A Brief Review of Quantum Information Aspects of Black...
Transcript of A Brief Review of Quantum Information Aspects of Black...
A Brief Review of Quantum Information Aspects of 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!?
Thermalradiation
Thermalradiation
Only thermal radiation?
Largeblack hole
Small black hole
Ψ
thermalρ
thermalUU ρ≠ΨΨ †ˆˆ
The Information Loss ProblemHawking (1976)
Why is the information loss problem so serious?
Too small energyto 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
HRn
HRnHR nnp∑=ρ
HRAHR
HRHR Ann
HRnHRAunp∑=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 of the Hawking radiation?
(4) Radiation
○ Black Hole Complementarity
InfallingParticle
From the viewpoint of free-fall observers, no drama happens across the horizon.
Large BH
Classical Horizon
(4) Radiation
○ Black Hole Complementarity
Hawking-likeRadiation
InfallingParticle
Stretched HorizonInduced byQuantumGravity
From the viewpoint of outside 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-fallobserver
Large BH
FIREWALL
This scenario iscriticized 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 of 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 (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 AdSBH’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 (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 (3)~
EI∝ρ
( ) 0~)exp( VOSS thermaltypical γ−≤−
You cannot use the microcanonical state in the typicality argument for evaporating black holes.
EBH I∝ρ
BHHR
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
sss uupρ
Γ
This can be proven in the following way.
=−
=+
10
,01
.−−++ −= ξξξξσ
( )1,0
,
=+≥
=
−=
−+±
+
−−
−
++
ppp
pp
pp
ξξ
For the up state , we first measure an observable given by
Then, eigenvalue of emerges with probability . ±pThus, the average state is given by
+
σ
[ ] −−−+++ +=++Γ ξξξξ pp1
( )0=−+ ξξ
1±
[ ][ ].
12
ρ
ξξξξ
=+=
+=++ΓΓ
−−−+++
−−−+++
uupuup
UUpUUp ††
.±± = uU ξNext, 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.
±±=± ρq±±⇒ρ with probability
If the down state emerges, the spin flip operation is performed so as to get the up state.
++⇒−− †
xx σσTherefore 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
++
[ ]ρρ Γ='
[ ]'' ρρ 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
AA ++ BB ++⊗
Let us assume that each state is pure and the up stateat the first stage.
A BBA ρρ ⊗
From the result proven previously, it turns out that local operation 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
AΓ BΓ
A B
( ) ( )∑∞
=
⊗=⇒1
)(µ
µρµρµρ BAAB 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.)
,, 21 AA ΓΓ ,, 21 BB ΓΓ
0 11
Local Operations and Classical Communication (LOCC)
AA ++ BB ++⊗
( ) 1)(,01
=≥ ∑∞
=µ
µµ pp 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 BAσ
AA ++ BB ++⊗
At first, Alice measures for the system A in the initial state and obtains the eigenvalue with probability .
AAA µµµ
ξξµσ ∑±=
=1 ( )1,0
,
=+≥
=
−=
−+±
+
−−
−
++
ppp
pp
pp
AA ξξ
Aσ1±±p
Alice Bob
A B1±=µAσ
AA µµ ξξ BB ++⊗
Next Alice announces the obtained eigenvalue to Bobvia a classical channel.
1±=µ
Alice Bob
A B BµΓ1±=µAσ
[ ]BBB ++Γ⊗ µ
AµΓ
Finally, Alice and Bob perform arbitrary local operations dependent on the eigenvalue for their systems.
Alice Bob
[ ] [ ]BBBBAAAA ++Γ=Γ= µµµµ µρξξµρ )(,)(
[ ]AAA µµµ ξξΓ )()( µρµρ BA ⊗
µ
A B BµΓ1±=µAσ
AµΓ
Then, the average state of the composite system becomesa separable state with correlation.
Alice Bob
)()( µρµρµ
µ BAp ⊗∑±=
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.
