Soft Mode and Interior Operator in the Hayden-Preskill Thought Experiment

PHYSICAL REVIEW D 100, 086001 (2019)

Soft mode and interior operator in the Hayden-Preskill thought experiment

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

(Received 2 May 2019; published 7 October 2019)

We study the smoothness of the black hole horizon in the Hayden-Preskill thought experiment by using two particular toy models based on variants of Haar random unitary. The first toy model corresponds to the case where the coarse-grained entropy of a black hole is larger than its entanglement entropy. We find that, while the outgoing mode and the remaining black hole are entangled, the Hayden-Preskill recovery cannot be performed. The second toy model corresponds to the case where the system consists of low energy soft modes and high energy heavy modes. We find that the Hayden-Preskill recovery protocol can be carried out via soft modes whereas heavy modes give rise to classical correlations between the outgoing mode and the remaining black hole. We also point out that the procedure of constructing the interior partners of the outgoing soft mode operators can be interpreted as the Hayden-Preskill recovery, and as such, the known recovery protocol enables us to explicitly write down the interior operators. Hence, while the infalling mode needs to be described jointly by the remaining black hole and the early radiation in our toy model, adding a few extra qubits from the early radiation is sufficient to reconstruct the interior operators.

DOI: 10.1103/PhysRevD.100.086001

I. INTRODUCTION reduced to very simple calculations based on Haar random unitary operators. Almost forty years since its formulation, the black hole The main idea of this paper centers around a tension information problem and its variants still shed new lights between the smooth horizon and the Hayden-Preskill on deep conceptual puzzles in quantum gravity, and also thought experiment. The no-firewall postulate asserts the provides useful insights to study strongly interacting presence of entanglement between the infalling and out- quantum many-body systems [1]. While the ultimate going Hawking pair. The recoverability in the Hayden- solution of the problem could be obtained only by Preskill thought experiment requires the outgoing Hawking experimental observations, progresses can be made by radiation to be entangled with a joint system of the early utilizing thought experiments based on simple toy models radiation and the reference qubits of the infalling quantum [2,3]. In the last decade, two particular thought experiments state. However, due to monogamy of entanglement, the no- on black hole dynamics have fascinated and puzzled firewall postulate and the recoverability look mutually theorists; the Almheiri-Marolf-Polchinski-Sully (AMPS) incompatible. thought experiment [4] and the Hayden-Preskill thought Here we seek for a resolution to this tension by arguing experiment [5]. The AMPS thought experiment suggests that the “outgoing Hawking radiation” in the Hayden- that the smooth horizon in an old black hole, which is a Preskill and AMPS experiments are actually different consequence of the equivalence principle, may be incon- degrees of freedom (d.o.f.). While it would be desirable sistent with monogamy of entanglement.1 The Hayden- to demonstrate such a separation of the Hilbert space of the Preskill thought experiment poses questions concerning the outgoing mode via direct calculations on actual models of absoluteness of the event horizon by suggesting that an quantum gravity, our goal is more modest. In this paper, object which has fallen into a black hole may be recovered. we will study a certain refinement of Haar random unitary While there have been refined arguments and counterargu- dynamics. The unitary operator U preserves the total global ments on these conclusions in more realistic physical Uð1Þ charge and acts as Haar random unitary operator settings, essential features of the original works can be in each subspace with fixed charge in a block diagonal manner. Although charges in black holes have led to 1A related argument has appeared in [6]. intriguing puzzles in quantum gravity [7], it is not our primary goal to study the effect of global charges on the Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. AMPS and Hayden-Preskill thought experiments. Our Further distribution of this work must maintain attribution to primary focus is on unitary dynamics which preserves the author(s) and the published article’s title, journal citation, energy. We utilize the block diagonal structure of Uð1Þ- and DOI. Funded by SCOAP3. symmetric Haar random unitary to capture ergodic

2470-0010=2019=100(8)=086001(15) 086001-1 Published by the American Physical Society BENI YOSHIDA PHYS. REV. D 100, 086001 (2019) dynamics which essentially acts as Haar random unitary on the initial state of the black hole) by incorporating the effect each small energy window. This is partly motivated from of backreaction by the infalling observer explicitly [29]. recent works where Uð1Þ-symmetric local random unitary The paper is organized as follows. In Sec. II, we warm up circuits successfully capture key properties of energy con- by studying the case where the black hole is entangled only serving systems such as an interplay of diffusive transport through a subspace and its evolution is given by Haar phenomena and the ballistic operator growth [8,9]. random unitary. A corresponding physical situation is that We will show that Uð1Þ-symmetric modes are respon- the entanglement entropy SE of a black hole is smaller than S sible for the Hayden-Preskill recovery whereas the non- its coarse-grained entropy BH. The eternal AdS black hole S S symmetric modes are responsible for correlations between corresponds to E ¼ BH whereas a one-sided pure state the infalling and the outgoing Hawking pair. With an actual black hole corresponds to SE ¼ 0. We will see that taking S quantum entanglement between the interpret the symmetric and non-symmetric modes as low outgoing mode and the remaining black hole, but the energy (soft) and high energy (heavy) modes, respectively. Hayden-Preskill recovery is no longer possible, highlight- Namely we claim that the Hayden-Preskill thought experi- ing their complementary nature. In Sec. III, we analyze ment can be carried out by using soft modes which are the case where the black hole is entangled through a distinct from the Hawking radiation. Such low energy Uð1Þ-symmetric subspace and its evolution is given by modes may be the pseudo-Goldstone mode which corre- Uð1Þ-symmetric Haar random unitary. Physically this sponds to the ’t Hooft’s gravitational mode [10]. Or perhaps corresponds to a black hole which is entangled with its they may correspond to soft gravitons due to spontaneous partner through the subspace consisting of typical energy breaking of supertranslation symmetries [11]. In our toy states at given temperature. In Sec. IV, we present concrete model, however, the correlation between the infalling and recovery protocols by following [12]. In Sec. V,we outgoing Hawking modes is found to be purely classical describe the procedure to construct the interior operators as opposed to quantum correlations in the Hawking pair of the outgoing soft mode. In Sec. VI, we conclude with which would be seen by an infalling observer. Namely, the discussions. outgoing soft mode is found entangled with a joint of the Before delving into detailed discussions, we establish remaining black hole and the early radiation, not with a few notations used throughout this paper. See Fig. 1. the remaining black hole itself. We will denote the Hilbert spaces for the input quantum We also discuss the construction of the interior partner state as A, the original black hole as B, the remaining black operators of the outgoing Hawking mode. While the hole as C, and the late Hawking radiation as D.Itis partners of the outgoing soft mode operators cannot be convenient to introduce the reference Hilbert space for the found in the remaining black hole, adding a few qubits from input quantum state. See [5,30] for detailed discussions on the early radiation to the remaining black hole is enough to the use of the reference system. The reference Hilbert space construct the interior operators. The key observation is that is denoted by A¯. The entangled partner of B is denoted by B¯. the reconstruction of the interior operators can be seen as The unitary dynamics U of a black hole acts on AB ≃ CD. the Hayden-Preskill thought experiment, and hence the The Hilbert space dimension of a subsystem R is denoted method from [12] can be used to explicitly write down the by dR while the number of qubits on R is denoted by nR. interior operators. This observation enables us to show that Entropies are computed as binary entropies. the black hole interior modes are robust against perturba- The Hayden-Preskill thought experiment with Haar tions on the early radiation due to scrambling dynamics. random unitary with various global symmetries was studied From the perspective of the AMPS puzzle, our result independently by Nakata, Wakakuwa and Koashi. They appears to suggest that interior soft operators need to be pointed out that, for Uð1Þ symmetry with generic input reconstructed in the early radiation instead of the remaining black hole. While this resonates with previous approaches bundled under “A ¼ RB” or “ER ¼ EPR” [13–17], these run into various paradoxes [18–20] (See [13,21] for summaries) as the construction may lead to apparent nonlocal encoding between the interior and the early radiation. Relatedly, construction of interior operators is state-dependent, which may suffer from a number of potential inconsistencies with quantum mechanics. For further discussion, please see a selection (but by all means not a complete set) of recent work [14,17–19,22–28]. In an accompanying paper, we will debug these problems and present constructions of interior oper- ators which are local (i.e., without involving the early radiation) and state-independent (i.e., no dependence on FIG. 1. The Hilbert space structure.

