Probing Hawking radiation through capacity of entanglement
Tatsuma Nishioka
YITP, Kyoto
May 19, 2021

Based on 2102.02425 and 2105.08396
with K. Kawabata, Y. Okuyama (UTokyo), K. Watanabe (Davis)

Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Overview

1. Introduction: entropy of Hawking radiation
2. Review: capacity of entanglement
3. Capacity of Hawking radiation in toy models
4. Capacity formula in 2d dilaton gravity

Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

2 / 36Outline

1. Introduction: entropy of Hawking radiation
2. Review: capacity of entanglement
3. Capacity of Hawking radiation in toy models
4. Capacity formula in 2d dilaton gravity

Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Hawking radiation from an evaporating BH• Suppose the initial state of matter is pure

• After the evaporation of BH, the system is in amixed state of thermal radiation:

nradiatioawking• BH starts to evaporate due to Hawking radiation

Hbut after gravitational collapse a black hole isformedI+

singularity

horizonρpure = |ΨihΨ|I−ρpure −−−−−−−−−→ ρmixed
BH evaporation

which appears to contradict with unitarity [Hawking 76]
Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Page curve for the radiation
To model an evaporating BH with radiation, suppose|Ψi ∈ HBH ⊗ HR ,HBH : BH system ,• For a pure state |Ψi Page showed [Page 93]
when dim HR  dim HBH the radiation system isalmost maximally entangled :HR : radiation systemSRSR ≈ log dim HR• In the opposite limit, dim HR  dim HBH ,from unitarity
tPaget ≡ log dimHR

SR ≈ log(dim Htot − dim HR )Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Island formula for the radiation entropy• To reconcile with the Page curve, the entropy ofradiation should be calculated by the island formula
Almheiri-Mahajan-Maldacena-Zhao 19]:
Area(∂ΣI )+ Smat (ΣR ∪ ΣI )SR = min ext
ΣIΣI4GN• ΣR : radiation region R• ΣI : island region II+

singularity

horizon[Penington 19, Almheiri-Engelhardt-Marolf-Maxfield19,ΣR

ΣII−• No island → linear growth at early time
• With island → saturation or decay at late time

Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Replica wormholes
I+

ΣI
I+

singularity
horizon• The island formula is a generalization of theRyu-Takayanagi formula for entanglement entropy[Ryu-Takayanagi 06], which has a gravitational path
integral derivation [Lewkowycz-Maldacena 13, · · · ]horizonsingularity• The island regions are accounted for by replica
wormholes [Almheiri-Maldacena-Hartman-Shaghoulian-Tajdini
19, Penington-Shenker-Stanford-Yang 19]

Tatsuma Nishioka (YITP, Kyoto)

ΣI

ΣRI−

ΣRI−

May 19, 2021 @ Kyoto Univ Seminar

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Goal of this talk
• We will examine if Hawking radiation (or replica wormholes) can be captured by capacityof entanglement, a quantum information measure other than entanglement entropy• Calculate the capacity for two toy models of radiating black holes:• End of the world (EOW) brane model [Penington-Shenker-Stanford-Yang 19]• Moving mirror model [Akal-Kusuki-Shiba-Takayanagi-Wei 20]• Derive a formula for the capacity in 2d dilaton gravity• Apply to an eternal AdS2 black hole coupled to a flat bath region at high temperature• The capacity has a peak or discontinuity at the Page time, showing a good probe of theradiation
Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Outline

1. Introduction: entropy of Hawking radiation
2. Review: capacity of entanglement
3. Capacity of Hawking radiation in toy models
4. Capacity formula in 2d dilaton gravity

Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Entanglement entropy
Divide a system to A and B = Ā: Htot = HA ⊗ HB

Entanglement entropy
SA = −trA [ρA log ρA ]t

• The reduced density matrixρA ≡ trB [ρtot ]AB• For a pure ground state |Ψiρtot = |Ψi hΨ|Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Replica trick and Rényi entropy
Entanglement entropy

φA2SA = lim Snn→1φA1φA
3φA2nthRényi entropy
Sn ≡11log trA [ρnA ] =log Z(n)1−n1−nφA1φAnZ(n): partition function on the n-fold cover branched over A

Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Analogy to statistical mechanics
We regard Z(n) ≡ trA [ρnA ] as a thermal partition function at an inverse temperature β ≡ n:Statistical mechanics

Rényi entropy

inverse temperature

β=nHamiltonianpartition function

HA = − log ρAhiZ(β) = trA e−β HAfree energy

F (β) = −β −1 log Z(β)

energyE(β) = −∂β log Z(β)

thermal entropyS̃(β) = β 2 ∂β F (β)

heat capacity

C(β) = −β ∂β S̃(β)

Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Capacity of entanglement• The “thermal" entropy is not the Rényi entropy
Sn = −1nlog Z(β) =F (β) 6= β 2 ∂β F (β)
n−1n−1but a refined one (improved Rényi/modular entropy [Dong 16, Nakaguchi-TN 16]):n−122S̃n ≡ S̃(β) = β ∂β F (β) = n ∂nSnn• The capacity of entanglement [Yao-Qi 10] is non-negative for a unitary theory:Cn ≡ C(β) = n2 h (HA − h HA in )2 in ≥ 0where h X in ≡ trA X e−n HA /Z(β)
Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Outline

1. Introduction: entropy of Hawking radiation
2. Review: capacity of entanglement
3. Capacity of Hawking radiation in toy models
4. Capacity formula in 2d dilaton gravityTatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Capacity of Hawking radiation in toy models of BH[Kawabata-TN-Okuyama-Watanabe 21]• We will examine if the capacity can probe the Hawking radiation, i.e., replica wormholes:

Capacity of entanglement (n = 1)
C ≡ Cn=1 = −∂n S̃n |n=1 (= −2 ∂n Sn |n=1 )• Two toy models of radiating black holes• End of the world (EOW) brane model [Penington-Shenker-Stanford-Yang 19]• Moving mirror model [Akal-Kusuki-Shiba-Takayanagi-Wei 20]

Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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EOW brane model [Penington-Shenker-Stanford-Yang 19]• A quantum mechanical model of a radiating black hole:ji
k1 X|Ψi = √|ψi iB |iiR
k i=1hψj |ψi iB |iihj|R =EOW
brane

hyperbolic
disk

Auxiliarysystem R
Black holesystem B• B: BH system of dimension eS0 (JT gravity + EOW brane)• R: auxiliary system of dimension k to measure Hawking radiation• ∃ replica wormhole: hψi |ψk iB = δij + e−S0 /2 RijtrR [ρnR ] =Tatsuma Nishioka (YITP, Kyoto)1(k eS0 )nkX

(Rij : random variable)hψi1 |ψi2 iB · hψi2 |ψi3 iB · · · hψin |ψi1 iB

i1 ,··· ,in =1May 19, 2021 @ Kyoto Univ Seminar

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Planar approximation• In the planar limit, eS0  1 with k e−S0 fixedtrR [ρnR ] ≈ nk Z2
k n−1 Zn
1+·+ ··· +k n−12(Z1 )2(Z1 )n1Zn (∝ eS0 ): replica wormhole partition function of disk topology with n boundaries
k (Z1 )3

k 2 Z2 Z1

k3 Z3

fully disconnected

partly connected

fully connected

Example (n = 3):Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Entanglement entropy at early and late times• dim HR = k ⇐⇒ # of radiation particles ≈ log k• log k: time of BH evaporation• Early time (log k  S0 ): fully disconnected solution dominates
trR [ρnR ] ≈1k n−1⇒SR ≈ log k• Late time (log k  S0 ): fully connected solution dominates
trR [ρnR ] ≈Tatsuma Nishioka (YITP, Kyoto)Zn(Z1 )n⇒SR ≈ lim (1 − ∂n ) log Zn
n→1May 19, 2021 @ Kyoto Univ Seminar

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Capacity and Page curve• The asymptotic behavior of the capacity:(k (ZZ12)2 ∝ elog kCR ≈limn→1 ∂n2 log Zn(early time)(late time)
SRCRtPage = S0t ≡ log kWhat happens for the capacity around the Page time?
Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Microcanonical ensemble• Replica partition functions Zn can be solved analytically in the microcanonical ensemble byfixing the energy of BH (in planar limit):• Entanglement entropy reproduces the Page curve for an eternal BH• The capacity shows a peak around the Page time and decays to zero at late time

