Soft-Hair Enhanced Entanglement Beyond Page Curves in Black-Hole Evaporation Qubit Models

Transcription

1 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:

2 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. Large Infalling Particle

3 This picture drastically changes, because black holes can emit thermal flux due to quantum effect. (Hawking, 1974) r 2GM c 3 T HR 8π k B GM Black hole ain t so black!

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

5 Why is the information loss problem so serious? Too small energy to leak the huge Small amount of information. (haronov, et al 1987; Preskill 1992.) If the horizon prevents enormous amount of information from leaking until the last burst of, only very small amount of energy remains, which is not expected to excite carriers of the information and spread it out over the outer space.

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

7 What is the purification partner of the Hawking radiation? Nothing, Information Loss (2) Exotic Remnant (haronov, Banks, Giddings, ) (3) Baby Universe (Dyson,..) (4) Radiation Itself (Page, ) Black Hole Complementarity ( t Hooft, Susskind, ) Firewall (Braunstein, MPS, )

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

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

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

11 What is the purification partner of the Hawking radiation? Nothing, Information Loss (2) Exotic Remnant (haronov, Banks, Giddings, ) (3) Baby Universe (Dyson,..) (4) Radiation Itself (Page, ) Black Hole Complementarity ( t Hooft, Susskind, ) Firewall (Braunstein, MPS, ) (5) Zero-Point Fluctuation Flow (Wilczek, Hotta-Schützhold-Unruh )

12 (5) Zero-Point Fluctuation Flow Zero-Point Fluctuation of Quantum Fields ψ Entanglement Sharing Energy Cost of the Partner entangled Hawking Particle voiding the planck-mass remnant problem. (Wilczek, Hotta-Schützhold-Unruh)

13 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 and Hawking particles is not able to be attained. popular conjecture The Page Curve Don Page, Phys. Rev. Lett. 71, 3743 (1993).

14 Page s Strategy for Finding States of Evaporation: Nobody knows exact quantum gravity dynamics. So let s gamble that the state scrambled by quantum gravity is one of TYPICL pure states of the finite-dimensional composite system! That may not be so bad!

15 HR dim H dim H HR HR, S EE <<Page Time>> Page Curve Maximum value of EE is attained at each time, and is equal to thermal entropy of smaller system. OLD S EE S thermal time ln HR 4G ln ln HR M Page 0.7M (0) Semi-classical regime!

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

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

18 Temperature of is measured by temperature of Hawking radiation. T T HR Note that has negative heat capacity! 1 T HR 8π GM T C 1 8πGT 2 < 0 HR T as 0 M

19 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 S >> S >> EE thermal horizon /(4GN )

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

21 I. Page Theorem and Firewalls

22 Lubkin-Lloyd-Pagels-Page Theorem ( Page Theorem ) Typical states of and B are almost maximally entangled when the systems are large. N Ψ B Typical State of B B N B N << N B Ψ B [ ] ρ Tr Ψ Ψ B B B S EE Tr [ ρ ln ρ ] S ln N EE ρ 1 N I

23 Maximal Entanglement between and B ρ 1 N I Max B 1 N N n 1 u n v~ n B Orthogonal unit vectors N B N

24 By using the theorem, MPS and other people proposed the firewall conjecture. OLD C Late radiation B Early radiation Ψ BC HR B C Page Time 1 <<, B, C B C << OLD

25 Proposition I means that and BC are almost maximally entangled with each other. C B ρ BC I BC I B IC BC B C NO CORRELTION BETWEEN B ND C! Harrow-Hayden

26 B C C B BC I C I B 1 1 ρ FIREWLL! ( ) [ ] BC x Tr ρ ϕ 2 ) ( ) ( ) ( ε ρ ε ϕ ϕ O x x Tr BC C B x C x B 0 ε ε

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

28 S EE B standard area law of entanglement entropy Ψ 0 B B B for low-energy-density states

29 Page curve states Ψ + HR 1 n 1 u n v~ n HR HR S ln EE V Not area law, but volume law!

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

31 H B H + H B 0 E + E B const. Ψ B B N B >> N [ ] ρˆ Ψ B S Tr Ψ Ψ B B B B Tr 1 ˆ ρ exp β Z [ ˆ ρ ln ˆ ρ ] ( H ) S S ( β ) B thermal, EE is almost equal to thermal entropy of the smaller system for typical states.

32 Remark: for ordinary weakly interacting quantum systems, entanglement entropy is upper bounded by thermal entropy, as long as stable Gibbs states exist. EE H H B Tr [ ρ H ] E Ψ B B [ ] ρ Tr Ψ Ψ B B B E + E B [ ] [ ] ln Tr ln S thermal S Tr ρ ρ ρ ρ E Gibbs state: ρ β exp( ( E) H ) / Z ( β ( E))

33 I Conventional proof : [ ρ ρ ] λ ( Tr [ ρ H ] E ) λ ( Tr [ ρ ] 1) Tr ln 2 1 δi ( ρ ) 0 exp( ρ β Tr ( E ) H ) / Z ( β ( E )) [ ρ H ] E β β ( E ) [ ] [ ] ρ ln ρ Tr ρ ln ρ S thermal Tr 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.

