Observer-Dependent Black Hole Interior from Operator Collision

PHYSICAL REVIEW D 103, 046004 (2021)

Observer-dependent black hole interior from operator collision

Beni Yoshida Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

(Received 6 April 2020; accepted 8 January 2021; published 5 February 2021)

We present concrete construction of interior operators for a black hole which is perturbed by an infalling observer. The construction is independent from the initial states of the black hole while dependent only on the quantum state of the infalling observer. The construction has a natural interpretation from the perspective of the boundary operator’s growth, resulting from the collision between operators accounting for the infalling and outgoing modes. The interior partner modes are created once the infalling observer measures the outgoing mode, suggesting that the black hole interior is observer dependent. Implications of our results on various conceptual puzzles, including the firewall puzzle, the Marolf-Wall puzzle, and the information problem, are discussed. Explicit counterarguments against the ER ¼ EPR approach by Maldacena and Susskind are also presented.

DOI: 10.1103/PhysRevD.103.046004

I. INTRODUCTION The key challenge is to understand the experience of the In the last decade, our understanding of the black hole infalling observer by examining the validity of semiclass- interior has significantly advanced by using ideas from ical bulk descriptions properly. What distinguishes our quantum information theory. It has led to a conjecture that approach from many of previous ones is to include the bulk operators within the entanglement wedge enclosed by infalling observer explicitly as a part of the quantum system the extremal surface of minimal area can be reconstructed (Fig. 1). Adding an infalling observer has a rather dramatic on boundary degrees of freedom in the AdS=CFT corre- effect which disentangles the outgoing mode from the early spondence [1–4]. The attempt to find concrete expressions radiation completely. This conclusion follows from a of bulk operators, however, has been only partially suc- certain quantum information theoretic relation, referred cessful. While bulk operators in the smaller causal wedge to as the decoupling theorem. It asserts separability of two were explicitly constructed [5], those outside the causal subsystems due to scrambling dynamics. Hence, the wedge remain elusive. And quite unfortunately black hole preexisting entanglement plays no role in describing the ’ interior operators are the prime example of those behind the infalling observer s experience. This is of course a counterargument against the “ER ¼ EPR approaches” to the fire- causal horizon. See a review [6] for a more complete list of 1 references on this subject. wall puzzle [9]. Recently, we pointed out that black hole interior operators The most salient feature of our construction is that it does can be constructed in boundary degrees of freedom by a not involve any degrees of freedom from the early radiation procedure similar to the Hayden-Preskill thought experiment and hence is independent of initial states. This is a direct that runs backward in time [7,8]. We envision that this consequence of the aforementioned decoupling theorem. It observation is crucial in developing a generic prescription for is however important to recognize that the construction reconstructing bulk operators behind the causal wedge and still depends on how the infalling observer is introduced will provide important insights on the physics of the black to the system and hence is observer dependent. Observer hole interior. The goal of the present paper is to write down dependence of the black hole interior is reflected in the fact concrete expressions of interior operators by explicitly implementing the ideas outlined in these works. We also 1The ER ¼ EPR slogan refers to the idea of identifying discuss their physical interpretations and implications to quantum entanglement as spacetime. Here, ER refers to the various conceptual puzzles concerning black hole interiors. Einstein-Rosen bridge and EPR refers to the Einstein-Podolsky- Rosen pair. On the other hand, the ER ¼ EPR approach does not refer to the idea of identifying quantum entanglement as a wormhole. It refers to a specific proposal for resolution of the Published by the American Physical Society under the terms of firewall puzzle by Maldacena and Susskind speculated in [9] as the Creative Commons Attribution 4.0 International license. well as a similar resolution due to Raamsdonk [10]. In this paper, Further distribution of this work must maintain attribution to we will essentially argue that the ER ¼ EPR approach is the author(s) and the published article’s title, journal citation, incorrect. On the other hand, there is nothing incorrect about and DOI. Funded by SCOAP3. the idea of identifying quantum entanglement as a wormhole.

2470-0010=2021=103(4)=046004(24) 046004-1 Published by the American Physical Society BENI YOSHIDA PHYS. REV. D 103, 046004 (2021)

outgoing mode This paper is organized as follows. In Sec. II, we present the problem setting and the decoupling theorem. In Sec. III, we obtain concrete expressions of interior partner operators. early radiation In Sec. IV, we relate the construction to the operator growth. black hole In Sec. V, we discuss observer dependence. In Sec. VI,we 1 discuss implications of our construction to conceptual puzzles. In Sec. VII, we provide counterarguments against infalling observer some of previous approaches. In Sec. VIII, we discuss the cases where the black hole does not satisfy underlying assumptions of the decoupling theorem. In Appendix A, we present the definition of scrambling and summarize diagrammatic tensor notations. In Appendix B, we present a quantitative version of the decoupling theorem in terms of L1 FIG. 1. A cartoon picture of the problem setting. A black hole is the distance. In Appendix C, we discuss the thought perturbed by an infalling observer. Our goal is to construct the experiment by AMPS. In Appendix D, we obtain expressions interior partner operator of the outgoing mode. of interior partner operators at finite temperature by imposing one technical assumption. In Appendix E, a certain calculation of OTOCs is presented. In Appendix F, we justify the that expressions of interior modes are not unique, a finite temperature assumption. manifestation of its quantum error-correction property.2 We will see that observer dependence provides resolutions and useful insights on various conceptual puzzles concern- II. BLACK HOLE DECOUPLING THEOREM ing black hole interior, including the firewall puzzle, the In this section, we present the problem setup which Page curve, and the Marolf-Wall puzzle (concerning explicitly includes the infalling observer as a part of the whether two observers can meet inside a two-sided black dynamics of the black hole. We then provide a precise hole or not). definition of interior partner operators in our setup. Finally, Our construction bears some similarity with a pioneering we present the black hole decoupling theorem which sug- work by Verlinde and Verlinde, written just 4 months after gests separability (decoupling) of the outgoing Hawking the AMPS (Almheiri-Marolf-Polchinski-Sully) [12] paper, mode and the early radiation when an infalling observer is which utilized the idea of quantum error correction to present. A summary of our notation is given in Fig. 2 for construct interior operators [13]. The new ingredient in our convenience. work is to relate reconstructability to the concept of The main result of this section is a mathematically scrambling as diagnosed by out-of-time order correlators rigorous proof that the interior partner operator can be (OTOCs). Indeed, the construction is closely related to the constructed in a state-independent manner when the black boundary operator’s growth in the AdS=CFT correspon- hole’s dynamics is scrambling. An explicit construction dence [14]. We will see that the interior partner of the will be presented in the next section. While we will phrase outgoing operator W can be constructed by time evolving the black hole decoupling theorem by using the black hole the infalling operator V and looking at the overlap with W. physics language, the theorem itself is a rigorous quantum This suggests that the interior modes are created once the information theoretic statement applicable to generic outgoing modes are measured by the infalling observer, scrambling systems. Our strategy in this paper is to care- highlighting its observer dependence. fully deduce the bulk semiclassical description (and its Throughout the main text of the paper, we will use the possible breakdown) by relying on predictions from infinite temperature approximation which treats a quantum n S n black hole as an -qubit quantum state where BH ¼ is the Bekenstein-Hawking coarse-grained entropy. Generaliza- time tion to finite temperature systems is discussed in the : infalling observer Appendix D. Unitarity and scrambling dynamics are : reference assumed in this paper. We assume tensor factorization into : initial black hole subsystems for the Hilbert space, but we believe that this is : early radiation not a crucial limitation of our argument. : remaining black hole BH : outgoing mode 2It is perfectly fine to construct interior operators in a state- dependent manner by using the early radiation as in [11], for instance. However, the decoupling theorem suggests that such construction is not relevant to the experience of the infalling FIG. 2. A summary of the notation. Horizontal lines with dots observer when the backreaction is taken into account. represent EPR pairs while K is an arbitrary unitary.

046004-2 OBSERVER-DEPENDENT BLACK HOLE INTERIOR FROM … PHYS. REV. D 103, 046004 (2021) rigorous quantum information theoretic results on boun- Hence, these “qubits” on B and B¯ should be thought of as dary quantum mechanical systems. coarse-grained degrees of freedom.3 Imagine that an infalling observer (or some measurement A. Black hole with infalling observer apparatus) A is dropped toward the black hole at time t ¼ 0 We will be primarily interested in a black hole which has (Fig. 2). The ultimate goal of this research program is to emitted more than a half of its entropy content. Such an old probe the degrees of freedom behind the black hole black hole is maximally entangled with the early radiation horizon. A much more modest, yet still important, goal that has been already emitted. Here we simplify the here is to examine the effect of including A by relying on problem by expressing the black hole as a discrete system rigorous quantum information theoretic concepts in an S d S unambiguous manner. Depending on initial states of the of BH ¼ log2 qubits where BH is the Bekenstein- Hawking coarse-grained entropy. infalling observer A, the black hole will respond in different Then, the initial state of the old black hole can be chosen ways. Rather than keeping track of the outcomes for all the as a generic maximally entangled state between the black possible input states on A, it is convenient to append a ¯ ¯ hole B and the early radiation B. Recall that a maximally reference system A which is entangledP with the probe A and 1ffiffiffiffi entangled state can be represented as forming EPR pairs jEPRiAA¯ ¼ p j jjiA ⊗ jjiA¯ . Being dA ¯ d entangled with A, the reference A effectively keeps track of 1 XB A A¯ ðI ⊗ KÞjEPRiBB¯ jEPRiBB¯ ≡ pffiffiffiffiffiffi jjiB ⊗ jjiB¯ ; ð1Þ different input quantum states on ;if is projected onto d B j¼1 jψ i (a complex conjugate of jψi), A is set to jψi. The use of entangled reference systems is a standard technique in where K is an arbitrary unitary operator acting on the early quantum information and communication theory, and often radiation B¯ . In general, the Hilbert space for the radiation B¯ used in black hole physics, dating back to the work by is larger than B (and is possibly unbounded). Here we think Hayden and Preskill [15]. S of focusing on a 2 BH -dimensional subspace of the radiation After the time evolution by scrambling unitary dynamics Hilbert space which is entangled with the black hole B. U ¼ UðtÞ, the system evolves to

