JHEP10(2019)132 Springer April 30, 2019 August 2, 2019 October 9, 2019 : : : September 18, 2019 : Revised Received Published Accepted Published for SISSA by https://doi.org/10.1007/JHEP10(2019)132 . 3 1902.09763 The Authors. c AdS-CFT Correspondence, Black Holes, Holography and condensed matter

” or “ER = EPR” proposals. We show that the gravitational backreaction by Recently we pointed out that the black hole interior operators can be recon- , B [email protected] R = A E-mail: Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada Open Access Article funded by SCOAP ArXiv ePrint: crossing the horizon. Keywords: physics (AdS/CMT) from the originally entangled qubitthe in early the radiation early cannot radiation.tion influence Namely, of the quantum experience the operation of interior on verification the mode of infalling does quantum observer not since entanglementInstead involve descrip- by it the the will early perform outside radiation the observer at Hayden-Preskill does recovery all. which not saves We an create also infalling a observer argue firewall. from that tion of interior operators. Ourthe construction “ avoids the non-locality problemthe which infalling plagued observer, whofrom simply falls the into early a radiation.quantum black entanglement hole, between the The disentangles outgoing the infalling mode outgoing and observer mode the crosses interior mode the which horizon is distinct smoothly and sees Abstract: structed by using the Hayden-Preskillpropose recovery protocols. a resolution Building of on the this firewall observation, we problem by presenting a state-independent reconstruc- Beni Yoshida Firewalls vs. scrambling JHEP10(2019)132 8 35 16 20 15 7 30 – i – 25 21 9 4 32 20 32 29 10 U 11 13 31 22 33 14 6 17 4 30 5 27 25 24 11 1 8 9.1 AMPS thought9.2 experiment Revisiting firewalls 9.3 Puzzle on non-local interactions (and its resolution) 7.3 Geometric interpretation 8.1 No-go arguments 8.2 Observer-dependence 6.2 State-independence 6.3 Code subspace 7.1 Including Alice 7.2 Sending probes 5.2 Young black5.3 hole Monogamy and scrambling 6.1 Long-throat AdS black hole 4.1 Scrambling 4.2 OTOCs in4.3 a black hole Reovery protocols 5.1 Hayden-Preskill for interior operators 3.2 State representation3.3 of Reconstruction of3.4 operators Quantum cloning puzzle 2.1 From the2.2 outside From the2.3 inside Interior operators 3.1 Setup 9 Resolution of AMPS puzzle 8 On definition of state-independence 7 Backreaction by infalling observer 6 State-independent interior operators 5 Reconstruction of interior operators 4 Scrambling and recovery 3 Review of Hayden-Preskill recovery Contents 1 Introduction 2 Review of firewall argument JHEP10(2019)132 ], 3 ]. 19 ] and the ] plays key 16 21 , , Relatedly, the ]. To overcome 1 20 15 , 14 , 5 manner that suffers 13 ]. The ultimate resolution of the 25 ]. Unfortunately this proposal runs – 8 state-dependent ]. Following earlier works, Almheiri, – 1 22 4 , 18 38 – 1 – ] for summaries). The reconstruction may lead to ], but these proposals have some caveats [ 37 12 18 , , 43 4 41 17 40 ]. According to Hawking’s semiclassical calculation [ 2 ]. (See [ 38 11 – 9 36 36 ” or “ER = EPR” where the interior operators are reconstructed in the B R = A Preliminary studies suggest that the idea of quantum error-correction [ The AMPS problem becomes most precise for the outside observer where the difficulty The fact that some description of interior operator requires the early radiation does not necessarily 10.1 Early time 10.2 Peeking in through10.3 the horizon Entanglement wedge reconstruction 1 roles in avoiding some of the identified problems [ mean that information canoperators be requires the sent early non-locally.interact radiation, non-locally However, then with if it the every is early possible radiation problematic by as reconstruction seeing it of interior implies the modes. that interior the infalling observer may previous proposals construct interiorfrom operators a number in of a potentialthese inconsistencies difficulties, with quantum further mechanics proposals [ proposal have was been made put inSachdev-Ye-Kitaev (SYK) forward. the model context [ A of concrete the state-dependent AdS/CFT correspondence [ under “ early radiation instead of theinto remaining various black paradoxes hole [ [ apparent non-local interaction between theinterior operators interior seem and to distant be radiation. sensitive to perturbations added to the early radiation. Also, to a violation ofThis quantum contradiction, called mechanics, the namely AMPSin the puzzle, describing monogamy sharply the illustrates of interior the of quantum fundamental a entanglement. difficulty quantum black hole. is constructing operators for interior modes. This viewpoint led to approaches bundled the most puzzling mysteriesMarolf, in Polchinski theoretical and physics Sully [ early (AMPS) Hawking radiation, argued one can that distillparticle by a qubit careful from that quantum is the perfectly processinghowever, entangled black that of with same an hole the particle outgoing is [ entangled with some field mode inside the black hole, leading 1 Introduction The question of how a quantum black hole forms and evaporates unitarily remains among B Verlinde-Verlinde proposal C Fault-tolerant entanglement D Alice’s quantum field theory A Energy measurement 10 Discussions JHEP10(2019)132 ]. 27 ] still 26 no matter ]. A quan- ˜ D 28 on the early . Namely, any ˜ D D ]. In parallel with 29 and that is entangled with R . The infalling observer D ˜ D ], suggested that verification of does not involve any degrees of 6 will be entangled with the interior D from the distilled qubit way while at the same time avoiding D ], a phenomenon of reconstructing an D influence the experience of the infalling Namely, we will find that construction ]. However, our observation in [ 26 2 on the infalling observer. 30 , 6 previous arguments which appear to suggest cannot – 2 – from the early radiation R depend ˜ D avoids state-independent . The outgoing mode from the originally entangled qubit . We show that an infalling observer leaves non-trivial gravitational D distinct . The disentangling phenomenon is a universal feature of a quantum black R which is D The aforementioned proposal provides an account for a physical mechanism where Based on the state-independent reconstruction, we suggest the following resolution of In this paper, we expand on the aforementioned observation and present a protocol to Recently we pointed out that the procedure of reconstructing the interior operators I thank Ahmed Almheiri and Suvrat Raju for useful discussions on this. 2 is intriguing to considerobserver can an see opposite quantum limitworks, entanglement of especially while the the the monogamy onequantum infalling relation by entanglement, observer Maldacena where does namely and the not. the Susskind outside distillation Previous [ of the entangled qubit observer since the constructionfreedom on of the interiorhole mode which can be studied quantitatively by using out-of-timethe order infalling correlation observer functions. sees quantum entanglement while the outside observer does not. It backreaction which disentangles the outgoing mode how she falls intomode a black hole crosses the horizon smoothlyquantum operation and on can the early observe radiation entangled modes of interior operators doesentangled not with depend the early on radiation), the but initial state ofthe the AMPS black puzzle. hole Let us (i.e.the distill how a outgoing it qubit mode is outgoing mode and thethe reconstructed early interior radiation. mode is Inoperators insensitive fact, is to we applicable any argue to perturbationWe that generic will on black our explain holes, construction how young of ourthat or state-independent construction interior old, interior operators in must quasi be thermal state-dependent. equilibrium. lative quantum circuit complexity argumentssuffers [ from the non-locality problem and state-dependence. reconstruct interior operators in a the non-locality problem. In particular, we show that quantum entanglement between the quantum information in a sensetum of black decay hole of exhibits out-of-time suchthe dynamics ordered conclusion in correlations by Hayden the [ and most Preskill,can dramatic this be manner observation suggests constructed [ that “almost” the insidethe interior early the operators radiation. remaining Also, black the holeon reason with the why a early interior few operators radiation are extra can insensitive qubits be to from explained perturbations by scrambling dynamics without relying on specu- interior operators which does not suffer from the aforementionedcan problems. be viewed asinfalling the quantum Hayden-Preskill state recovery fromInput [ an quantum old states black can hole be by recovered collecting when the the Hawking time-evolution radiation of [ the system scrambles AMPS puzzle, however, would be obtained only by finding an explicit way of reconstructing JHEP10(2019)132 , we 9 from the R , we discuss C This proposal , we show that 3 5 at all. Here, we argue R is distinct from the distilled , we make comments on the , we briefly review the firewall D B 2 , we conclude with discussions. , we discuss how our construction 10 and the early radiation 8 D – 3 – ] for details. ]. In appendix 31 22 ]. In fact, these recovery protocols proceed by protecting , we discuss geometric interpretations of the Hayden-Preskill , we discuss the notion of quantum information scrambling 31 D 4 , we briefly review the Hayden-Preskill recovery and the quantum 4 3 , we show that measurement of energy density suffices to entangle the in- A , we present a state-independent reconstruction of the interior operators. In , generates a high-energy gravitational shockwave which would become a firewall 6 R , we discuss the effect of backreaction by the infalling observer. In section and its construction does not involve the early radiation 7 ˜ D The rest of the paper is organized as follows. In section While we rely on intuitions from the AdS/CFT correspondence, our arguments are Here the infalling observer should be thought of as some light probe particle. If itOur were attitude a is macroscopic to treat a black hole as a quantum system and try to explain the AMPS puzzle without 3 4 object with largeoperations. energy, Also the the outside amountassociated observer of with entanglement wouldn’t the the infalling be outside observer. able observer See needs to [ to save verifygiving depends it up with on by unitarity, the low smoothness entropy (equivalence quantum principle) complexity or locality. falling observer with theproposal outgoing by radiation. Verlinde and Inavoids Verlinde previous appendix [ arguments for the necessitythe of state-independent state-dependence. reconstruction In from appendix thebody perspective systems. of In dynamics appendix inrecovery quantum protocols. many- the interior operators can beIn reconstructed by section using the Hayden-Preskillsection recovery protocol. propose the resolution ofIn the appendix AMPS puzzle. In section simplified toy models. argument. In section cloning puzzle. Inand section its relation to out-of-time ordered correlation functions. In section around/after the scrambling time. Theoperators proposed work state-independent for construction of generic interior quantumof black our holes argument, that however, satisfyof is these a two based assumptions. quantum on black Much simplethe hole. quantum scope information While of theoretic detailed the description calculations paper, on we realistic will black try holes to are be beyond clear about our assumptions behind the use of backreaction by the infalling observer. applicable to generic quantumproposals. black First, holes. the outside There observerquantum has are state a on two quantum his description Hilbert main of space. assumptions a Second, black behind out-of-time hole ordered with our correlation a functions pure decay observer can retrieve quantum entanglement back fromthe the Hayden-Preskill infalling observer recovery by on performing not the cross infalling the observer. horizon,resonates she with Since will the the recovery not protocols infalling be forby observer the able the Hayden-Preskill does to thought author experiment see proposed andquantum quantum entanglement entanglement. Kitaev between [ the outgoing mode and prevent thehowever, infalling can observer be refuted from by observingqubit crossing that the the interior horizon mode that smoothly. the verification This ofthe proposal, quantum infalling entanglement observer from will falling generate into “safety a black barrier” hole. which The stops key observation is that the outside radiation JHEP10(2019)132 GM (2.1) 3 r > corresponds : the outgoing  in the outside : the reference. : the remaining 0 i + D C A Ψ | GM ]. The argument has can be factorized into 2 = 2 r outside H . . Here R (b) : the early radiation. GM ⊗ H 3 R D represents the unitary time-evolution of the to include only modes with Schwarzschild : the remaining black hole. EPR EPR D U is of order the Planck length. This region is ⊗ H C H   < r < C – 4 – H + = GM . Throughout the paper, we will denote the infalling and outside 1 H EPR : the initial black hole. (a) B at any given time. The Hilbert space . i j i ⊗ | outside j | H : the outgoing mode. Horizontal lines represent that two subsystems are in the EPR =1 d j D are the modes confined in 2 consists of the radiation field outside of the black hole with roughly ) : the early radiation. (b) The Hayden-Preskill thought experiment. P . The notation used in the paper. Here D d R 1 R 2a √ H H The AMPS thought experiment relies on the following assumption on the Hilbert The notation in the present paper is slightly non-standard. For convenience, our : the input quantum state. Here while to the location of theoften stretched called the horizon, zone. and Itenergy is common less to restrict than the black hole temperature since higher energy modes are not confined. described by a unitary operator.Hilbert Namely, space there is a(figure pure quantum state 2.1 From the outside We first discuss the descriptionoutside of observer. a quantum black hole from the perspectivespace of structure. Bob, In the Bob’s description, the formation and evaporation of the black hole is in reconstructing the interior operators. notation is summarized in figure outside observers by Alice and Bob respectively. We begin with a briefa number overview of of technical the assumptions originalgoal and argument is fine of to details, remind AMPS and readers [ space we of will structure the not form cover postulated all ofsummarize by of quantum some them. entanglement AMPS, and of Our as the the associatedconcerns well proposed that Hilbert as may resolutions arise to of from introduce them. the puzzle, Our some goal and notation. is potential to motivate difficulties We some and also of the desired properties black hole. pairs: 2 Review of firewall argument Figure 1 black hole. (a)mode. The AMPS thought experiment. A JHEP10(2019)132 (2.2) (2.3) ). We with a i 2.2 . Finally, Ψ  | . to be a finite − and the double R A H GM GM 2 = 3 r . S GM < r < . ⊗ H (b) D I R is a Haar random quantum state in , but this detail does not matter in the ⊗ i S . Of course, approximating ⊗ H C Ψ | I H . It is common to restrict R | D  is approximately proportional