086001-2 SOFT MODE AND INTERIOR OPERATOR IN THE … PHYS. REV. D 100, 086001 (2019) states, the recovery requires collecting extensive number of qubits. A similar conclusion is obtained in the Appendix for ð4Þ nonsymmetric input states. Upon completion of this work, we became aware of an independent work [31] which addresses the black hole evaporation process with distinc- tion between hard and soft modes. This diagram has one input leg (index) and one output leg. One may bend the line and construct the following A. Summary of diagrammatic techniques diagram: In this paper, we will extensively use diagrammatic tensor notations in order to express wave functions and ð5Þ operators as well as physical processes. Here we provide a brief tour of key properties for readers who are not familiar with these techniques. which is the same trivial line, but with two output legs Wave functions and operators are represented by instead of one in and one out. This diagram represents an unnormalized EPR pair defined on H⊗2. Another important example involves a transpose of an operator:

ð1Þ ð6Þ which can be also explicitly written as X X X where jψi¼ Tjjjihψj¼ Tj hjj O ¼ Oijjiihjj: ð2Þ j j ij X X T O ¼ Oijjiihjj O ¼ Ojijiihjj: ð7Þ i;j i;j By using these tensors as building blocks, one can express various physics in a graphical manner. For instance, an OT O expectation value can be represented by Here the transpose exchanges the input and output of . The original diagram with O represents a physical process where an arbitrary input wave function jψi is acted by O and Ojψi appears as an output. The second diagram with OT describes a physical process which involves three ⊗3 Hilbert spaces of the same size H ¼ H1 ⊗ H2 ⊗ H3: ð3Þ

ð8Þ

In order to associate a physical process to an equation like hψjOn O1jψi,oneneedstoreaditfromtherighttothe Here H1 supports an arbitrary input wave function whereas left, i.e., the initial state jψi is acted by O1;O2; T H2 ⊗ H3 starts with the EPR pair. Then, the transpose O sequentially and then is projected onto jψi.Inthedia- acts on H2, and then the system is projected onto the EPR grammatic notation, one needs to read the figure from the pair on H1 ⊗ H2. The outcome on H3 is Ojψi. One may bottom to the top, i.e., the time flows upward in the diagram. represent this explicitly as the following equation: A key (yet sometimes confusing) feature of tensor diagrams is that the same tensor can represent different ⊗ I I ⊗ OT ⊗ I ψ ⊗ ∝ O ψ : physical processes depending on which tensor indices are ðhEPRj12 3Þð 1 2 3Þj i1 jEPRi23 j i3 used as inputs and outputs. Let us lookP at a few important ð9Þ examples. An identity operator, I ¼ j jjihjj, can be expressed as a straight line (i.e., a trivial tensor) since its As the above examples suggest, one may interpret an inputs and outputs are the same: operator as a quantum state and vice versa. To assign

086001-3 BENI YOSHIDA PHYS. REV. D 100, 086001 (2019) physical interpretations to the diagram, we simply read it where summations are implicit with k ¼ 1; …dA¯ , l ¼ from the bottom to the top. 1; …;dB¯ , m ¼ 1; …;dC, and o ¼ 1; …;dD. Triangles re- present normalized isometries. For instance, the input state AA¯ II. HAYDEN-PRESKILL WITH in is given by CODE SUBSPACES XdA¯ In this section, we revisit the Hayden-Preskill thought 1 pffiffiffiffiffi jkiA¯ jkiA: ð11Þ d ¯ experiment and the AMPS problem for quantum black A k¼1 holes which are entangled through “code subspaces.” Following the previous works, we will model a black Here jkiA spans only a subspace of A. The choice of jkiA is hole as an n-qubit quantum system with n ¼ S where BH not important as the system evolves by Haar random S is the coarse-grained entropy which is proportional to BH unitary. the area of the black hole. We will also assume that the Each subsystem, AAB¯ BCD¯ , admits the following physi- dynamics of a black hole is given by a Haar random S cal interpretation in the Hayden-Preskill thought experi- unitary operator. Let E be the entanglement entropy A A¯ between the black hole and all the other d.o.f. including ment. and correspond to Hilbert spaces for the input quantum state and its reference system respectively. An the early radiation. Previous discussions on the Hayden- d Preskill thought experiment and the AMPS thought input quantum information is drawn from A¯ -dimensional A¯ d A experiment concern maximally entangled black holes subspace and is encoded into A-dimensional subspace S S before thrown into a black hole. B corresponds to the with BH ¼ E. Here we will investigate a black hole ¯ S >S coarse-grained Hilbert space of a black hole whereas B with BH E. Black holes before the Page time satisfy this condition. Similar situations have been previously corresponds to the subspace where the black hole is S >S considered by Verlinde and Verlinde [32]. In particular, entangled. The initial black hole satisfies BH E since S >S they pointed out that taking BH E enables us to S d S d : construct partner operators of the outgoing mode in the BH ¼ log B E ¼ log B¯ ð12Þ d.o.f. of the remaining black hole. Furthermore, they proposed a scenario to resolve the AMPS puzzle by using Finally, C and D corresponds to the remaining black hole this effect. and the outgoing Hawking radiation respectively. The aim of this section is to study the consequence of U U† S >S The Haar average formula with two s and two sis taking BH E in the Hayden-Preskill thought experi- S ≫ S ment. We find that, if a black hole with BH E evolves Z under Haar random unitary operator, the Hayden-Preskill dU U U U U i j i j i0 j0 i0 j0 recovery cannot be performed unless one collects OðnÞ 1 1 2 2 1 1 2 2   qubits from the outgoing Hawking radiation. In fact, 1 ¼ δi i0 δi i0 δj j0 δj j0 þ δi i0 δi i0 δj j0 δj j0 ð13Þ we find that the Hayden-Preskill recoverability and the d2 − 1 1 1 2 2 1 2 2 1 1 2 2 1 1 1 2 2 smoothness of the horizon are mutually incompatible phenomena within the applicability of toy descriptions 1   based on Haar random unitary operator. − δi i0 δi i0 δj j0 δj j0 þ δi i0 δi i0 δj j0 δj j0 ð14Þ dðd2 − 1Þ 1 1 2 2 1 2 2 1 1 2 2 1 1 1 2 2 A. Haar integral where d ¼ 2n. Approximating d2 − 1 ≈ d2, this lets us The quantum state we are interested in is the following 2 2 compute the Haar average of TrfρCg and Trfρ ¯ g: with dA¯ ≤ dA and dB¯ ≤ dB: BD Z Z 1 dU ρ2 dU U U U U Trf Cg¼d2 d2 klmo klm0o k0l0m0o0 k0l0mo0 A¯ B¯ ð15Þ