ð2Þ

where D is the outgoing mode and C is the remaining black asymptotic anti–de Sitter (AdS) boundary (or, somewhere pffiffiffiffiffiffi hole. Here black dots represent factor of 1= dR in a sufficiently away from the black hole horizon). We assume subsystem R for proper normalization of jEPRiRR¯ .A that the observer A simply falls into the black hole and does precise definition of scrambling unitary U is presented not perform any particular measurement during her fall; she in Eq. (A2) in Appendix A. The unitary operator U is just attempts to cross the black hole horizon via free fall. defined from AB to CD and we assume dAdB ¼ dCdD.We After time t (which is longer than the scrambling time, as also assumed that the black hole Hilbert space factorizes measured in the boundary clock), the outgoing mode D into HA ⊗ HB ≃ HC ⊗ HD for simplicity of discussion. reaches the AdS boundary while, in the intermediate time, This is not an essential restriction of our treatment as the A and D would have encountered with each other near the decoupling phenomena occur for systems with nonfactor- black hole horizon [15].EvenifA and D do not interact izable Hilbert space too; see [7] for instance. directly with each other in their local frames, their The physics we hope to capture is as follows. An encounter necessarily introduces an indirect interaction infalling observer A jumps into the black hole from an mediated by gravitational backreaction. Namely, no matter how small the infalling observer A is, its effect is t 0 3 amplified due to the blueshift (relative to the ¼ slice) In the end, we will prove the decoupling from the early t radiation B¯ , so the finiteness of B¯ is not an essential assumption. and creates a gravitational shockwave when becomes A more crucial assumption is the finiteness of the black hole order of the scrambling time [16]. Our goal is to examine Hilbert space B. the effect of such gravitational interactions on the

046004-3 BENI YOSHIDA PHYS. REV. D 103, 046004 (2021) entanglement structure of the black hole. It is worth operators ΩðODÞ, supported on CAB, are defined to be emphasizing that what we find in this section must be a those which satisfy universal property of gravitational systems with a black hole horizon. O ⊗ I Ψ t ≈ I ⊗ Ω O Ψ t ; Now we are interested in constructing interior operators ð D CABÞj ð Þi ð D ð DÞCABÞj ð Þi ð3Þ for this perturbed black hole state jΨðtÞi. For an arbitrary operator OD acting on the outgoing mode, interior partner or graphically

ð4Þ

Here “≈” in Eq. (3) is evaluated by the overlap (fidelity) B. Decoupling theorem between the lhs and rhs. It cannot be emphasized enough, however, that con- Ω O The existence of a partner operator ð DÞ is guaranteed struction of interior partners is not unique [1].4 Here we since the mutual information IðD; CABÞ is maximal, i.e., D hope to find another expression of an interior partner is maximally entangled with CAB. One possible form of operator which is independent of the initial state ðI ⊗ partner operators can be immediately constructed on AB by KÞjEPRiBB¯ of the black hole. evolving OD backward in time and using the following The following (informally stated) decoupling theorem, relation: proven in a different context [17,18], will be useful. A more precise version is presented in Appendix B. Theorem 1: (Informal) Suppose that the system is O ⊗ I I ⊗ OT ; ð ÞjEPRi¼ð ÞjEPRi ð5Þ scrambling in a sense of Eq. (A2) and dA ⪆ dD. Then, the subsystems D and B¯ are decoupled (not entangled), or graphically ρDB¯ ≈ ρD ⊗ ρB¯ ; ð8Þ

2 dD where the error is Oð 2 Þ. Furthermore, for an arbitrary dA operator OD on the outgoing mode D, a partner operator ð6Þ ¯ ΩðODÞCA¯ can be constructed on CA without using any degrees of freedom from the early radiation B¯ . We present a brief review of the definition of scrambling in Appendix A. Loosely speaking, it says that OTOCs where OT represents the transpose of O. Namely, we will decay to small values. This is a universal property of find the following expression: quantum black holes whose background geometries have a well-defined event horizon [16]. I ⊗ K O t T I ⊗ K† : ð A¯ B¯ Þ ð ÞAB ð A¯ B¯ Þ ð7Þ 4One may interpret the interior partner operators as logical operators of a quantum error-correcting code where quantum It is worth noting that this construction is state dependent as information on D is encoded into CAB. It is then a well-known it depends on the initial state ðI ⊗ KÞjEPRiBB¯ through its fact that logical operators have multiple equivalent expressions dependence on K. Such state-dependent constructions have when their actions are restricted to a codeword subspace. The aforementioned state-dependent expression in Eq. (7) is just one been discussed in a number of previous works; see [11] for possible expression of an interior operator among many others. instance. See Sec. V for further discussions.

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The theorem suggests a rather surprising fact; an that interior partner operators must be state dependent. This infalling observer’s attempt to see the black hole interior folklore knowledge is no longer valid once the perturbation will always disentangle the outgoing mode D from the from an infalling observer A is taken into consideration.5 early radiation B¯ due to the scrambling dynamics of the The theorem also suggests that, due to the monogamy of black hole. Note that, in the absence of A, the outgoing quantum entanglement, the outgoing mode D is (almost) mode D would be entangled with the early radiation B¯ , and maximally entangled with the remaining black hole C and the hence the partner operator would be state dependent. reference A¯ , which implies the existence of a partner operator Indeed, a large body of previous works ([9,11] for supported on CA¯ . Here it is useful to represent the partner instance), without considering the effect of including an operator from theorem 1 graphically as follows: infalling observer A, have reached to a naive conclusion

ð9Þ

Such construction ΩðODÞ is state independent as it does not independent. In fact, the construction works not only for depend on the unitary K. In the next section, wewill construct an old (maximally entangled) black hole, but also for a young the interior operator ΩðODÞ explicitly. Namely, we will find (pure state) black hole since the expression does not involve that the construction is state independent, but is observer any degrees of freedom from the early radiation B¯ at all. dependent. Our construction of interior operators utilizes a certain It is worth emphasizing again that the disentangling quantum operation which distills EPR pairs between D and phenomena occur universally due to gravitational back- CA¯ as this enables us to identify which degrees of freedom reaction, no matter how the infalling observer A jumps into D is entangled with. The entanglement distillation pro- a black hole. As long as the infalling object carries a small but cedure is inspired by a probabilistic recovery protocol for finite positive energy (including its rest mass), it eventually the Hayden-Preskill thought experiment [18]. The key introduces significant gravitational backreaction, disentan- observation is that the interior reconstruction problem gles the outgoing radiation D from the early radiation B¯ , and can be interpreted as the information retrieval (the enables the state-independent interior operator construction. Hayden-Preskill or the information loss) problem running 6 Our conclusion relies only on a widely accepted hypothesis backward in time [7]. This is the reason why recovery that a black hole scrambles quantum information unitarily. protocols for the Hayden-Preskill thought experiment can 7 This is of course a counterargument against the so-called be utilized in reconstructing interior partner operators. A ER ¼ EPR approach [9] to the firewall puzzle by deterministic version of the recovery protocol from [18] Maldacena and Susskind. Their central idea builds on works equally well for this purpose, but we will focus on the presence of preexisting entanglement (or a wormhole) the probabilistic one for simplicity. between the outgoing mode D and the early radiation B¯ , which is negated explicitly by the decoupling theorem. We will return to this issue in Sec. VII. 6Imagine that we reverse the flow of the time so that the outgoing mode D “returns” back to the black hole at time t and the system evolves backward to t ¼ 0. Then, constructing the partner mode of III. STATE-INDEPENDENT INTERIOR D is equivalent to recovering the quantum information on D that OPERATOR has been thrown into a black hole in the reverse time flow. 7It is worth emphasizing that the interior reconstruction and the In this section, we present an explicit construction of Hayden-Preskill thought experiment are different problems. interior partner operators. The construction is supported Namely, the interior reconstruction works for both old and young exclusively on the remaining black hole C and the reference black holes, whereas the Hayden-Preskill recovery works only for system A¯ of an infalling observer, and hence is state old black holes. A common mistake, frequently found in the literature, is to view the information loss problem and the firewall problem as the same thing. They are related by time reversal, but are 5A few infalling quanta is enough to invalidate this belief. different problems.