to the identity D + 1 || . – 5 – . If ⊗ H | C | CD ) D D GM corresponds to the zone. (a) The outside Hilbert space inside || ≈ H 2b H C D = 2 = CD ]. For the time being, we proceed with eq. ( ] although these are purely classical ones resulting from r ρ 26 (a) 22 [ inside |  | R H | CD ρ . Roughly, are often included in  C  H correspond to the confined modes (the zone) and the radiation fields GM is maximally entangled with R = 2 ], the density operator in H r D are the remaining degrees of freedom which can be interpreted as some entities 33 , and C . The Hilbert space structures. The dotted line represents 32 can be factorized into (figure [ H D . (b) The infalling Hilbert space H are the modes just inside the horizon roughly with Following AMPS, we assume that Alice has quantum mechanical description of her Following AMPS, we consider an old black hole that has already emitted more than is all the remaining degrees of freedom that is not accessible to Alice. The modes at D S outside outside outside Here respectively as defined above. TheseH are subsystems which bothH Alice and Bob can access. the stretched horizon experiences in termsH of a quantum state on her time slice. Namely, the Hilbert space emphasize that this detail does not matter for discussions2.2 except appendix From the inside Next we discuss the description from the perspective of Alice, the infalling observer. This implies Haar random state is ancorrelations oversimplification. are A present in more carefulthe analysis conservation suggests of that the thermal energy [ half of itsH initial entropy: operator: Finally, sitting at the stretched horizondimensional at Hilbert space for conveniencesubsystems of are discussions. not Precise essential distinctions for among the these AMPS puzzle. Figure 2 lines represent H JHEP10(2019)132 . , C D ρ H (2.4) ⊗ ) = 0. do not , which D ρ R differ up to D C,D = ( for Alice and I (Bob) DR ρ DD DR ρ ρ and ] for instance, but these are is interpreted as degrees (Alice) DR 34 ρ R must be entangled. On the D . H 0. This suggests D i ≈ . This viewpoint is often advocated n and R are identical for two observers. D . The key assumption for us is that non-local D ⊗ | R R D S U contain Rindler modes which are close to D H i ⊗ − and n D | D D H n U D cannot be described as an operator inside S . To avoid the non-local signalling problem, the 2 . It may be possible that – 6 – βE R + D ” in accordance with the standard notation in the − and e B (Bob) DR D ρ is maximally entangled with the radiation S R D n = X H ≡ D = requires degrees of freedom from a far distant boundary ) ∝ A D (Alice) DR ρ are not entangled with each other due to the monogamy of D, 7 ( , always D 6 I ” (or “ R and = D D 5 ], arrived at (and vice versa). To be concrete, one may consider a large AdS black hole with 35 R , 34 First, it may lead to non-local signalling from the interior of the black hole to the The AMPS puzzle concerns an apparent inconsistency between Alice and Bob’s de- The term “firewalls” generically denote various forms of high energy densities that may appear near By “identical”, we mean that More precisely, this suggests a strong breakdown of the equivalence principle (or effective quantum field 5 6 7 the horizon. Inquantum addressing entanglement the between AMPS pairsprimary puzzle, focus of we is modes on will anproperly inside focus apparent understanding and on violation the breakdown of outside firewalls of monogamytheory. the that semiclassical of Note arguments may quantum black through that entanglement arise the hole. there and lensbeyond from may how of be the to the In quantum other scope restore information loss other physical it of mechanisms of by words, this for paper. our firewalls, see [ local unitary transformations withproperties factorized such form as quantum entanglement between theory description) in a regime of low curvature. very long wormhole throat. The descriptionwould of be the interior able mode, to which an measure, of infalling observer the black hole,referred which to sounds as highly thecommute with unphysical. commutator operators problem in The the sincereconstruction radiation non-local of operators the signalling for interior problem operator the in is interior the outside mode should be insensitive to perturbations Namely, its description mustunder involve the the slogan radiation “ literature) and also “ER = EPR”. This proposal, however,radiation runs into various paradoxes. The AMPS puzzle can bebecomes also most stated precise in in theof outside the observer’s freedom AdS/CFT description. on correspondence This where theor viewpoint other in side the of boundaryThis a suggests CFT, large that the two-sided the AdS puzzle interior mode black can in hole. be identified In with Bob’s the description, fact that entanglement. In particular, byworks [ using the strongthe subadditivity, existence AMPS, of as a well firewall. as earlier 2.3 Interior operators Alice’s Hilbert space may containimportant some degrees point of for freedom otherother us than hand, is Rindler in modes. that Bob’s The would two description, imply modes that in Bob are identical. scriptions. In Alice’smaximally description, entangled: AMPS argument. Finally, we postulate that the quantum description of JHEP10(2019)132 . 9 ]. ]. ]. ]. is . R R i 25 27 37 , 30 – (2.5) , Ψ ]. To | 36 6 22 and depends , 16 , R 18 R . i 15 CD , R 5 i, j ⊗ | CD i and the early radiation . Such a state can be , hence the AMPS thought | is fine as long as information R C i, j D | R || C i,j | X 2 | 10 ]. D on the early radiation by relying on computational complexity || 1 19 D C = log BH , | are EPR pairs between S and without the non-locality problem. p 14 and the radiation CD , R via some sort of quantum error-correcting S ≡ D CDR 13 i R , 10 CDR – EPR i – 7 – 8 Φ since the measurement of the interior mode by the infalling | EPR R Φ | ]. Namely, they conjectured that quantum circuit complexity of veri- , i 30 . But these state-dependent interior operators are problem- EPR K Φ | ) K ⊗ I = ( i ]. This viewpoint has been considered seriously by several authors [ ]. Also, the argument does not apply to the eternal AdS black hole where the verification of Ψ 8 is some unitary operator and | 21 38 , K 20 . The problem may be resolved by employing the idea of quantum error-correction To summarize, our goal is to reconstruct interior operators in a state-independent man- Second, it leads to state-dependent construction of interior operators [ Again, we think that this is a serious problem if all the possible representations of the interior mode Harlow and Hayden suggested that there is a certain obstruction for the infalling observer to verify Some previous works argue that the involvement of the early radiation R 9 8 10 fying the quantum entanglement istheoretical exponential assumptions. in the This entropy viewpointHowever, has the been further complexity strengthened theoreticputation by obstruction Aaronson [ itself and Susskind can [ entanglement be can avoided be by performed performing via simple the quantum so-called operations. precom- are state-dependent. observer would require non-localIn coupling this between paper, the we remaining willobserver black show can that hole be the achieved construction without of ever the involving interior degrees mode ofthe and freedom quantum its on entanglement measurement between the byexperiment the early the cannot radiation outgoing infalling be mode carried out [ our goal is to remindthe readers idea of of how backreaction the within quantum the cloning context puzzle of can the be AdS/CFTcannot resolved correspondence. be by sent using non-locally.interior We think mode that require degrees this of is freedom a from serious problem if all the possible representations of the 3 Review of Hayden-Preskill recovery In this section, weWe provide try a to brief be review cleartheoretic of about terms. the what Hayden-Preskill the We thought will recoverabilityhole of experiment mostly quantum while [ focus states the on means scope a in of situation information- the which original mimics argument the by AdS Hayden eternal and black Preskill is broader. Part of atic in describing quantuminconsistencies physics with of quantum the mechanics infalling [ observer due to aner number with of fault-tolerance potential against perturbations on where Then construction of operators foron the the interior modes unitary operator see the origin ofgiven state-dependence, by suppose some maximally thatwritten entangled a as state quantum with state of a black hole where the interior operator may becode encoded [ in Relatedly, arguments based onThe quantum key circuit question, complexity however, have isoperators. been the physical attempted origin [ of fault-tolerant encoding of the interior on JHEP10(2019)132 . A and (3.1) (3.2) B i ∈ H . We will ψ | A -qubit quantum (1) suffices. By be the remaining A O for all that, for any given n D = DR  V and i is an ” is measured in terms of . The surprising result is C i ≈ R ψ . | R i ): j . Let ). Bob’s goal is to reconstruct 3 something 3 ⊗ | , and “ AB | B ⊗ | A i | represents the time-evolution of the black into a black hole, it is more convenient j (figure | i out U = i (i.e., the expectation value of the output A ψ j | | i ψ i EPR X | ψ ψ | out ). | B A | ≈ | 1 | out – 8 – that acts on A
p | copies of EPR pairs between the black hole which has the same dimensionality as with U ψ | 0 BR = i with ih B A | out ψ | BR A to the black hole to account for the Hilbert space of the EPR i into a black hole and Bob, the outside observer, attempts U DR Φ A i qubits of the outgoing mode with = log ψ EPR | ⊗ |  B Φ | . Following Hayden and Preskill, let us assume that the black n A i + i ψ ψ having access to the early radiation A | | n ) R = I and ⊗ D D : n is sufficient to achieve this goal. Namely, if , reconstructs the original state | AB i R A U ψ | | represents a recovery unitary. )( is some subsystem in ' . The Hayden-Preskill thought experiment. V DR | out V D A | Let us append a subsystem ⊗ by catching C i I ψ ( 3.2 State representationInstead of of throwing an unknownto quantum state introduce an ancillarysee subsystem that this trick enables us to discuss the recoverability in a quantitative manner with Here the fidelity of thestate output with state respect on to the projector that state, catching “reconstruction”, we mean the existenceinput of state some recovery unitary infalling quantum state hole evolves by a Haarblack random unitary hole and the| outgoing mode respectively (figure to reconstruct it bywe collecting treat the the initial Hawking state radiation.the as radiation To simplify the argument further, 3.1 Setup As in the firewallemitted argument, more Hayden than and half PreskillAlice of considered throws its a an initial quantum old state entropy. black In hole the which Hayden-Preskill has thought experiment, Figure 3 hole and JHEP10(2019)132 is on D (3.6) (3.4) (3.5) (3.3) out A i ψ | that acts on EPR on the original U i , it is possible to 0 , e.g., the mutual ψ | A RD . and throw qubits from i out will generate A A : 0 0 and A 0 A ) is often called the “state i DR A acting on and V 3.3 . 0 0 EPR A A A EPR i | , we will have O something ) ψ i ∗ ]. To remind readers of the notation, is the remaining black hole and ⊗ | out ψ ∝ | | 11 T A 28 = C [ ). A out O 0
A A 0 U out ⊗ i A onto BR i A I 0 0 i ) is nearly maximal (i.e., OTOCs are small). is that we do not need to keep track of how A A 0 EPR = ( | – 9 – A EPR EPR ) ,DR 0 Φ A out I A A 0 ( ⊗ | i ≈ | ⊗ A I , the above projection on i 0 A Ψ ) is close to its maximal value, nearly perfect EPR pair 0 | if A ) | out A ∗ i A 0 . The wavefunction in eq. ( ψ EPR DR | ,DR DR h , implying successful reconstruction of the input state. One 0 A ) V ( is the early radiation, I A EPR ⊗ ( | out R ⊗ I CDR A C
0 0 0 . After the time-evolution by a unitary operator R A A A I since B I O ( ). If ( ⊗ A AB ,DR 0 U is distilled on A ( ⊗ I 0 out A A 0 ]. I A 31 ” is measured in terms of the fidelity with the perfect EPR pair (i.e., the expectation i ≈ = i Distillation of EPR pair can be related to recovery of infalling quantum states as Information theoretically, the recoverability of unknown quantum states can be quan- , the system is given by The fidelity of the distilled EPR pair can be quantitatively related to out-of-time ordered correlation EPR Ψ into the black hole is the initial black hole, | | 11 identify a partner operator on Recall the following relation: function [ 3.3 Reconstruction ofWe operators will use theAMPS Hayden-Preskill problem. recovery For protocols this to purpose,terms construct it of is interior operators. convenient operators to Given in state an the the arbitrary Hayden-Preskill unitary recovery operator in If the output Hilbert space merit of introducing thethe reference recovery system operation works for each choice of input states. value with respect to the EPR projector on follows. By projecting theinput reference Hilbert system space information can be distilled by applying some recovery unitary operator Here “ representation” (or the “Choi representation”) of B the late radiation. titatively addressed by the amount of entanglement between which is supported on A AB information-theoretic measures. Prepare an EPR pair on JHEP10(2019)132 i ψ | (3.7) (3.8) (3.9) (3.10) , Alice U as follows: , catch the U T DR . e O in the Schwarzschild EPR s r . Define is instantaneous once the log out i i s T A ψ r Ψ | | O ) ∼ t T DR . e O EPR ]. Hence, the quantum cloning puzzle DR ⊗ EPR V 43 C ]. In order to verify the quantum cloning, 0 out A T A 42 I ( – O ≈ = 39 † DR i ≈ and its transpose – 10 – V satisfies Ψ 0 | A ) ≡ O DR EPR , the quantum cloning puzzle can be stated as potential violation T 0 DR CDR e A I O on EPR ⊗ , and jump into a black hole to meet Alice who possesses the 0 i T DR A e ψ O | O ( A traditional approach to resolve this puzzle is to resort to the 12 at the stretched horizon) for the black hole to implement EPR s r in a clock at the stretched horizon [ log inside the black hole whereas Bob possesses the reconstructed copy of s r i ∼ ψ | t log ∼ t While the above viewpoint provides a tentative resolution of the quantum cloning By introducing the reference system 12 puzzle, a refined resolution has been recently proposed within the contextof of monogamy the AdS/CFT ofindeed entanglement. related to This the observation firewall problem. hints that the Hayden-Preskill thought experiment is outgoing mode is collected.time Then (or if itwill takes reach more the than blackstudies hole suggest singularity that before time meeting scalethan Bob. to “delocalize” Quantum quantum information information must theoretic is be avoided. indeed longer idea of complementarity; we decide not tolong be as bothered there by is violation no of observer quantum whoBob mechanics can needs as verify to it [ wait foroutgoing a mode, black reconstruct hole to implementother a copy. highly-complicated unitary For the time being, assume that reconstructing 3.4 Quantum cloning puzzle The Hayden-Preskill recovery leadspossesses to an apparent cloningoutside of the quantum black states hole. since Alice or graphically which holds for any unitary operator Then, the partner operator or graphically JHEP10(2019)132 . ]: 13 as R i 53 j (4.1) U – ⊗ | 51 , B i ]. The key j 44 | 48 j – P 44 | , are considered.
