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Black holes as clouded mirrors: the Hayden-Preskill protocol with symmetry

The Hayden-Preskill protocol is a qubit-toy model of the black hole information paradox. Based on the assumption of scrambling, it was revealed that quantum information is instantly leaked out from the quantum many-body system that models a black hole. In this paper, we extend the protocol to the case where the system has symmetry and investigate how the symmetry affects the leakage of information. We especially focus on the conservation of the number of up-spins. Developing a partial decoupling approach, we first show that the symmetry induces a \emph{delay of leakage} and an \emph{information remnant}. We then clarify the physics behind them: the delay is characterized by thermodynamic properties of the system associated with the symmetry, and the information remnant is closely related to the symmetry-breaking of the initial state. These relations bridge the information leakage problem to macroscopic physics of quantum many-body systems.


Introduction
Black holes are the most peculiar objects in the universe. While macroscopic properties of black holes can be fairly understood by general relativity, finding microscopic descriptions of black holes has been a central problem in fundamental physics. A significant step was made by the discovery of Hawking radiation [1,2]: due to a quantum effect, a quantum black hole emits thermal radiation. This discovery raises a question about whether the radiation carries away the information in the black hole. Although information leakage is unlikely in the classical case due to the no-hair theorem [3,4,5], the holographic principle indicates that information should leak out as a black hole evaporates. This conflict is known as the information paradox.
A novel approach to the information paradox was proposed from the theory of quantum information, known as the Hayden-Preskill (HP) protocol [6]. In the protocol, a quantum black hole is modeled by a quantum many-body system consisting of a large number of qubits. After scrambling dynamics of the system, which is typically the case in quantum chaotic systems, the system is split into two random subsystems, one is a remaining black hole and the other is the 'Hawking radiation'. Due to the scrambling dynamics, it was shown that information stored in the original many-body system can be recovered from the 'radiation' even if the size of the corresponding subsystem is small. This phenomena is often expressed as that information is quickly leaked out from the black hole. This result spiked various research topics over quantum gravity, quantum chaos, and quantum information [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28], including the trials of experimentally simulating quantum gravity in a lab [29,30,31].
In this paper, we study the HP protocol when the quantum many-body system has symmetry. Since quantum many-body systems typically have symmetry, it is important to clarify how symmetry affects the protocol. We especially consider the system with conservation of the number of up-spins. The conservation law makes it difficult to fully investigate the protocol by the technique used in [6]. In [14,15], the difficulty was avoided by considering restricted cases with fixed conserved quantities, and possible connections to other conceptual puzzles of quantum black holes [32,33,34,35] were argued. Here, we develop a quantum information-theoretic technique to simultaneously deal with quantum information and symmetry, and investigate the HP protocol in general cases.
We first show that the symmetry changes when and how the information leaks out from the black hole. More specifically, we show the existence of a delay of information leakage and an information remnant, which are numerically shown to be macroscopically large for certain conditions of the initial many-body sys-tem. Then, we clarify the physics behind these phenomena: the delay and the information remnant are, respectively, characterized by thermodynamic properties and symmetry-breaking of the many-body system. These relations unveil non-trivial connections between quantum information in a many-body system and the macroscopic physics associated with the symmetry. Although we show the connections for the systems conserving the number of up-spins, we expect that they hold for a wider class of symmetries as far as they are global and abelian.
The paper is organized as follows. For clarity of the presentation, we provide a brief overview of our results in Sec. 2. We start the main analysis with introducing our notation in Sec. 3. In Sec. 4, we overview the original HP protocol, and propose a modified protocol with symmetry. The partial decoupling is explained and is applied to the HP protocol with symmetry in Sec. 5. The physics behind the delay and the information remnant is discussed in Sec. 6. We conclude this paper with discussions in Sec. 7. Technical statements are proven in Appendices.