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A. Entanglement distillation The outgoing mode D is (nearly maximally) entangled with CA¯ , so it should be possible to distill EPR pairs between D and CA¯ by applying some quantum operation on CA¯ . The goal of this subsection is to present the entanglement distillation procedure. Given the time-evolved state jΨðtÞi in Eq. (2), let us construct the following state:

ð10Þ

0 ¯ 0 ¯ Here ancillary subsystems D D are introduced, qubits on D are moved to D , jEPRiDD¯ is prepared on DD, and the time- reversal U† is applied on DC. Some readers may find the time-reversal U† unphysical in a real world, but this is just a mathematical trick to obtain the interior construction. So far, all the operations are deterministic. We now apply a projection onto EPR pairs on AA¯ . The unnormalized outcome is

ð11Þ

ΠðEPRÞ where the projector is denoted by AA¯ . The probability amplitude for this event is

ð12Þ

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A line connecting two B’s on the top and bottom of the systems in Eq. (A2) in Appendix A. After some calculation, diagram is omitted for ease of drawing. Finally, the normal- we obtain ized outcome after the postselection is given by 1 d2 1 Δ 1 O D d ⪆ d : ˜ ðEPRÞ ¼ 2 þ 2 ð A DÞ ð15Þ jΦ i ≡ pffiffiffiffi Π ¯ jΦ i: ð13Þ d d back Δ AA back D A

Some readers may have already noticed a relation We now show that EPR pairs have been already distilled Δ 0 ¯ ðEPRÞ between and OTOCs by staring at the diagrammatic on D D after the postselection by Π ¯ . The probability of Δ U’ U†’ AA representation of where s and s appear twice. The observing EPR pairs on D0D¯ is given by relation between OTOCs and Δ can be explicitly seen by the following identity [17]: 1 ZZ Φ˜ ΠðEPRÞ Φ˜ Φ ΠðEPRÞΠðEPRÞ Φ h backj D0D¯ j backi¼Δ h backj AA¯ D0D¯ j backi Δ O 0 O t O† 0 O† t dO dO ; ¼ h Að Þ Dð Þ Að Þ Dð Þi A D ð14Þ 1 1 ¼ Δ d2 O O D where the integrals over A and D represent uniform ≈ 1: averaging over all the basis operators on A and D, ð16Þ respectively. Hence, the amplitude Δ can be evaluated by using the asymptotic form of OTOCs for scrambling In the second step, we have used the following identity:

ð17Þ

B. Construction of interior operator Ω O To construct interior partner operators ð DÞ for the ð19Þ perturbed black hole state jΨðtÞi, we need to “undo” the † projection ΠAA¯ and the time-reversal U . Partner operators are explicitly given by

1 1 ΩðODÞ ≡ Tr½ðOD ⊗ ICA¯ ÞðUBA ⊗ IA¯ Þ Δ dD ðEPRÞ † Most readers will not find this expression particularly × IB ⊗ Π U ⊗ I ¯ ; 18 ð AA¯ Þð BA AÞ ð Þ illuminating. In the next section, we will further simplify the expression and discuss its relation to the boundary or graphically operator’s growth.

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It can be verified that these are appropriate partner operators by evaluating the overlap between ðOD ⊗ IÞjΨðtÞi and ðI ⊗ ΩðODÞÞjΨðtÞi,

ð20Þ

where a part enclosed by a dotted square corresponds to ΩðODÞ. Taking average over basis operators on D,wehave

Z † dODhΨðtÞjðOD ⊗ ΩðODÞÞjΨðtÞi 1 1 1 Φ ΠðEPRÞΠðEPRÞ Φ ≈ 1; ¼ h backj AA¯ D0D¯ j backi¼ 2 ð21Þ Δ Δ dD

R ð22Þ where we used dOO ⊗ O ¼jEPRihEPRj. Hence, † 0 hΨðtÞjðOD ⊗ ΩðODÞÞjΨðtÞi ≈ 1 for each OD.Itisworth The probability amplitude for the projection onto j i can be emphasizing that our construction has a concrete opera- computed from the OTOC asymptotics, tional meaning: It is constructed by distilling EPR pairs between the outgoing mode D and the remaining black hole C plus the reference A¯ . Indeed, the average overlap in Eq. (21) is equal to the EPR distillation probability in Eq. (16).

C. Pure state observer Some readers may find the use of a reference system A¯ a bit uncomfortable. Instead of introducing a reference system A¯ entangled with A, let us consider a scenario where a measurement probe A,preparedin a particular pure state (e.g., j0i), is dropped toward the black hole. The procedure to distill EPR pairs and construct interior operators proceeds in an analogous manner except that this time we need to project A onto j0i, ð23Þ

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2 Note that the reconstruction criteria become dA ⪆ dD previous works, including the ER ¼ EPR approach by instead of dA ⪆ dD. In the qubit count, A needs to contain Maldacena and Susskind, then speculated that interior more than twice qubits on D. operators must be state dependent in order to avoid the The probability of observing EPR pairs on D0D¯ is firewall. Here we have avoided the no-go result by Marolf 1 Φ Πj0ih0jΠ Φ 1 1 ≈1 and Polchinski by considering the effect of an infalling Δh backj A D0D¯ j backi¼Δd2 . Interior partner oper- D observer and explicitly demonstrated that state-independent ators are given by interior operators can be written down. It is again worth emphasizing that our conclusion relies only on a widely accepted hypothesis that a black hole scrambles quantum information unitarily. In later sections, we will see that the possibility of state- independent interior construction not only reveals funda- ð24Þ mental flaws in the ER ¼ EPR approach to the firewall puzzle, but also suggests a rather different scenario for resolving the firewall puzzle.8

E. Seeing the interior We conclude this section by presenting a possible bulk quantum gravity interpretation of the entanglement distillation. Imagine that the entanglement distillation protocol is D. Young black hole interior actually implemented on the boundary quantum mechani- So far, we have focused on an old black hole that is cal system. The fact that an EPR pair is distilled between D˜ ¯ maximally entangled with the early radiation B. Rather and the outgoing mode D suggests that the partner mode D˜ surprisingly (but in some sense quite naturally), our has been extracted as some simple degrees of freedom on construction remains valid for a young black hole which the boundary. A possible physical interpretation is that the has not yet emitted the Hawking radiation and can be interior partner mode D˜ is brought to the AdS boundary represented as a pure state. from the black hole interior by the entanglement distillation This is simply because the expression of interior operator protocol. Also, the fact that the EPR entanglement between ¯ does not involve the early radiation B at all. To be explicit, ¯ ðEPRÞ A and A has been restored via the EPR projection Π ¯ let us consider the initial state of the following form: AA suggests that the infalling observer A has returned to the exterior from her journey to the black hole interior. As such, the infalling observer A, who crossed the black hole horizon, may return to the black hole exterior with the interior partner mode D˜ in her hand by implementing the ð25Þ entanglement distillation.

IV. OPERATOR GROWTH INTERPRETATION In this section, we will relate the aforementioned construction to the operator growth interpretation of scram- ψ bling [14]. The operator growth picture relies crucially on where j BHi represents the pure state of a young black hole the finiteness of the Hilbert space. It is an interesting future on B. Then, in evaluating Eq. (20), we simply insert 1 problem to generalize our construction to quantum field jψ ihψ j, instead of the maximally mixed state IB, BH BH dB theories. See [21] for discussions on oscillator systems. to the line corresponding to B. This replacement does not affect the outcome of Eq. (21), showing that the same A. Truncation by local operator construction ΩðODÞ works as an interior operator for a Scrambling phenomena can be associated with the young black hole as well. Hence, we conclude that our boundary operator’s growth. Given a single qubit operator construction of interior partner operator is state independent and works for both young and old black holes. Our construction provides a possible resolution to a 8There are two versions of the firewall puzzle: (i) for an old version of the firewall puzzle for black hole microstates due black hole and (ii) for generic black hole pure states [19,20].Itis rather strange to try to resolve the puzzle by treating black holes to Marolf and Polchinski [19]. They argued that, if an with preexisting entanglement [the case (i)] in a special manner as interior operator is state independent, then there must exist in the ER ¼ EPR approach. Our approach resolves the puzzle for a firewall for typical black hole microstates. Much of both cases on an equal footing as we will see in Sec. VI.

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V n V in a system of qubits, consider its time evolution by the Note that truncated operator =WD acts nontrivially only on −iHt iHt Hamiltonian VðtÞ¼e Ve , C. In the Choi representation jOi ≡ ðO ⊗ IÞjEPRi, the truncation can be defined by jV=W i ≡ jIDihWDjVi. t2 D V t V − iHt H; V − H; H; V To gain someP intuition, let us expand V in terms of Pauli ð Þ¼ ½ 2! ½ ½ V α P D operators: ¼ P∈Pauli P . Assume that consists of 3 it just a single qubit and choose WD to be a Pauli operator þ ½H; ½H; ½H; V þ : ð26Þ 3! fID;XD;YD;ZDg. Define four sets of Pauli operators on H, depending on the Pauli operator supported on D, Commutators ½H; ½H; ½H; V generically increase the Pauli PD ⊗ Pauli Operators onC number of supports and VðtÞ becomes a high weight jPD ¼f g t V t operator for large . The growth of ð Þ can be probed PD ¼ ID;XD;YD;ZD: ð28Þ by OTOCs. When W is some other single qubit operator on V a distant qubit and VðtÞ has nontrivial support on that qubit, The truncation =WD is constructed by picking up terms hVðtÞWð0ÞVðtÞWð0Þi decays from its initial value. with Pauli operators containing WD on D, The operator growth can be related to interior X V α P: reconstruction by considering truncation of operators. =WD ¼ P ð29Þ P∈ For an arbitrary operator V on a Hilbert space PaulijWD H ¼ HC ⊗ HD, its truncation by WD is defined by In other words, the truncation gives rise to nontrivial operators when the original operator V and the target operator WD “collide” with each other.