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

35 Thus, a system of a black hole and Hawking radiation is not in a typical state, at least in the sense of the LLPP theorem. Because we have no stable Gibbs state, thermal entropy of Schwarzschild is not needed to be a upper bound of entanglement entropy. S EE /( 4G )

36 II. Soft Hair Evaporation at Horizon

37 What is the purification partner of the Hawking radiation? Nothing, Information Loss (2) Exotic Remnant (haronov, Banks, Giddings, ) (3) Baby Universe (Dyson,..) (4) Radiation Itself (Page, ) Black Hole Complementarity ( t Hooft, Susskind, ) Firewall (Braunstein, MPS, ) (5) Zero-Point Fluctuation Flow (Wilczek, Hotta-Schützhold-Unruh ) (5) Soft Hair without Energy (Hotta-Sasaki-Sasaki (2001), Hawking-Perry -Strominger (2015))

38 Gravitational vacuum degeneracy with zero energy plays a crucial role. Hawking, Perry and Strominger call it soft hair. Hawking Particles T > 0 Horizon soft hair ω 0 Bondi-Metzner-Sachs soft hair Information E 0 M. Hotta, K. Sasaki and T. Sasaki, Class. Quantum Grav. 18, 1823 (2001). M. Hotta, Phys.Rev. D (2002).. Strominger, JHEP07,152 (2014). S. W. Hawking, M. J. Perry and. Strominger, PRL 116, (2016). M. Hotta, J. Trevison and K. Yamaguchi, Phys. Rev. D94, (2016). S. W. Hawking, M. J. Perry and. Strominger, JHEP 05, 161 (2017). (Feb 3 rd, 2017 at Cambridge)

39 Horizon soft hair of black holes comes from would-be gauge freedom of general covariance, diffeomorphisms (Hotta et al, 2001). This is a similar mechanism of emergence of momentum eigenstates orthogonal to each other. y Black Hole at Rest y' y Running Black Hole y' p 0 x' x x' t' x coshω + t sinhω, xsinhω + t coshω We must have Lorentz covariance in the theory. p ' msinhω x x'

40 Lorentz transformation, one of general coordinate transformations, generates an infinite number of physical states with different values of momentum. Horizon soft hair of black holes comes from would-be gauge freedom of general covariance, diffeomorphisms (Hotta et al, 2001). This is a similar mechanism of emergence of momentum eigenstates orthogonal to each other. y' y y' y Running Black Hole p p msinhω x x' 0 p msinhω 0 p ' msinhω x x'

41 Soft hair of black holes comes from would-be gauge freedom of general covariance. Horizon τ Supertranslation: ' τ + T ( θ, φ Superrotation: θ ' Θ( θ, φ ), ) Horizon Q horizon 0 φ ' Φ( θ, φ ) Q' horizon q We must have horizontal asymptotic symmetry as a part of general covariance in the theory.

42 Soft hair of black holes comes from would-be gauge freedom of general covariance. Supertranslation and superrotation generate an enormous number of physical states with different values of holographic charges. Horizon Horizon Q horizon q Q' horizon q Q horizon 0 Q q 0 horizon

43 Symmetries Null Future Infinity Poincare Symmetry: x ' Λx + c including time translation generated by Hamiltonian H BMS Supertranslation: u ' u + C( θ, φ (BMS Superrotation?) ) Near Horizon Horizon Supertranslation: τ θ ' φ τ + ' Θ( ' Φ( T θ θ (,, θ φ φ, ) φ ), ) Horizon Superrotation: [ H I, I Q ] [ H I, I Q ] 0 horizonst horizonsr Near-horizon symmetry provides degeneracy of Hamiltonian.

44 Horizon Hawking radiation ψ lnw O ( rea /( 4G) ) Soft hair of O soft hair 4 G Collapsing Matter µ µ J ( i) Collapsing Matter n infinite number of supertranslation charges at the horizon might store whole quantum information of absorbed matter. Unitarity? (Hotta-Sasaki-Sasaki, 2001) (Hawking Perry-Strominger, 2017) 0

45 III. Multiple Qubit Model of Black- Hole Evaporation and Breaking of Page Curve Conjecture

46 Hawking Temperature Relation: Key Idea: T 8π G T 1 Ψ H Ψ 8π 1 GM Construct many-body systems which reproduces this relation. This condensed matter model yields negative heat capacity. Bekenstein-Hawking entropy can be defined even if the systems do not have actual horizons. S d Ψ H T Ψ 4 π ( ) 2G Ψ H Ψ 4G 2 4 π r 4G 2 horizon 4G

47 Decaying qubits of into qubits of HR in Page Model Hawking Radiation Black Hole S EE S EE S thermal () entropy ρ [ Ψ ] TrB Ψ B B Page time S EE [ ( t)ln ( t) ] duˆ ( ) ( t) Tr ρ ρ t N / 2 Implicitly assumed that partition function of the total system is well defined. Thus, the heat capacity is always positive. N n(t)

48 In order to incorporate negative heat capacity, we consider not only emission of Hawking particles but also decreasing of degrees of freedom (soft hair)! (Hotta-Nambu-Yamaguchi 2017) Zero-Energy Soft Hair Evaporation Hawking Particle Emission Cf. Negative heat capacity in D0 particle model, E. Berkowitz, M. Hanada, and J. Maltz, Phys. Rev. D94, , (2016).

