Reflected entropy for an evaporating black hole Yang Zhou ( ) - PowerPoint PPT Presentation

Reflected entropy for an evaporating black hole Yang Zhou ( 周洋 ) Based on arXiv:2006.10846 with Tianyi Li and Jinwei Chu

Introduction • Quantum gravity is the key to understand the origin of our universe • A simpler object involving quantum gravity is black hole. They have a temperature that leads to Hawking radiation. • Black holes also have entropy, given by the Area of the horizons. • The question is whether black holes behave like ordinary quantum systems. People believe they do (string theory, AdS/CFT) but do not know how.

• Importantly, there is a paradox if they do: consider a black hole formed by a pure state, after evaporation it becomes a thermal state (according to Hawking)-> information is lost • You may argue that strange things can happen at the end of the evaporation. But the paradox already shows up near the middle age of BH. • To understand this, we first introduce 2 different notions of entropy: fine-grained entropy and coarse-grained entropy.

Fine-grained < coarse-grained [Review: arXiv:2006.06872] • 1 st : Fine-grained entropy is simply the von Neumann entropy. It is Shannon’s entropy with distribution replaced by density matrix. It is invariant under unitary time evolution. • 2 nd : Coarse-grained entropy is defined as follows. We only measure simple observables . And consider all possible es A i . density matrices which give the same result as our system. g, Tr [˜ ρ A i ] = Tr [ ρ A i ]. We then choose the maximal von Neumann entropy over all possible density matrices . It increases under unitary time y S (˜ ρ ). evolution. -> entropy in thermodynamics.

Information paradox • Bekenstein-Hawking entropy is coarse-grained entropy. • The thermal aspect of Hawking radiation comes from separating entangled outgoing Hawking quanta and interior Hawking quanta. Each side is a mixed state. • As the entropy of radiation gets bigger and bigger, we run into trouble because, the entangled partners in black hole should have the same entropy, which exceeds the horizon entropy. • In fact, the constantly increasing result was made by Hawking. Page suggested that the outgoing radiation entropy should follow Page curve

Page curve [Fig from arXiv:2006.06872]

How to reproduce Page curve?

QES formula for BH [Penington, Almheiri-Engelhardt-Marolf-Maxfield] • The fine-grained entropy of black hole surround by quantum fields is given in terms of semiclassical entropy by ⇢ Area( Q ) � S B = ext Q + S (˜ ρ B ) , 4 G N a( Q ) S (˜ ρ B

Island formula for radiation [Almheiri-Mahajan-Maldacena-Zhao] • Similarly, the fine-grained entropy of radiation is given in terms of semiclassical entropy by ⇢ Area( ∂ I = Q ) � S ( ρ R ) = ext I + S (˜ ρ R ∪ I ) 4 G N (˜ ρ R a( Q ) an island

Quantum extremal surface [Engelhardt-Wall,RT,HRT] • QES origins from holographic entanglement entropy in AdS/CFT with bulk matter ⇢ Area( Q ) � " " + S bulk ( aa ⇤ ) AA ⇤ ) = AA ⇤ ) = S ( AA ⇤ ) = ext Q 4 G N ( aa " B

Motivations • So far we only consider BH + radiation is pure, but what if BH + radiation is a mixed state? • Are there other quantities which can have island formula? • Can we read more information about the island? • Can we compute the correlation between Hawking radiation A and B?

Von Neumann entropy vs Reflected entropy (See also arXiv:2006.10754 by V.Chandrasekaran,M.Miyaji,P.Rath)

Outline • Reflected entropy and the holographic dual • Quantum extremal cross section • Gravitational reflected entropy • Eternal black hole + CFT model