B. Interior operator from collision The notion of truncation simplifies the expression of interior operators. Using Z dO O ⊗ O ; A A A¯ ¼jEPRihEPRjAA¯ ð30Þ

the interior partner operator in Eq. (19) can be expressed ð27Þ concisely as

ð31Þ

1 † T A O −t † ⊗ O with the integral over basis operators on . The reason why a partner operator Δ Að Þ=O A¯ without taking O −t D the backward evolution Að Þ appears can be understood averages since statistical variances are suppressed in a O† t O† 0 O t O 0 by rewriting OTOCs as h Dð Þ Að Þ Dð Þ Að Þi ¼ scrambling system. Then the relation between the operator † † hODð0ÞOAð−tÞODð0ÞOAð−tÞi which evaluates the overlap growth and interior operator construction becomes between OAð−tÞ and ODð0Þ. transparent; when OAð−tÞ collides with the target Some readers may feel uncomfortable about taking an outgoing operator OD, interior partner operators can be average over OA. It turns out that we can safely ignore constructed. averaging in most cases. When A is sufficiently large, we For the cases with a pure state input j0i on A, we obtain may pick one Pauli operator OA at random to construct an even simpler expression

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probe target dependent. This suggests that the black hole interior should be considered as observer dependent.9 In fact, we have already seen observer-dependent nature of the black hole interior rather vividly in the operator growth interpretation. We found that the interior operator can be constructed by examining the collision between the outgoing mode W and FIG. 3. The partner operator as the upper half of the ladder the infalling mode Vð−tÞ. The implication is that the diagram (q ¼ 4 SYK). interior mode was created dynamically once the infalling observer measures the outgoing mode. This interpretation 1 further hints observer dependence of the black hole interior. ΩðODÞ¼ OAð−tÞ=O† OA ¼ IB ⊗ j0ih0jA: ð32Þ We may formalize the notion of observer dependence by Δ D borrowing a quantum information theoretic language. Let The physical interpretation is clearer in this case. Recall us consider a maximally entangled black hole and recall that j0i was the initial state of the infalling observer and OA how the interior modes are realized for the outside and is its projection. Hence, the interior partner operators are infalling observers, respectively. In the absence of the constructed due to the gravitational collision between the infalling observer, the entangled partner of the outgoing infalling observer and the target outgoing OD. This mode can be found as some degrees of freedom in the early suggests that the interior modes are created once the radiation. This is what the outside observer perceives as the outgoing modes are measured by the infalling observer. interior mode. Yet, from the perspective of the infalling observer, this thing in the early radiation is not the interior C. On SYK model mode anymore since it has been already decoupled from the outgoing mode. The infalling observer will perceive the Finally, we make a brief comment on interior operators mode which we have identified in previous sections as the for the SYK model. Since the SYK model exhibits true interior partner mode. scrambling dynamics, our prescription can be applied to As is evident from this argument, observer dependence obtain the entangled partner. It has a simple Feynman of the black hole interior is closely related to the fact that diagrammatic interpretation modulo one technical subtlety. expressions of interior partner operators are not unique. ψ 0 ψ t ψ 0 ψ t Consider the OTOC h 1ð Þ Nð Þ 1ð Þ Nð Þi where This property can be better understood by using an analogy ψ ; ψ 1 N are interpreted as infalling and outgoing modes, with quantum error-correcting codes. In the language of respectively. Here we would like to construct a partner of quantum error-correcting code, partner operators can be ψ N. The dominant contribution to OTOCs comes from the interpreted as logical operators where quantum information ladder diagram; see Fig. 3.In[22], it has been pointed out on the outgoing mode D is encoded into CAB. Consider the that the ladder diagram characterizes the collision between Γ∶H → H following embedding map D code defined by ψ 1ð−tÞ and ψ Nð0Þ. In this interpretation, the upper half of the ladder diagram can be viewed as the dominant con- ψ −t ψ Γ tribution in the growth of 1ð Þ. Hence, by truncating N jψi ∈ HD ! ΓðjψiÞ from the upper half of the ladder diagram, we obtain the ψ Ψ t ∈ H ⊆ H ; interior operator (Fig. 3). ¼h jDj ð ÞiDCAB code CAB ð33Þ It should be however noted that the above diagrammatic interpretation has a problem. The ladder diagram accounts where we omitted the normalization factor for simplicity. for the dominant piece in the OTOC decay until the For readers familiar with tensor diagrammatic notations, it scrambling time. Around the scrambling time, the sub- is convenient to represent the embedding map graphically leading terms become important. Since the fidelity of our as follows by rotating the original diagram for jΨðtÞi reconstruction becomes better when OTOCs have decayed 180 degree: to small values, the diagrammatic expression in Fig. 3 is a good description of interior partner operator only for some 9Observer dependence of a black hole itself might sound like time window. an old and standard idea from the perspective of its thermody- namic property. For instance, the no hair theorem tells that black V. OBSERVER-DEPENDENT INTERIOR holes in thermal equilibrium are characterized by a few macro- scopic observables. This suggests that whatever happened in the We have seen that the black hole interior operator can be past will not affect the present observer as far as outgoing modes constructed in a way independent of the black hole’s initial are concerned. A typical time scale for this is order of the thermal state due to the infalling observer’s backreaction. Yet, it is time. What we advocate here is distinct from this conventional wisdom based on the thermodynamics of black holes. We argue important to recognize that the construction is dependent that the black hole interior, which is understood as entangled on how the infalling observer A is introduced to the system. partners of outgoing modes, must be observer dependent too. A Hence the construction is state independent but is observer typical time scale for this is order of the scrambling time.

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between the firewall problem and the information loss problem since two problems are often (wrongly) treated as the same issue in the community.

ð34Þ A. Firewall puzzle Let us briefly recall the firewall argument [12,24,25] (Fig. 4). Given an old black hole B which is maximally entangled with the early radiation R, let us consider an outgoing mode D which was just emitted from the black hole. Since the black hole is maximally entangled, the Here partner logical operators are those which may trans- outgoing mode D must be entangled with some degrees of H R form wave functions in the code subspace code in a freedom in . Then the outside observer, Bob, may distill a nontrivially manner. In the quantum error-correction inter- qubit RD from R that is entangled with the outgoing mode D pretation, logical operators on AB is what the outside and hand it to the infalling observer Alice who is going to observer perceives as the entangled partner whereas those jump into a black hole. After crossing the smooth horizon, D˜ on CA¯ is what the infalling observer will measure when Alice will see an interior mode which is entangled with D approaching the horizon. the outgoing mode . This, however, leads to a contra- D R We have argued that quantum error-correcting nature of diction because is also entangled with D. If we believe ˜ interior reconstruction gives rise to observer-dependence. what Bob says, D has to be entangled with RD,soD and D At this point, it is interesting to recall that the holographic are not entangled, leading to possible existence of some mapping from bulk operators to boundary operators in the high energy density at the horizon. AdS=CFT correspondence can be understood as a quantum The aforementioned reconstruction of interior partner error-correcting code. The canonical example of holo- operators naturally guides us to a resolution of the firewall graphic quantum error correction is the empty AdS space puzzle [8]. The resolution is immediate from a physical where the bulk operator at the center can be reconstructed interpretation of the aforementioned reconstruction of the on any interval which is longer than the half of the system interior partner operators. When Alice falls into a black D via the AdS/Rindler reconstruction. Our interpretation is hole, the outgoing mode is disentangled from the R that each boundary expression corresponds to what the distilled qubit D due to the backreaction which makes D boundary observer, who has an access to the given OTOCs decay. In fact, the decoupling theorem 1 tells that boundary interval, perceives as the bulk operator. is decoupled from any degrees of freedom in the early R We conclude this section by recalling a certain argument radiation . Hence, the monogamy of quantum entanglement is not violated. Alice will be able to observe the by Hayden and Preskill [15]. Let us consider two observers ˜ who jump into the one-sided black hole and try to meet interior mode D which is distinct from the original partner inside the black hole. If the time separation Δt between mode RD. them is longer than the scrambling time, an observer needs It is worth discussing how this proposal resolves the large amount of energy, larger than the Planck energy, to puzzle in the form of the AMPS thought experiment. In the send a signal to the other observer. Hence, two observers AMPS thought experiment, the outside observer hands will not be able to meet. This was the crucial observation by distilled qubits RD to the infalling observer who attempts to Hayden and Preskill which led to the lower bound on the verify the violation of the monogamy of entanglement. So, scrambling time. Our interpretation of their argument is as follows. When two observers are separated by more than black hole the scrambling time, the black hole interiors they observe early radiation are unrelated. They can enjoy their own interiors without interior mode being perturbed by others. See [23] for further discussions. 1

VI. ON BLACK HOLE PUZZLES In this section, we shall discuss implications of state independence and observer dependence on several important conceptual puzzles concerning the black hole interior, such as the firewall puzzle [12,20,24,25] and the Marolf- outgoing mode Wall puzzle [26]. Namely, we will propose novel resolutions of these puzzles by relying on the lessons we have FIG. 4. Summary of the firewall argument. The outgoing mode learned from the interior reconstruction. We also make D cannot be entangled with the early radiation R and the interior some comments on the relevance (and the irrelevance) mode D˜ simultaneously.