49 Negative heat capacity induced by soft particle decays E t ) E tot o ( 1 tot o E ( t ) 2 E E 0 E 0 E 0 E 0 E 0 E 0 Large Number of Particles Low T 1 T < T 1 2 Small Number of Particles High T 2 If radiation coupling is switched on, the temperature of emitted radiation rises!

50 Our Model: aˆ 0 + aˆ 0 + N qubits of at Initial Time N-1 N Ψ ( t 0) U N 1 N ( N ) ˆ One of typical pure states of N qubits Energy conserved random unitary operation (Fast Scrambling)

51 aˆ 0 Our Model + a+ 0 Fast-Scrambled Pure-State Black Hole at Initial Time ˆ 0 Evaporating Soft Hair Hawking Particle Soft Hair Emission H o ω

52 SH HR o H H H H + + ( ) + N n z H o n 1 1 ) ( 2 σ ω ( ) Ψ + + Ψ + + Ψ Ψ dx x x g dx x x g dx x i x H R R R x R HR ) ( ˆ ) ( ) ( ˆ ) ( ) ( ˆ ) ( ˆ ( ) Ψ + Ψ + Ψ Ψ dx x x f dx x x f dx x i x H S S S x S SH ) ( ˆ ) ( 0 ) ( ˆ ) ( 0 ) ( ˆ ) ( ˆ Initial State: S R Qbits vac vac t Ψ Ω 0 ( 0 ) ( ˆ ) ( ˆ Ψ Ψ S S R R vac x vac x ~ internal dynamics ~ Hawking radiation emission ~ Soft hair emission

53 One-site reduced state is described by ρ Tr2,3, N, R, S [ ] Ω( t) Ω( t) ( 1 p( t) ) p( t) ρ( T ( t)). ρ( T ) + e ω T ω T 1+ e Qubit Gibbs state at temperature T Hawking particle is emitted by flipping the up state to the down state. So the radiation temperature is equal to T! This model actually reproduces the Hawking temperature relation: 1 M ( t) N H, o T ( t) Qbit Survival Probability 8π G M ( t) N + + ( 4πGωT (0)) 1

54 Entanglement entropy between a single site and other subsystems: S Tr EE [ ] ρ ln ρ Thermal entropy per site: ω Sthermal psth p ln 1 + exp T Bekenstein-Hawking Entropy per site: + ω T 1+ 1 dm 1 S horizon N T N 4G S 1 ω exp T N Page Curve Conjecture EE thermal S S S after Page time

55 ctually, EE thermal S >> S >> S (Hotta-Nambu- Yamaguchi 2017) ( 1) ( 1) Before Page Time fter Page Time ω T (0) 0.1 Page time Last Burst of

56 T (0) : Initial Temperature High temperature regime: S EE T (0) T ln T T (0) S thermal S N >> due to Soft Hair Contribution 4T T >> ω (0) T (0) T ln 2 (0) T 2 due to Time-Dependence of Survival Probability p(t)

57 First Law: ds d ( ) NpEth deth NEth + dp T One-body Gibbs State Relation: veraged Thermal Entropy: Np T de dsth T S th thermal NpS th T ds ds thermal N S th E th T dp S 1/ T ( p') ( ) S Np ln 2 + N dp' dβ E ( 1/ β ) thermal Large discrepancy is generated by this term 0 p 0 th

58 First Law: ds d ( ) NpEth deth NEth + dp T One-body Gibbs State Relation: veraged Thermal Entropy: Np T de dsth T S th thermal NpS th T ds ds thermal N S th E th T dp Large discrepancy is generated by this term S T (0) T (0) T (0) 0 + O 2 O T T T 2 2 2

59 Feedback to Firewall Paradox C B Late radiation Early radiation C B Late radiation S Soft hair Early radiation Early Hawking radiation is entangled with zero-energy BMS soft hair, and that late Hawking radiation can be highly entangled with the degrees of freedom of the, avoiding the emergence of a firewall at the horizon.

60 Summary We construct a model of multiple qubits that reproduces thermal properties of 4-dim Schwartzschild s. fter Page time, entanglement entropy, thermal entropy and Bekenstein-Hawking entropy are not equal to each other. Entanglement entropy exceeds thermal entropy and entropy. Our result suggest that early Hawking radiation is entangled with zero-energy soft hair, and that late Hawking radiation can be highly entangled with the degrees of freedom, avoiding the emergence of firewalls at horizons.