity. Bekenstein-Hawking entropy can be defined even if the systems do not have actual horizons.
( )
G A G r G H G T H d S
horizon BH BH
4 4 4 4 2 4
2 2
= = Ψ Ψ = Ψ Ψ = ∫ π π
Implicitly assumed that partition function of the total system is well defined. Thus, the heat capacity is always positive.
EE
S ) (t n
2 / N N
Page time
Decaying qubits of BH into qubits of HR in Page Model
[ ]
AB AB B A
Tr Ψ Ψ = ρ
[ ]
) ( ˆ ) ( ln ) ( ) ( t U d t t Tr t S
A A A EE
− = ρ ρ
Hawking Radiation Black Hole
) (A S S
thermal EE =
BH entropy
In order to incorporate negative heat capacity, we consider not only emission of Hawking particles but also decreasing of BH degrees of freedom (soft hair) ! (Hotta-Nambu-Yamaguchi 2017)
Hawking Particle Emission Zero-Energy Soft Hair Evaporation
≈ E ≈ E ≈ E ≈ E ≈ E ≈ E
2
T
1
T
2 1
Negative heat capacity induced by soft particle decays
E t E = ) ( 1
E t E ≈ ) ( 2
High Low
Large Number of Particles Small Number of Particles If radiation coupling is switched on, the temperature of emitted radiation rises!
1 2 3 4 N-1 N
N N N
−1 3 2 1 ) (
N qubits of A at Initial Time
Energy conserved random unitary operation (Fast Scrambling)
Our Model:
ˆ †
+
= + a
ˆ †
−
= − a
One of typical pure states of N qubits
Our Model
ˆ †
+
= + a
ˆ †
−
= − a
Hawking Particle Evaporating Soft Hair
) 1 (
+ − − + + + = ω
Fast-Scrambled Pure-State Black Hole at Initial Time
Soft Hair Emission
SH HR
H H H + + =
( )
=
+ =
N n z
H
1
1 ) ( 2 σ ω
( )
Ψ − + + Ψ + − + Ψ ∂ − Ψ = dx x x g dx x x g dx x i x H
R R R x R HR
) ( ˆ ) ( ) ( ˆ ) ( ) ( ˆ ) ( ˆ
† †
( )
Ψ − + Ψ − + Ψ ∂ − Ψ = dx x x f dx x x f dx x i x H
S S S x S SH
) ( ˆ ) ( ) ( ˆ ) ( ) ( ˆ ) ( ˆ
† †
Initial State:
S R Qbits
vac vac t Ψ = = Ω (
) ( ˆ ) ( ˆ = Ψ = Ψ
S S R R
vac x vac x
~ BH internal dynamics ~ Hawking radiation emission ~ Soft hair emission
)). ( ( ) ( ) ( 1 ) ( ) (
, , , 3 , 2 ) 1 (
t T t p t p t t Tr
S R N
ρ ρ + − = Ω Ω =
T T
e e T
ω ω
ρ
− −
+ + + + − − = 1 ) ( Qubit Gibbs state at temperature T
One-site reduced state is described by Hawking particle is emitted by flipping the up state to the down state. So the radiation temperature is equal to T! This model actually reproduces the Hawking temperature relation:
( ) ( )
t M G t T
BH
π 8 1 =
( ) ( )
= =
−1 ) 1 (
) ( 4 , T G N H N t M
ω π
Qbit Survival Probability
[ ]
) 1 ( ) 1 ( ) 1 (
ln ρ ρ Tr SEE − =
+ + − + = = T T T p pS S
th thermal
ω ω ω exp 1 1 exp 1 ln
) 1 (
N S G A N T dM N S
BH horizon BH BH
= = =
4 1 1
) 1 (
Entanglement entropy between a single site and other subsystems: Thermal entropy per site: Bekenstein-Hawking Entropy per site: Page Curve Conjecture ⇒
after Page time
) 1 ( ) 1 ( ) 1 ( BH thermal EE
S S S = =
Before Page Time After Page Time Last Burst of BH
) 1 ( ) 1 ( ) 1 ( BH thermal EE
( )
1
( )
1
Page time
1 . ) ( = T ω
Actually,
(Hotta-Nambu- Yamaguchi 2017)
) ( ln ) (
) 1 (
T T T T SEE ≈
High temperature regime:
2 ln ) (
) 1 (
T T Sthermal ≈
2
) ( ) ( 4 ≈ T T T N SBH ω
←due to Soft Hair Contribution ←due to Time-Dependence
) ( T
: Initial Temperature
( )
dp T NE T dE Np T NpE d dS
th th th BH ) 1 ( ) 1 ( ) 1 (
+ = =
dp T E S N dS dS
th th thermal BH
− − =
) 1 ( ) 1 (
T dE dS
th th ) 1 ( ) 1 (
=
( ) ( )
β β / 1 ' 2 ln
) 1 ( ) ' ( / 1 th p T p thermal BH
E d dp N Np S S
+ − =
Large discrepancy is generated by this term
One-body Gibbs State Relation: First Law: Averaged Thermal Entropy:
) 1 ( th thermal
NpS S =
( )
dp T NE T dE Np T NpE d dS
th th th BH ) 1 ( ) 1 ( ) 1 (
+ = =
dp T E S N dS dS
th th thermal BH
− − =
) 1 ( ) 1 (
T dE dS
th th ) 1 ( ) 1 (
=
= + =
2 2 2 2
) ( ) ( ) ( T T O T T O T T SBH
Large discrepancy is generated by this term
One-body Gibbs State Relation: First Law: Averaged Thermal Entropy:
) 1 ( th thermal
NpS S =
B A C
Feedback to Firewall Paradox
Late radiation Early radiation BH
B A C
Late radiation Early radiation BH
S
Soft hair
Early Hawking radiation is entangled with zero-energy BMS soft hair, and that late Hawking radiation can be highly entangled with the degrees of freedom of the BH, avoiding the emergence of a firewall at the horizon.
We construct a model of multiple qubits that reproduces thermal properties of 4-dim Schwartzschild BH’s. After Page time, entanglement entropy, thermal entropy and Bekenstein-Hawking entropy are not equal to each other. Entanglement entropy exceeds thermal entropy and BH entropy. Our result suggest that early Hawking radiation is entangled with zero-energy soft hair, and that late Hawking radiation can be highly entangled with the degrees of BH freedom, avoiding the emergence