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(a) (b)

FIG. 5. (a) The bulk interpretation of the proposed resolution of the firewall puzzle. The infalling observer A is treated as a gravitational shockwave. (b) A schematic picture of previous proposals. This is not consistent with unitarity or scrambling of the black hole dynamics.

the infalling observer should carry more qubits than RD energy. The bulk interpretation is straightforward from a when she crosses the black hole horizon. This effectively reasoning similar to Fig. 5(a) where an infalling observer A creates a situation with dA ≥ dD. Then, from the decou- creates her own interior geometry by her backreaction. pling theorem, one can easily show that D is no longer Below is a side comment. Scrambling phenomena have entangled with RD due to scrambling dynamics. See to do with entanglement dynamics which can be measured Appendix C for more details. either by preparing two entangled copies of the same The bulk interpretation of the aforementioned resolution system or by evolving the system forward and backward can be obtained by treating the backreaction from Alice as a in time. It is quite interesting that the firewall problem can gravitational shockwave [Fig. 5(a)]. For simplicity of be resolved in a way which crucially relies on entanglement drawing, let us consider the AdS eternal black hole. dynamics, but does not appear to involve time reversal or Given the outgoing mode D, one possible representation two copies of the system. The key to understand this puzzle of interior partner operators can be constructed by time is that the infalling observer, not the outside observer, will evolving a corresponding mode on the right-hand side. This be the one to measure quantum entanglement by seeing the mode, constructed exclusively on the degrees of freedom smoothness of the horizon. To communicate her infalling on the right-hand side, was called RD. We now include the experience back to the outside observer, she needs to be effect of the infalling observer A as a gravitational pulled back to the outside. In our construction of interior shockwave and draw the backreacted geometry where operators, this was done by performing the EPR distilla- the horizon is shifted as depicted in Fig. 5(a). If the time tion. We may use the deterministic variant from [18] if we separation between the outgoing mode D and the infalling want to save her for sure. This procedure does involve time observer A is longer than or equal to the scrambling time, reversal which is required for verification of entanglement ˜ the interior mode D, which is distinct from RD, can be from the perspective of the outside observer. found across the horizon. We also address a version of the firewall puzzle for B. Rendezvous in black hole? typical state black holes due to Marolf and Polchinski [19]. Observer dependence due to gravitational backreaction O Their argument can be summarized as follows. If D˜ is a also provides a possible resolution of another interesting O ψ O ⊗ partner operator of D, they should satisfy h BHj D puzzle concerning the Hilbert space structure of two-sided O ψ ≃ 1 ψ ’ D˜ j BHi where j BHi is the black hole s pure state. black holes. Consider the two-sided AdS black hole O ψ Suppose that D˜ is a partner operator for all j BHi and is geometry which is usually interpreted as the thermofield ψ independent of j BHi. Then we have double state of the left and right CFTs whose Hilbert space Z H H ⊗ H 1 has a factorized form ¼ L R. In a naive semi- d ψ ψ O ⊗ O ˜ ψ O ⊗ O ˜ 0 classical treatment on a fixed background geometry, it j BHih BHj D Dj BHi¼d Trð D DÞ¼ appears that two observers from the left and the right can be ð35Þ arranged to meet inside the black hole if they depart sufficiently before t ¼ 0. See Fig. 6. But the conformal for traceless OD, leading to a contradiction. Thus, Marolf field theory (CFT) Hamiltonian HL þ HR does not couple and Polchinski concluded that either (i) a partner operator two sides, so two observers should not be able to meet. O ψ D˜ does not exist (i.e., the firewall exists for typical j BHi), How do we make sense of these? O ψ or (ii) D˜ depends on j BHi. We have avoided this puzzle A possible resolution of this puzzle was suggested by due to Marolf and Polchinski by noting that a partner Marolf and Wall [26]. This puzzle itself was referred to as O ψ operator D˜ can be independent of j BHi when an infalling the Marolf-Wall puzzle in [27]. Their proposal to resolve observer is treated as an actual physical object with finite this puzzle was that we need additional degrees of freedom

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C. Page curve and state dependence Finally, we discuss the implication of our state- independent reconstruction to the information loss problem. There are many variants of the black hole information puzzle, but we shall focus on the particular question of why the entanglement entropy starts to decrease after the Page time [29]. In the literature, we often see that the Page curve problem and the firewall problem are discussed on the equal footing as if these have the same origin; see [11,30] for instance. While both problems are concerned about entangled FIG. 6. Classical trajectories from the left and right sides can partners of outgoing modes, their resolutions come from intersect inside the black hole. Can two observers meet? entanglement properties of completely opposite nature as we clarify below. The initial increase of the entanglement entropy results beyond the original CFTs, some sort of superselection from the fact that the outgoing mode is entangled with the sectors, in order to describe the meeting. This proposal is remaining black hole degrees of freedom. After the Page closely related to the tensor factorization problem as time, however, the situation will change drastically. If pointed out in [28]. ’ Here, however, we arrive at a different resolution of the unitarity is assumed from the outside observer s descrip- puzzle by considering the effect of backreaction. In fact, tion, the outgoing mode after the Page time must be the structure of the problem is identical to the one in the somehow entangled with the early radiation since otherwise firewall puzzle. Suppose that two signals were sent at the entanglement entropy will not decrease. This observation seems to suggest that the interior partner of the late t ¼ −t0 so that their semiclassical trajectories would intersect in the black hole interior (Fig. 6). For definiteness, outgoing mode must be constructed by using the early we focus on the left signal L as the one we start with and radiation degrees of freedom and hence must be state ask if it receives the right signal R in the interior or not. In dependent. It appears to be in tension with our view that the AdS=CFT correspondence, the right signal R can be interior partner operators can be reconstructed in a state- interpreted as an excitation on the right CFT. Then, in the independent manner. absence of the left signal, one can find some mode R0 on the Our proposal to resolve this apparent tension is that in an evaporating black hole, a large portion of outgoing modes left side at time t ¼þt0 which is entangled with the mode R. In the unperturbed geometry, the right signal, started at can be still reconstructed in a state-independent manner, but a tiny portion accounting for the black hole evaporation t ¼ −t0, would travel along the mode R which is realized as the entangled partner of R0. However, due to the back- must be state dependent. reaction by the left signal L, the near-horizon geometry will As a warm-up, let us recap the interior reconstruction be shifted in the same manner as in the firewall problem. problem for a large AdS black hole which does not From the reasoning similar to Sec. VI A, we see that the evaporate. The time evolution of this quantum system t ⪆ t can be depicted as follows: right signal will not reach the left signal if 0 scr. The original entangled partner mode R, where the right signal was supposed to travel along, is no longer entangled with the left side R0 due to the backreaction. Inside the black hole, the left signal L will come across the newly created R˜ t ≪ t partner mode where the right signal is absent. If 0 scr, two signals would meet near the singularity where we ð36Þ should doubt the validity of semiclassical treatment from the beginning. The central idea behind this resolution is again that the black hole interior is observer dependent. Left and right observers jump into their own black hole interiors which are separated degrees of freedom. Given a possible relation between the Marolf-Wall puzzle and the tensor factoriza- where A, D can be interpreted as infalling and outgoing tion problem, we speculate that observer dependence of modes, respectively, while B, C are all the other degrees of black hole interiors may also provide useful insights on the freedom. State-independent reconstruction is possible as latter problem. long as dA ⪆ dD. This condition is satisfied in the AdS

046004-14 OBSERVER-DEPENDENT BLACK HOLE INTERIOR FROM … PHYS. REV. D 103, 046004 (2021) black holes where the outgoing modes are reflected back at entanglement entropy between the black hole and the 10 11 the boundary and hence dA ¼ dD. environment, leading to the Page curve behavior. For evaporating black holes, however, there will be more The smoothness of the horizon requires the interior outgoing modes than infalling modes, corresponding to the partners to be state independent and the emergence of situation with dD ⪆ dA. Let us examine how the state- the Page curve requires those to be state dependent. In an independent reconstruction ceases to work when the sub- evaporating black hole, two types of interior operators can system D becomes larger than A. A toy model which coexist. In the firewall problem, we typically focus on mimics this situation is depicted below. situations where state-independent modes are dominant as evaporation process is assumed to be slow.