0.6

2.5

0.5

2.0

0.4

1.5

0.3

1.0

0.2

0.5

0.1

24

Tatsuma Nishioka (YITP, Kyoto)

6810

2

4

68

10

May 19, 2021 @ Kyoto Univ Seminar

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Canonical ensemble• Numerically calculate Zn in the canonical ensemble by fixing the inverse temperature β ofBH (in planar limit):• Entanglement entropy reproduces the similar Page curve as in the microcanonical ensemble• The capacity shows a peak around the Page time and approaches to a constant at late time

50

1540
30

1020

5

10

1020

Tatsuma Nishioka (YITP, Kyoto)

30

40

50

1020

30

4050

May 19, 2021 @ Kyoto Univ Seminar

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Moving mirror model of radiating BH [Davies-Fulling 76, Birrel-Davies 84, · · · ]• CFT2 on flat space with reflecting boundary condition at a moving mirror• Known to have thermal energy flux (Hawking radiation) at null infinityusingularitytv
I+horizonI+x
insidethe mirror

I−I−
Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Conformal map to BCFT2• After a conformal map, the model becomes Boundary CFT2 on the right half plane:
utv

ũṽ

t̃
horizontrajectory ofv=−p(u)insidethe mirrorxinsidethe mirror

conformalmapx̃

mirror trajectoryv=p(u)Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Measuring Hawking radiation• Take an interval R at a fixed distance from themirror and measure the radiation• Replica partition functions can be calculated usingtwist operators:trR [ρnR ] ∝ h σn (t0 , x0 ) σ̄n (t1 , x1 ) iBCFT

utσnv
σ̄nRx

• Two-point functions in BCFT can be fixed by
conformal block [McAvity-Osborn 93]

Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Holographic CFT• Two-point functions greatly simplify in holographic CFT with large central charge[Takayanagi 11, Sully-Van Raamsdonk-Wakeham 20]:˜n (t̃1 , x̃1 ) iRHPh σ̃n (t̃0 , x̃0 ) σ̄(˜n (t̃1 , x̃1 ) iR1,1h σ̃n (t̃0 , x̃0 ) σ̄
1= maxQ˜n (t̃i , x̃i ) i 2 1,1e2(1−n)Sbdy · i∈{0,1} h σ̃n (t̃i , x̃i ) σ̄R(connected OPE channel)(disconnected OPE channel)• Sbdy ≡ log h0|Bi: boundary entropy for a boundary state |Bi• Twist correlator in flat space:c1˜n (t̃0 , x̃0 ) iR1,1 = (t̃0 − t̃0 )2 − (x̃ − x̃0 )2 − 12 (n− n )h σ̃n (t̃, x̃) σ̄
Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Entanglement entropy and Page curve [Akal-Kusuki-Shiba-Takayanagi-Wei 20]• Entanglement entropy can have two phases
corresponding to the two OPE channels:hicondis
SR = min SR, SR

1210

connected

8

disconnected6• This model has a phase transition between thetwo phases and reproduces the Page curve for a
non-evaporating BH

Tatsuma Nishioka (YITP, Kyoto)

42-1

01234

56May 19, 2021 @ Kyoto Univ Seminar

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Capacity in the moving mirror model• The capacity takes a universal form in eachphase:(conSR(connected channel)
CR =disSR − 2 Sbdy
(disconnected channel)1086• ∃ discontinuity at the Page time:
C con − C dis tPage

4
2= 2 Sbdy
-1012

34

56

• The capacity captures a phase transition
between the two phases
Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Outline

1. Introduction: entropy of Hawking radiation
2. Review: capacity of entanglement
3. Capacity of Hawking radiation in toy models
4. Capacity formula in 2d dilaton gravity

Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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2d dilaton gravity + large-c CFT• A general dilaton gravity in two dimensions (4GN = 1):
Igrav = IEH + Idil
• Einstein term (topological):
S0IEH = −4π