Canonical purification [Dutta-Faulkner] • Consider a mixed state on a bipartite Hilbert space ρ AB . discu • Flipping Bras to Kets for the basis = | i i h j | . i ⌘ | i i ⌦ | j i ample 𝜍 𝐵𝐶 = 1 • A canonical purification 𝜍 𝐵𝐶 = 1 2 ( ↑↑ ⟨↑↑ | 𝐵𝐶 + ↓↓ ⟨↓↓ | 𝐵𝐶 ) 2 ( ↑↑ ⟨↑↑ | 𝐵𝐶 + ↓↓ ⟨↓↓ | 𝐵𝐶 ) | p ρ AB i 2 Make a purification ) = ( H A ⌦ H ? A ) ⌦ ( H B ⌦ H ? B ) ⌘ Make a purification 1 𝛀 (𝟐) | p ρ AB i 2 1 B | p ρ AB i hp ρ AB | = ρ AB ( ↑↑↑↑ 𝐵𝐵 ′ 𝐶𝐶 ′ + ↓↓↓↓ 𝐵𝐵 ′ 𝐶𝐶 ′ ) 𝐵𝐵 ′ 𝐶𝐶 ′ = 𝛀 (𝟐) ( ↑↑↑↑ 𝐵𝐵 ′ 𝐶𝐶 ′ + ↓↓↓↓ 𝐵𝐵 ′ 𝐶𝐶 ′ ) Tr H ? 𝐵𝐵 ′ 𝐶𝐶 ′ = 2 A ⌦ H ? 2 EE (1) = log2 • Reflected entropy 𝑇 𝐵𝐵 ′ (1) = log2 𝑇 𝐵𝐵 ′ S R ( A : B ) ⌘ S ( AA ? ) p ⇢ AB 1 = log 2 𝐹 𝑄 𝐵: 𝐶 = min 𝑇 𝐵𝐵 ′ ≤ 𝑇 𝐵𝐵 ′ 1 = log 2 𝐹 𝑄 𝐵: 𝐶 = min 𝑇 𝐵𝐵 ′ ≤ 𝑇 𝐵𝐵 ′ 𝐹 𝑄 𝐹 𝑄

Reflected entropy • Properties pure state : S R ( A : B ) = 2 S ( A ) , factorized state : S R ( A : B ) = 0 , bounded from below : S R ( A : B ) � I ( A : B ) , bounded from above : S R ( A : B )  2min { S ( A ) , S ( B ) } understand re- • Graph description te ψ ABc ∈ H ABc c | p ρ AB i = | reduced density p Tr c | ψ ih ψ | i 2 ( H A ⌦ H A ∗ ) ⌦ ( H B ⌦ H B ∗ ) " " * B # * S R ( A : B ) = S ( AA ⇤ : BB ⇤ ) p ρ AB = Entanglement Entropy of Red Curve c

Holographic reflected entropy [Dutta-Faulkner] Σ %&' " 2 " * B A B # * AdS 𝑒 +1 t CFT 𝑒

 { Multipartite reflected entropy  ∆ W ( A : B : C ) [Chu-Qi-YZ,2019]

Replica trick • Replica trick in canonical purifications H H : ψ ( m ) m 2 i , p = | (Tr c ρ 0 ) ψ 1 = | Tr c | ψ ABCabc ih ψ ABCabc | i 1 : ψ ( m ) = | (Tr bb 0 ρ ( m ) p m 2 i , ψ 2 = | Tr bb 0 | ψ 1 ih ψ 1 | i ) 2 1 : ψ ( m ) = | (Tr aa 0 a 00 a 000 ρ ( m ) m 2 i p ) ψ 3 = | Tr aa 0 a 00 a 000 | ψ 2 ih ψ 2 | i 3 2 • Replica trick in Renyi index n ln Tr R (Tr L ρ ( m ) n ) n n 1 3 S S n n = ∆ R ( A : B : C ) = lim n ! 1 S n n , n n (Tr ρ ( m ) 1 � n n n n n ) n n en m ! 1. 3

Quantum corrected reflected entropy A B S R ( A : B ) = 2 h A [ ∂ a \ ∂ b ] i ˜ + S bulk ρ ab ( a : b ) + O ( G N ) ( R 4 G N

FLM on double replicas 1 h A [ m ( AA ⇤ )] i + S bulk ( aa ⇤ ) + O ( G N ) S ( AA ⇤ ) = 4 G N h A [ m ( AA ⇤ )] i = 2 h A [ ∂ a \ ∂ b ] i S bulk ( a : b ) = S bulk ( aa ⇤ ) R