VII. CRITICAL REMARKS ON PREVIOUS PROPOSALS

ð37Þ The aforementioned reconstruction of interior partner operators reveals certain fundamental flaws in previous proposals for resolutions of the firewall puzzle, including the ER ¼ EPR approach [9,10], the state-dependence approach [11], and the quantum computational complexity approach [31]. These approaches, namely, those bundled as the ER ¼ EPR approach, appear to be widely accepted where jDj > jAj. Small portion of the outgoing modes will without much scrutiny in the current mainstream research escape to the environment while the rest returns to the black community. In this section, we revisit these previous hole. For black holes in a flat space, those returning modes approaches critically and point out explicitly what went are typically confined in the radiation zone. wrong. A more comprehensive set of criticisms are pre- Let us examine where D is entangled with. When U is sented in [23]. scrambling and dA ≤ dD, we obtain [7] A. Remarks on ER = EPR ¯ ¯ IðD; CAÞ ≈ 2log2dA IðD; BÞ ≈ 2ðlog2dD − log2dAÞ: Let us begin with proposals, often bundled under the ð38Þ slogan ER ¼ EPR due to Susskind and Maldacena [9]. These proposals roughly go as follows. The distillation of the qubit RD by the outside observer Bob generates This suggests that there exists ≃dA-dimensional subspace ¯ perturbations which will become a high-energy density (≃ log2 dA qubits) of D which is entangled with CA.For near the horizon as shown in Fig. 5(b). This perturbation operators acting on this subspace, entangled partners can be spoils the quantum entanglement between D and D˜ and reconstructed on CA¯ in a state-independent manner as these B¯ ≈d =d creates a firewall (or Alice may be killed by the shock- are not entangled with . However, the remaining D A- wave). As such, Alice is not able to observe entanglement dimensional subspace (≃ log2 dD − log2 dA qubits) is between D and D˜ . One may draw the Penrose diagram entangled with the early radiation B¯ . Hence, their entangled corresponding to these proposals by explicitly treating RD partners must be reconstructed in a state-dependent manner. as a gravitational shockwave. This subtle distinction between state-dependent and The ER ¼ EPR approach to the firewall puzzle, how- state-independent partner operators provides an interesting ever, is inconsistent with the implication of the black hole insight on the emergence of the Page curve in the decoupling theorem which suggests that the outgoing mode evaporating black hole. Imagine that the outgoing mode D is no longer entangled with the early radiation B¯ . A new is made of log2 dD qubits and release them one by one to partner mode D˜ after perturbation by an infalling observer the environment which joins the early radiation B¯ . When is distinct from the original partner mode RD, and the we release the first log2 dA qubits, entanglement entropy construction of D˜ does not depend at all on how the black between the black hole and the early radiation will increase hole was initially entangled with its early radiation B¯ . Here since those qubits are entangled with CA¯ (plus qubits in D the observer dependence of the black hole interior plays a which have not been released). On the other hand, the key role. The infalling observer will see an interior which is remaining log2 dD − log2 dA qubits are entangled with prepared by her own backreaction. Her infalling experience the early radiation B¯ , so releasing them will decrease the

11 d2 A similar argument for the state dependence in the Page 10 O D Strictly speaking, the reconstruction error scales as ðd2 Þ,so A curve derivation has been recently made in [30]. In our under- A needs to contain a few more qubits than D. We will ignore this standing, however, we disagree on its implication to the firewall subtlety in the discussions. problem.

046004-15 BENI YOSHIDA PHYS. REV. D 103, 046004 (2021) as well as her black hole interior will not be affected by any unitarily scrambling dynamics. Advocates of these pro- quantum operations localized on the early radiation. In fact, posals, however, often argue that such inconsistencies may whether the black hole is young or old does not matter for not be detected from the outside observer and thus one does crossing the horizon. not need to be bothered by them seriously. Ideas along this Another counterargument against the ER ¼ EPR line of philosophy originate from the black hole comple- approach is presented in later subsections too. mentarity approach due to Susskind who postulated that an apparent violation of quantum mechanics is acceptable if B. Remarks on state dependence the violation cannot be verified by a single observer. Here, we demonstrate fundamental flaws in complementaritylike Next, let us discuss the state-dependence approach arguments. which postulates that interior operators may be dependent Let us briefly recall how the complementaritylike argu- on the initial state of the black hole. Such a viewpoint has been promoted largely by Papadodimas and Raju as a ment would arise in the ER ¼ EPR approach. Based on potential resolution of the firewall puzzle; see [11] for naive predictions from the bulk geometry, the ER ¼ EPR A instance. In the absence of an infalling observer, it is approach postulates that the infalling observer from one R certainly true that interior partner operators depend on the side will be hit by a shockwave D from the other side even initial state of the black hole. In the presence of an infalling though two sides are not coupled on the boundaries. This observer, however, the construction of interior operator D˜ conclusion is of course incorrect as we have already seen in dynamically changes and becomes state independent due to Sec. VI B, but let us tentatively assume that A and RD will the decoupling theorem. Hence, unfortunately the state- encounter in the black hole interior. Since the encounter dependence approach also is not consistent with the fact between A and RD is nonlocal from the perspective of the that the black hole scrambles unitarily. boundaries, such nonlocal interactions should be hidden from the perspective of the outside observer. One possible C. Remarks on computational complexity argument for avoiding such nonlocality, which will turn out Finally, let us discuss the quantum circuit or quantum to be incorrect, is based on the following observation. In A R computational complexity approach. The central idea is to order for the outside observer to verify whether and D accept the violation of the monogamy of entanglement. Yet, have encountered inside the black hole or not, he needs to A by postulating that highly complex interior operators save from the black hole interior and pull her back to the cannot be measured in physically reasonable time scales, black hole exterior. Then he may ask A if she has one may argue that the violation is essentially not detect- encountered RD or not. There are a few methods of saving able. Ideas along this line were originally proposed by a quantum state, which has fallen into a black hole, and Harlow and Hayden [31]. In the absence of an infalling reconstructing it in the black hole exterior. One method is to observer, it is certainly true that interior operators can perform the traversable wormhole protocol [32] or the potentially have high quantum circuit complexity. When Hayden-Preskill recovery protocol [18] by coupling two the unitary operator K in Eq. (1) is highly complex, the sides of the black hole. interior partner operator will be also highly complex. In the The complementaritylike argument for avoiding the presence of an infalling observer, however, the complexity nonlocality then goes roughly as follows. Since saving A of interior operator D˜ , which was dynamically created by from the interior appears to require nonlocal coupling the gravitational backreaction, does not depend on the between two boundaries, the nonlocality resulting from initial state of the black hole at all. Furthermore, one can the encounter between A and RD cannot be detected unless construct D˜ explicitly in an efficient manner as we have we actually make two boundaries coupled. Hence, from the already seen in the present paper. Hence, we claim that the perspective of the outside observer, the locality violation quantum complexity approach does not resolve the firewall will not be detected. puzzle.12 The above speculation is based on a false assumption that saving the infalling observer A from the black hole D. Remarks on complementarity interior requires coupling two sides. As we have mentioned We have discussed why previous proposals, such as the in Sec. III E, an infalling observer in a two-sided black hole ER ¼ EPR approach, fail to resolve the firewall puzzle. can be saved from the black hole by quantum operations The crux of our counterarguments was that these which act solely on one side of the black hole [8]. The approaches are inconsistent with predictions from the black rescued observer will tell us that she was not hit by the hole decoupling theorem which holds universally for shockwave RD as the whole procedure is localized only on one side of the black hole and has no dependence on the R 12The quantum circuit complexity of the interior partner mode other side where D is supported. It is thus incorrect to say D˜ , in the presence of an infalling observer, has an interesting that an inconsistency cannot be verified by an outside holographic interpretations on the bulk; see [23]. observer since no inconsistency has occurred.

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VIII. MORE ON DECOUPLING THEOREM Next, let us justify the condition of dA ⪆ dD by presenting three observations. The black hole decoupling theorem is applicable only (iv) AMPS: For an application to the firewall puzzle, when the time separation Δt between an observer A and the namely, the AMPS thought experiment, we can outgoing mode D satisfies Δt ≥ t and dA ⪆ dD. In this scr easily justify this requirement. In the thought experi- section, we discuss the cases which do not satisfy these two ment, the outside observer distills RD and hand it to assumptions. the infalling observer A who jumps into the black Let us begin with the cases with Δt ≤ t . We present scr hole to verify the entanglement between D and D˜ . three observations supporting our conclusions concerning 2 This effectively realizes a situation where dA d the firewall puzzle and the Marolf-Wall puzzle. ¼ D with a pure state input on A as discussed in (i) Singularity: Let us look at this problem in the two- sided AdS black hole where the system is in the TFD Appendix C. Δt (v) Metaphysical explanation: In order to experience (thermofield-double) state. If the time separation A between A and D is much shorter than the scram- some physics, the infalling observer herself should bling time Δt ≪ t , the gravitational backreaction carry some amount of entropies, at least as much as scr the objects D she is going to measure. While being from A is negligible. Then, A and RD might appear to encounter inside the black hole as the unperturbed metaphysical, this explanation can be further justi- Penrose diagram seems to suggest. However, the fied from the aforementioned observation (iii) which suggests the decoupling occurs as much as the trajectories of A and RD will intersect near the D singularity [Fig. 5(b)]. It is then not surprising if observer measured . A D effective bulk descriptions break down severely (vi) Effective QFT: One may interpret and as there. Also, the encounter location will be outside infalling and outgoing modes of low energy QFT B C the region where the Rindler approximation is on the lhs wedge, respectively, while and can be applicable. As such, we can safely ignore the cases viewed as all the other high-energy degrees of Δt ≪ t freedom (d.o.f) associated with the future and past scr. This observation, however, does not provide a satisfactory justification for intermediate horizons, respectively. It is thus natural to assume time scales. dA ¼ dD as also discussed in Sec. VI C. Here A and Δt ⪅ t D (ii) Entanglement quality: If scr, an infalling are d.o.f which an outside observer can easily see. observer will encounter the outgoing mode D away On the other hand, the infalling observer, who from the horizon. Recall that the proper temperature travels nearly at the speed of light, will propagate 1 along the infalling mode A. Hence, it is natural to near the Rindler horizon is given by T ¼ 2πρ where ρ expect that the initial state of A may become is the proper distance from the horizon. As one goes irrelevant as long as the initial state is chosen from away from the horizon, the density of thermal low energy subspace of the QFT. Namely, regardless entropy becomes smaller and the quality of quantum of the initial state of A, she will see the same bulk entanglement, which an infalling observer may be effective QFT description near the horizon. This is able to detect via simple free fall, will deteriorate. D˜ Hence, there is no inconsistency even if the decou- indeed the case as the creation of occurs for any A pling phenomena are weak. initial state of . However, it is worth emphasizing Measurement strength: that the map from the bulk QFT to the boundary is (iii) Recall that the decoupling A phenomena occur as a result of the interaction dependent on the initial state of . Here the infalling A D observer will have an access to the outdoing mode D between the observer and the outgoing mode , ˜ which is quantified by the decay of OTOCs. This and the interior mode D instead of A. suggests that the decoupling, as well as creation of D˜ , We have presented justifications of our conclusions for occurs as much as the observer A measures D. Then, the cases where the system does not satisfy underlying even if D and D˜ are not perfectly entangled, there is no assumptions of the decoupling theorem. Perhaps a better contradiction with the monogamy of entanglement question to ask is whether we actually need to include the ˜ since A is observing D (and its partner D)only infalling observer as additional degrees of freedom or not in weakly. deriving the bulk description from the boundary descrip- A D If wishes to measure more directly or strongly, tion. From the outside viewpoint, the infalling and outgoing she may increase the size of herself (dA) to enhance modes influence each other in a dynamical manner. When gravitational scattering or perform actual physical ’ measurement (e.g., number operator) on D upon switching to the infalling observer s viewpoint, infalling encountering it. We expect that such strong inter- modes somehow decohere, perhaps due to the choice of the actions or direct measurements will lead to significant infalling trajectory. Ultimately, we hope to understand why decay of OTOCs and create D˜ which is nearly the interior operator is “automatically decoded” simply by perfectly entangled with D. switching to the infalling observer’s frame.

