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 conventionally described merely by classical geometry, and nothing cannot go out of the event horizon. Black hole is black. Infalling Large BH Particle

This picture drastically changes, because black holes can emit thermal flux due to quantum effect. (Hawking, 1974) = 2 r GM BH BH 3  c = T π HR 8 k GM B BH Black hole ain’t so black!

Thermal The Information Loss Problem radiation Hawking (1976) Ψ ˆ ˆ Ψ Ψ ≠ ρ † U U thermal Large black hole Thermal radiation Small black Only hole thermal radiation? Unitarity breaking? ρ thermal Information is lost!?

Why is the information loss problem so serious? Too small energy to leak the huge amount of information. Small BH (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.

From a modern viewpoint of quantum information, Information Loss Problem = Purification Problem of HR ∑ ρ = p n n HR n HR HR n Mixed state Hawking Partner radiation system Entanglement A system HR HR What is the partner after the last burst? ∑ Ψ = p n u n n HRA HR A HR HR n Composite system in a pure state

What is the purification partner of the Hawking radiation? (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, …)

(4) Radiation ○ Black Hole Complementarity From the viewpoint of free-fall observers, no drama happens across the horizon. Classical Horizon Infalling Large BH Particle

(4) Radiation ○ Black Hole Complementarity From the viewpoint of Hawking Radiation outside observers, Large BH the stretched horizon absorbs and emits quantum information so as to maintain the unitarity. Stretched Horizon Infalling Induced by Particle Quantum >> T E Gravity Unruh Planck

(4) Radiation ○ Firewall Large BH A FIREWALL at the horizon burns out FIREWALL free-fall observers. The inside region of BH does not exist! Free-fall observer

What is the purification partner of the Hawking radiation? ( 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) Zero-Point Fluctuation Flow of Quantum Fields ψ Zero-Point Fluctuation entangled Hawking Entanglement Sharing ≠ Particle Energy Cost of the Partner → Avoiding the planck-mass remnant problem. (Wilczek, Hotta-Schützhold-Unruh)

The partner entangled with a Hawking particle remains elusive due to the lack of quantum gravity theory to date. 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).

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!

Page Curve S EE Maximum value of EE is attained at each time, and is equal to thermal entropy of smaller system. HR BH A = = S S = dim , EE thermal BH H 4 G BH OLD BH = dim HR H time HR ln HR <<Page Time>> ≈ ln ln BH HR ○ ≈ 0 . 7 ( 0 ) M M Semi-classical regime! ○ Page BH

Page Curve Conjecture for BH Evaporation: Proposition I : After Page time, BH is maximally entangled with Hawking particles. No correlation among BH subsystems. BH Maximal Entanglement No correlation Hawking particles due to quantum monogamy Proposition II : = = /( G 4 ) S S A after Page time. EE thermal horizon

[ ] ∏ N After Page time, ρ = Ψ Ψ = ⊗ ρ ( 1 ) Tr BH HR n = 1 n N= # of BH degrees of freedom BH = ( 1 ) S NS No correlation among EE EE BH subsystems due to ( ) quantum monogamy = ( 1 ) S NS thermal thermal After Page time, 1 A = = ( 1 ) ( 1 ) horizon S S Lemma: EE thermal 4 N G This relation is criticized in this talk.

Temperature of BH is measured by temperature of Hawking radiation. = T T BH HR Note that BH has negative heat capacity! 1 1 = π = = − 2 < 0 T T C HR BH BH π 8 GM 8 GT BH BH HR BH → → ∞ 0 as T M BH BH

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 >> >> ( 1 ) ( 1 ) /( 4 ) S S A GN EE thermal horizon

Outline I. “Page Theorem” and BH Firewalls II. Soft Hair Evaporation at Horizon III. Multiple Qubit Model of Black- Hole Evaporation and Breaking of Page Curve Conjecture

I. “Page Theorem” and BH Firewalls

Lubk ubkin-Lloyd-Pagel els-Page T e Theo eorem em ( (“Page e Theo eorem em”) Typical states of A and B are almost maximally entangled when the systems are large. N A B N A Ψ B AB N << N Typical State of AB A B [ ] ρ = Ψ Ψ Tr Ψ AB A B AB AB [ ] = − ρ ln ρ S Tr EE A A A 1 ≈ ρ ≈ ln I S N A A EE A N A

Maximal Entanglement between A and B N 1 1 ∑ A ~ = ρ = ⇒ Max u v I A A n n AB A B N N = 1 n A A Orthogonal unit vectors N ≥ N B A

By using the theorem, AMPS and other people proposed the BH firewall conjecture. OLD BH Late radiation B Early radiation C A Ψ = ∪ HR A B ABC BH = C Page Time 1 << , , A B C << B C A → OLD BH

Proposition I means that A and BC are almost maximally entangled with each other. B C A     1 1 1     ρ = = ⊗ I I I     BC BC B C BC B C     NO CORRELATION BETWEEN B AND C! Harrow-Hayden

B     1 1 C     ρ = ⊗ A I I     BC B C B C     x C ε x B   2 ϕ − ϕ     ( ) ( ) 1 x x ρ =       B C Tr O ε ε BC    2      ε → 0 [ ] ( ) ∂ ϕ ρ = ∞ 2 ( ) Tr x BC FIREWALL!

Flaw of Page Curve Conjecture: Area law of entanglement entropy is broken, though outside-horizon energy density in BH evaporation is much less than Planck scale.

∝ ∂ = ∂ ← standard area law of | | | | S EE A B entanglement entropy A Ψ ≈ 0 AB AB B for low-energy-density states

| | BH 1 ∑ ~ Ψ = u v Page curve states ⇒ + n n BH HR BH HR | | BH = 1 n BH HR ← Not area law, = ln ∝ S BH V EE BH but volume law!

This is because zero Hamiltonian (or high temperature regime) is assumed in Page curve conjecture. = 0 . H AB

≈ + ≠ 0 H H H + = AB A B . E E const A B A B N >> N Ψ B A AB [ ] ρ = Ψ Ψ ˆ Tr Ψ A B AB AB AB [ ] = − ρ ρ ˆ ˆ ln S Tr AB A A A 1 ( ) ρ ≈ − β ˆ exp H A A Z A ≈ β EE is almost equal to thermal entropy of ( ) S S , AB thermal A 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. [ ] H ρ = H Tr H E B A A A A A A B + = Ψ E E E AB [ ] A B ρ = Ψ Ψ Tr A B AB AB [ ] [ ] = − ρ ρ ≤ − ρ ρ = ln ln S Tr Tr S EE A A A A thermal ρ = − β β exp( ( ) ) / ( ( )) E H Z E Gibbs state: A A A

Conventional “proof”: [ ] [ ] [ ] ( ) ( ) = − ρ ρ − λ ρ − − λ ρ − ln 1 I Tr Tr H E Tr 1 2 A A A A A A A A A ( ) δ I ρ = 0 A ( ) ( ) ) ρ = − β β exp( ) / ( E H Z E A A A A A [ ] ( ) ρ = ⇒ β = β Tr H E E A A A A A [ ] [ ] − ρ ρ ≤ − ρ ρ = ln ln Tr Tr S A A A A thermal 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.

No stable Gibbs state for Schwarzschild BH due to negative heat capacity! (Hawking –Page, 1983) If there exists a stable Gibbs state, the heat capacity must be positive. [ ] ( ) β = − β ( ) exp Z Tr H BH BH ( ) 2 − ( ) E E β = d E 1 / T = > 0 2 dT T