Z Z 1 dU ρ2 dUU U U U ; Trf BD¯ g¼d2 d2 klmo kl0mo0 k0l0m0o0 k0lm0o A¯ B¯ ð16Þ ð10Þ and after simple calculations of delta functions, we find

086001-4 SOFT MODE AND INTERIOR OPERATOR IN THE … PHYS. REV. D 100, 086001 (2019) Z 1 1 d 1 dU ρ2 − C − The mutual information increases by two as the Trf Cg¼ þ 2 D dC dDdA¯ dB¯ d ddCdA¯ dB¯ number of qubits in increases by one. In order d2 ≫ 1 1 1 for this regime to be present, we need A¯ . ≈ 2 dA¯ þ ð17Þ (c) (large dD)FordD ≫ d , we find dC dDdA¯ dB¯ dB¯ Z 1 1 1 1 dDdA¯ dB¯ ≫ dC;dDdB¯ ≫ dCdA¯ ð24Þ dU ρ2 − − Trf BD¯ g¼ þ dDdB¯ dCdA¯ ddCdA¯ ddDdA¯ 2Ið2ÞðA;¯ BD¯ Þ ≈ d2 ; 1 1 A¯ ð25Þ ≈ þ ð18Þ dDdB¯ dCdA¯ implying that the correlation is nearly maximal. where terms with 1=d factors are ignored. C. Smoothness B. Recoverability Next let us turn our attention to the AMPS problem. Here The Hayden-Preskill thought experiment concerns we are interested in whether partners of outgoing mode recoverability of the input quantum state on A¯ by having operators on D can be reconstructed on the remaining black C C D an access to both the early radiation B¯ and the outgoing hole or not. This is possible if and only if and retain mode D. As such, the recoverability of a quantum state can strong quantum correlations. Hence, as a measure of 2 be studied by asking whether one can distill EPR pairs smoothness of the horizon, we will use Ið ÞðC; DÞ. between A¯ and DB¯ . One can analyze the recoverability by We compute the Haar average of the mutual information C D studying quantum correlations between A¯ and BD¯ in jΨi in between and : Eq. (10). Hence, as a measure of recoverability, we will use Z d d2 2 ¯ ¯ ð2Þ C D the R´enyi-2 mutual information, defined by Ið ÞðA; BDÞ¼ dU 2I ðC;DÞ ≈ : ð26Þ d d ¯ d ¯ d Sð2Þ Sð2Þ − Sð2Þ 2 D A B þ C A¯ þ BD¯ A¯ BD¯ by following [30]. We can compute the R´enyi-2 mutual information We also derive the mutual information between A¯ B¯ and D A¯ BD¯ between and : in order to illustrate the monogamy of entanglement. Z The two mutual information are related as follows d d ¯ d ¯ d Ið2ÞðA;¯ BD¯ Þ D A B þ C dU 2 ≈ dA¯ : ð19Þ d d d d ð2Þ D B¯ þ C A¯ SD 2 ð2Þ ¯ ¯ 2 dD 2I ðAB;DÞ ¼ ≈ ð27Þ 2Ið2ÞðC;DÞ 2Ið2ÞðC;DÞ Let us look at recoverability in three regimes. 1 2 (a) (small dD)Ford ≫ dD, we find dA¯ dB¯ where we approximated ρD by a maximally mixed state. 1 2 (a) (small dD)Ford ≫ d , we find dA¯ dB¯ D dDdA¯ dB¯ ≪ dC;dDdB¯ ≪ dCdA¯ ð20Þ Ið2ÞðC;DÞ 2 Ið2ÞðA¯ B;D¯ Þ 2 ≈ dD; 2 ≈ 1 ð28Þ leading to implying that Ið2ÞðA¯ B;¯ DÞ ≈ 0 and does not increase as Ið2ÞðA;¯ BD¯ Þ 2 ≈ 1; ð21Þ dD increases. 2 1 (b) (intermediate and large dD)FordD ≫ d , we find dA¯ dB¯ implying that A¯ and BD¯ are not correlated. d 2 A¯ 2 1 d d d ¯ d ¯ (b) (intermediate dD)Ford d ≫ dD ≫ d d d , we find Ið2ÞðC;DÞ Ið2ÞðA¯ B;D¯ Þ D A B B¯ A¯ B¯ 2 ≈ ; 2 ≈ ð29Þ dA¯ dB¯ d dDdA¯ dB¯ ≫ dC;dDdB¯ ≪ dCdA¯ ð22Þ implying that Ið2ÞðA¯ B;¯ DÞ becomes large as the 2 leading to number of qubits in D increases while Ið ÞðC; DÞ remains unchanged. 2 d d ¯ d ¯ 2Ið2ÞðA;¯ BD¯ Þ ≈ D A B : d ð23Þ D. Physical interpretation S d Recall that the initial black hole satisfies BH ¼ log B 2Strictly speaking, the R´enyi-2 mutual information is not an and SE ¼ log dB¯ in our toy model. For the recovery in the entanglement monotone in general. See [33] for details. Hayden-Preskill thought experiment, we need