Overview of the Results
In the HP protocol, a quantum black hole is modeled by a quantum many-body system. After unitary time evolution of the system, it is split into two random subsystems. One is identified with a remaining black hole, and the other is with the Hawking radiation escaping away from the black hole. Following this modelling, we refer to the original many-body system as a BH and to the random subsystems as a remaining BH and a radiation. Analogously to actual black holes, the size of the radiation increases as time passes. Consequently, the size of a remaining BH decreases. Eventually, the whole BH becomes radiation, which corresponds to the complete evaporation of a quantum black hole.
A brief description of the HP protocol is as follows [6]: 1. Quantum information source A of k qubits is thrown into an initial BH B in composed of N qubits, which has already emitted some radiation B rad . The quantum information in A is kept track of by a reference system R.
2. The new BH S := AB in of N + k qubits undergoes scrambling dynamics and gradually emits new radiation, i.e., a random subsystem S rad of S. The number of qubits in S rad is denoted by . The BH shrinks to the remaining BH S in of N + k − qubits. As time goes on, increases.

The quantum information originally stored in
A is tried to be recovered from the radiations B rad and S rad .
It was shown in Ref. [6] that, if the size of the new radiation S rad exceeds a certain threshold, the remaining BH S in is decoupled from the reference R. As a result, the error in recovering the information from the radiations B rad and S rad decreases exponentially quickly as increases. The threshold depends on the condition of the initial BH B in , and can be the same order of the number k of qubits in the quantum information source A.
The main concern in this paper is to clarify what would occur if the system has symmetry. Since the dynamics cannot be fully scrambling in the presence of symmetry, information should be leaked out from the system in a different manner. We particularly consider conservation of the number of up-spins, or equivalently, the Z-component of the angular momentum (Z-axis AM for short). Since the corresponding symmetry is a Z-axial symmetry, we call the system the Kerr BH in analogy with actual black holes. Throughout the paper, we denote by L the mean of the Z-axis AM in the initial Kerr BH B in and by δL its standard deviation.
Assuming that the dynamics of the system is scrambling that respects the axial symmetry, we first show that partial decoupling occurs in the Kerr BH, rather than the full decoupling as in the case of BH without any symmetry. This results in two substantial changes, the delay of information leakage and the information remnant. The delay means that, compared to the BH without any symmetry, it is needed to collect more radiation in order to recover quantum information of A, while we mean by the information remnant that a certain amount of the information of A remains in the Kerr BH until it is all radiated.
Depending on the Z-axis AM L and its fluctuation δL of the initial Kerr BH B in , the delay ranges from O( √ k) to macroscopically large values, and the information remnant varies from infinitesimal to constant. Thus, with certain initial conditions, the symmetry radically changes the behavior of how radiation carries information away from the BH. In particular, when the Kerr BHs has extremely large Z-axis AM or fluctuation, there is a possibility that the Kerr BHs do keep nearly all information of A until the last moment. This is in sharp contrast to the HP protocol without symmetry that predicts an instant leakage of information.
To physically understand the delay and the information remnant, we further investigate them from different perspectives. We argue that the delay is ascribed to the structure of the entanglement generated by the scrambling dynamics with symmetry. To capture the information-theoretic consequence of such entanglement, we introduce a new concept that we call clipping of entanglement, and propose the following condition for the information recovery from the radiation to be possible: for most of possible values of the Z-axis AM n in the new radiation S rad , where H(B in ) is the entropy of the initial Kerr BH B in , H S rad n is the subspace of the Hilbert space of S rad with the Z-axis AM n, and H Sin L−n is that of the remaining Kerr BH S in with the Z-axis AM L − n. This condition indeed reproduces the results based on the partial decoupling. Moreover, the results of the BH without symmetry [6] can be also obtained from the same idea.
One of the merits of the entanglement clipping is that the condition for the information leakage, Eq. (1), consists only of the entropy of the initial system and the dimension of the Hilbert spaces with respect to the conserved quantity. Hence, for quantum many-body systems with any extensive conserved quantity, we can easily compute the delay of information leakage.
Apart from the conciseness, the argument based on entanglement clipping has another merit that it unveils a non-trivial relation between the delay of information leakage and thermodynamic properties of the system. We first show that the delay is characterized by up to the second-order derivatives of the entropy in B in in terms of the Z-axis AM. The derivatives are then connected to thermodynamic properties of B in as follows. Let ω(T, λ) be the state function of B in conjugate to the Z-axis AM, and α(T, λ) be thermal sensitivity of the conserved quantity, where T is the temperature of B in . We show that, when the initial Kerr BH B in has a small Z-axis AM, where k B is the Boltzman constant, and S is the entropy of the initial Kerr BH. Note that both ω(T, λ) and α(T, λ) depend on the temperature T , but their product is temperature-independent. The formula for a large Z-axis AM is also obtained. This allows us to understand the delay in terms of thermodynamic properties of the initial system. The information remnant can also be intuitively understood. The fact that partial decoupling occurs instead of full decoupling implies that the remaining Kerr BH S in keeps some correlation of the conserved quantity with the reference R. For this reason, it should not be possible to recover all coherence of the quantum information that is kept track by R. By investigating how much classical information about the reference R can be inferred from the remaining Kerr BH S in , we quantitatively show that where ζ(S in ) represents the degree of symmetrybreaking in S in . This indicates that the symmetrybreaking in the Kerr BH is the dominant factor of the information remnant. The relations, Eqs. (2) and (3), establish quantitative connections between the information leakage problem and macroscopic physics. We expect that they universally hold for any quantum chaotic systems with abelian symmetry. In those cases, the quantities related to the Z-axis AM in Eqs. (2) and (3) should be replaced by the corresponding conserved quantities. A primary instance is energy conservation, in which the relations similar to Eqs. (2) and (3) can be obtained. In particular, the delay of information leakage when energy is conserved should be given in terms of the heat capacity C V of the quantum system. That is, we have for energy conservation, where E and S are the energy and the entropy of the initial BH, respectively.

