1. No category

Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Black holes and reversibility
Matteo Smerlak
Perimeter Institute for Theoretical Physics
ILQGS
March 24, 2015
1 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Setup: gravitational collapse
▶
A black hole forms from
ingoing matter.
▶
Trapping horizon forms and
peels off outgoing geodesics.
▶
Thermal Hawking radiation
is emitted.
▶
Breakdown of predictability?
2 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Information loss as a physical problem
“Information loss violates a basic tenet of quantum mechanics.”
▶
Information loss happens all the time:
▶
▶
with open systems (decoherence)
with non-Cauchy “out” surfaces
[Wald 13]
▶
Information loss does not mess up with conservation laws.
[Banks, Peskin, Susskind 84; Unruh, Wald 95]
▶
The only real question is: what difference would it make?
What physical effects relate to the information loss problem?
3 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
The Hawking effect
Squeezed vacuum
Open questions
Outline
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
4 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
The Hawking effect
Squeezed vacuum
Open questions
The Hawking effect
5 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
The Hawking effect
Squeezed vacuum
Open questions
Two-mode squeezed vacuum
|ψAB ⟩ ∝
∞
∑
(tanh r )n |n, n⟩
=⇒
ρA ∝
n=0
Stronger squeezing, higher temperature
∞
∑
(tanh r )n |n⟩⟨n|
n=0
(e −ℏω/kT
= tanh r ).
6 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
The Hawking effect
Squeezed vacuum
Open questions
Observing TMSV
BEC
SQUID
Nonlinear optics
Hydrodynamics
7 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
The Hawking effect
Squeezed vacuum
Open questions
Open questions
The evaporation problem is a runaway problem
radiation =⇒ mass loss =⇒ smaller hole =⇒
higher squeezing =⇒ more radiation...
The questions for us are
▶
does this lead to an explosive behavior?
▶
does thermality break down at late times?
▶
what astrophysical signatures should we look for?
8 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Entanglement entropy
The Page curve
In two dimensions
Outline
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
9 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Entanglement entropy
The Page curve
In two dimensions
Entanglement in finite systems
In finite dimensions, entanglement entropy
S[ρA ] ≡ −trA [ρA ln ρA ] with ρA ≡ trB [ρAB ]
is unitarily invariant and satisfies the triangle inequality
|S[ρA ] − S[ρB ]| ≤ S[ρAB ] ≤ S[ρA ] + S[ρB ].
10 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Entanglement entropy
The Page curve
In two dimensions
Page’s conjecture
Page time?
Hawking phase
purification?
[Page (93,13)]
11 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Entanglement entropy
The Page curve
In two dimensions
Reversibility : three open questions
Is the evaporation process
1. unitary, viz. is purity preserved ?
SvN [ρout ] = SvN [ρin ] ?
[Hawking (76)]
2. cyclic, viz. does entanglement return to its initial value?
lim SP (u) = lim SP (u) ?
u→+∞
u→−∞
[Page (93)]
3. conservative, viz. do energy input and output match ?
lim M(u) = 0 ?
u→+∞
12 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Entanglement entropy
The Page curve
In two dimensions
Working assumptions
Neglect
▶
angular momentum (of spacetime and fields)
▶
backscattering
▶
non-conformal interactions
but not
▶
semiclassical backreaction (even strong).
Reduces field dynamics to 2d CFT:
∫
2
ϕ(t, r ) = r
dΩ2 Φ(t, r , Ω)
S2
13 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Entanglement entropy
The Page curve
In two dimensions
Renormalized entanglement entropy
In QFT, entanglement entropy is UV-divergent. Substract vacuum
contribution
Defines renormalized entanglement entropy
SP (u) = [ρψ (u)] − S[ρ0 (u)]
[Holzhey, Larsen, Wilczek (94)]
14 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Entanglement entropy
The Page curve
In two dimensions
The Page curve
Starting from the (non-covariant) CFT formula for a segment
S[ρ(R)] =
1
L(R)
log
3
ϵ
[Holzhey, Larsen, Wilczek (94)]
we obtain the geometric formula
S(u) =
1
ln χ(u)
12
[Bianchi, MS 14]
with χ = ω+ /ω− the in-out redshift factor.
15 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Entanglement entropy
The Page curve
In two dimensions
Vaidya spacetime: the Hawking phase
1
S(u) =
log
12
(
(
))
1 + W e −u/4M
u
(
)
∼
−u/4M
48M
W e
[Bianchi, de Lorenzo, MS 14]
16 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Entanglement entropy
The Page curve
In two dimensions
“Hawking spacetime”: thunderbolt
S(u) ∼
1
log
12
(
4M
u − uH
)
[Bianchi, de Lorenzo, MS 14]
17 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Entanglement entropy
The Page curve
In two dimensions
From spacetime to the Page curve
More examples illustrate the connection between geometry and
entanglement...