B | D 1 U √ † D , a recovery operation O (the reference system) ⊗ H = † 0 U C U A † BR A H i UO EPR and D Φ O | B † U A ⊗ H O A H Tr 1 d – 11 – , into i ≡ ) H t ( † D O from the interior to the exterior. A concrete physical (0) i † A ψ denotes the total Hilbert space dimension. Two different | O | ) t ]. In particular, for the eternal AdS black hole, they identified D ( ]. || D 49 C 50 O | , 31 (0) = A ). | O h B 3.3 || A | = (the late radiation and the early radiation) in the state representation of is an arbitrary unitary operator that accounts the time-evolution of the system. d U DR While Hayden and Preskill considered Haar random unitary Since their interaction couples two boundaries, null geodesics through the wormhole is no longer 13 where bipartitions of the totalHere Hilbert space achronal. Hence the average null energy condition (ANEC) is obeyed for achronal geodesics. can be performed in strongly-interacting quantummation systems over which the delocalize whole quantum system. infor- bling, This can phenomena, be often probed called by quantum the information out-of-time scram- ordered correlation (OTOC) function [ quantum black hole whose initial stateWe is are represented primarily by interestedand in quantum entanglement between defined in eq. ( (OTOC) functions. Wethought also experiment. discuss concrete recovery protocols for4.1 the Hayden-Preskill Scrambling Let us continue our discussion on the Hayden-Preskill recovery. Again, we will focus on a We have reviewed the Hayden-Preskill thought experimentfrom and the the perspective quantum cloning of puzzle namely backreaction within for the the context eternal ofgeneric AdS the quantum black AdS/CFT hole. systems correspondence, byery In and reviewing this quantum section, the information we relation scrambling extend as between the diagnosed the argument by Hayden-Preskill to out-of-time more recov- ordered correlation The relation between traversable wormholesis and also the discussed Hayden-Preskill in thought [ experiment 4 Scrambling and recovery insight is that Bob’s operationand from the pull outside the may shiftrealization quantum the of state location this of scenario the isGao, event horizon the Jafferis so-called and traversable Wall wormholecertain [ phenomena forms discovered by of interactionsenergy between to CFTs the bulk on and opposite shift boundaries the event which horizon send so that negative signals can traverse the wormhole. horizon is an absolute entity andapplied its from location never the moves outside. regardlessrecovery of operation quantum A operations may modern have viewpointviewpoint a on is non-trivial largely this backreaction motivated puzzle to fromAdS/CFT the is recent correspondence developments black to and in hole related postulate studies toy geometry. that of models backreactions This such Bob’s as in the the SYK model [ correspondence. The original argument via complementarity assumes that the black hole JHEP10(2019)132 α 4 (4.2) whereas R . See figure = 2 Sandwiched ⊗ H D α D respectively. For ) is defined for the ⊗ H i D ) ⊗ H t C ). 4.2 ( C H † D and O = ,RD 0 ⊗ H A A B 0 (0) ( A I † A represents the mutual R´enyi-2 H O ⊗ H ) t A ( RD 0 D (2) H A S O = − (0) H A (2) RD O S h + D 0 – 12 – (2) A dO S A ) in the above setting, the mutual R´enyi-2 information does ≡ dO ) ,RD )) which is supported on 0 Z . take averages over all the basis operators supported on ], a certain average of OTOCs is expressed as the A ( U to be basis operators (such as Pauli operators) supported = 3.3 ,RD 56 D 0 divergence. A similar duality relation is known for two of them with ) (2) D A I α ( ,RD , dO ≥ ]. In [ ,O 0 (eq. ( . Out-of-time ordered correlation (OTOC) functions. (2) A ) A A I 55 are operators that act on subsystems ( U O dO (2) D I ,RD 0 = 2 [ O − with no overlap, the OTOC starts at 1 and then decays under chaotic A 2 1 β ( ]. Specifically, the following equality was proven: I Figure 4 + It is worth reminding that the left hand side of eq. ( and 54 1 α , A, D A 14 28 O . For satisfying respectively. Here The decay of OTOCs implies that the Hayden-Preskill recovery can be β Whilt we have A, D 14 not lower bound theinformation theory, ordinary the mutual mutual R´enyi-2 information informationmust is in use not the general. a Sandwichedand proper R´enyi- In entanglement fact, monotone. from divergence InsteadR´enyi and the one is perspective proven of to lower quantum bound the mutual information state representation of OTOCs on the right handfor side graphical are representation calculated of on holds OTOCs. for any We unitary emphasize operator that the aforementioned relation where integrals with A, D information. concreteness, we take on time-evolution. performed [ Note that JHEP10(2019)132 ]. 31 ). As I ]. This ], and 1 d 57 = , 60 , ρ 43 59 is the Bekenstein- BH 16 S in the Schwartzshild black s r ]. 58 [ log s s . Essentially, these states are simplified r r -qubit system. We will discuss its validity BR ≈ i log BH s -qubit system where S r EPR BH ≈ Φ ]. S | ) Namely, smallness of OTOCs implies large 63 K ⊗ ]. For the AdS black hole, OTOCs have been 15 – 13 – I ). ( 44 ]. Kitaev pointed out that these pioneering works ≈ ]. Shockwave geometries in the Reissner-Nordstr¨om 4.2 ]. Recently shockwave geometries in the Kerr black to be a weak perturbation. Under the gravitational 61 52 62 A O are small. For this reason, we do not view the simplification with D . 6.3 and C as fundamental limitation of our argument. For simplicity of discussion, we proceed as well as BR i B ) which implies the existence of reconstruction procedures. EPR and Φ | ) A ,RD K 0 ] by using OTOCs. Recently Shor presented an argument showing that scrambling in the alternative A ⊗ These calculations suggest that OTOCs start to decay from its initial value signifi- The Hayden-Preskill recovery and quantum information scrambling can be explicitly ( Note that the above formula is restricted to quantum systems at infinite temperature (i.e., the system An alternative definition of quantum information scrambling was previously considered [ I 31 ( 16 15 (2) between OTOCs and theeralization recoverability of for this more relation realisticNamely it black for was holes black shown at holesbetween that finite with the temperature. decay more of generic Further≈ OTOCs gen- quantum implies states the recoverability haswith as been this long simplified as discussed description quantum in offrom correlations a [ other quantum perspective black in hole section as a is finite-dimensional and the quantumsuch, the state conclusion in on OTOCs the isapproximated recoverability given by is by maximally applicable the entangled only maximallydescriptions to states mixed quantum of state black holes maximally whoseHawking entangled initial (or black states the are coarse-grained) holes entropy. as Some an readers might be worried about the validity of the relation definition considers initial quantum states withbecome low entanglement thermalized. (e.g., product We states)tion think and of asks that if quantum the the information states experiment definition scrambling is as based clear. its on relation OTOCs Forin to interested [ is black readers, a hole a more physicsdefinition definition appropriate and cannot of Hayden-Preskill characteriza- be scrambling thought achieved at within finite the temperature scrambling time was proposed a black hole with ascrambling time. pure quantum state on his Hilbert space, and OTOCs decay after the cantly around the scrambling time,hole. which As is such, of after order thecan scrambling be time, retrieved a as quantum longargument state as is that the not has black fallen hole restricteddence. into is to a maximally Namely, black black entangled. we hole holes We only that emphasize assumed that are that this described the in outside the observer AdS/CFT has correspon- a quantum description of actually computed OTOCs incomputed disguise by Shenker [ and Stanfordblack [ hole were studied byhole Sfetsos were studied [ by BenTov and Swearngin [ experiment, let us discussbe OTOCs explicitly for computed black by holes.blueshift, taking such OTOCs a for perturbation gravitational isas systems greatly it can amplified falls and into becomesshockwaves a a on black gravitational correlation hole. shockwave functionsalso For by were the Kiem, discussed Schwarzschild Verlinde black and by hole, Verlinde ’t [ the Hooft effect and of Dray gravitational [ related by the aboveI formula in eq. ( 4.2 OTOCs inHaving a black seen hole that scrambling implies the recoverability in the Hayden-Preskill thought JHEP10(2019)132 to the can be . Both i ∗ D ψ U | ]. The input were initially 31 D D . In particular, to reconstruct | are small. t A ]. Here we briefly review i | ) t ≈ 31 ( ' to be EPR pairs. In the diagram, D | are interpreted as outgoing modes. O D D | ), and then projects the outgoing D D (0) EPR will be teleported to Bob’s register A D 5 are applied on the entangled partner , it takes i O ) ψ t T U | ( to be original EPR pairs. The protocol U D to be EPR pair in a deterministic manner. . Here O | D D in figure D D and | (0) D A ∗ B = ] for instance. While this paper focuses on the physics – 14 – O | U h 65 D EPR EPR , | upon postselecting projection 0 64 to implement A t , and | do not overlap), this recovery operation can be interpreted C | ) provides a formal proof of the existence of recovery pro- D = 4.2 | C | for summary of the whole process. Noting that and , , the recovery protocol works when | 5 A U is teleported to B (which is denoted as | A = R | from both sides into EPR pairs. Upon postselecting the measurement on B i 17 | = 0 (if D , ψ | t | 0 . See figure . Hence, if it takes time A 0 | (1), the quantum circuit complexity of both recovery protocols is roughly equal and U A . A recovery protocol for the Hayden-Preskill thought experiment from [ = O | D A | | ∼ The second protocol is more involved and works deterministically. It incorporates the The first protocol works probabilistically. It applies the complex conjugate These recovery protocols work well after the scrambling time whereas the traversable wormhole picture = A | | 17 A is valid before thefrom scrambling that time. before The theafter nature scrambling the of time, scrambling scrambling time, see aftermade we [ the by believe relying scrambling that on time a the is qualitatively physics quite similar of different implication scrambling before on the the scrambling AMPS time. puzzle can be sequence of unitary operators which restores If to that of protocols enable us to reconstruct partner operators, although the first protocol requires achieves high recovery fidelity when OTOCs decay. quantum Grover search algorithm where multiple times with Grover rotationoriginal operators paper inserted. for details. Interested readers Unlike should the see first the protocol, the second protocol applies a careful modes outcome to be EPRqubits pairs, on Alice’s quantum state EPR pairs at as the outside observer’s attempt to restore tocols in a scramblingrecovery system. protocols The which author workthese and for protocols. Kitaev any proposed scrambling two systems particular “simple” [ entangled partner chosen as any subsystem as long as OTOCs 4.3 Reovery protocols The above relation in eq. ( Figure 5 quantum state | For scrambling unitary JHEP10(2019)132 ) 0. 2.5 → 1 and 1 → as in eq. ( ). In particular, BR i ]. We will focus on is a separate problem. ,DR 0 26 U EPR A ( Φ | I ) when the Hayden-Preskill may be avoided by the idea K 6 ⊗ K I ( . Performing recovery protocols Geometric interpretations of these ≈ BR i operator, her bit is flipped; 0 is a prerequisite to treat the information loss . This essentially cancels the unitary 18 ∗ X U EPR forces OTOCs to decay, one can prove the Φ is a highly complex unitary operator. One | UK U -dependence and achieve state-independent . A generic quantum state for a maximally K K – 15 – BR with i -dependence in section U . K EPR D Φ . Here we would like to clarify the applicability of our ≈ | K ]. Yet, it is not clear if such a procedure is physically plausible 38 does not affect the mutual information K or not. Then it is not possible for Bob to know Alice’s original bit. As this simple since X K and reduces the system back to , however, may be problematic if K ∗ UK We will return to this issue of the As a matter of principle, the quantitative argument on the recoverability remains valid So far we have mostly focused on a quantum black hole whose quantum state can Some readers may be worried about the fact that Bob needs to understand the dynamics of quantum 18 fact, a similar problem0 can or be 1 identified andimmediately in Bob know tries classical the to answer. systems. find IfThen, Suppose Alice out Bob has what that needs applied is Alice to aAlice written prepares flip Pauli- has in her a Alice’s applied bit classical bit. back bit classical to If observation find suggests, Alice’s the Bob bit originalproblem knowing remains in information. the unchanged, a dynamics Bob Now rigorous will suppose information Bob theoretic does setting, not and our know if little knowledge of black hole for reconstruction.valid concern. Given our However we little emphasize understanding that of this quantum is gravity, not this the might issue sound intrinsically related like to a quantum gravity. In a slightly reorganized reprintcases from where section the 5 initial ofdiscussions state on our generic of previous entangled work the states,next [ old as section. well black as hole the is issue of given the by state-dependence, EPR to the pairs while deferring 5 Reconstruction of interiorIn operators this section, weby point using out a that protocol similar the to black the hole Hayden-Preskill interior recovery operators procedure. can This be subsection reconstructed is in realistic systems. recovery is used toAMPS reconstruct interior puzzle, operators. we Wereconstruction can will of actually see interior operators. that, avoid in the the context of the operator with might argue that recovery protocolscompletely. cannot The be difficulties performed associated before withof the highly the black “precomputation” complex hole [ evaporates argument. regardless of as long as the unitaryexistence time-evolution operator of recovery protocols. Also,to the aforementioned generic recovery protocols initial can states be applied by replacing protocols are discussed in appendix be approximately modeledentangled by black hole, however, shouldwith be some modeled unitary by operator us to normalize reconstructed operators properly. JHEP10(2019)132 ) (5.3) (5.4) (5.1) (5.2) D,CR ( I . That is, as possible. is the early C R R by using only qubits 19 . . Here T CA U e O . We have seen that, from i R Ψ | EPR ) be the complementary subsystem T CR B e O ⊗ by using as few qubits from EPR D is the outgoing mode. The goal is to find , and I on the outgoing mode: R T CR D e ≈ = ( O D = i O – 16 – . Ψ | ) AB CR ) = 0, partners cannot be supported on I = ) is the origin of the firewall problem (or relatedly the i ⊗ EPR Ψ T D,C | CR D ( e O O I ( must have non-trivial support on is not unique. be a small subsystem of ). Here we would like to construct a partner T CR e O T CR A in writing e of an arbitrary operator O AB R = T CR is the remaining black hole and , as graphically shown below: e R O R C is split into two subsystems (so R R from Here our aim is to construct the partner Construction of A 19 where inside on the perspective of theearly outside radiation quantum mechanicalstate-dependence description, problem this and fact the (the non-locality problem). use of the To be explicit, let Existence of such partner operatorsis is maximal. guaranteed since However, thepartner since mutual operators information the partner or graphically where the black holeradiation, started in EPR pairs and time-evolved by The quantum state of interest is as follows: 5.1 Hayden-Preskill for interior operators JHEP10(2019)132 D ]. It (5.6) (5.7) (5.5) 31 and can be D A O . BD CA. , which is maximally for a given R from the early radiation