( ) ( ) .0)(1
=
⊗∑
∞
=µ
µρµρµ BAAB pENT
( ) 0=++⊗++ BBAAABENT
Entangled states are defined as non-seperable states:
( ) ( ) 0)(1
>
⊗≠∑
∞
=µ
µρµρµρ BAABAB pENT
This definition leads to a fact thatany 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 BQuantum Channel
Ψ1,0
Increasing entanglement requires non-local operations directlyconnecting A to B, or quantum channels which can transport not only classical but also quantum information (quantum states).
A BNon-Local Operation
interaction
Example: entanglement of pure state of two systems
[ ] [ ]ABABABABABBA ΨΨ=ΨΨ= Tr,Tr ρρ
[ ] [ ]BBBAAAABABABEE ρρρρ lnTrlnTr)( −=−=ΨΨ
Entanglement Entropyas one of entanglement indices
Contracted State:
[ ]−−+++=2
1Bell
−−−Ψ+++Ψ +=Ψ BABAAB vupvup
Schmidt Decomposition of Pure State:
'''' ,,1,0 ssVsBsssAsAss
s vvuupp δδ ===≥ ∑±=
Ψ±Ψ
( ) 2ln=⇒ BellBellEEAB
The Bell states attain the maximum value of entanglement entropy.
( ) ss
sABABAB ppEE Ψ±=
Ψ∑−=ΨΨ⇒ ln
II. Lubkin-Lloyd-Pagels-Page Theorem, Page Curve Hypothesis and BH Firewall Conjecture
The BH firewall conjecture is based on the Page curve hypothesis, and the hypothesis was inspired bythe Lubkin-Lloyd-Pagels-Page(LLPP) theorem.
A BANBN
ABΨ
ABΨ
Typical State of ABBA NN <<
[ ]ABABBA Tr ΨΨ=ρ
[ ]AAAEE TrS ρρ ln−=
AEE NS ln≈
Typical states of A and B are almost maximally entangled when the systems are large.
AA
A IN1
≈ρ
Lubkin-Lloyd-Pagels-Page Theorem:
Maximal Entanglement between A and B
AB NN ≥
∑=
=AN
nBnAn
AAB
vuN
Max1
~1
Orthogonal unit vectors
⇒= AA
A IN1ρ
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
HHR
HBH
dim
,dim
=
=
HRln
EES
bhpage MM 7.0≈
BHHR
GABHSHRBH BH
EE 4ln1 =≈⇒<<<<
HRBH lnln ≈○
○
<<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:
thermalEE SS = of the smaller system.
BAHR ∪=CBH =
BA
Late radiation
C Early radiation
CBA ,,1<<
ACB <<
ABCΨ
Page Time
OLD BH
OLD BH ⇒
BAC
⊗
== CBBCBC I
CI
BI
BC111ρ
Proposition I means that A and BC are almost maximally entangled with each other.
Harrow-Hayden
NO CORRELATION BETWEEN B AND C!
BAC
⊗
= CBBC I
CI
B11ρ
FIREWALL!
( )[ ] ∞=∂ BCxTr ρϕ 2)(
=
−
2
2 1)()(ε
ρεϕϕ OxxTr BC
CB
CxBx
0→ε
ε
III. Canonical Typicality for Non-Vanishing Hamiltonian,and No Firewalls
Problem for Proposition I of 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.
|||| BASEE ∂=∂∝
A
Bfor low excited states
ABAB0≈Ψ
← standard area law of entanglement entropy
AVA 2=
∑=
=Ψ||
1
~||
1 A
nBnAnAB
vuA
AEE VAS ∝= ln
Not area law, but volume law forhighly 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.
.0=ABH
In BH physics, we have to treat canonical typicality with non-vanishing H in a precise manner.Then non-maximal entanglement emergesand 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)
jjjAB EEEH =
[ ]{ }EEEjE j ,|)( δ−∈=∆
= ∑∆∈ )(
)(Ej
jjES EcEVMicrocanonicalEnergy Shell:
[ ]
+=
ΨΨ=
∑−
=
1
1
2
1 AN
nnnA
A
ABABBA
TTIN
Trρ
A B )(EVESAB∈ΨAN BN
[ ] [ ] '',0, nnAnnnnn NTTTrTTrTT δ===†
Bloch Representation of higher-dim quantum states
[ ]Ann TTrT ρ=
.)(
∑∆∈
=ΨEj
jjABEc
== ∑∆∈ )(
dim)(dimEj
jjES EcEVD
1))(exp( >>∝ BB NVD γ
BA NN <<
for ordinary systems.