086001-5 BENI YOSHIDA PHYS. REV. D 100, 086001 (2019) sffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi accessed easily from the outside whereas C corresponds to dB dD ≥ dAdA¯ : ð30Þ complex ones. dB¯ III. HAYDEN-PRESKILL WITH Uð1Þ SYMMETRY When a black hole is maximally entangled with dB ¼ dB¯ , the above lower bound reduces to In this section, we study the Hayden-Preskill thought pffiffiffiffiffiffiffiffiffiffi experiment in the presence of conserved quantities. For dD ≥ dAdA¯ ð31Þ simplicity of discussions, we will consider systems with Uð1Þ global symmetry where the total spin in the reproducing the result in [12]. z-direction is preserved. We will find that the Hayden- Here we are interested in cases with dB ≫ dB¯ where the Preskill recovery is possible only if an input quantum state coarse-grained entropy is larger than the entanglement is embedded into a subspace with fixed charges. entropy. Namely, if the number of qubits in B and B¯ U satisfy nB − nB¯ ∼ OðnÞ, then Bob needs to collect an A. ð1Þ-symmetric system extensive number of qubits. As such, if dB ≫ dB¯ , simple Our motivations to study Uð1Þ-symmetric systems are recovery is not possible (unless one collects an extensive two-fold. The first obvious motivation is that we want to number of qubits). If we take dB¯ ¼ 1,wehavedD ≥ address the Hayden-Preskill recoverability and the AMPS pffiffiffiffiffiffi pffiffiffi dA dB ∼ Oð dÞ implying that Bob needs to collect more problem in the presence of symmetries. The second, less than a half of the total qubits for reconstruction as pointed obvious, motivation is to study the same set of questions for out by Page [2]. quantum systems which conserve total energy. The failure of recovery for dB ≫ d ¯ can be understood Let us illustrate the second point. In the discussions from B B Ið2Þ C; D d the previous section, we treated qubits on as coarse- from the calculation of ð Þ. When D is small, most S d D grained d.o.f. of a black hole with BH ¼ log B¯ . Instead, of qubits in are entangled with the remaining black hole B C A¯ B¯ one might want to interpret as physical qubits on the and do not reveal any information about . Once B¯ Ið2ÞðC;DÞ d boundary quantum system and the subspace as a typical 2 reaches a stationary value of 2, entanglement ðdA¯ dB¯ Þ energy subspace of a black hole at finite temperature. between D and A¯ B¯ starts to develop. From the perspective Namely, if we consider the entangled AdS black hole, this ð2Þ S S d d of the firewall puzzle, the increase of I ðC; DÞ appears to amounts to assuming that E ¼ BH ¼ log B¯ while log B suggest that partner operators of the outgoing mode D can qubits are placed at UV. In this interpretation, our calcu- be found on the remaining black hole C. Verlinde and lation would suggest that the recovery in the Hayden- Verlinde employed this mechanism as a possible resolution Preskill thought experiment is not possible for a maximally of the firewall puzzle and made an intriguing relation to entangled black hole at finite temperature. However, this theory of quantum error-correction [32]. These observa- conclusion is weird as a physical process akin to the tions illustrate that Hayden-Preskill and the smoothness of Hayden-Preskill recovery has been recently found [34]. the horizon are mutually complementary phenomena when It has been also argued that scrambling in a sense of decay U is a Haar random unitary which thoroughly mixes the of out-of-time order correlator is sufficient to perform Hilbert space AB. Namely, the presence of partner oper- recovery protocols even at finite temperature [12]. ators in C requires that D is correlated with C whereas the Hence, something must be wrong in this interpretation. recoverability of an input quantum state in the Hayden- The error in the aforementioned argument can be traced Preskill thought experiment requires that D is entangled back to the approximation of the black hole dynamics by with A¯ B¯ . In fact, the tradeoff between the smoothness and Haar random unitary. Since Haar random unitary does not recoverability is strikingly sharp; once D becomes large conserve energy, it brings quantum states on the input enough to start releasing information about A¯ via increase of Ið2ÞðA;¯ BD¯ Þ, the growth of Ið2ÞðC; DÞ stops. Finally, we make a comment on the complexity of performing recovery protocols in the Hayden-Preskill thought experiment. For dB ¼ dB¯ , a simple recovery protocol is known to exist [12]. When applying this method to the case with large dD, it is crucial to identify d.o.f. in D which is not entangled with C. Under chaotic dynamics of a black hole, it is plausible to expect that such d.o.f. become non-local inside D and require complex operations. Hence, performing the Hayden-Preskill recovery may be unphysical when dB ≫ dB¯ . In fact, it may be more correct FIG. 2. Uð1Þ-symmetric Haar random unitary as a toy model of to say that D corresponds to simple d.o.f. which can be energy conserving dynamics.

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Hilbert space AB to outside of the window of typical energy ðmAÞ ðmBÞ Define dA¯ ≡ dimðHA Þ and dB¯ ≡ dimðHB Þ. The input states, and hence recovering quantum states becomes harder. state in AA¯ is given by This observation motivates us to consider the AMPS problem d and the Hayden-Preskill thought experiment by considering 1 XA¯ pffiffiffiffiffi k k some version of Haar random unitary dynamics which j iA¯ j symiA ð35Þ d ¯ captures physics of energy conserving systems. A k¼1 One can mimic such a situation by considering Uð1Þ- k symmetric Haar random unitary where the total Uð1Þ where j symi are states with fixed charges. For instance, one 100 010 001 n 3 m 1 charges are conserved. To be concrete, a Uð1Þ-symmetric may consider j i, j i, j i for A ¼ and A ¼ . system can be modeled as a set of n qubits where basis After a Uð1Þ-symmetric unitary evolution, the total states can be expressed as an n-binary string and its total charge m ¼ mA þ mB is conserved. The output Hilbert charge is defined as the number of 1s: space is (assuming nC ≥ m ≥ nD)

nD Xn H ⨁ Hðm−QÞ ⊗ HðQÞ: i ; …;i m i i 0; 1: out ¼ C D ð36Þ j 1 ni ¼ k k ¼ ð32Þ Q¼0 k¼1 It is convenient to define We will consider the Uð1Þ-symmetric Haar random unitary: ðm−QÞ ðm−QÞ ðQÞ ðQÞ n n dC ¼ dimðHC Þ dD ¼ dimðHD Þ U ⨁ Um H ⨁ Hm 33 ¼ ¼ ð Þ XnD m¼0 m¼0 d H dðm−QÞdðQÞ: ¼ dimð outÞ¼ C D ð37Þ Q 0 where Um is independently Haar random acting on each ¼ H fixed-charge subspace m (see Fig. 2). m−Q Q ð Þ nC ð Þ nD With energy conservation in mind, our goal was to We have dC ¼ðm−QÞ and dD ¼ðQ Þ. consider a toy model of scrambling dynamics which mixes Much of the analysis resembles the one in the previous eigenstates with roughly equal energies. By viewing each section. The only complication is the treatment of delta fixed-charge subspace Hm as eigenstates with roughly functions when the Hilbert space does not have a direct equal energies, a Uð1Þ-symmetric Haar random unitary product structure. The quantum state jΨi of the Hayden- can capture dynamics which throughly mixes quantum Preskill thought experiment can be expressed as follows states from the small typical energy window. In a realistic 1 quantum system, the Hilbert space structure does not Ψ pffiffiffiffiffiffiffiffiffiffi U k l s; t j i¼ klðs;tÞj iA¯ j iB¯ jð ÞiCD ð38Þ decompose into a diagonal form in an exact manner. dA¯ dB¯ Here we hope to capture some salient feature of energy conserving systems in this simplified toy model. where (s, t) indicates that summations over (s, t) should be taken according to Eq. (36). We find B. Haar integral Z R m 2 Let us denote the local charge on by R (i.e., the total dU Trfρ g ’ R A C number of 1 sin ). We will consider the cases where and Z B m m 1 have fixed charge A and B respectively. The quantum ¼ dU Ukl s;t U 0 Uk0l0 s0;t0 U 0 0 0 ð39Þ state of our interest is d2 d2 ð Þ klðs ;tÞ ð Þ k l ðs;t Þ A¯ B¯

ρ2 and a similar equation for Trf BD¯ g. In using the Haar formula, we need to apply delta functions to (s, t). For 2 instance, the first term of the Haar integral in TrfρCg is ð34Þ 1 k; l;k0; l0 s; t d2 d2 · ðnumber of Þ · ðnumber of ð Þ and A¯ B¯ ðs0;t0Þ with s ¼ s0Þð40Þ where filled triangles represent normalized isometries 1 XnD XnD d2 d2 dðm−QÞ dðQÞ 2 dðm−QÞ dðQÞ 2: onto fixed-charge subspaces. The input quantum states ¼ 2 2 · A¯ B¯ · C ð D Þ ¼ C ð D Þ d ¯ d ¯ AB H HðmAÞ ⊗ HðmBÞ A B Q¼0 Q¼0 on can be spanned in in ¼ A B where ðmRÞ ð41Þ HR represents a subspace of HR with fixed charge mR.