T . In the original Hayden- CR e O A is rotated upside down. In U and OTOCs between D reconstruction on reconstruction on EPR EPR physically distinct A D ≈ – 17 – is bigger than as the diagram of can be constructed by using the Hayden-Preskill A U T CA EPR e O and just a little bit of qubits on ] is that the task of reconstructing ], as long as C 26 is interpreted as an “input” while its partner is constructed on 31 D HP recovery: input EPR ) by 180 degrees and bent some arrows, we obtain the following relation: Interior operator: input 5.4 represents the transpose of It is worth summarizing the discussion so far as follows: T 20 U As such, the partner operator A key observation in [ The partners of interior operators are often called mirror operators in the literature. It is an interesting 20 . It is immediate to see that the structure of the diagram is identical to the one for the coincidence that Hayden and Preskill referred to their findings as “black hole as mirror”. entangled with the black hole,As was essential such, for the the recovery recoverabilityother of was hand, input possible the quantum aforementioned only states. construction forblack of holes old the too interior black as operators holes we is after will possible the see for below. Page young time. In fact, On the the age of the black hole is not so essential. the construction of interiorquantum operators information theoretic outlined technique, above namelyis, the can reconstruction however, be protocol worth achieved from emphasizing by [ and that using the these the construction two of same phenomena,Preskill interior the operators, thought Hayden-Preskill are experiment, recovery the presence of the early radiation We will study actualfuture expressions work. of reconstructed interior operators more concretely5.2 in a Young black hole The Hayden-Preskill recovery, as in the original Hayden-Preskill thought experiment, and recovery protocol from [ decay. the rotated diagram, the remaining black hole R Hayden-Preskill thought experiment. Here performed by using the Hayden-Preskillfigures recovery in protocol. eq. ( Namely, by rotating the above JHEP10(2019)132 (5.8) (5.9) (5.10) -dimensional R d and some small C R i j ⊗ | . . DC is supported on a i j | R R = 1 would correspond to a one-sided =1 d j X EPR R d R EPR . Hence, even when a black hole is young, d 1 D √ -dimensional (in a sense that the Bekenstein- n = – 18 – is 2 . Taking R ). EPR CD = D i EPR is bigger than Ψ | A ) while the early radiation . The triangle in the figure denotes an isometric embedding, n n 2 = < is bigger than BH R S d DC , as long as R itself is bigger than of R A To see this explicitly, let us consider a young black hole which is not yet maximally the interior operator can be(as long reconstructed as by using the Hayden-Preskill recovery protocol Then it is immediatesubsystem to see that the partner can be reconstructed on For such a time-evolved young black hole, let us rotate the diagram to obtain emphasizing that black hole. The quantum state of interest is obtained by time-evolving the above state: where the black hole HilbertHawking entropy space is Hilbert space with entangled. One may model such a black hole simply as follows: JHEP10(2019)132 R and , the A i (5.11) where . D EPR CA Φ | ) K ⊗ ), and project a ]. The above ob- leads to different albeit very little. I 6 A 5.1 R when OTOC between to be any subsystem of works as good as before the CA A CA is said to be decoupled from 22 . However, our construction does . B In qubits count, the reconstruction . Fault-tolerance is a quantum in- R R 23 . at all. . Let us apply a projection onto some R , so the reconstructed operators depend ∗ B B ⊂ B UK qubits from the early radiation are damaged as – 19 – on →  . The existence of multiple equivalent expressions U − as before. Recall that we were able to reconstruct 21 T CA : e O BH B . i AB S ∗ without using ψ is (nearly) maximally entangled with into i ⊗ | CA . In a sense, our construction is only “slightly” non-local as all the qubits D ψ R R | against perturbations on . The reconstructed operator on on = A i . We will remove these undesirable features in the next section by T CA supported on K e O EPR B | i ) | ∗ onto some fixed quantum state. Starting from a maximally entangled ψ ψ | . It is worth emphasizing that one can choose ih | R ∗ fault-tolerant ψ D | ⊗ | ' I  . Relatedly, the quantum circuit complexity of reconstruction can be huge depending So far we have shown that the partner operators can be constructed on From these observations, we notice that the aforementioned reconstruction of interior Another (perhaps more illuminating) way to see that the reconstruction method works Note ( More precisely, it is because More precisely, it is an erasure error. K is a small subsystem in decay. In quantum information theoretic term, the subsystem 21 22 23 on the complexity of considering the effect of including the infalling observer. D not resolve some ofFirst, important the problems reconstruction identified still inSecond, the requires the reconstruction context an is of state-dependent. access therecovery For to a protocols firewall generic distant should initial problem. radiation state be ( on run with servation provides a physical accountdynamics on of how fault-tolerance black is holes. achieved due to scrambling A it uses only a small portion from the early radiation long as as long as OTOCs areexpressions small. of the Moreover, interior choosing operators differentof subsystems operators is a keyfault-tolerance characteristic was of quantum previously error-correcting pointed code. out by The necessity Maldacena of and such Susskind [ operators is formation theoretic term tosome say types that of encoding errors. ofnoises quantum Here, which information the would is black damage resilient holeis against interior robust operators even are if immune any to subsystem rather of drastic After the projection, thewith a system small reduces subsystem toprojection a since its young expression black does hole not which involve is entangled only for a young blacksubsystem hole of is toold start black with hole, an leta old us partner split black operator hole,quantum as state in eq. ( JHEP10(2019)132 24 . | D | ' . Namely, | . Note that will be able A D | AB R (i.e., the distilled from Bob. We D R . R from the early radiation . as long as . In fact, we will show due to scrambling property B A | 7 R ) is nearly maximal. Since ) is close to zero. Namely, D | is left untouched, the outgoing ' D,B | ( A D, AC A ( | (2) ever accessing I (2) of the early radiation I and section A 6 as long as without R R Alice Bob – 20 – as qubits which are entangled with A can be any subsystem of and Bob is able to see an EPR pair from the outside. A , this suggests that while Bob can distill an EPR pair by accessing . On the other hand, if AB (2) D S AC , together with a little bit of qubits AC = while Bob cannot reconstruct it on C R and AC ) = 2 D as degrees of freedom which Alice can access, then she is able to see on D,B C D ( from the early radiation (2) D I for schematic illustration of the situation. ) + 6 is entangled with . The AMPS puzzle from the perspective of the monogamy of entanglement. Alice can D is included to D, AC In this section and the next, we discuss the effect of explicitly including the infalling It is worth emphasizing that These statements can be made quantitative by using the monogamy relation of the . ( A common misunderstanding is to think U can be any subsystem of the early radiation A 24 . This construction, however, still suffers from the state-dependence and non-locality (2) observer to the firewallto problem. the system, We no will matter show how that itqubits inclusion is in of the done, AMPS the leads thought experiment). infalling to They observer significant are not. gravitational backreaction 6 State-independent interior operators In the previous section, weon have the pointed remaining out black that hole theR interior operators can be constructed problems. pair from thewill outside make observer this as intuitionthat she generic more needs perturbations precise to to inoutgoing take the mode section black a hole few by qubits the infalling in observer will disentangle the mutual information. Decay ofI OTOCs implies that Alice reconstructs Hence, from the perspective of the infalling observer, it is rather “easy” to steal the EPR if an EPR pair between mode See figure 5.3 Monogamy andOur scrambling finding in thisquantum information section scrambling. sheds astruction The is new important that lesson light whoever from onto possesses the the see a aforementioned some AMPS tiny recon- degrees portion puzzle of through freedom the which are lens entangled of with the outgoing mode Figure 6 distill an EPR pair byA accessing of JHEP10(2019)132 . , t U D R B (6.1) represents K . Discussions R state-independent is highly complex). . This decoupling on how the infalling R K and time-evolves by BR i depends EPR Φ | ) . Here intermediate Hilbert spaces 7 K is far distant on the right-hand side. . ⊗ 1 R I U ··· EPR T from the early radiation U – 21 – D = U is depicted in figure corresponds to the modes on the AdS boundary while T t is discretized into small time steps: = A t are the confined modes. U and construct the interior partner without using t A R . where t ,B t A . Imagine that B . Discretized time evolution of a large AdS black hole with long throat. In this section, we will demonstrate that a certain small perturbation to the black hole observer-dependent We emphasize that details ofsion. the The discretization quantum are state at notare important labelled in by the followingcorresponds discus- to other degreesstretched of horizon freedom as well including as modes other between entities the living boundary at the and stretched the horizon. The AdS assumption isimagine that used the to black ensureacting hole on that initially starts the with black ( hole does not evaporate. Let us In the next section,in we a will rather see universal that manner, a regardless similar of phenomena how occurs perturbations6.1 without are any added fine-tuning, to the Long-throat black AdS hole. blackLet us hole consider a large two-sided AdS black hole with long throat (i.e., is sufficient to disentanglethis the section outgoing is mode to fromfrom simply the the point early early out radiation. radiation thaton The it main physical is interpretations goal possible of of tomay such disentangle find a the the perturbation particular outgoing scenario are mode we given consider in in the this section next rather section. fine-tuned and Readers artificial. Hence, the inclusion ofand the resolves infalling the observer non-locality enables problem.observer state-independent is interior Yet, introduced the operators to construction but the system. In this sense, our construction is some complex unitary and which disentangles the outgoingphenomena enables mode us tostate construct of interior the operators black in hole a without way using independent any of degrees the of initial freedom from the early radiation Figure 7 JHEP10(2019)132 . . 1 E E at R U C t (6.5) (6.6) (6.2) (6.3) (6.4) A ··· and can be on . t
T E E E U A as long as T C T C E t . Since the i e is entangled e = O O U n ⊗ U E , let us discuss . . i i ···
by applying the 0 i R EPR | E T E U ⊗ | i . Hence, the interior EC i ⊗ | = . j 5 | , the reconstruction of 0 BR E onto i EPR U E 7 i 8b ) = . i ⊗ | i j ) EPR . The resulting state can be | . We find that ) T EPR BR ( 0 i in section . K 8a i ⊗ | 8b can be reconstructed by quantum ⊗ | i Ψ in figure | | A ⊗ ) BR : D ever. t i 1 and time-evolve it by . Namely, the partner EPR E R BR | i U T C ) R n R e ⊗ O . Importantly, it has no dependence on K i ··· t EPR | 0 ⊗ 1 | ) EPR U ⊗ | − I ) 1 t K K ( ··· U U K ( ⊗ , which have the same dimensionality as T , the quantum state of the black hole is given by ) can be reconstructed on – 22 – 1 ⊗ i ≈ ··· U E ) T 1 U E 1 D t as shown in figure T = U − E, , one can construct a partner operator A ( t = 0 0 ··· t U E . D 1 Ψ ··· can play the role of U ( | system and the SWAP operation, the outgoing mode E − 1 ) t BR is added to the black hole Hilbert space by hand, and is E on I − i t E U E t SWAP ( E ⊗ U D E A
EPR O D E, = ( Φ O ER | ( i ) E ) I t K ) quantum circuit complexity, but also independent from the choice ⊗ | Ψ( ⊗ = SWAP t : | t i I U − ) 8a t ( T 0 ··· | ( Ψ T , and hence is state-independent. As such, the reconstruction is not only | O U K . More explicitly, at time = scr i t . Namely, we notice that ) t , and prepare EPR pair on ' T t A ( The aforementioned protocol works for the case of one-sided AdS black holes. Let us The outcome of these operations is depicted in figure We now show that the partner of interior operators t 0 is no longer entangled with the early radiation Ψ − | efficient with of the initial state ( represent the initial state ofThis the black is hole equivalent by to projecting the right-hand side 6.2 State-independence Having constructed a partner operatorthe without problem using of the the earlyinterior state-dependence. radiation operators depends As only isthe on evident unitary from figure Due to the additionalD of the reconstructed without involving the early radiation Given an arbitrary operator such that with operators (the partner ofrecovery unitary operators from on the Hayden-PreskillT recovery protocol with not a part ofas the shown original in Hilbert figure space. Let us apply SWAP operator between Here SWAP is a unitary operator which acts as SWAP( time written explicitly as follows: Note that the EPR pair on operations which areConsider strictly the localized quantum state on of the the black left-hand hole side at time without ever accessing To begin, we append two subsystems JHEP10(2019)132 i R is ∗ ψ 6.3 t to a | The . (a) joins R B t E 25 R E onto as t R R . R , be it unitary or non- R The aforementioned pro- , contradicting with our claim. In the next section, we will brings the black hole on R 26 are entangled. R 27 t EPR A (b) . EPR pair is prepared on decay for one-sided black holes too. t A and D E copies of EPR pairs, projecting disappears. n t and R . Here it is crucial for an infalling observer as certain measurement apparatus or physi- t A A rather than the escaping modes E t E are the confined mode inside the zone whereas – 23 – A t and A E . Naively, this appears to suggest that it might be possible to EPR i ψ | . (b) The outcome. . Again, the interior operators can be reconstructed in a t -system plus SWAP can be realized by performing informationally . Here E ¯ B SWAP (a) E t 9 -dimensional subspace where the black hole is initially entangled with onto E n A and B t and its interior by non-unitary operation on A B . One may interpret 5 , and the entropy of the black hole decreases. R . A protocol for an infalling observer to access Readers might find that the above construction of the interior operator is rather ar- Finally let us return to the discussion of evaporating black holes as originally con- The AdS black holeSituations can be analogous viewed to as a the limit where When the initial state of the black hole is given by are the modes which escape from the zone and never return. As time passes, to interact with the confined modes picks some quantum states of the black hole in thermal equilibrium within the typical energy window, 26 27 25 t R reconstructed interior operators willquantum behave state well. with much Yet, higher/lowerfor if temperatures, further projection then discussion on reconstruction on won’t this work point. well. See section complete measurements (POVMs). will project the blackinfluence the hole black on hole on The important point is thatshould the be 2 interpreted as a coarse-grained Hilbert space with roughly equal energy. As long as projection on tificial and fine-tuned. Obviously,considered this in section setup was devisedcal probe in which order is to introducedconsider to mimic the corresponding the black physical situation hole situations Hilbert and space. how our reconstruction