Volume of B
Hilbert space dimension of B
nTEvaluate for
∫= pdcpcff D)()(
( )kjjkkkjjkjkj
jjjj
DDcccc
Dcc
''''''
''
)1(1
,1
δδδδ
δ
++
=
=
∗∗
∗
−∝ ∑
∆∈
1)()(
2
Ejjccp δ 1)( =∫ pdcp D
Uniform Ensemble on Mircrocanonical Energy Shell:
( )1
11 1
1
222
+
≤− ∑
−
= DT
NTr
AN
nn
AAAA ρρ
( )( ) 110exp)exp( 23 >>=∝ OVD Bγ
Max eigenvalue
))(exp( BAA VO γρρ −≤−
( )1
22
+≤−
D
TTT
nnn
BN 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 .
++= BA HHHNegligibly small
.constEEH BA =+=
Harlow argued a canonical typicality in a weak interaction limit.
ABΨ [ ]ABABBA Tr ΨΨ=ρ
[ ]AAAAB TrS ρρ ln−=
)(, βAthermalAB SS ≈
( )AA
A HZ
βρ −≈ exp1
A BAB
Ψ
.constEE BA =+
AB NN >>
Without any proof, Harlow argued these only in the weak interaction limit.
BAC
FIREWALL?
( )[ ] ?)( 2 ∞=∂ BCxTr ρϕ
Harlow pointed out a possibility that BH firewalls may exist even after canonical typicality with non-zero Hamiltonian.
,BAHR ∪= CBH =
CBA ,,1<< ACB <<
( )( )( ) ( )CB
CBBC
HHHHββ
βρ−⊗−=
+−∝expexp
exp
No Correlation, just like CB II ⊗
arXiv:1409.1231Harlow,
However, the worry is useless.We can prove nonexistence of firewalls for general systems by using the general theory of canonical typicality.
( )( )BCCBBC VHH ++−∝ βρ expACB <<
Irrespective of the strength of the interaction between B and C,
M. Hotta and A. Sugita, Prog. Theor. Exp. Phys, 123B04 (2015).
( )( )BCCBBC VHH ++−∝ βρ exp
BCVBH CH
[ ] ?!lim ∞=BCBCVTr ρ0→BCV
Harlow’s worry:
Actually, a correlation exists between B and C for small interactions.
( )( )'''exp' '''''' CBCBCBBC VHH ++−∝= βρρ
[ ] [ ] ∞<= BCCBBCBC VTrVTr '''ρρ
BCV''BH ''CH
'''CBV
''' '''' CBCBBCCB VHHVHH ++=++
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 BAB
ΨEEE BA =+
[ ]ABABBA Tr ΨΨ=ρ
))((/))(exp( EZHE AAA ββρ −=
[ ] [ ] thermalAAAAEE STrTrS =−≤−= ρρρρ lnln
Arbitrary state:
Gibbs state:
BHAH
[ ] [ ]( ) [ ]( )1ln 21 −−−−−= AAAAAAAAA TrEHTrTrI ρλρλρρ
If a stable Gibbs state exists, it attains the maximum of the von Neumann entropy with average energy fixed.
0=Iδ
)(/)exp( ββρ AAA ZH−=
[ ] [ ]AAAA TrTr ρρρρ lnln −≤−
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)
GTME BH π8
1== 0
81
2 <−=GTdT
Edπ
( )[ ]BHBH HTrZ ββ −= exp)(
( )02
2
>−
=T
EE
dTEd
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/( GA
)4/( GASS thermalEE ≈≤
ABΨ
ABU
MicrocanonicalEnergy Shell
.constEtotal =
ABΨ is a typical state with almost certaintyafter a relaxation time.