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It is convenient to define When Q ≥ mA and nD ≥ nA, we have m − Q ≤ mB and nC ≤ nB. So, the right-hand side (RHS) is smaller than XnD XnD unity. Hence it suffices to take W ≡ dðm−QÞ 2 dðQÞ W ≡ dðm−QÞ dðQÞ 2: C ð C Þ · D D C · ð D Þ   Q 0 Q 0 ¼ ¼ nD ≫ dA¯ ð50Þ ð42Þ Q

We find in order to satisfy Eq. (48). Z While one cannot easily satisfy Eq. (48) by taking large W W nD for very small Q, contributions from such cases are dU ρ2 ≈ D C Trf Cg 2 þ 2 negligibly small. In fact, for large nC, we have d d dA¯ dB¯ Z     WC WD 2 dU ρ2 ≈ Q 0 nD Q 0 nD Trf BD¯ g 2 þ 2 ð43Þ Wð Þ ≃ Wð Þ ϵ2Q Wð Þ ≃ Wð Þ ϵQ 51 d dB¯ d dA¯ C C Q D D Q ð Þ after ignoring terms suppressed by 1=d. p ðQÞ ðQÞ with ϵ ¼ 1−p. Since both WC and WD are (approxi- mately) proportional to a binomial distribution and its C. Recoverability square respectively, contributions to WC and WD are Let us find the criteria for recovery. If nD is large enough dominated by Q ∼ OðnDÞ. For such Q, it suffices to take such that nD ≫ nA in order for Eq. (48) to hold. Hence, we conclude that nD ≫ nA is sufficient for the recovery to be possible. WD WC We will present concrete recovery protocols in Sec. IV. ≫ ; ð44Þ dA¯ dB¯ D. Smoothness we will have Next, let us compute the mutual information IðC; DÞ. For simplicity of discussion, we focus on the case where 2 1 2 WD 2 WD ρ ρ ≈ ρ ≈ dA ¼ 1, i.e., with no input state. In this case, ρCD is a Trf A¯ g¼d Trf Cg d2 Trf BD¯ g d2d ð45Þ A¯ A¯ maximally mixed state with charge m:

Ið2Þ A;¯ BD¯ ≈ 2 d n implying ð Þ log2 A¯ . Hence the recovery will 1 XðmÞ n ρ k k : be possible. As such, we need to find the condition on D CD ¼ n j symih symj ð52Þ such that Eq. (44) holds. ðmÞ k¼1 Let us write WC and WD as follows: This density matrix can be decomposed into a black XnD XnD diagonal form: ðQÞ ðQÞ WC ¼ WC WD ¼ WD ð46Þ Q¼0 Q¼0 XnD ðQÞ ðQÞ ðm−QÞ ðQÞ ρCD ¼ PrðQÞρCD ρCD ¼ σC ⊗ σD ð53Þ where Q¼0        2 2 m−Q Q Q nC nD Q nC nD ð Þ ð Þ ð Þ ð Þ where σC and σD represent maximally mixed states of WC ≡ WD ≡ : m − Q Q m − Q Q charge m − Q and Q respectively. The probability weight is ð47Þ given by

nD nC We look for the condition for the following inequality: ð Q Þðm−QÞ PrðQÞ¼ n : ð54Þ ðmÞ WðQÞ WðQÞ D ≫ C : ð48Þ nC Q dA¯ dB¯ For large ,Prð Þ can be approximated by a binomial distribution with p ¼ m=n. The mutual information is By writing it down explicitly, we have given by

n nD nC XD 1 ð Q Þ ðm−QÞ I A; B − Q Q ≃ n : ≫ n : ð49Þ ð Þ¼ Prð Þ log Prð Þ log D ð55Þ d ¯ B 2 A ðmBÞ Q¼0

086001-8 SOFT MODE AND INTERIOR OPERATOR IN THE … PHYS. REV. D 100, 086001 (2019) pffiffiffiffiffiffi Since the variance of the binomial distribution is ∼ nD, only. While we will study Haar random dynamics, we pffiffiffiffiffiffi we can approximate it as a distribution over nD-level states believe that similar conclusions hold for any “scrambling” 1 systems in a sense of [12] where scrambling is defined with which can be encoded in ∼ 2 log nD bits. The above argu- ment suggests that these “charge-bits” are strongly correlated respect to out-of-time order correlation functions. with those on C. However, as the block diagonal form We have discussed the recoverability in the Hayden- suggests, the correlation between C and D is purely classical. Preskill thought experiment without presenting explicit recovery protocols. The original work by Hayden and Preskill was essentially an existence proof of recovery E. Physical interpretation protocols when the dynamics is given by Haar random Let us interpret the Uð1Þ-symmetric Haar random unitary. Recently the author and Kitaev have constructed unitary as an energy conserving dynamics. Then, the simple recovery protocols which work for any scrambling Hilbert space B corresponds to physical qubits of a systems whose out-of-time order correlation functions decay ¯ quantum system while B corresponds to the coarse-grained [12]. Similar recovery protocols can be applied to our Uð1Þ- Hilbert space which is also the typical energy subspace symmetric toy model. For simplicity of discussion, we will at finite temperature. The black hole is maximally focus on a probabilistic recovery protocol. A deterministic entangled with the early radiation in this interpretation; protocol can be also constructed by following [12]. Since the S S d D BH ¼ E ¼ log B¯ . The Hawking radiation contains analysis in this section is a simple extension of the original symmetric and nonsymmetric modes which can be inter- work, we keep the presentation brief. preted as soft and heavy modes respectively. When the Consider the following quantum state: input is symmetric with fixed charge, we found that the recovery is possible by collecting Oð1Þ qubits. However, as shown in Appendix, when the input is nonsymmetric with variance in charge values, we found that the recovery ð57Þ requires nD to be extensive. With energy conserving systems in mind, this implies that input quantum states should be encoded in soft modes for recovery. We also see that the correlation between C and D is purely classical and A0A¯ 0 results from charge conservation. With energy conserving where Bob prepared a particular quantum state on AA¯ systems in mind, it corresponds to correlations of heavy which is identical to the one on , and applied the U B0 A0 modes under energy conservation. Therefore, we interpret complex conjugate sym on and . The initial quantum these heavy modes as Hawking quanta whereas soft modes state on BB0 is a maximally entangled symmetric state: Pd 1ffiffiffiffi B¯ are some entity responsible for the Hayden-Preskill recov- p l ⊗ l 0 d l¼1 j symiB j symiB . In the diagram, the unfilled ery and scrambling dynamics. B¯ dot with mB represents a normalized projection onto the From the perspective of the AMPS problem, calculations m B0D in this section motivate us to consider two different kinds of subspace with total charge B. Bob has an access to DD0C0A¯ 0 A¯ A¯ 0 operators on D. The off-diagonal operators are the ones (or ), and his goal is to distill EPR pairs on . D Bob’s strategy is to perform a projection onto EPR pairs which change the local charge in . The partners of those 0 DD0 ΠðDD Þ off-diagonal operators can be identified in C as operators on . Denoting the projector by EPR , the probability which decrease and/or increase the total charge. This is due of measuring EPR pairs is to classical (diagonal) correlations between C and D.On the other hand, the diagonal operators are the ones which leave the total charge unchanged up to phases. Such operators can be explicitly written as follows: X W ¼ eiθn jnihnj: ð56Þ n