[Bianchi, de Lorenzo, MS 14]
... but in this approach, where
spacetime =⇒ entropy,
backreaction is an input. Next best thing after blind guess!
Importance of other, less narrow approach.
18 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Example of acyclic processes
Consequences of cyclicity
An open problem
Outline
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
19 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Example of acyclic processes
Consequences of cyclicity
An open problem
Unitarity violations?
Several authors propose that evaporation is non-unitary (in the
QFT sector):
▶
decoherence without
dissipation: spin bath model
[Unruh, Wald 95; Unruh 12]
▶
quantum gravity
decoherence: defects in
spacetime weave
[Perez 14]
Here I’ll explore another possibility: unitary but acyclic evaporation.
20 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Example of acyclic processes
Consequences of cyclicity
An open problem
The moving mirror
▶
Mirror starts at rest...
▶
... then accelerates...
▶
... then is inertial again.
∆S ∝ (relative rapidity)
Unitary but acyclic.
What does cyclicity imply?
21 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Example of acyclic processes
Consequences of cyclicity
An open problem
Outgoing energy flux
Other natural observable at I + : energy flux
F (u) ≡ 4πr 2 ⟨in|Tuu |in⟩
and Bondi mass
∫
M(u) ≡ M0 −
u
du ′ F (u ′ ).
−∞
In the 2d approximation,
F (u) = −
1
24π
( ...
)
p (u) 3 p¨(u)2
−
p(u)
˙
2 p(u)
˙ 2
[Fulling, Davies, Unruh (76)]
22 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Example of acyclic processes
Consequences of cyclicity
An open problem
The it from bit equation
2
˙
¨
+ S(u)
2πF (u) = 6S(u)
▶
“Page curve” S(u) determines energy flux F (u)
▶
Energy flux F (u) determines Page curve S(u), via
¨
−ψ(u)
+ 12πF (u) ψ(u) = 0
where
ψ ≡ e 6S
▶
Flux F (u) is “exceptional”: F (u) + δF (u) not a flux
▶
Implies quantum inequality: |F |τ 2 ≲ 1
23 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Example of acyclic processes
Consequences of cyclicity
An open problem
It-from-bit and the GSL
2
˙
¨
2πF (u) = 6S(u)
+ S(u)
Generalizes GSL in two ways:
▶
Includes non-adiabatic term (identity rather than ineq.)
▶
Does not require special causal structure (event horizon)
▶
¨ ≪ S˙ 2 . For a Schwarzschild black
Gives back GSL when |S|
hole, with
S˙ =
1
48MB
and
˙
˙ B = − SBH
F = −M
32πMB
you get
dSBH + dS =
u
> 0.
96M
24 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Example of acyclic processes
Consequences of cyclicity
An open problem
A black hole’s last gasp
2
˙
¨
2πF (u) = 6S(u)
+ S(u)
At the “Page time” u∗ , the flux is negative: F (u∗ ) < 0.
Mass law
Entropy law
1.0
1.2
Hawking
0.8
0
Page
Hawking
MHuL ‘ M
SHuL ‘ S
max
1.0
Page
0.8
0.6
0.4
0.6
0.4
0.2
0.2
0.0
0.0
0.0
0.5
1.0
1.5
0.0
uΤ
0.5
1.0
1.5
uΤ
Black hole’s “last gasp”.
25 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Example of acyclic processes
Consequences of cyclicity
An open problem
Time scales
26 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Example of acyclic processes
Consequences of cyclicity
An open problem
Lifetime of a black hole
From the it-from-bit equation we get that if
▶
the evaporation process is cyclic
▶
energy is conserved: MB (u) > 0,
then the purification time must be large:
{
O(M04 /mP3 ) if M1 = O(mP )
(M02 − M12 )2
τP ≥ ξ
=
M1 mP2
O(M03 /mP2 ) if M1 = O(M0 /2)
[Carlitz-Willey 87; Bianchi, MS 19]
Recent nonsingular black hole models fail to respect this bound.
[Frolov, Vilkoviski 91; Hayward 06; Bardeen 14; Rovelli, Vidotto, Haggard 14]
27 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Example of acyclic processes
Consequences of cyclicity
An open problem
An open problem
1. Is there a nonsingular black hole spacetime such that
evaporation is cyclic and (sub)-conservative?
2. What kind of spacetime does Page’s curve describe?
28 / 29
Black holes as squeezers
Past/future entanglement
(A)cyclic processes
Example of acyclic processes
Consequences of cyclicity
An open problem
Conclusions
▶
Focus on asymptotic observers (us).
▶
In field theory, unitarity is not equivalent to cyclicity.
▶
From a (guessed) geometry, can compute the Page curve.
▶
Inverse problem seems insightful, thanks to it-from-bit.
Thanks to
A. Ashtekar, E. Bianchi, T. de Lorenzo, A. Perez, C. Rovelli.
29 / 29