In ordinary quantum systems,
HRBH +Ψ
HRBH IU ⊗
)(emissionU
Sub-Hilbert space of non-typical states
The state of BH evaporation can be non-typical until the last burst.
Fast scrambling of BHdoes 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
mirrortrajectory
)( −+ = xfx
)(ˆ +xinϕ
)(ˆ −xoutϕ
)( −+ = xfxMirror Trajectory:
))((ˆ)(ˆ −− = xfx inout ϕϕ
( ) ))((ˆ)(ˆ,ˆ −+ −= xfxtx inin ϕϕϕ
0|ˆ)( =−+= xfxϕBoundary
Condition:
Solution:
Scattering Relation:
( )xtx ±=±
∂∂
−∂∂
−= −−
−−
−−
−−−
−−
223
)()(
23
)()(
2410)(ˆ0
xfxf
xfxfxT inin π
:ˆˆ::ˆˆ:ˆoutoutT ϕϕϕϕ −−−−−− ∂∂=∂∂=
))((ˆ)(ˆ −− = xfx inout ϕϕ
Out-going energy flux:
derived from
(Dynamical Casimir Effect)
( )−−−+ +−== xexfx κ
κ1ln1)(
Moving Mirror Model in 1+1 dim. mimics 3+1 dim. spherical gravitational collapse.
−− ≈−∞≈ xxf )(
( )−− −−≈∞≈ xxf κκ
exp1)(
The mirror does not move in the past.
The mirror accelerates and approaches the light trajectory,
.0=+xacceleration
2
120)/1(ˆ0 TxT inin
πκ =>>−−−
πκ2
=Tacceleration
The mirror emits thermal flux in the late time.
Temperature:
12exp
10ˆˆ0 )()(
−
∝ω
κπ
ωω inoutout
in aa †
Hawking Radiation!
+x−x
Hawking Radiation
Mirror Trajectory:
( )[ ]−+ −+−= xx κexp1ln
1+1 dim Moving Mirror Modelas analogue of Hawking radiation emission
πκ2
=T
At rest → Gradually accelerated→ Uniform acceleration
+x−x
mirrortrajectory
)( −+ = xfx
−1x
−2x A
B
Bε−−
2x
ε+−1x
[ ]ABABBA Tr ΨΨ=ρ
[ ]AAAB TrS ρρ ln−=
Entanglement Entropy of Emitted Radiation
( )
∂∂−
= −−
−−
−−
)()()()(ln
121
122
2
12
xfxfxfxfSAB ε
Entanglement entropy:
Lattice spacing (UV cutoff)
( )( )
−∂∂
−=
−=
−−−−
−−
−−
2
1212
2
12
)(
)()(
)()(ln121
xxxfxf
xfxf
SSS ABvac
ABren
Renormalized entanglement entropy:
Holzhey-Larsen-Wilczek
+x−x
Hawking Radiation
Mirror Trajectory:
( )( )( )
−+
−+−= −
−+
hxxx
κκ
exp1exp1ln
BH Evaporation Trajectory
At rest → Uniform Acceleration → At rest
0 100 200 300 400 500 600 7000.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.010
x
y
)( −−− xT
−x
500,1 == hκ
0 100 200 300 400 500 600 7000
5
10
15
20
x
y
EES∆
−2x
2,500,1 1 −=== −xhκ
Thanks to Daniel Harlow
Page time
Page Curve for BH Evaporation Trajectory
BHMBHM
HRBH0⊗Ψ
HRBH +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
( )( )( )
−+
−+−= −
−+
hxxx
λκ
exp1exp1ln
Planck-energy last burst with a tiny amount of information
Possible modificationof the Page curve,assuming local dynamics.
κλ >>Planck energy scale
BH scale
0 100 200 300 400 500 600 7000
10
20
30
40
50
x
y
EES∆
−2x
2,500,100,1 1 −==== −xhλκ
Modified Page Curve for BH Evaporation
“Page time”
Contributionof zero-point fluctuationwithout energy cost for information storage
All of the information comes out at the end by zero-point fluctuation flow.