Quantum extremal cross section [Li-Chu-YZ,2020] ⇢ Area( Q ) � + S bulk ( aa ⇤ ) S ( AA ⇤ ) = ext Q 4 G N ⇢ 2Area( Q 0 = ∂ a \ ∂ b ) � + S bulk S R ( A : B ) = ext Q 0 ( a : b ) R 4 G N

Eternal black hole + CFT [Almheiri-Mahajan-Maldacena] • AdS black holes do not evaporate. • Information paradox can be realized in AdS spacetime joined to a Minkowski region, such that black hole can radiates into the attached Minkowski region • Consider 2d Jackiw-Teiteboim gravity in AdS plus a CFT 2 (also in Minkowski), with a transparent boundary condition • Explicit computations can be done in this model

 ˆ � I total = − S 0 ˆ ˆ ( R + 2) � 4 ⇡ − � b ˆ R + 2 K 2 K + S CFT − 4 ⇡ 4 ⇡ Σ ∂ Σ Σ ∂ Σ

QES for a single interval in = 4 ⇡ 2 dyd ¯ y out = 1 ds 2 ds 2 y ) , ✏ 2 dyd ¯ y , sinh 2 ⇡ � 2 � ( y + ¯ y = � + i ⌧ , y = � − i ⌧ , ¯ ⌧ = ⌧ + � . S gen = S 0 + � ( � a ) + S CFT ([ � a, b ]) • –a is determined following QES condition ⇣ ⌘ π sinh β ( b + a ) ✓ 2 ⇡ a ◆ = 12 ⇡� r @ a S gen = 0 sinh ! ⇣ ⌘ � c � π sinh β ( a � b )

Gravitational reflected entropy (B-R) ⇢ A ( Q ) � S ( B L ) = min ext Q + S (˜ ⇢ B L ) 4 G N ⇢ A ( Q = @ I L ) � S ( R L ) = min ext Q + S (˜ ⇢ R L [ I L ) 4 G N ⇢ A ( @ I ) � S R ( R L : B L ) = S ( R ) = min ext I + S (˜ ⇢ R [ I ) 4 G N ⇢ 2 A ( Q 0 = @ ˜ I L ∩ @ ˜ � B L ) S R ( R L : B L ) = min ext Q 0 + S R (˜ I L : ˜ B L ) ⇢ R L [ ˜ ⇢ ˜ 4 G N

b = 0 . 01 , � r = 100 , S 0 = c = 20000 and ✏ UV = 0 . 01.

B-B reflected entropy ⇢ 2 A ( Q 0 = ∂ ˜ B L \ ∂ ˜ B R ) � S R ( B L : B R ) = min ext Q 0 + S R (˜ B L : ˜ B R ) ρ ˜ ρ ˜ 4 G N S R ( B L : B R ) = 2 S 0 + 2 � r + c 3( b − ln cosh t + ln 2) S R (˜ B L : ˜ B R ) ∼ c ( − 0 . 15 ⌘ ln ⌘ + 0 . 67 ⌘ ) ⇢ ˜ ⇢ ˜

L ck b = 1 , � r = 100 , S 0 = c = 1000

R-R reflected entropy ⇢ 2 A ( Q 0 = @ ˜ I 1 ∩ @ ˜ � I 2 ) S R ( R 1 : R 2 ) = min ext Q 0 + S R (˜ I 1 : ˜ I 2 ) ⇢ R 1 [ ˜ ⇢ R 2 [ ˜ 4 G N

} 1 2 k b 1 = 0 . 01 , b 2 = 5 , � r = 10 , S 0 = c = 2000 and ✏ UV = 0 . 001.

Summary • Reflected entropy curve is the analogy of Page curve for a globally mixed state • Reflected entropy can be computed for R-R, B-B and B-R • Reflected entropy has island cross-section as its area term • Future direction: multipartite generalization (in progress)

Thank You! 24