ZS0R−2πΣ2ZK = −S0 χ[Σ2 ]∂Σ2

• Dilaton term (constraint):
1Idil = −4π

Z


 Φb
Φ R + U (Φ)(∇Φ)2 + V (Φ) −2π
Σ2ZK∂Σ2• For JT gravity on AdS2 :Φ=φ,Tatsuma Nishioka (YITP, Kyoto)

U =0,V =2
May 19, 2021 @ Kyoto Univ Seminar

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Replica calculation in gravitational path integral• Two descriptions of the replica geometry [Lewkowycz-Maldacena 13]:
fn : no singularity in gravity region• n-fold cover M
fn /Zn : conical singularities at Zn fixed points in gravity region• orbifold Mn ≡ M

Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Replica partition function• The on-shell actions related by [Dong 16, Nakaguchi-TN 16]

11fIgrav [Mn ] = Igrav [Mn ] + 1 −nn• The "area" term:i
XhA(n) =S0 + Φ(n) (wi )

(n)
A|{z}

localized on singularities

wi : conical singularities in gravity regioni• In the semiclassical limit, 1  c  1/GN :11fn ] − 1 log ZCFT [Mfn ]− log Tr ρn = Igrav [Mnnn11fn ]= Igrav [Mn ] + 1 −A(n) − log ZCFT [Mnn
Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

31 / 36Island and capacity formulas in 2d dilaton gravity• The island formula [Almheiri-Maldacena-Hartman-Shaghoulian-Tajdini 19, Penington-Shenker-Stanford-Yang19]:
S≡X
1=[S0 + Φ(wi )] + SCFTlog Tr ρn1−nn=1i∈∂I• The capacity formula [Kawabata-TN-Okuyama-Watanabe 21]:
X1nC ≡ −2∂nlog Tr ρ=−∂n Φ(n) (wi ) n=1 + CCFT
1−nn=1i∈∂I

Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Implication of the formula: discontinuity at Page time
• The capacity should be calculated
fn6=1 !):
• with the values wi fixed by the QES condition (which does not depend on M
∂wi [S0 + Φ(wi ) + SCFT ] = 0• on the dominant saddle with least entropy (dictated by the island formula)• When there are two competing saddle solutions (with and without island)• the entropy is continuous at the Page time: ∆S ≡ Sno-island − Sisland |Page = 0• the capacity typically shows a discontinuity (we have no rigorous proof):∆C ≡ Cno-island − Cisland |Pagei
Xh=S0 + Φ(wi ) + ∂n Φ(n) (wi )|n=1 6= 0i∈∂I
Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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Example: eternal AdS2 BH at high temperature• Technical difficulties in calculating the capacity:• For ∂n Φ(n) (wi )n=1

FlatAdS2

solve the dilaton EOM onRadTM̃n6=1• For CCFT solve the conformal welding problem[Almheiri-Maldacena-Hartman-Shaghoulian-Tajdini 19]⇒ Both problems are hard to solve in general!
• We can circumvent these problems for thesemi-infinite radiation region in high temperature
limit

Tatsuma Nishioka (YITP, Kyoto)

Flat
I

AdS2
PRad
T

May 19, 2021 @ Kyoto Univ Seminar

34 / 36Comparison with thermodynamic quantities• The island phase always favored (no phasetransition)

FlatAdS2P• S and C coincide with thermodynamic entropyand capacity:
Sisland = Sth (β) ,Cisland = Cth (β)• We speculate this coincidence is universal in
the island/replica wormhole phase

Tatsuma Nishioka (YITP, Kyoto)

TbAdS2TFlat

May 19, 2021 @ Kyoto Univ Seminar

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Summary and future direction• The capacity of entanglement can be a good probe of Hawking radiation• EOW model: sensitive to the dominant replica wormhole saddle, dependent on the choice ofensembles• Moving mirror model: discontinuous at the Page time (in holographic CFT)

• The capacity formula implies• a discontinuity at the Page time
• coincidence with thermal capacity in the island phase

Tatsuma Nishioka (YITP, Kyoto)May 19, 2021 @ Kyoto Univ Seminar

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