resolves the AMPS the radiation tocol works successfully by swapping E completely decoupled from state-independent and low-complexity manner without ever accessing unitary, the reconstructed interior operatorsimportant are point for independent us from is the that OTOCs initial between state. sidered by AMPS. We modelprocedures the depicted evaporation in dynamics figure asR a sequential application of the Figure 8 SWAP operation between recovery protocol does not depend on any quantum operation on JHEP10(2019)132 and charge J -qubit systems. BH S can be interpreted as code H . Strictly speaking, this con- R -dimensional Hilbert space, de- evolves. As such, it is important can be interpreted as the coarse- t BH S U , angular momentum BH -qubit toy models, and hence is an 2 represents the modes which escape and S t M BH ≈ R S which is determined by the classical black code H – 24 – . code H , which have an identical classical black hole geometry. This subspace is are the confined modes. t will influence the Hamiltonian under which A code t H A . The dynamics of an evaporating black hole. while ], Almheiri, Dong and Harlow proposed that the subspace These interpretations become sharp in the context of the AdS/CFT correspondence. The very motivation in reconstructing interior operators in a state-independent manner To understand the validity of the approximation, let us return to the underlying as- R 20 . Those black hole microstates, however, differ in a subtle manner. They may differ in approximate statement inside the subspace hole geometry. In [ a codeword subspace of a quantum error-correcting code. Their insight, as well as concrete is to understand quantum fieldobserver. theory This on suggests a that curved reconstructedblack space-time operators hole experienced must by geometry, be the but state-independent infalling state-independence on can is a be fixed restricted dependent to on the validity choices of of geometries. Our claim on the spanned by quantum statesthe which same correspond thermodynamic to parameters, blackQ such holes as in mass thermal equilibriumterms with of matter contentstretched on horizon. a Black fixed holes geometrygeometry in and span thermal the other equilibrium subspace high with energy the same objects background living classical at the sumption behind using simplified descriptions ofIn quantum black static holes geometries, as the Bekenstein-Hawking entropy grained entropy. Namely itnoted suggests that by there are We have presented state-independentpletely reconstruction insensitive of to interior perturbationsclusion on operators on the that state-independence early are is radiation added com- a to bit ofto exaggeration. note that For instance, the state-independence large is perturbations an approximate statement. puzzle. We will alsogravitational argue systems that and the does underlying not physical require phenomenon fine-tuning. is quite universal6.3 in Code subspace Figure 9 join JHEP10(2019)132 . t ∆ from − in the R = ˜ D t at . To minimize t A be the outgoing ∆ . We also show by 180 degrees as − D D D = t from the early radiation . As an infalling observer, D will become clear later. Here assumes the smallest possible 10a , provided that the black hole , the Hayden-Preskill recovery phe- t D M 28 . code H code H and the interior mode instead of ˜ D D travels along the infalling mode can be viewed as Rindler modes in the opposite – 25 – ]. ˜ M D 56 We also believe that the above claim includes cases = 0 as shown in figure is spanned by these wavefunctions contained in the which departs the boundary at t , it is convenient to draw the partner mode and 29 . M by relying on the Eigenstate Thermalization Hypothesis M 1 code T D ], suggests that bulk operators can be interpreted as logical H ≈ 21 code S r H is insensitive to any perturbations to the early radiation ≈ D , one may consider a limit where M can be constructed by rotating the trajectory of ˜ . Here the apparatus D in the Schwartzshild black hole whereas the thermalization time for small S /β r 1 . Near the horizon, ≈ log ]. In particular, for strongly interacting quantum many-body systems, eigen- S 66 r 10a ≈ To understand the effect of Keeping this caveat in mind, our claim in a more precise term is that the reconstruction From the perspective of the boundary Hilbert space, one may also interpret the origin When perturbations being the system completelyBy outside thermal equilibrium, we mean that time-ordered correlation functions reach their stationary values. as well as any quantum operations added before time 28 29 . As a result, the infalling observer crosses the horizon smoothly and observes quantum the trajectory of in figure nomenon may not be possible as discussed in [ mode that reaches the boundaryone may at consider some apparatus the effect of adding energy, say Penrose diagram. The reason why we used Our reconstruction protocol requiredthat an we access to have the justinfalling earlier observer traded mode. can the access Readers earlierbe spatial might modes concrete, think non-locality by let simply with us falling focus the into the on temporal black the hole non-locality. two-sided earlier. eternal To AdS The black hole. Let entanglement between the outgoingthat the mode interior mode geometric perspective. Our conclusionsOTOCs. will be supported quantitatively by using decay of 7.1 Including Alice 7 Backreaction by infallingIn observer this section,simply we fall show into that aR gravitational black backreaction hole, disentangles by the the outgoing infalling mode observer, who of evaporating black holestypical where time the scale systemorder for reaches our thermal reconstruction equilibriumperturbation procedure is quickly. of is order The the scrambling time which is of may think that thesmall energy subspace window. of interior operators is fault-tolerantR against perturbations added tois the entangled initially partner in thermal equilibrium. operators which label different codeword states in of the codeword subspace (ETH) [ states from a tiny energy window are expected to look thermal in small subsystems. One toy model of such a scenario [ JHEP10(2019)132 . R M de- (7.1) ˜ D to the . by simply , Bob can M and 6 ˜ . Under the scr D i D t . 0 | ' and t ]. Here the interior D 67 and the early radiation (or degrees of freedom in D , Bob’s quantum entangle- EPR R (b) M in the presence of a shockwave ˜ D becomes a gravitational shockwave ]. As such, when ∆ and M 52 . (b) The protocol from section D M ) = BAR – 26 – can be viewed as entangled degrees of freedom i , we obtain the following state: ˜ D U EPR Φ and ⊗ | starts with some particular pure state, say D M i M 0 , Bob can verify the entanglement between | ) by using the Ryu-Takayanagi formula [ (a) . This hints that Bob’s quantum entanglement is destroyed by ˜ )( D M ˜ . We then append the subsystem for the apparatus D R t is disturbed. Precise form of interactions between the apparatus I D, ∆ ˜ ( D ⊗ I − and . (a) The effect of an apparatus = . Suppose that the quantum state of a black hole is given by EPR pairs D and BAM t is entangled with some other degrees of freedom in the early radiation M D U ( D at is constructed exclusively on the early radiation ˜ D Figure 10 ABR i One might think that this observation does not fully deny a possibility that the out- However, in the presence of the measurement apparatus by measuring EPR Φ | black hole. We assumetime-evolution that by a unitary operator the infalling observer who has crossed the horizon. going mode In order to make athe more apparatus quantitative argument, let us explicitly study the effect of adding which shifts the geometry nearcays. the The horizon, two-point the correlation correlation functionsis function between identical between to OTOCsno defined for longer a verify thermal theR state entanglement [ between the outgoing mode the right hand side). ment between and other degrees of freedomsense is that not it is so a important. gravitational In effect. fact, Namely, the since interaction is universal in a in the boundary conformalabsence of field the theory apparatus whichmeasuring is the in correlation the functions thermofieldmutual between information double them. state. Also, oneoperator In may the explicitly compute the regions. Near the boundary, JHEP10(2019)132 . 2 E at | . was D A | B probes M with ) is not ' is that from the | E 6 A ) is nearly including its M C,D D | ( . Again, this M ) as discussed I C,D 6 is Haar random ( We assume that I U . Let the outgoing C,D ( 30 E . I ]. E ) for the state defined as in section M 31 , E 28 C,D E is the energy of ( I E , which Bob would have seen, R ) becomes nearly maximal which is a Haar random unitary, one can where and U for the illustration. Here the earlier EC D E/T D, ( 10b (or scrambling time). If (2) I are indeed entangled. This also implies that U ) becomes nearly maximal when which we constructed on section – 27 – EC , we think of replacing the infalling mode remain outside the boundary whereas C,D also disentangled the outgoing mode D ( M E I . We emphasize that this argument on E and E M D due to the monogamy of quantum entanglement. Instead, R escape from the boundary and replace the infalling mode . Here we consider a situation which exactly mimics our reconstruction ], and we will make further comments on their work in appendix t and the reference M 22 ∆ . One merit of introducing the probe mode as a reference qubit. See figure is sufficient to restore nearly maximal correlation in A − R | E is the temperature. . = Also, further analysis with energy conservation into consideration suggests D T | A . Decay of OTOCs implies that in a subtle detail. Also we had to change the size of the black hole Hilbert t ). Again, generic decay of OTOCs is sufficient to show that 6 6 ' 31 with some “probe” mode. Let us prepare the EPR pair on at | t 7.1 A M ∆ | We have seen that including Instead of adding the apparatus A similar setting was previously consideredWhile by the Verlinde aforementioned estimate and assumes Verlinde that in a slightly − The increase of the entropy is approximately givenThis by statement can be shown by a slight generalization of results from [ is no longer entangled with 30 31 = protocol differs from theintroduced. scenario in Nevertheless, the we previous subsection believe where that an theserest apparatus mass differences, and which result from how we D it is entangled withargument the is interior not mode particularly limited to the AdS/CFT correspondence. early radiation the size of the Hilbert space for the black hole itself does not change. Note that this while keeping outgoing mode the bulk. Thisin section is essentially identicalsuggests that to the the outgoing protocol mode with SWAP operation discussed space by including method. t mode 7.2 Sending probes In the previous subsection, wemeasurement discussed apparatus). the The effect situation, ofin however, including section differs the from infalling the observer (or reconstruction a method relate certain average ofin OTOCs eq. to the ( mutualmaximal. information that in appendix disappears as a result oflimited including to the AdS/CFThole correspondence to as be it maximally only entangled. requires the quantum statedifferent of context a [ black the increase of thethis coarse-grained extension entropy of due the Hilbert tomuch space shorter the via than inclusion equilibration the of occurs timeunitary scale in operator, of thermalization applying time one which can is This show suggests that that quantum entanglement between Here we increased the Hilbert space dimension of the black hole in order to account for JHEP10(2019)132 M can D supported and which lives on ˜ D R β . . The point is that physics CFT T E H = which is supported on the S D . Hence, the disentangling phenomenon decay after the scrambling time. From . Hence, we propose that reconstruction R A D 32 – 28 – ) and A is disentangled from the early radiation (or 34 D M without involving In principle, the infalling observer may choose to fall into a C do not necessarily commute. , uncomfortable. The reason why we need to introduce extra 33 . in the black hole Hilbert space, do not affect our main conclusion t and the infalling mode AB EE ∆ and M C , the interior operator is realized as − consisting of microstates at the same temperature. Inclusion an infalling observer A and A = corresponds to quantum mechanical operators for the infalling observer t 6 CFT AC on ⊂ H D β (1) value after a very long time (of the order the quantum recurrence time). But in time scales H be the full Hilbert space of the CFT and consider a black hole at temperature O CFT from section . On the other hand, from the perspective of the infalling observer who travels along Finally we speculate on non-uniqueness of reconstruction of interior operators. From Regardless of how the probes or the apparatus are introduced, what is important H Although we have included an observer or an apparatus, one canIn discuss the aforementioned the scenarios, monogamy infalling of observers were entangle- introduced to the system by appending ancilla One might wonder whether an observer can be introduced in a way which introduces no backreaction. R D 33 34 32 Hilbert spaces. Readers mightgling wonder phenomena if or addition not. ofphysical ancilla realistic One systems can description are actually of necessaryLet avoid black to the holes, achieve use the at disentan- ofa least ancilla subspace within systems the simply contextor by of measurement considering the apparatus a AdS/CFT increases more of correspondence. the adding entropy infalling of observers the can black be hole discussed by in ∆ the full CFT Hilbert space by utilizing ETH. Onerecovery possible protocol way by accessing to both undo sides the of effect the ofment black purely backreaction hole from is as the we tooperators operator discuss perform for later. algebraic the perspective. Hayden-Preskill The important point is that representations of basis From the outside quantumadded mechanical viewpoint, on this one appears sidefeature to of of be the interacting not black quantumreturns possible hole. many-body to if systems. the Our perturbations argumentwhich It are are relies is relevant on to in decay the principlevalue AMPS of possible and puzzle, OTOCs we that then which believe the is oscillates that value OTOCs a around decay of rather monotonically it. OTOCs generic to a We small stationary believe that such a conclusion can be mathematically proven We speculate that different infallinginterior trajectories operators. correspond An to explicit differentis relation reconstructions beyond between of the trajectories scope and of reconstructions, this however, paper. the infalling mode remaining black hole of who departed at black hole at any time by following any trajectory as long as it does not violate the causality. be reconstructed on occurs universally no matter how she falls. the perspective of theon outside observer, the interior operator is realized as “probe mode” as an excitation. for us is thatgravitational OTOCs perspective, this between is aand universal shockwave phenomenon geometries. resulting from From quantum the informationto surface theory gravity prove perspective, that this the is sufficient outgoing mode significantly. Some readers mayand find the inclusion ancilla of qubits extradegrees objects, of freedom such is as thatqubits, the a and apparatus black hence, hole is thereformulation, treated is one as no may a simply room system turn for of on discrete adding the finite-dimensional source probes. term which In creates a the more “apparatus” or realistic the field theoretic include the apparatus JHEP10(2019)132 R . It is 35 D . We now is outside a ˜ D D . If the time 11 is longer than or . D M in a backreacted geometry D . According to the causal struc- D and can be found across the horizon and is D D . ˜ as a gravitational shockwave and draw the D is disentangled from the early radiation D M and the infalling observer – 29 – , Alice eventually enters a region which may be D D , one possible representation of interior partner operators can be constructed D . The infalling observer as a gravitational shockwave. The interior mode Here it is important to note that the interior mode emerges from the right hand side, and is not longer entangled with , Alice may be hit by a gravitational shockwave created by the outside observer, ˜ D 36 This geometric interpretation enables us to deduce the experience of the infalling ob- For simplicity of discussion, let us consider the AdS eternal black hole. Given the out- I thank Xiaoliang QiHere and we Ying treated Zhao the for effect useful of discussions on the this. perturbation by Alice first on the left hand side, and then discussed 11 35 36 the effect of theeffect perturbations of added the on perturbation the onimplied right the by hand right decay hand side. of side OTOCs,addressing One first. we the might According should experience wonder to treat why of the the we the structure effect did infalling of of not quantum observer the entanglement from treat perturbation the the on left the hand left side. hand side first when ture, after observing theaffected interior by mode quantum operationsure on the right handaccording side. to the Inside backreacted geometry. thison It region, the is, shaded however, right unclear in hand to fig- side us can whether actually perturbations influence the experience of the infalling observer at all rium. is different from the original partner operator server. From the backreactedhorizon smoothly geometry, and we observe see the that entangled pair the infalling observer can cross the separation between the outgoingequal mode to the scrambling time,outside the the causal interior influence mode ofexplanation any of operations on the the fault-tolerance right andworth hand state-independence noting side. of that This the a is interior the similar operator geometric observation applies to generic black holes in thermal equilib- going mode by time-evolving a corresponding modeexclusively on on the the right degrees hand side. ofinclude freedom the This on operator, effect the constructed ofbackreacted right the hand geometry infalling side, where observer was the denoted horizon by is shifted as depicted in figure We have argued that(or the the outgoing right mode Our hand primary side) focus due wasmechanical to on description. assessing the Here the decay we effect present of of geometric OTOCs interpretation backreaction of caused from our the by discussion. outside the quantum infalling observer. Figure 11 shaded region that may bemode influenced by quantum operations on the right hand side.7.3 The original Geometric interpretation JHEP10(2019)132 , . | is R C and and R (8.2) (8.3) (8.4) (8.1) must C C |  | CR ˜ D O | is the early radiation. R are entangled with . . For the sake of discussion, . . However, this construction is also a partner operator for D i . i CD i C , a partner operator Ψ CD | i ˜ ) O D and EPR | CR young ) C CDR ˜ young O Ψ C i † . The answer is clearly no since Ψ | ⊗ K . CD ≈ | T I D CD D D i EPR I 0 | acting on and ask if ( ⊗ ) CD does exist: i KO C D T D K such that and C ˜ O O i ≈ O = young – 30 – ˜ ⊗ O observer-dependent D D Ψ Ψ young is the outgoing mode and | I | ) CR = ( Ψ | ˜ D i O ) = ( CR and , but D I i is drawn randomly from the Haar measure. If O C Ψ ⊗ | EPR ) is nearly maximal (for typical states). Hence, given a , let us fix | ⊗ D ) D CD C O i O ( ˜ CD O C,D ( C ( I I young and ⊗ Ψ | D , a partner operator CD i D O state-independent O influences Alice’s infalling experience at all if it is strictly localized on young R Ψ | is the remaining black hole, can be decomposed into C R Next, let us recall the argument for a young black hole where the early radiation respectively. Given a unitary operator are entangled in a way which depends on each quantum state. For a given some other typical pure quantumD state absent. Consider a “typical”one pure may assume quantum that state the mutual information unitary operator by using the factclearly ( depends on the initial state of the black hole. D satisfy One possible representation can be constructed as follows: old black hole. Consider a maximally entangled black hole: where Here introduced, being 8.1 No-go arguments Let us briefly recall some versions of the no-go arguments for state-independence for an 8 On definition ofSome state-independence previous studies suggest thatIn black this hole section, interior operators weour must revisit construction be these avoids state-dependent. them. no-go The argumentson main for the message state-independence is initial and that state our discuss construction of how does the not depend black hole, but does depend on how the infalling observer is the remaining black holeturbation and on distant early radiation).In In any cases, fact, we quantumof do operations crossing not on a think the smooth that right horizon per- hand and side seeing do not affect Alice’s experience in the absence of non-local interactions between the left and right hand sides (or between JHEP10(2019)132 i ]. D 9 O (8.8) (8.7) (8.6) Φ(0) | . Let as a physical 7.1 E ) (8.5) D O . ⊗ C ˜ O were partner operators of C Tr( . ˜ , not for the unperturbed original O observers n i 1. But the r.h.s. becomes zero for . 1 M . 2 ) i i ) t i ( 0 ≈ 0 = (0) 0 Φ | Φ CD | i ⊗ | Φ(0) CD i ) | i t Ψ ( Ψ | | ) U ) Φ(0) | D = Ω( D – 31 – 37 = i O O ), the expression of interior operators would be = i ) i ⊗ t ⊗ (0) 8.6 0 ( 0 C C (0) , it does depend on Φ 0 ˜ Φ | ˜ i O | O ( Φ ( | CD CD Φ(0) | | | Ψ Ψ ) in eq. ( h i 1 i h | Ψ | d . As such, previous no-go arguments do not apply to our construction. i Z ) t Φ( | , we explicitly appended an infalling observer to the system: , we would have : 7.1 U CD i Ψ Similar justifications hold for other scenarios of introducing infalling observers to the It is, however, worth emphasizing that we are not entirely free from dependence on A similar conclusion can be also obtained by considering an average over typical quan- | Arguments along this line are also used as an evidence for firewalls in typical black hole microstates [ 37 inclusion of the infallingbe observers expressed on as a an unified isometric footing. quantum operation: But formally, this procedure can initial state of the black hole black hole. For instance,probe we to have the illustrated blackthe the hole. ones use for of perturbed In an states this ancilla with case, such mode the ancilla modes. reconstructed It interior is operators unclear are to of us how course to treat the quantum state specifics of the system.observer Namely, is our introduced construction to manifestly theobserver system. depends on For is instance, how chosen if the adifferent. infalling (say different initial state Hence, of the while infalling our construction of interior operators does not depend on the What we have constructed,interior by partner the operators use for of the the quantum Hayden-Preskill state recovery protocol, are the In section After the inclusion of theunitary infalling observer, the whole system time-evolves by some chaotic simply due to theblack fact hole, that but it for doesTo a see not slightly this point give perturbed explicitly, partner state letincluding operators us the after recall infalling for the how observer, the inclusion specifically the the interior initialbe of scenario operators state the the discussed were initial in of infalling reconstructed section after state the observer. of the black hole, which can be two-sided or one-sided (old or young). traceless unitary operators, such asoperators Pauli in operators. a state-independent Hence manner. one cannot choose8.2 the partner Observer-dependence Our construction of the black hole interior operators avoids the above no-go arguments tum states. Observe the following relation where the integral isfor taken over the Haar measure. If JHEP10(2019)132 which D depends on the which is distinct i D (0) 0 Φ | is entangled with some D have no relevance at all as . This, however, leads to a ˜ D D due to the backreaction which does not depend on the initial ˜ D i is disentangled from the distilled qubit and ) t ( D 0 by performing some careful quantum D Φ | , consider an outgoing mode D as the same physical degrees of freedom. R . ˜ D D from the black hole or hand it to Alice who . Our claim is that construction of interior i ˜ and D . The outside observer Bob distills a qubit (or (0) – 32 – ˜ R 0 D can be constructed in the same manner for young Φ | ) D t ( and is independent of initial states of the black hole. U ˜ D = as well as how the infalling observer is introduced, namely i i ) t ( 0 Φ(0) Φ | is also entangled with | ” try to interpret B D R = is disentangled from the distilled qubit which is entangled with the outgoing mode A D . Bob may isolate the qubit D R , but depends on the map Ω. Discussions on actual expressions of interior 38 i that is entangled with the outgoing mode ˜ D Φ(0) | The fallacy of the above argument is clear. When Alice falls into a black hole, the The AMPS argument relies on three key assumptions; a) unitarity b) no drama (i.e., smooth horizon) is state-independent. Furthermore, 38 . This conclusion was obtained from decay of OTOCs. Calculations of OTOCs generically go beyond ˜ while b) and c)quantum result evolution and from argued general that relativity, theThe namely horizon must essential the be point equivalence smooth. principle. in Hence,D the our We assumption have argument c) assumed must is be unitary effective that violated. n¨aive quantum the field outgoing theory. In mode gravitational its shockwave simplest and form leaves significant at backreaction theeffective to massless the quantum geometry. limit, field the Such theory a perturbation on calculation becomes a goes a beyond fixed background geometry. Here we argued that theyD must be different.black In holes. fact, and c) effective quantum field theory outside the horizon. Note that a) results from quantum mechanics outgoing mode makes OTOCs decay. Furthermore, Alice willfrom observe the the interior original mode partnerHence, the mode monogamy of quantum entanglementbundled is under not “ violated. Recall that the approaches operation on is about to fallinterior into mode a blackcontradiction hole. because After crossing the horizon smoothly, Alice will see an Let us consider ais version of maximally the entangled AMPS with thoughthas the experiment. been early just Given radiation an emitteddegrees old from of black the freedom hole black in that hole.qubits) the The early outgoing radiation mode 9 Resolution of AMPS puzzle In this section, we present a resolution of the9.1 AMPS puzzle. AMPS thought experiment operators for such astate slightly perturbed blackoperators hole for each possiblefuture work. scenario of introducing infalling observers will be given in a of the infalling observer.unitary Generally operations as speaking, well Ω as additionchoice consists of of of ancilla the some systems. initial simple state Obviously, the (low map complexity) Ω.some After chaotic the dynamics: inclusion of the infalling observer, the black hole time-evolves by Here the isometry Ω accounts for all the quantum operations associated with the inclusion JHEP10(2019)132 ˜ D are ˜ D and D . from Alice and bring ˜ and creates a firewall. t D A ˜ D and and D D , Alice may be (or may not be) hit D is realized as a mode different from the , Alice still crosses the horizon smoothly ˜ D D are not entangled due to perturbations, the and ˜ D since – 33 – D D and D and . D D and , generates a high-energy gravitational shockwave which prevents D R . Bob’s strategy, then, would be to retrieve t A . After observing the interior mode , we summarized the original AMPS argument, previous proposals along ˜ D 12 , as well as quantum operations afterwards by Bob, creates perturbations which ˜ D Here we propose a physical mechanism for Bob to protect quantum entanglement from In figure In summary, we propose the following resolution of the AMPS puzzle: While we also concluded that It is worth comparing our proposal with the previous proposal by Maldacena and follows. dom that were originallyentanglement entangled is with not the violated. outgoing mode. Hence the monogamy of interior operators correspond to different infalling trajectories that the infalling observer falls. This disentangles the outgoing mode from the early radiation. between the outgoing mode and the interior mode. Reconstructions of interior operators are not unique. Different reconstructions of the The infalling observer crosses the horizon smoothly and observes quantum entanglement The interior mode which the infalling observer sees is distinct from degrees of free- An infalling observer leaves non-trivial gravitational backreaction no matter how she on the early radiation the infalling observer from crossing the horizon smoothly. the backreaction by Alice. Recallwith that earlier Alice mode can see quantum entanglement by interacting We have argued that theobserver does infalling not observer due sees toconsider quantum the a entanglement possible backreaction while scenario by where the the thethe infalling outside outside infalling observer observer. observer can does see It not. quantum is entanglement Wethat while also have verification already intriguing of refuted to quantum previous entanglement, proposals namely which the suggested distillation of the entangled qubit the lines of ER = EPR, and our proposal.9.2 Revisiting firewalls 4) 2) 3) by a gravitational shockwave whichentanglement Bob between creates, but this has nothing to do with quantum 1) underlying physical mechanism differs indisentangled due a to crucial the backreaction manner. bydespite the In the infalling our observer loss herself, of scenario, not entanglement dueand between to observe Bob. the Also, entangledoriginal pair mode Susskind, an approach oftenqubit called “ER = EPR”.will They become argue a that high-energy theperturbation distillation density spoils of near the the the quantumAs entanglement horizon. such, between Alice is Furthermore, not they able argue to that observe entanglement this between (1) (2) (3) (4) (6) (7)