Unlike off-diagonal operators, the interior partners of diagonal operators W cannot be found in C. Hence, the nonlocality problem remains for partners of symmetric (soft) modes. ð58Þ

IV. RECOVERY VIA SOFT MODE where the filled dots in the middle represent projectors onto The goal of this section is to show that the Hayden- EPR pairs. Recovery is successful if Bob can distill EPR Preskill recovery can be performed via symmetric modes pairs on A¯ A¯ 0. Let us denote the fidelity of the distillation,

086001-9 BENI YOSHIDA PHYS. REV. D 100, 086001 (2019) conditioned on the measurement of EPR pairs on DD0,by classical correlations between C and D. As such, we will F DD0 EPR. The probability of measuring EPR pairs on both focus on the partner of symmetric (soft) operators. and A¯ A¯ 0 is The main result of this section is the observation that construction of interior partner operators can be interpreted as the Hayden-Preskill recovery problem in disguise. This observation enables us to show that the black hole interior modes are robust against perturbations on the early radi- ation due to scrambling dynamics. Namely, even if almost all the qubits, except a few, in the early radiation are damaged, interior partner operators can be still constructed. We assume dA ¼ 1 although our construction works well for cases with dA > 1 too.

A. Interior from Hayden-Preskill ð59Þ Interestingly, reconstruction of the interior operators can be performed by using a procedure similar to the Hayden- P P F Both EPR and EPR EPR can be explicitly computed. Preskill recovery. Let us begin by defining what we mean From the above diagrams, we notice by interior partner operators. The quantum state of our d d interest is as follows: P ρ2 B¯ P F ρ2 B¯ : EPR ¼ Trð BD¯ Þ EPR EPR ¼ Trð CÞ ð60Þ dD dDdA¯ For Uð1Þ-symmetric Haar random unitary, we obtain W d P ≃ P F ≃ D B¯ ð64Þ EPR EPR EPR ð61Þ dA¯ dD where subleading terms are suppressed for nD ≫ nA. F ≃ 1 Hence, upon postselection, we have EPR implying nearly perfect recovery. The aforementioned recovery protocol can be modified where D is the outgoing mode and C is the remaining to use only the symmetric mode. Namely, by applying a black hole. Our task is the following; given a symmetric projection onto maximally entangled symmetric states on (diagonal) operator OD, find the partner operator VCB¯ such DD0, EPR pairs can be distilled on A¯ A¯ 0. Let us denote the that DD0 projector onto entangled states with charge Q by Πð Þ. Q O ⊗ I Ψ ≃ I ⊗ V Ψ : This projector is related to the EPR projector by ð D CB¯ Þj i ð D CB¯ Þj i ð65Þ   XnD Graphically the above equation reads ðDD0Þ 1 nD ðDD0Þ Π ¼ ΠQ : ð62Þ EPR 2nD Q Q¼0

Let us denote the probability amplitude for measuring ðDD0Þ ΠQ by PQ.ForUð1Þ-symmetric Haar, we find    n n d P ≃ P F ≃ C D B¯ Q Q EPR 2 ð63Þ m − Q Q d dA¯ ð66Þ for large nD which satisfies Eq. (49). Hence, the Hayden- Preskill recovery can be carried out via symmetric modes. Note that the existence of a partner operator is guaranteed. The question concerns how to write it down. V. CONSTRUCTION OF INTERIOR OPERATOR To address this problem, it is convenient to interpret the Finally, we discuss the construction of partner operators above quantum state jΨi as a map from D to CB¯ where that would describe the interior mode in our toy model. The (symmetric) quantum states on D are encoded into CB¯ . partner of nonsymmetric (heavy) operators on D can be To make this interpretation more concrete, let us deform the easily constructed on the remaining black hole C due to the diagrams in Eq. (66) in the following manner:

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UT U Here sym represents the transpose of sym, which results from flipping the diagram upside down. The circle on B¯ represent a projection onto a symmetric subspace with fixed charge. An elongated circle also represents a projection onto a symmetric subspace with fixed charge. Note that the projector does not factor on D ⊗ C¯ . Also note that the O UT projector commutes with symmetric operator D and sym. T ¯ Our task is to reconstruct a partner of OD on BC.Of ð67Þ course, it is possible to find a partner operator on the early B¯ OT UT radiation by simply time-evolving D by sym. However, “ ” the construction of a partner operator VCB¯ is not unique Here the original diagrams had three upward output legs ¯ whereas the deformed graphs use the D-index as an “input” since BC is larger than D. The nonuniqueness of the by bending it downward. The new diagrammatic equation interior partner operator is closely related to the fact that the OT V above quantum state jΨi can be interpreted as a quantum represents a map from D¯ to a partner operator CB¯ . The map itself can be explicitly expressed as follows: error-correcting code where (symmetric) quantum states on D are encoded into CB¯ . Here we want to find an alternative representation which involves as few qubits on B¯ as possible. A key observation is that, in the above diagram, the outgoing mode D can be interpreted as an input Hilbert space for the Hayden- Preskill thought experiment where the unitary evolution UT B¯ ð68Þ is given by sym. To be explicit, let 0 be some small B¯ n ¯ ∼ subsystem of the early radiation which contains B0 O 1 n ¯ ≫ n B¯ ð Þ qubits such that B0 D. Let 1 be the complement ¯ ¯ of B0 in B. The recoverability in the Hayden-Preskill thought experiment for symmetric modes implies that there V ¯ B¯ C exists a partner operator B0C supported on 0 :

This is an isometric embedding (preserving inner products) from a smaller Hilbert space D¯ onto a larger one CB¯ .Itis worth recalling that B¯ and C correspond to the early radiation and the remaining black hole respectively. We can associate a physical process to the deformed diagram by reading it from the bottom to the top. Namely, it begins with a system consisting of an arbitrary initial state on D¯ and a maximally entangled symmetric state on BB¯ . Then U the system evolves by sym, and is projected onto the EPR pair on DD¯ to obtain an output wave function on CB¯ . ð70Þ One can further simplify the above map (and make the relation to the Hayden-Preskill problem more explicit) U by rotating the box of the unitary operator sym by While we assumed dA ¼ 1, the above procedure works for ¯ 180 degrees. Carefully redrawing the diagram, we obtain dA > 1 cases since B0 can be chosen arbitrarily. the following(s): We have observed that the reconstruction of the interior operators is essentially the Hayden-Preskill thought experi- ment. This suggests the following result: (i) While the partner operator of the outgoing soft mode D cannot be found in the remaining black hole C, ¯ adding a few extra qubits B0 from the early radiation ð69Þ B¯ to C is enough to construct the interior operator. UT Due to the scrambling nature of sym, one may ¯ choose any set of a few qubits B0 in order to construct the interior operator as long as out-of-time ¯ order correlation functions between D and B0 decay.