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ACKNOWLEDGMENTS temperature OTOCs in an analogous manner by using thermal expectation values, I thank Yoni BenTov and Yingfei Gu for discussions. Research at the Perimeter Institute was supported by the α β Government of Canada through Innovation, Science and hOi ≡ TrðρOÞhO1O2i ≡ Trðρ O1ρ O2Þ Economic Development Canada and by the Province of α; β > 0 α β 1: Ontario through the Ministry of Economic Development, þ ¼ ðA5Þ Job Creation and Trade. Again, specifics of α, β does not affect the main result, and APPENDIX A: REVIEW OF QUANTUM we will choose α ¼ β ¼ 1=2. The asymptotic form of INFORMATION CONCEPTS OTOCs at finite temperature can be also derived from ETH [35]. 1. Scrambling Let us present some intuition behind the aforementioned Here we recall the definition of scrambling in terms definition of scrambling. The thermal ensemble ρ has of OTOCs. significant contributions from eigenstates within typical At infinite temperature, OTOCs are defined with respect energy subspace at given temperature. Hence, one may 1 to the maximally mixed state ρ ¼ d I, interpret each insertion of powers of ρ as projection onto typical energy subspace. The asymptotic form of OTOC 1 U hWðtÞZð0ÞYðtÞXð0Þi ≡ Tr½WðtÞZð0ÞYðtÞXð0Þ; ðA1Þ suggests that the time evolution resembles Haar random d unitary within the typical energy subspace. Here we are particularly interested in operators X, Y, Z, W which do not O t where time-evolved operators are defined by ð Þ¼ change the energy of the system much so that the system U†O 0 U ð Þ . The phenomenon of scrambling is often asso- remains within the typical energy subspace. ciated with the decay (decorrelation) of OTOCs from their It is worth emphasizing that our main focus is on systems W Z Y X initial values when , , , are basis operators such as after the scrambling time when OTOCs have decayed to Pauli and Majorana operators. In a more genetic language, asymptotic values. The famous Lyapunov behavior is t the system is said to be scrambling at time if the following typically observed only before the scrambling time. asymptotic decomposition holds for all the local operators W, Z, Y, X [18]: 2. Diagrams hWðtÞZð0ÞYðtÞXð0Þi In this paper, we extensively use diagrammatic tensor ≈hWYihZihXiþhZXihWihYi−hWihZihYihXi; ðA2Þ notations in order to express wave functions and operators as well as physical processes. While these techniques have 1 become standard in the past several years, it would be still where expectation values are defined by hOi ≡ Tr½O. d beneficial to have a brief summary to make the presentation We immediately see that the asymptotic value for a large self-consistent. Wave functions and operators are repre- system size is zero when X, Y, Z, W are traceless. This sented by asymptotic form has been derived for Haar random unitary [33] and from the eigtenstate hermalization hypothesis (ETH) [34]. Corrections to the asymptotic form are polynomially (exponentially) suppressed with respect to the system size n for systems with (without) conserved quantities. At finite temperature, OTOCs are defined with respect to ðA6Þ a thermal mixed state ρ ¼ e−βH=Trðe−βHÞ, hWðtÞZð0ÞYðtÞXð0Þi ≡ Tr½ραWðtÞρβZð0ÞργYðtÞρδXð0Þ; which can be also explicitly written as ðA3Þ X X X jψi¼ Tjjjihψj¼ Tj hjj O ¼ Oijjiihjj: where j j ij ðA7Þ α; β; γ; δ ≥ 0 α þ β þ γ þ δ ¼ 1: ðA4Þ

Specifics of α, β, γ, δ do not affect the main result, and we By using these tensors as building blocks, one can express will focus on α ¼ β ¼ γ ¼ δ ¼ 1=4 unless otherwise various physical processes in a graphical manner. For stated. We can infer the asymptotic form of finite instance, an expectation value can be represented by

046004-18 OBSERVER-DEPENDENT BLACK HOLE INTERIOR FROM … PHYS. REV. D 103, 046004 (2021) X X † O ¼ Oijjiihjj O ¼ Oijjjihij i;j i;j X X T O ¼ Oijjjihij O ¼ Oijjiihjj; ðA12Þ i;j i;j ðA8Þ where O† ¼ðOTÞ. In the tensor diagrams, the transpose can be represented as follows:

ðA13Þ In order to associate a physical process to an equation like hψjOn O1jψi, one needs to read it from the right to the left, i.e., the initial state jψi is acted by O1;O2; sequentially and then is projected onto jψi. In the dia- The following relation is particularly useful in constructing grammatic notation, one needs to read the figure from the partner operators: bottom to the top, i.e., the time flows upward in the diagram. O ⊗ I I ⊗ OT ; H CdR ð ÞjEPRi¼ð ÞjEPRi ðA14Þ Consider a Hilbert space R ¼ whoseP dimension- d ≡ H I dR j j ality is R j Rj. An identity operator, R ¼ j¼1 j ih j, can be expressed as a straight line (i.e., a trivial tensor) or graphically since its inputs and outputs are the same,

ðA9Þ ðA15Þ

This diagram has one input leg (index) and one output leg. One may bend the line and construct the following diagram: 4. State representation Given an operator O acting on a Hilbert space H, its state H ⊗ H ðA10Þ representation (Choi representation), supported on , is defined by which is the same trivial line, but with two output legs instead of one in and one out. This diagram represents an ðA16Þ unnormalized EPR pair defined on HR ⊗ HR¯ . We utilize the following notation for normalized EPR pairs: In this regard, the EPR pair is a state representation of an identity operator; jEPRi¼jIi. See [17] for further discussions on the state representation. Given a density matrix ρR on HR, consider a purification (thermofield double state) on HR ⊗ HR¯ , ðA11Þ ffiffiffiffiffiffi X X ffiffiffiffiffi p ρ p ψ ψ → pp ψ ⊗ ψ : where the black dot carries a factor of 1= dR. ¼ jj jih jj j j ji j j i ðA17Þ j j 3. O; O†; OT; O The purified state can be viewed as a state representation of It is also beneficial to define O; O†;OT;O and review ρR. By introducing an unnormalized state some useful relations associated with them. Given an operator O, the complex-conjugate transpose O†, the ρ˜ ≡ d ρ ρ˜ d ; transpose OT, and the complex-conjugate are defined by R R R Tr½ R¼ R ðA18Þ

046004-19 BENI YOSHIDA PHYS. REV. D 103, 046004 (2021) we have Using the OTOC asymptotics, we obtain the claimed X ffiffiffiffiffi bound. 1=2 p jρ˜R i¼ pj jψ ji ⊗ jψ j i: ðA19Þ A similar bound can be derived for the cases with a pure j state input by analogous analysis. We obtain 2 2 1 dD ρBD¯ − IB ⊗ ID ≤ O þ OðϵÞ: ðB4Þ APPENDIX B: STATEMENT OF DECOUPLING dB¯ dD 1 dA THEOREM For a finite temperature generalization of the decoupling Below we sketch a proof of ρBD¯ ≈ ρB¯ ⊗ ρD. When two theorem, see [18] and the Appendix in [36]. states are close in L1 distance, they cannot be distinguished by any measurement. The strategy is to show that ρBD¯ is APPENDIX C: MORE ON AMPS PUZZLE 1 1 close to a maximally mixed state d d IB ⊗ ID in the L B¯ D Consider an old black hole that is maximally entangled distance. More precisely, we will prove the following with the early radiation. For simplicity of discussion, we bound: assume that the initial state is EPR pairs. Assume that the black hole time evolves by some unitary dynamics U. 1 2 d2 ρ − I ⊗ I ≤ O D O ϵ ; Consider a scenario where an outside observer distills some BD¯ d d B D d2 þ ð Þ ðB1Þ B¯ D 1 A qubits RD that would be entangled with the outgoing mode ffiffiffiffiffiffiffiffiffiffi D by applying some distillation unitary operator D, 1 p † where the L norm is defined by kOk1 ¼ Tr O O. Here, ϵ is the error from the asymptotic form of OTOCs which is typically suppressed polynomially in the system size n. The derivation is straightforward. Observe ðC1Þ