JHEP10(2019)132) (5) (1) (2) (3) (4) (6) (7) (1) (2) (3) (4) (6) (7) A, BD ( i i 0 (8) n n (2) 2 2 (2) (2) BD AC I | | (1) (2) (3) (4) (6) (7) S S = + + i! ) | t 0 (10) i 0 (11) ( 0 (9) ) ··· ··· t (2) (2) D C D ( S + + S ) = 0. ! ! O D i! D i i ) ) ) + + t O ] work, ( m m (0) ˜ | | D ˜ A 1 A (2) (2) D B (0) A A, C A, D + D O 31 O S S A ( ( ) I I t D, AC O ( (0) ··· ··· ) D ( t )= )= A D I ( ) (5) earlyradiation + + Beni Yoshida ) April 22, 2019 O O ) D i i ) = 0 suggesting h 2 2 O | | ) (5) A, C B,D (0) (1) (2) (3) (4) (6) (7) ( ( A + + (0) A, BD (2) i i (2) O A ( to be EPR pairs. h 1 1 I I i i | | O (1) (2) (3) (4) (6) (7) 0 (8) n n ) (5) (2) h A, BD D, AC | 2 2 ( (2) (2) BD AC I ( | | D i i 2 S S I 0 (8) n n (2) same? = + + i! 2 2 (2) (2) BD AC I D ) log | | | (1) (2) (3) (4) (6) (7) t 0 (10) i S S A, BD 0 (11) ( 0 (9) ) shockwave? = ( ··· ··· + + i! ) = max and t (b) (2) (2) D i i C ) | D ( t 0 (10) 0 (8) i n n S + + S ! (2) ! 0 (11) D ( entangled 0 (9) ) O D i! ··· ··· 2 2 (2) (2) D BD AC I t i i | | ) (2) (2) ) ) D C D ( + + t O S S S + + S ! ! ( m m (0) = O + + D i! i! ˜ D, | | D D i i ) ) ) ) | ˜ A 1 A ( (2) + (2) Center for Theoretical Physics, Massachusetts Institute of Technology + D B (0) A t t O 0 (10) i A, C A, D 0 (11) ( ( + D 0 (9) m m (0) ) O I O BH S ˜ max, Alice traverses the wormhole. S ··· ··· A | | ( ( t D (2) (2) ) D C ˜ A 1 A D ( (2) (2) I D B Universal quantum phases in stabilizer codes (0) A I t O S + + A, C S ! A, D ! + D ( O (0) ··· ··· ) O O D S S A D i! ≈ ( ( D t i i ) ) ) ) )= )= A D I entangled I ( t + + t O O + + ) ( Beni Yoshida (0) ··· ··· ( ) m m D (0) ) April 22, 2019 O O ˜ ) D | | t i i D )= )= h A ˜ A D 1 A ( A (2) ) (5) (2) D B (0) A 2 2 earlyradiation + + O Beni Yoshida | | A, C A, D + D A, C ) April 22, 2019 O O O ) B,D D O S S (0) A i i ( ( ) = max, we have , and steal the EPR pair from Alice ( h ) ( 2 2 I I A t O + + O A, (0) | | A, C ( B,D ) (5) (0) ··· ··· (0) ) D D ( (2) i i ( R (2) O (1) (2) (3) (4) (6) (7) ( A t )= )= A D A + + h 1 1 I ( I I (0) | | + + A, BD Beni Yoshida O (2) i i ) April 22, 2019 O (2) O O ) D A ( h i i outgoing mode h h 1 1 D, I | I i i | | 2 2 O O ( 0 (8) | | 2 n n ) (5) A, C (2) h B,D (0) A, BD | ( I ( 2 2 (2) (2) BD AC I and ( | | A + + (0) 2 ), we still have i i S S log (2) i i 0 (8) (2) O i A n n = (2) + + i! h 1 1 I I 2 2 (2) | | ) (2) log BD | AC I O | | D t 0 (10) (1) (2) (3) (4) (6) (7) i h A, BD 10 S S 39 0 (11) ( 0 (9) | ) ( ··· ··· | = t + + i! (c) (2) (2) D 2 i i C D ( ) | 0 (8) n n t 0 (10) S + + S ! (2) i ! entangled O + D 0 (11) i! ( 0 (9) – 34 – 2 2 ) (2) (2) D BD AC I ··· ··· i i log | | ) ) ) t (2) (2) D shockwave C + + D i t O S S ( ( . This is essentially Bob’s attempt to observe the Center for Theoretical Physics, Massachusetts Institute of Technology m m (0) S + + = S ! ! + + i! ˜ | | O D i! D D ) | ˜ A i i 1 A ) ) ) (2) (2) Center for Theoretical Physics, Massachusetts Institute of Technology 01 D B (0) A t 0 (10) i + + t O | A, C A, D 0 (11) D ( + D 0 (9) Universal quantum phases in stabilizer codes ) O O BH ( S S ··· ··· m m A (0) ( ( ( ˜ t | | (2) (2) ) D C D D entangled ( ˜ A I Universal quantum phases in stabilizer codes I 1 A t O (2) (2) D 2 B (0) A S + + S ! ! ( (0) ··· ··· ) A, C O D A, D 1 + D D i! O O S D S A t i i ( ( ) √ ) ) )= )= ) A D Center for Theoretical Physics, Massachusetts Institute of Technology ( + + t O I I t + + O Beni Yoshida ( m m (0) ( ) April 22, 2019 Alice O O (0) ··· ··· ˜ ) ) D D | | i i D t h ˜ A 1 A )= )= (2) (2) Universal quantum phases in stabilizer codes A D D B (0) A 2 2 ( ) (5) O | | earlyradiation + + A, C A, D + D A, C Beni Yoshida O B,D O S S (0) A ) April 22, 2019 ( O O ) ( D ( i i ) ( h I I A t + + O 2 2 (0) O | | ( (0) ··· ··· ) D A, C B,D (2) i i (0) (2) O onto EPR pair. Since A t ( )= )= ( A D h 1 1 I ( I A + + | | + + (0) Beni Yoshida O A, BD ) April 22, 2019 O O ) D (2) i i h i i by applying the EPR projection operator. The second protocol (2) O A ( outgoing mode D h | h 1 1 I 2 2 I i i O | | | | 2 O A, C 0 (8) B,D (0) n n ) (5) (2) h D ( ( | R 2 2 (2) (2) A BD AC + + I (0) | | 2 log (2) i i S S (2) O A h 1 1 = I + + I i! | | O ) log | and its partner h t 0 (10) i | A, BD 0 (11) ( 0 (9) ) ( ··· ··· and . The first protocol proceeds by verifying the entanglement between (a) 2 t (2) (2) D i i C D ( 0 (8) n n firewall? S + + S ! (2) D ! entangled O D i! 3 log 2 2 (2) (2) D BD AC I i i | | ) D ) ) + + t O S S Center for Theoretical Physics, Massachusetts Institute of Technology ( m m (0) = + + i! ˜ | | D ) | ˜ A 1 A (2) (2) Center for Theoretical Physics, Massachusetts Institute of Technology D B (0) A t 0 (10) i Universal quantum phases in stabilizer codes A, C A, D 0 (11) ( + D 0 (9) ) O O BH S S ··· ··· A ( ( t (2) (2) ) D C D ( I Universal quantum phases in stabilizer codes I t ) is not necessarily large. Hence, we conclude that Alice does not see the EPR pair. And O S + + S ! ! ( (0) ··· ··· ) O D D i! D t i i ) Center for Theoretical Physics, Massachusetts Institute of Technology ) ) )= A )= A D entangled ( + + t O + + Beni Yoshida ( m m (0) ) April 22, 2019 O O ˜ ) D | | i i D Universal quantum phases in stabilizer codes h ˜ A 1 A (2) . Schematic pictures of (a) the AMPS argument, (b) some previous proposals and (c) (2) A, D B (0) A 2 2 O | | A, C A, D + D ( A, C O B,D O S S (0) A ( ( ( ) ( I I I A t + + O (0) ( (0) ··· ··· ) D (2) i i (2) O A t )= )= cannot be reconstructed on the left hand side. Since A D h 1 1 I ( I | | + + Beni Yoshida O ) April 22, 2019 O O ) D h i i outgoing mode h | At this point, it is pretty satisfying to recall how the recovery protocols of [ 2 2 D O onto other entangled states, such as | | 2 A, C The first protocol projects B,D (0) ( ( A + + (0) 39 log D (2) i i (2) O A h 1 1 I I | | that So, we conclude thatD Alice does notHowever, see the EPRunfortunately, pair she and does simply not return returns to to the the outside. outside. If we project the outgoing mode EPR pair between achieves this goal inBoth a protocols deterministic verify manner the by EPRby unitarily preventing pair her restoring between from crossing the horizon. is, by preventing Aliceentanglement between from the falling outgoingcross into mode the a and horizon, black she the will hole, early not radiation. Bob see quantum is Since entanglement. reviewed able Alice in to does section not protect quantum Figure 12 our proposal. it back toPreskill his recovery possession. protocol from One the concrete outside (and and peaceful) pull method Alice is herself to to run the the outside! Hayden- That O h | 2

log

Center for Theoretical Physics, Massachusetts Institute of Technology Universal quantum phases in stabilizer codes Center for Theoretical Physics, Massachusetts Institute of Technology Universal quantum phases in stabilizer codes JHEP10(2019)132 ˜ D on the ˜ and the D D exclusively on ] for problems does not lead to which consists D 68 ˜ D D Our viewpoint is that the black hole 40 and create a new interior mode ˜ D can be seen by coupling two boundaries in a suitable ˜ – 35 – D from D . R and the Hilbert space factorizes into the left and right. (See [ R H + L H The resolution of this non-locality puzzle can be obtained straightforwardly. By falling We have encountered an analogous problem in the context of the AMPS puzzle where The backreaction from Bob’s verification of quantum entanglement is different from Based on this observation, we make the following proposal as physical interpretation of A common misguided approach to the non-locality problem in the two-sided AdS black hole goes as black hole recklessly. entanglement. ifying (or protecting) quantumearly entanglement radiation between the outgoing mode = However, it does not kill an infalling observer. Instead, it savesSince her an infalling from observer falling does not into cross a the horizon, she will not be able to see quantum An outside observer creates a non-trivial backreaction for an infalling observer by ver- 40 follows. “From the boundaryother side perspective, or whether not Alice cannothorizon. be has verified Yet, interacted from the with eithermanner, interaction boundary the between because e.g. partner Alice the via mode interaction and the wouldcontradiction happen traversable because behind wormhole two the effect. boundariesthink need The this to interaction is be between apaper, Alice correct coupled the resolution and for infalling of verification observer the of will non-locality the disentangle problem. interaction”. As We we do have not rigorously demonstrated in this into a black hole, Alice leaves aexclusively backreaction of and creates the a replicated left mode on hand the side right degrees hand of side, freedom. there is Since no she contradiction. does not interact with the infalling observer jumpsevery into possible a representation black ofhand hole interior and side. modes sees Suppose requires themode. that degrees interior This mode. she of suggests crosses freedom that Supposethe the on there left that horizon the and must right smoothly be right hand some and sides interaction can of between the observe degrees AdS the of black interior freedom hole, on leading to a contradiction. and right boundaries respectively andprinciple by meet starting inside at the sufficiently blackwhen early hole. time. they meet. This Two observers can From may the behowever, interact this perspective achieved with seems of in each to the other boundary imply quantum some mechanical non-local descriptions, coupling between left and right. 9.3 Puzzle on non-localOne mysterious interactions feature of (and the its AdS/CFTthe correspondence resolution) horizon is in an the apparent two-sided non-locality blackH behind hole. The CFT Hamiltonianrelated on to the boundary the factorization is of given by the Hilbert space). Imagine two observers who start from left either high-energy barrier orhorizon. some hard It wall is whichwormhole, worth would can noting terminate be that the interpreted the as geometry a Hayden-Preskill at shockwave recovery, the with as negative well energy. as the traversable 2) 3) firewalls that may arise from the outside observer’s1) verification of quantum entanglement: JHEP10(2019)132 10.2 when R . There are three scr t ' t in the Schwartzshild black . S r scr t and the early radiation log  S t r D ' t . Hence the validity of quantum field theory is via a careful quantum operation acting only on one side pair with ∆ – 36 – scr D t D D  t has nothing to do with her infalling experience. Furthermore, we ˜ D . t is the proper distance from the horizon. As one goes away from the near the horizon, she needs ∆ ρ D even if Alice has fallen into a black hole. This, however, suggests that Alice where scr t 1 πρ 2 The second explanation concerns the quality of quantum entanglement that Alice may The first explanation concerns the applicability of effective quantum field theory near In the reminder of the paper, we present discussions and speculations on relevant While we have focused on the AMPS puzzle in black hole horizons, it will be interesting  = very close to the singularity for ∆ t grain a larger volumeoutgoing in order mode to distill a singleher EPR side pair. of the Inthink boundaries. order that it for Namely, is Alice possibleof to to the see black meet hole, the the namely interiorfor by mode further performing discussions. the Hayden-Preskill recovery protocol on one side. See section entanglement for small ∆ be able to see.T Recall that thehorizon, proper the temperature density near of the thermal Rindler entropy horizon becomes is smaller. given by Namely, one needs to coarse- possible physical explanations for the the singularity. In the PenroseD diagram of the AdS blackquestionable. hole, one This notices observation seems that to Alice suggest meets that Alice may not be able to see quantum Since OTOCs startquantum to entanglement decay between significantly the∆ around outgoing the mode scramblingshould time, not be Bob able canreason, to verify observe our quantum proposal entanglement when resolves crossing the the AMPS horizon! puzzle For this only when ∆ recedes the horizon away so that she cannot actuallytopics. cross the cosmological horizon. 10.1 Early time to ask about cosmological horizons.opposite In direction, the making dS space,negative the a energy Penrose shockwave shifts shockwave diagram the in “taller”,firewall horizon the in in problem AdS the a in space. wayhorizons. black similar hole As to One such, horizons sending possible our does a scenario proposed not resolution may apply for be to the that the the problem backreaction in by cosmological an existing observer senting state-independent reconstructions ofproblem interior operators. to verify It the isare proposed an currently scenario interesting working future in on concretethe writing SYK models the model. of interior quantum operators black in holes. an explicit We manner, namely in entanglement between the left andelsewhere. right hand sides. We hope to address this issue10 further Discussions In this paper, we argued that the AMPS puzzle does not lead to inconsistencies by pre- interior for an infalling observer emerges due to scrambling dynamics, not due to quantum JHEP10(2019)132 , . , , t is S S A D 0 r r scr t S A r log log S S to be a r r log ≈ S ' r t DD scr t ' t is entangled with D would remain invariant and This is not surprising as we A 42 is almost state-independent even . After the black hole evolves by A D 41 , Bob can verify the EPR pair when ∆ is kept outside the black hole. We prepare S r onto the EPR pair. This leaves D . This suggests that the infalling observer, who log – 37 – A 0 A S . The initial state of the black hole is given by r A 4 ≈ summarizes the reconstruction of the interior operator 13 is the reference system of . 0 S A r , is kept outside the black hole. We then implement the inverse . Recalling that the Hayden-Preskill thought experiment says D log S . What is surprising is that the outgoing mode S , the outgoing mode r † r for further discussions on the reconstruction of interior operators for U where U ≈ log C which is left outside the black hole. One possible interpretation is that, S scr r BR t i D / scr t EPR is small. We think that this phenomena may be of independent interest in the | . The important point is that construction of A t t 0 A i Let us ponder over possible geometric interpretation of these protocols. Since See appendix The third explanation is to assert that the loss of quantum entanglement for modes to the system. Finally, we project More precisely, we applied a projection operator to pick an event where the infalling observer returns This observation enables us to obtain “upper bound” on the scrambling time. If the scrambling time 42 41 † EPR in a sense of decayHowever, of Alice OTOCs, can is also longer see than we the arrive EPR at pair, leadingone to may the conclude violation of monogamy of entanglement. Assafely. such, This works onlyrecovery probabilistically. protocol to One make may this apply happen the deterministically. deterministic version of the Hayden-Preskill due to the backreactionis of the somehow pulled Hayden-Preskill recovery to protocol,to the cross the outside the interior of horizon operator the smoothly black and hole. returns safely Hence, with the the infalling interior observer mode appears in hand! is projected to thereappear EPR as pair, the an samefell quantum input state into quantum on a state black on applied hole, the returns inverse toanother the mode outside again safely. an additional EPR pair.other The half, half denoted by ofU the EPR pair isnearly thrown ideal into EPR a pair.as black a hole Figure physical while process. the reconstruct interior operators, iton is a intriguing black to holestruction think as protocol of an outlined physically actual in| implementing physical section them process.a Let unitary us operator focus on the probabilistic recon- context of quantum many-body physics. 10.2 Peeking in throughWhile the we horizon have used the Hayden-Preskill recovery protocols as a mathematical tool to however, the Rindler modesloss in of question entanglement are cannot separateddeviations be by may detected be a superseded by long by simple distance, e.g., local and curvature physical hence corrections. observables. the small ∆ Such tiny when ∆ necessary for Alice to see a good quality EPRaway pair. from the horizontheory does descriptions. notwill lead The create to loss high significant of energy violation entanglement density as of in entanglement effective the is quantum Rindler at field short modes distance near scale. the For horizon small ∆ hole which is of order the scrambling time. As such, we speculate that ∆ JHEP10(2019)132 which E E without involving D were to be measured in EPR pair, D D . If EPR D – 38 – ]. Since operators in the entanglement wedge must be in- states same 18 . Our protocol requires EPR pairs in the ancilla system R . Reconstruction of the interior operator must be identical quantum states. A As this speculation suggests, reconstruction of the interior operators by the Hayden- and A Energy measurement We have presented athe protocol early to radiation reconstruct the interior operator version of the paper,Xiaoliang I Qi, Suvrat would Raju, like Lenny to Susskindedgment and thank of Ying Ning discussions Zhao Bao, for with usefulResearch Raphael a discussions. at Bousso, person Acknowl- the Juan does Perimeter Maldacena, notInnovation, Institute Science necessarily is imply and supported the Economic bythrough person’s Development the the Canada agreement. Ministry Government and of of by Economic Canada the Development, through Job Province Creation of and Ontario Trade. Acknowledgments I would like to thankVerlinde Ahmed for Almheiri, useful Yoni discussions BenTov, oncareful Isaac related reading Kim, topics. of Rob the I Myers and preliminary thank Herman draft Yoni BenTov and and a Rob number Myers of for useful feedbacks. For the second reconstructing operators in therecently entanglement wedge. made by Progresses Almheiri alongterpreted this [ as line quantum have field been theorystructions operators must on be a possible. fixed geometry, We state-independent expect recon- that our results are useful in achieving this goal. Wall traversable wormhole. 