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B. Fault-tolerance Hayden-Preskill thought experiment by using a toy model From the outside quantum mechanical viewpoint, the with energy conservation. Within the validity of the toy AMPS problem (or the nonlocality problem) originates model, our calculation suggests that the Hawking radiation from the fact that IðC; DÞ is small and the partner operator corresponds to heavy modes whereas the Hayden-Preskill of D cannot be constructed in the remaining black hole C. thought experiment must concern soft modes only. The The construction of interior operators via the Hayden- correlation between the remaining black hole and the Preskill recovery protocols enables us to construct a partner outgoing radiation is found to be classical. The classical ¯ ¯ correlation remains due to the energy conservation and due operator on CB0 where B0 is a small subsystem with ¯ to the fact that the black hole in our toy model is entangled jB0j⪆jDj. While we still need to include a few extra qubits with the early radiation only through soft modes. Our toy from the early radiation B¯ , the construction is “almost” model suggests that the off-diagonal correlation decoheres inside the remaining black hole C. since the phases of heavy modes are scrambled by chaotic In fact, this observation sheds a new light on questions dynamics in soft modes. Finally, we observed that the concerning the robustness of the black hole interior procedure of reconstructing the soft part of the infalling against perturbations on the early radiation. Some previous mode can be interpreted as the Hayden-Preskill recovery works attempted to resolve the AMPS puzzle by using the protocol. As such, while the description of the infalling concept of quantum circuit complexity by drawing dis- mode may require the early radiation, only a few extra tinctions between simple and complex quantum operations qubits will be sufficient. In the reminder of the paper, we [15,35]. Namely, it has been argued that action of simple present discussions on relevant topics. quantum operations on the early radiation should not disturb the black hole interior operators [15]. Heuristic explanations for such fault-tolerant “encoding” of interior A. AMPS puzzle and scrambling operators have been presented by using an analogy with Our construction of interior operators via the Hayden- quantum error-correcting code [32,36–40]. However con- Preskill recovery phenomenon sheds a new light on the crete physical origin of robustness of the black hole interior AMPS puzzle through the lens of quantum information mode was not discussed from the outside quantum scrambling. For simplicity of discussion, let us consider a mechanical perspective. maximally entangled black hole at infinite temperature Our construction of interior operators suggests that robust- which is represented by n copies of EPR pairs. To recap ness of the black hole interior arises from scrambling briefly, the AMPS puzzle concerns an apparent tension dynamics. The essential point is that the small subsystem between descriptions by the infalling observer and the ¯ ¯ B0 in B can be chosen in an arbitrary manner as long as the outside observer. From the perspective of the outside ¯ out-of-time order correlation functions between B0 and D observer, the outgoing mode D is entangled with some ¯ decay [12,30]. In particular, let us consider a scenario where d.o.f. in the early radiation B. From the perspective of the some large perturbations are added on the early radiation B¯ infalling observer, the same outgoing mode D must be ¯ ¯ which would damage all the qubits on B1 (n − log jB0j entangled with some interior mode, leading to violation of qubits). The construction of the interior operator is immune the monogamy of quantum entanglement. to such a drastic error since it does not involve any qubits The intriguing lesson from our construction of interior ¯ B¯ from B1. In this sense, our construction is naturally fault- operators is that whoever possesses a tiny portion 0 of ¯ tolerant against perturbations on the early radiation B¯ . This the early radiation B will be able to distill a qubit which is ¯ unusual robustness of the encoding of interior operators entangled with the outgoing mode D. Namely, if B0 is results from the very fact that they can be constructed via the included to C as d.o.f. which the infalling observer can ¯ Hayden-Preskill recovery protocols. touch, she can distill an EPR pair between D and B0C. ¯ The remaining question concerns how to write down the On the other hand, if B0 is left untouched by the infalling ¯ ¯ ¯ interior operators explicitly. On a formal level, if the time observer, the outgoing mode D is entangled with B ¼ B0B1 UT D evolution sym is a scrambling unitary with the decay of and the outside observer can distill an EPR pair between out-of-time order correlation functions, the interior oper- and B¯ . See Fig. 3 for schematic illustration. In fact, these ators can be explicitly constructed by running the recovery statements can be made quantitative. Decay of out-of-time ð2Þ ¯ protocol proposed in [12]. We are currently working on order correlation functions implies that I ðD; B0CÞ is ð2Þ ¯ ð2Þ ¯ specific quantum systems and observe that the concept of nearly maximal [30]. Since I ðD; B0CÞþI ðD; B1Þ¼ operator growth perspective plays important roles in the ð2Þ ð2Þ ¯ 2S , this suggests that I ðD; B1Þ is close to zero. construction. This will be presented elsewhere. D Namely, the infalling observer may reconstruct a partner ¯ operator on B0C while the outside observer cannot recon- VI. DISCUSSIONS ¯ struct it on B1. ¯ In this paper, we addressed the tension between the It is worth emphasizing that B0 can be any subsystem ¯ ¯ smoothness of the horizon and the recoverability in the of B as long as jB0j⪆jDj. Hence, from the perspective