2 1 ρBD¯ − IB ⊗ ID ≤ dB¯ dD ρBD¯ d ¯ d B D 1 1 2 − I ⊗ I ρ2 d d − 1: B D ¼ Trð BD¯ Þ B¯ D ðB2Þ ¯ dB¯ dD 2 A natural choice of D is to take D ¼ U so that D and D form EPR pairs, but we will keep it unspecified. The rhs can be evaluated by noticing Imagine that the outside observer hands these distilled d qubits to the infalling observer who is going to jump into a ρ2 D Δ: black hole. After the time evolution, we will obtain the Trð BD¯ Þ¼ ðB3Þ dB¯ following quantum state:

ðC2Þ

T where the black hole time evolves by some unitary VðD ⊗ ID¯ Þ as the total dynamics of the black hole, the dynamics V. Here U and V are different unitary operators above diagram corresponds to the interior reconstruction in general as the size of the Hilbert space changed by problem with the infalling observer being prepared in EPR 0 ¯ T adding the distilled qubits to the black hole. By viewing pairs on D D. As long as VðD ⊗ ID¯ Þ is scrambling, one

046004-20 OBSERVER-DEPENDENT BLACK HOLE INTERIOR FROM … PHYS. REV. D 103, 046004 (2021) can construct interior partner operators on the remaining APPENDIX D: FINITE TEMPEARTURE black hole C. We expect that this is indeed the case for We construct partner operators at finite temperature. We generic choices of D.13 consider the following quantum state at finite temperature:

ðD1Þ

where the black hole was initially in the thermofield double 1. Distillation ρf1=2 BB¯ state j B i on and then was perturbed by the meas- Having imposed the aforementioned assumption, gener- ρe 1=2 urement probe prepared in some entangled state j A i. alization to finite-temperature systems is straightforward; we The construction procedure be applied to finite temper- simply replace all the dots in the previous diagrammatic pffiffiffiffiffi ature systems which satisfy a certain technical assumption calculations with ρR. One subtlety is that, since the out- ρ ρ ⊗ ρ [18]. Given the input ensemble AB ¼ A B, we will going mode D is thermal, partner operators ΩðODÞ must assume that the output ensemble factors into C and D, satisfy

† † † 1=2 † 1=2 ρCD ≡ UρABU ≈ ρC ⊗ ρD: ðD2Þ hΨðtÞjOD ⊗ ΩðODÞjΨðtÞi≈hODODi≡Tr½ρD ODρD OD; ðD3Þ Unfortunately, this condition is problematic for physical reasons; if ρCD is thermal, C and D are entangled. We will reproducing thermal expectation values. This relation can be present a justification for such an assumption in viewed as the definition of interior partner operators for finite Appendix F. For our purpose, it suffices to coarse-grain temperature systems. 1=2 and remove preexisting correlations between C and D in a Hence we consider distillation protocol of jρfD i from ¯ thermal ensemble ρCD. D and CA, which proceeds in the following manner:

ðD4Þ

13 T Whether there exist special choices of D which make VðD ⊗ ID¯ Þ nonscrambling or not is an interesting question. If such D exists, it would be performing the Hayden-Preskill recovery which would protect an infalling observer from crossing the horizon. See [8] for details.

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1=2 ¯ 1=2 1=2 where jρfD i is inserted on DD and projection ΠAA¯ ≡ Let us decompose ΠAA¯ ¼jρeA ihρeA j in terms of basis 1=2 1=2 ¯ jρeA ihρeA j is applied on AA. operators (such as Pauli operators), 1=2 The probability amplitude to measure jρeA i can be X evaluated via the asymptotic form of OTOCs at finite Π α V ⊗ W temperature. After some calculations (see Appendix E), we AA¯ ¼ V;W A A¯ V;W obtain α ρe 1=2 V† ⊗ W† ρe 1=2 : V;W ¼h A j A A¯ j A i ðD9Þ Δ ≡ Φ Π Φ ≈ ρ3 h backj AA¯ j backi Tr½ DðD5Þ

14 Each term in the decomposition contributes to ΩðOÞ as for dA ≫ dD. The following lower bound is useful: follows: Φ Π Π Φ h backj D0D¯ AA¯ j backi ≥ Φ Π Π Π Φ ρ2 2: h backj D0D¯ AA¯ BB¯ j backi¼Tr½ D ðD6Þ

1=2 Hence, the probability amplitude to measure jρfD i after the postselection is lower bounded by ðD10Þ

2 2 1 Tr½ρD 2 Sð3Þ−Sð2Þ Φ Π Π Φ ⪆ 2 ð D D Þ; h backj D0D¯ AA¯ j backi 3 ¼ ðD7Þ Δ Tr½ρD

ðαÞ ð3Þ ð2Þ where SD is the R´enyi-α entropy of ρD.IfSD − SD ≈ 0, 1=2 0 ¯ jρfD i can be distilled on D D. Here it is convenient to consider thermally dressed operators [37], 2. Dressed operator growth The distillation procedure constructs the following f 1=4 1=4 OR ≡ ρfR ORρfR : ðD11Þ interior partner operators:

Then the diagram in Eq. (D10) can be interpreted as f † f † truncation of VA ð−tÞ by OD . Hence, we see that interior partner operators can be constructed due to the thermally dressed operator growth. ðD8Þ

APPENDIX E: OTOC CALCULATION Let us represent OTOCs graphically as follows:

ðE1Þ

14 3 3=2 1=2 This is the result for α ¼ β ¼ γ ¼ δ ¼ 1=4 in Eq. (A3). When α ¼ 1, we obtain Δ ≈ Tr½ρDTr½ρA Tr½ρA . If the variance of the 3=2 1=2 spectrum of ρA, normalized by its mean, approaches zero asymptotically, Tr½ρA Tr½ρA ≈ 1.

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Due to linearity, we can generalize the above OTOCs as a function of P, Q which act on H⊗2. We then have

ðE2Þ

Using the OTOC asymptotics, we obtain entropy of the black hole. These UV divergent pieces, however, are often ignored in the Bekenstein bound [38]. 3 3 3 3 3 Δ ¼ TrðρDÞþTrðρAÞ − TrðρDÞTrðρAÞ ≈ TrðρDÞ: ðE3Þ 2. Soft mode Even if we remove short-range entanglement between C and D; however, there still remains some classical corre- APPENDIX F: ON ρCD ≈ ρC ⊗ ρD lation between C and D due to the presence of conserved Here we present a justification of the assumption quantities (such as the energy, charge, and angular momen- 15 tum). The Hayden-Preskill recovery problem for systems ρCD ≈ ρC ⊗ ρD. with conserved quantities (symmetries) has been discussed in [7]. Here it is convenient to consider two types of 1. Coarse-graining physical modes (hard and soft modes) on C, D. Roughly, For simplicity of discussion, we approximate ρCD as a hard modes are those which are associated with change of e−βHCD conserved quantities on C, D, whereas soft modes are those thermal ensemble ρCD ¼ −βH . Consider the thermo- Tr½e CD which do not change them. It is convenient to imagine an ρg1=2 C; D; C;¯ D¯ field double state j CD i supported on . The approximate decomposition of the Hilbert space into a assumption ρCD ≈ ρC ⊗ ρD asserts that C (D) is entangled α α block-diagonal form: H ≈ ⨁HC ⊗ HD where α labels the only with C¯ (D¯ ), respectively. In general, however, this is α not the case. conserved quantity. Soft mode operators are block diago- Here we are interested in what degrees of freedom the nal, while hard mode operators may change α-labels by outgoing mode D is entangled with. Eigenstates of the acting on both C and D. Here the thermal density matrix ρβ Hamiltonian HCD are entangled between C and D.Atlow can be interpreted as an approximate projection onto states temperature, a fewer number of states contribute to the within a typical energy window. Such a subspace corre- thermal ensemble and thus entanglement between C and D sponds to some single label of α in the above decom- survives in ρCD. Then the “partner” operators of D can be position, and soft mode operators do not change the energy found on C due to the preexisting entanglement in ρCD.In of the system much. Examples of soft modes include this case, the system does not need to be perturbed in order gravitational shockwaves which only shift the horizon, to construct the partner operator from the beginning. commuting with ρβ. When α-label in C changes, α-label in Indeed, such modes on D do not lead to the firewall puzzle D must change too due to the conservation law. This since, in order for the puzzle to exist, D needs to be suggests that hard modes on D are correlated classically entangled with CD in the first place. See Fig. 7. with hard modes on C. Hence we can focus on entangle- There is another reason why entanglement between C ment associated with soft modes only. In [7], it has been and D may not be an interesting one to focus on. In discrete shown that soft modes on D can be reconstructed by using spin systems, entanglement between C and D is mostly of the Hayden-Preskill recovery protocols. short-range nature. In quantum field theories, such entanglement is UV divergent and hence is cutof-dependent property which should be removed by some proper coarse- graining. We may simply discard short-range entanglement C D C D between and and construct subspace sub and sub on ρ ⊗ ρ which we may have a factor structure Csub Dsub . Due to the preexisting entanglement between C and D, releasing D to the environment appears to increase entanglement

15I thank Alexei Kitaev for discussions on this. FIG. 7. Short-range entanglement between C and D.

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