10.3 Entanglement wedge reconstruction We have presented a protocol to reconstructrecovery. interior operators Hence by it using the is Hayden-Preskill reasonable to expect that the Hayden-Preskill recovery is useful in Preskill recovery protocols enablesoutside us even to when probe theobservation, the it black physics may hole be behind is interesting the one-sided. to horizon consider from To a turn the similar process the by speculation using into the a Gao-Jafferis- concrete Figure 13 A JHEP10(2019)132 . D (A.2) (A.3) (A.4) (A.1) . Let 2.1 should look as operators to multi-level D C . According to X D sustains diagonal whose dimensionality CD ρ E i ∈ H n | are correlated. An analysis ) originates from the total n by D 2.2 βE Z − D an E e i n C n E . = | | A X n n i . If we start the system in an arbitrary in eq. ( ih n E ⊗ | i n D n | A n ρ 43 c | | n . n iθ C , w n e | ih ]. X n n | n – 39 – 26 ih X → n n | E X = . Unlike off-diagonal operators, the interior partners n i n θ 0 w = | θ O A n i W in a particular manner. Here we argue that much simpler X ψ | A = D , the measurement leads to the evolution: ρ A is the Weyl operator (a generalization of Pauli- cannot be found in i | n θ j | -basis while off-diagonal correlations decohere due to scrambling . The interior partners of those can be identified in i n O ih . The discrepancy with c n D n | n + 1 βE P − j | e = j n i . Denote the orthonormal basis by P P ψ | A = = . The procedure of quantum measurement can be formulated as a unitary process represents all the data about X A Z θ Off-diagonal correlations can be restored by simply measuring the energy of the earlier The above observation prompts us to consider two different classes of operators on To set the stage, we elaborate on the issue briefly mentioned in section . The diagonal operators are the ones which leave energy eigenstates unchanged up to Off-diagonal partners exist due to the existence of diagonal correlations. Diagonal partners do not exist 43 CD where systems). due to the absence of off-diagonal correlations. under the unitary operator mode in the following manner. Weis adjoin equal the system to to apure pointer system state Here of diagonal operators particle number) in which decrease and/or increase theρ energy density. This isphases. due Such to operators diagonal can correlations be in explicitly written as follows: with energy conservation takencorrelations into in consideration the indicatesdynamics (i.e., that correlations are classical) [ The off-diagonal operators are the ones which may change the energy eigenstates (or the with energy conservation. In particular, energy densities on and generic interaction can achieve this goal. us denote the orthonormalthe standard basis rules of of thethermal: statistical outgoing physics, mode the reduced density operator on interact with the earlier mode JHEP10(2019)132 . . . . | ], R C E B ) is 26 | (B.1) to (A.5) . The < A A | C,D 0 ( I B | from θ as a process of O ) resembles black A B.1 only in the codeword R . Indeed, they performed a C whereas the eternal AdS black R i and the remaining black hole j ˜ ent E ⊗ | > S B i allows Alice to “copy” j ˜ BH | | S 0 M be the coarse-grained entropy of the black =1 B j | X . The situation of eq. ( = | can be used for the Hayden-Preskill recovery, 0 ent BH – 40 – S S B D . Specifically, consider a quantum state given by 1 ]. To illustrate the idea in a simplified setting, . Hence, from the outside description, all Alice | | 0 A = . =1 22 p B σ | j C i} = BH j ˜ can be reconstructed on S {| BR D i = Ψ 0 | is entangled with the early radiation B B H ] in a slightly different context. 69 with can be reconstructed in the pointer B . Here, by the “codeword subspace”, we simply mean that | D B be the entanglement entropy between the black hole and the radiation ⊂ H 0 B ent |  | S H 0 B | Nevertheless, we will make an interesting link between our proposal and their view- In our interpretation, however, this observation itself does not resolve the AMPS puz- The important point here is that applying this proposal runsperformed. into another Indeed, if problem thethen where outgoing it the mode should Hayden-Preskill not be recovery entangled cannot with be point. Let us view the energy measurement discussed in the appendix hole and Then the black holes under considerationhole satisfy corresponds to theholes case before with the Pageexplained. time. Second, the Ifof proposed such the reconstruction a codeword of subspace, codeword interior and subspace operators as emerges, depends such, its is on state-dependent. origin choices should Third, as be pointed out in [ by using the error recovery procedureidea known was in considered quantum error-correction [ theory. A related zle. First, this proposalsuch does as not the apply eternal to AdS black black holes hole. which have Let maximal entanglement where If the black holenearly evolves maximal, by implying a that careful Haar statistical physics random analysis and unitary, presented the explicit constructions mutual of information interior operators suppose that a black hole subspace interior operator B Verlinde-Verlinde proposal Verlinde and Verlinde proposed an intriguingthe resolution idea of of the AMPS quantum puzzle error-correction by employing [ which holds for anyneeds density to operator do is measure the energy (or the particle number) of the earlier mode This can be most explicitly shown by noting the following identity: JHEP10(2019)132 , is A BH B (B.2) < S . In other ent | S B | 2 = log suffices. The important . ent | | S . ]. By having D D , and does not require any | || 22 M leads to nearly maximal mu- . We have treated a similar D C ' 2 | | D | D < A | | and | ' B and | C | is restored. . For simplicity of discussion, let us A C | D D D and and C ⊗ H C – 41 – C code subspace → H B H : : ]. When thermal correlations are taken into account, only the A i reduced the entanglement entropy of the black hole. Here 0 CD 26 | A U )[ to qubits, but its entanglement entropy is is measured, the postselected black hole dynamics can be interpreted | by including a measurement apparatus B i B C,D 7 n ( || | I A is Haar random. An explicit calculation based on Haar random unitary | into some pure state without introducing the pointer system. If a particular 2 U A . A black hole entangled through a codeword subspace. An infalling observer measures = log , and quantum entanglement between A BH The work by Verlinde and Verlinde did not provide concrete account for the physical In fact, collapsing the input Hilbert space by generic projective measurements on S on i n C Fault-tolerant entanglement The state-independent reconstruction of interiorfrom operators the may perspective be of quantum ofof many-body independent physics quantum interest as entanglement it in reveals chaotic a quantum certain dynamics. universal aspect fine-tuning. origin of the codeword subspace.creates We think situations that inclusion similar of to the theirs. infalling observer effectively tual information off-diagonal correlations need toimplication be of restored, this so taking observationserver is effectively creates that a (essentially) situationsubspace. any where measurement the black This by hole an restores is infalling quantum entangled through ob- entanglement a in codeword situation in section restores quantum entanglement between assume that suggests that projective measurements on A with by words, measurement on viewed as the subspace whereresembles the the black scenario hole considered isquantum by entangled entanglement with Verlinde the and can partner. Verlinde be [ This restored situation between energy eigenstate as an isometry from An alternative interpretation of this dynamics is to think that the black hole is represented Figure 14 | “collapsing” JHEP10(2019)132 | D D ψ O O ih in a . As (C.4) (C.5) (C.1) (C.2) ψ B | C . Here σ B = σ A O is fault-tolerantly . In other words, ) and are entangled. t B ( D σ is a typical pure state D D i O φ | and : C U are entangled only weakly, we α , decay to small values. From 1 (C.3) , but is independent of i D = A . Note that the expectation value i | α  (0) φ ψ out . Hence, if | is a universal feature of the chaotic B A ih ≈ B and ρ I φ B | . | σ O B 1 φ † D ) C = d t ih ⊗ and U ( B on is entangled with some fixed degrees of , let us reconstruct a partner operator = φ σ | in A D i B i | D = U σ and φ D O D i | ψ B h Uρ is an arbitrary quantum state, pure or mixed, σ on D O ih C i B = O – 42 – ψ D D ⊗ | σ O O ⊗ D is nearly maximally entangled with its complement are only weakly correlated, the construction of = out ⊗ D ρ O in D and D O D ρ i h O A O φ , regardless of the choice of the initial state | h ≡ h d were nearly maximally entangled. C is universal even when OTOCs have not decayed to small α and is exponentially suppressed. As such, the structure of en- Z D D D α O inside as initial states. We then typically have and as the initial state. Since . This implies that and . Here our primary interest is in how B 2 B B C D | σ I D D B 1 | on d | ' by using the Hayden-Preskill recovery protocol. Consider the maximally φ = | ih C A B | φ σ | . is a fixed pure state on U = i B ψ | σ . The system evolves under a chaotic unitary dynamics Despite the fact that The above observation assumed that OTOCs have decayed to small values. Next, let In this paper, we argued that the partner operators can be constructed on Let us consider an initial quantum state of the following form: B may be interpreted as a code subspace where entanglement with as long as where the deviationtanglement from between values yet. states where the integral is takensampled uniformly from over all the Haar measure (or an ensemble forming 2-design), then we have where the expectation value iswould taken be with close respect to to 1 if is (almost) state-independent in the following sense. To be specific, let us consider pure the scrambling time. Given ansupported operator on mixed state will have this subspace is determinedsuch, by the the entanglement dynamics structuredynamics between us consider situations where OTOCs have not decayed to smaller values yet, e.g., before way independent of freedom, denoted by D stored in such a way that is protected from any perturbations on Here we make two assumptions.decay First, to we assume small that values. OTOCstime-ordered for correlation Second, functions, we suchthese assume assumptions, as that we the can find systemC that thermalizes, in a sense that where on JHEP10(2019)132 D D into the EPR D D . Here two modes (b) 15b traverse across the black hole A ] which projects 31 (c) – 43 – EPR . The backreaction by the Hayden-Preskill recovery A . This cartoon picture enables us to depict the recovery 15a (a) . (a) The Hayden-Preskill recovery protocol. (b) Backreaction from the Hayden-Preskill Let us focus on the first recovery protocol from [ recovery procedure in accordance withdiagram the convention is of the shown Penrose inprotocol diagram. figure The schematically flipped inon the two Penrose sides diagram areand as reach collapsed in the into figure other EPR boundary pairs. at This lets admits any classical geometricthe description backreacted or geometry. not. Below we present some speculationpair. on To gainthe some eternal insights, two-sided it AdSdownward is black on useful hole. left to Let draw and us the right take process the hand in time sides directions the respectively. to Penrose be diagram One upward of may and flip the diagram of the circumvent this, we maythe think of outside performing so the thatsingularity. Hayden-Preskill Alice To recovery understand protocol can quantum from haverecovery, field we effective must theory quantum understand under field itsare the unsure theory effect effect about on without of what the reaching is the black the Hayden-Preskill the hole corresponding geometric geometry. picture. At It this is moment, even we unclear to us if it We discuss effective quantum fieldcording theory to that the the equivalence infallingchanical principle, observer descriptions we may expect until experience. that she Ac- dressing Alice reaches this will the have question singularity. normal lies The quantum in me- fundamental difficulty the in very ad- fact that Alice never returns to the outside. To Figure 15 recovery. (c) Alice’s quantumare field identified. theory near the horizon. Two surfaces inD the opposite regions Alice’s quantum field theory JHEP10(2019)132 ] A 31 , and A be the which we wish ρ A D 14 15b . If quantum A (1975) 199 . So, even when | 0 43 A || . An analysis from [ Phys. Rev. A | scr , t ' ' 2 | 0 t An Apologia for Firewalls D | . If a signal is sent between ]. . She will enter the shaded 0 scr t t − eventually becomes of order the ≈ SPIRE 0 0 ρ is small compared to the black hole IN t t < 0 ][ Black Holes: Complementarity or ρ ]. Commun. Math. Phys. , with 0 t SPIRE − IN is small. In a qubit count, the sufficient condition would – 44 – ][ = 0 ]. t A and glue two regions together. If the time separation arXiv:1207.3123 SPIRE 0 [ ρ IN ), which permits any use, distribution and reproduction in suffices if are to be transmitted, we only need = as well as the amount of quantum information in | A ρ A D · | ⊂ 0 arXiv:1304.6483 (2013) 062 A ]. [ = 0 on the left and right hand sides respectively. The geometry const becomes of order the scrambling time. In the Penrose diagram, let so that signals entering the diamond can traverse. Precise value of . The recovery protocol prevents Alice from seeing EPR pairs at t 02 (1976) 206] [ 0 CC-BY 4.0 suffices to transmit all the quantum information contained inside ' ρ D | 2 , it will traverse to the other side. These observations suggest that the Breakdown of Predictability in Gravitational Collapse Particle Creation by Black Holes at | is of order the scrambling time, 46 SPIRE 0 15b This article is distributed under the terms of the Creative Commons A | = t D IN | = 0, it will cross the horizon and reach the singularity. If a signal is D D − [ ' JHEP ρ t (2013) 018 | , = D 44 | t . and and 09 A and n A and lets her traverse to the opposite region. This situation can be realized by D 1 2 0 t 0 ' Erratum ibid. ρ − JHEP Firewalls? [ (1976) 2460 Suppose that Alice jumps into a black hole at D A. Almheiri, D. Marolf, J. Polchinski and J. Sully, S.W. Hawking, A. Almheiri, D. Marolf, J. Polchinski, D. Stanford and J. Sully, S.W. Hawking, For simplicity of discussion, consider a deeply scrambling regime with n will depend on the size of ≤ = 44 [2] [3] [4] [1] 0 suggests that information in the subspace is a big subsystem, be References can be simply treatedPlanck as scale. identification of opposite sidesOpen by neglecting Access. physics belowAttribution License the ( any medium, provided the original author(s) and source are credited. between radius, but is still largequantum compared field to the theory Planck is length.separation defined We becomes speculate on larger that this Alice’s thanPlanck effective truncated the length. and scrambling In time, traversable this Rindler regime, space. we If expect the that the effect of the Hayden-Preskill recovery region near the horizondistance from which the is horizon, approximatedat Alice by the sees the the surface Rindler backreactionρ from space. the Hayden-Preskill Letting recovery cutting the Rindler space at sent before portion inside the diamond effectively disappears,as and shown antipodal in points figure should bet identified to transmit. the outgoing mode us take will be significantly modifiedis only constructed inside by the consideringt modes shaded at diamond shown in figure becomes significant when the separation of the time between the infalling observer JHEP10(2019)132 ] ]. ]. , 11 ] ]. 88 , , 61 ]. , ] SPIRE SPIRE SPIRE JHEP IN IN [ gravity Phys. 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