+x
−x
MirrorTrajectory
HawkingParticle
Zero-PointFluctuation
Entangled Pairin 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
HawkingParticle 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.
0=r
Zero-point fluctuation
Hawking Radiation
in0
Apparenthorizon
Collapsing Shell
Black Hole EvaporationZero-point fluctuation
BA
Late radiation
Early radiationC
AABCBC SSS == ,0
BCABCBAB SSSS −+≥
Strong subadditivy:
Strong Subadditivity “Paradox”
ABAB SSS +≥Page Curve Hypothesis ABA SS >
No Drama:
BA
Late radiation
Early radiationC
Strong Subadditivity “Paradox”
ABABA SSSS +≥>
Page Curve Hypothesis ABA SS >
BS>0
BA
Late radiation
Early radiationC
Strong Subadditivity “Paradox”
Remnant& Zero-Point Fluctuation Flow
ABA SS <
until the last burst.
Thus, no strong subadditivity paradox!
0 100 200 300 400 500 600 7000
10
20
30
40
50
x
y
EES∆
−2x
We don’t care the no drama condition breaks at the last burst, because the horizon is affected by quantum gravity.
0=> + flcptzrABA SS
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 horizonin an average meaning.
InfallingParticleLarge BH
Classical Horizon
The strong subadditivity paradox has been resolved.
However, we have another possibility of 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 .
in0
in0
inn E
nnnnn xdxOxOxxa 0)(ˆ)(ˆ),,(
'111∑∫≈Ψ∀
E’ L’
Ψ⇒in0
∑∫n E
nnnnn xdxOxOxxa
'111 )(ˆ)(ˆ),,(
in0
f
'L
'E
L
E
−x
fOEˆ−−T
EL
EO
'ˆ
EO
EO
.0 fin →
Note that the Reeh-Schlieder property is maintained in the time evolution:
xdxOxOxxaO n
n EnnnnE ∑∫= )(ˆ)(ˆ),,(ˆ
111
Even in the future infinity, we may remotely generate any excitation with some probability smaller than 1.
L
E
Li
?FW
)( fwh xgx =+
∑=i
LEi if 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 .
fwxx =−
If measurement operator is constructed from Reeh-Schlieder operation, an arbitrary post-measurement state including firewalls can emerge.
fwxx =−
)( fwh xgx =+
Firewall Measurement Paradox:
Eiψ
EiE OM ˆˆ ∝
V. Informational Cosmic Censorship Conjecture
“Resolution of the Paradoxfrom a viewpoint of
Quantum Measurement Energy Cost”
M. Hotta, J. Matsumoto and K. Funo, Phys. Rev. D89, 124023 (2014)
'ˆ
iEM
'Li 'L
'E
Li
LBecause the mirror merely stretches the modes of the field, the future measurement is equivalent to a past measurement for the in-vacuum state.
'ˆˆ
iEiE MM ⇔
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.
Li
in0
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.
'' ˆˆiEiE MM†
Hotta, Matsumoto and Funo,Phys. Rev. D89, 124023 (2014).
( ))( fwhfw xgxE −+δ
planckfw El
OE <<
<
π121
'E 'L
+x
++T
Lil
⇒No Firewall appears!
)( fwh xgx =+
If we assume FW appears at with less-than-1 probability in a finite-energymeasurement, then
Hotta, Matsumoto and Funo,Phys. Rev. D89, 124023 (2014).
)( fwh xgx =+
'E 'L
+x
++T
l+E
fw
fw
rlErE
Eπ121−
≥+
∞→+ErlE fw π12
1→
Energy cost of measurement:
Energy cost of FW measurement diverges!
fwE
Hotta, Matsumoto and Funo,Phys. Rev. D89, 124023 (2014).
)1(Or =
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
0110101011iEM
⇒ Firewall Information Censorship
Informational extension of Cosmic Censorship
M. Hotta, J. Matsumoto and K. Funo, Phys. Rev. D89, 124023 (2014)