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quantities. Each subspace HE;Q defines a Hilbert space for the classical geometry determined by a set of E, Q. Heavy operators correspond to those which moves between differ- ent subspaces whereas soft modes correspond to d.o.f. inside HE;Q. In realistic situations, such a decomposition into the block diagonal form will be an approximate one. Also there are ambiguities on which d.o.f. should be treated as soft modes. For instance, depending on the problems of interest and energy/time scales as well as dimensionality, FIG. 3. The AMPS puzzle and scrambling. Alice, an infalling matter on the bulk may be considered as either soft or hard ¯ mode. Our toy model aims to capture the idealistic limit observer, can distill an EPR pair by accessing B0C while Bob, an ¯ ¯ ¯ outside observer, can distill an EPR pair by accessing B ¼ B0B1. where the decomposition becomes exact with sharp dis- ¯ ¯ Note that B0 can be any subsystem of the early radiation B as long tinction between heavy and soft modes. ¯ as jB0j⪆jDj due to scrambling property of U. It is worth recalling that separation of soft and heavy modes plays important roles in a number of problems in quantum gravity. To add a more speculative comment, the of the infalling observer, it is rather easy to disentangle the quantum error-correcting property in the AdS=CFT corre- outgoing mode D from the early radiation B¯ as she needs B¯ spondence is a manifestation of such separation of energy to touch only a few qubits in . In fact, it turns out that scales [41,42]. In this interpretation, the geometry (E, Q) quite generic perturbations to the black hole by the determines the codeword subspace HE;Q while the low infalling observer will disentangle the outgoing mode D B¯ without energy modes correspond to different codeword states in from the early radiation ever accessing the a quantum error-correcting code determined by (E, Q). early radiation B¯ . In an accompanying paper, based The “errors” in this quantum error-correcting code HE;Q are on this disentangling phenomena, we will propose a heavy operators which moves the system to the outside of possible resolution of the AMPS problem in a manner the codeword subspace HE;Q. In this sense, our toy model which is free from the nonlocality problem and the state- is an attempt to apply the idea of quantum error-correction dependence problem [29]. to dynamical problems in quantum gravity. Hence, we believe that our approach of using the Uð1Þ-symmetric toy B. Soft modes ≃ codewords model, despite being very simple, is applicable to a wide Our toy model crucially relies on the assumption variety of interesting questions in quantum gravity. that there is a clear separation between heavy and soft At this moment, however, it is unclear to us how the modes. The key insight behind this simplification is that black hole evaporates and eventually gets entangled only a few thermodynamic quantities, such as energy and through soft modes. On one hand, if we assume that the charge, determine the underlying classical geometry. underlying geometry changes adiabatically during evapo- This is essentially the statement of the no hair theorem ration, then it is reasonable to assume that the fluctuation of of classical black holes. However, there are many black energy, or more generically heavy modes, is greatly sup- hole microstates which are consistent with the given pressed. On the other hand, as is clear from the calculation classical geometry. The soft mode, discussed in the toy of IðC; DÞ in our toy model, the process of emitting the model, aims to capture all of these extremely low energy Hawking radiation does introduce fluctuations whose d.o.f. While it is unclear to us to what extent this toy energy scale is much larger than soft modes by definition. model can capture the actual physics, it is concrete Hence, in order for our toy model to be applicable, there and simple enough to make theoretically verifiable needs to be some physical mechanism to suppress the predictions. energy variance in a dynamical manner. One interesting point is that a simple toy model with Uð1Þ symmetry can be interpreted as a energy conserving C. Factorization of Hilbert space system and naturally gives rise to heavy and soft modes. Throughout the paper we used a toy model that repre- In a more generic setting, we may imagine an approximate sents a black hole as a system of qubits. This is a drastic decomposition of the full Hilbert space into a block simplification building on two nontrivial assumptions. diagonal form: First, it is assumed that the black hole Hilbert space is discrete. Second, it is assumed that the Hilbert space of the H ≈ ⨁HE;Q ð71Þ E;Q black hole is factorizable. The first assumption is relatively well justified as it stems from the finiteness of the black where E represents the energy and Q represents the hole entropy. On the other hand, the second assumption is charge, angular momentum and other relevant macroscopic incorrect in a strict sense.

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AA¯ Whether the Hayden-Preskill recovery (as well as whereP the red triangles on corresponds to 1 Q the AMPS argument) is applicable to systems with non- pffiffiffiffiffiffiffi j ¯ ⊗ j˜ Q 1 j¼0 j iA j iA. Let us denote the Hilbert space factorizable Hilbert spaces is an interesting problem. We þ with total charge mB þ a by Ha and the Haar random envision that this question can be ultimately answered by unitary acting on it by Ua. using operator algebraic approaches, defining Hilbert 2 2 By directly computing Trfρ g and Trfρ ¯ g, one can spaces from operators instead of starting from a given C BD Hilbert space. For such an extension, we would need to show that the reconstruction does not work for some define entanglement and recoverability in operator alge- states. To understand which states can be reconstructed, braic languages. While providing a full-fledged answer to however, we need an additional argument. We will find j this question is clearly beyond the scope of this paper, our that the diagonal information about the total charge can be ¯ ð2Þ analysis may be viewed as a first step toward this question. reconstructed from both C and BD. This implies I ðA; ¯ 2 2 ¯ In this paper, we discussed the Hayden-Preskill recovery BDÞ;Ið ÞðA; CÞ⪆ logðQ þ 1Þ. Noting that Ið ÞðA; BDÞþ problem in the presence of symmetries. A symmetric Ið2ÞðA; CÞ¼2 logðQ þ 1Þ, we conclude Ið2ÞðA; BD¯ Þ; subspace, even if it is embedded on a factorizable Ið2ÞðA; CÞ ≈ logðQ þ 1Þ. This suggests that the off-diagonal Hilbert space, is known to be nonfactorizable. We studied phase information (anything except the total charge) cannot conditions under which the recovery is possible and saw be reconstructed from either C or BD¯ . that distinction between symmetric and nonsymmetric Below we describe how to distinguish different values p operators is crucial. This observation hints that the criteria of j. Our argument works for ϵ ¼ 1−p < 1, but we believe on recoverability may be stated purely in terms of operators that a similar conclusion applies to cases with ϵ ≃ 1. without direct use of entanglement and the structure of a Consider the following quantum state given Hilbert space.

ACKNOWLEDGMENTS I am grateful to Nick Hunter-Jones for collaboration during the initial phase of the project. I would like to thank ðA3Þ Ahmed Almheiri, Nick Hunter-Jones, Naritaka Oshita, John Preskill, Brian Swingle and Herman Verlinde for useful discussions on related topics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario ðjÞ 2 through the Ministry of Research and Innovation. For sufficiently large nD, the Haar average of TrfðρC Þ g is given by P APPENDIX: U(1)-SYMMETRIC HAYDEN- Z nD nC nD 2 q 0ð Þð Þ dU ρðjÞ 2 ≈F F ¼ mBþj−q q PRESKILL; NONSYMMETRIC INPUTS Trfð C Þ g j j ¼ n 2 ðA4Þ ðm jÞ Here we consider the Hayden-Preskill thought experi- Bþ ment with Uð1Þ-symmetric time-evolution when an input for sufficiently large nD. Noting that ϵFj ≈ Fj 1, the quantum state is nonsymmetric. To be specific, we will þ R´enyi-2 entropy in C differs by log ϵ when the total charge consider the input states which are superpositions of the differ by one. Hence, measuring the R´enyi-2 entropy in C following states with different charges: (or DB¯ ) is sufficient to learn the value of j. We would ˜ like to note that the above calculation concerns the Haar jji¼j1 10 i j ¼ 0; ;Q ðA1Þ ðjÞ 2 ðjÞ 2 average of TrfðρC Þ g. In order for the value of TrfðρC Þ g to reliably distinguish values of j, the statistical variance where the first j entries are 1’s. So we have nA ¼ Q and j of Trfðρð ÞÞ2g must be small. For Haar random ensemble, nB ¼ n − Q. The quantum state of our interest is C the variance is suppressed by the total Hilbert space dimension.3

3Readers might think that measurement of total charge on C could reveal the value of j. Indeed, if the value of j differs by one, ðA2Þ nC the expectation value of the total charge on C differs by n since the charge will be uniformly distributed after the time-evolution. However, since the variance of the charge is large, distinguishing different values of j may require a large number of charge measurements.

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