Preliminaries
Throughout the paper, we denote the sequence {i, i+ 1, . . . , j − 1, j} of integers between two natural numbers i and j (i ≤ j) by [i, j]. The logarithm is always taken in base two.
The basic unit of quantum information is a qubit represented by a two-dimensional Hilbert space. The number of qubits in a system S is denoted by |S|. We often write the relevant systems in the superscript, such as a Hilbert space H S of a system S, an operator X SR on SR, and a superoperator E S→B from S to B. For superoperators from S to itself, we denote it by E S .
The partial trace over a system X is denoted by where I is the identity operator, by M S ρ SR M S † . The identity superoperator on S is denoted by id S .

Linear Operators and Superoperators
We denote a set of linear operators from H A to H B by L(H A , H B ), and L(H A , H A ) by L(H A ). We also use the following notation for the sets of positive semi-definite operators, quantum states, and subnormalized states.
A maximally entangled state between S and S , where H S ∼ = H S , is denoted by Φ SS . The completely mixed state on H S is denoted by For instance, since the marginal state of a maximally entangled state is the completely mixed state, we have The fundamental superoperators are the conjugations by a unitary, an isometry, and a partial isometry. An isometry V H→K is the linear operator such that When dim H = dim K, the isometry is called a unitary and also satisfies V H→K (V H→K ) † = I K . A partial isometry is a linear operator from H to K such that it is an isometry on its support. Projections, isometries, and unitaries are special classes of a partial isometry.
A quantum channel T S→B is a completely-positive (CP) and trace-preserving (TP) map. A map is called CP if (id S ⊗T S→B )(ρ S S ) ≥ 0 for any ρ S S ≥ 0 and is TP if Tr[T S→B (ρ S )] = Tr[ρ S ]. We also say a superoperator T S→B is sub-unital and unital if T S→B (I S ) ≤ I B and T S→B (I S ) = I B , respectively.
For a quantum channel T S→B , a complementary channel is defined as follows. Let V S→BE T be a Steinspring dilation [36] for a quantum channel T S→B , i.e., V S→BE T is an isometry satisfying for any ρ S ∈ L(H S ). The mapT S→E defined bȳ for any ρ S ∈ L(H S ) is called a complementary channel. Any superoperator T S→B from S to B has a Choi-Jamiołkowski representation [37,38] defined by This is sometimes called the channel-state duality.

Norm and Entropy
The Schatten p-norm for a linear operator X ∈ L(H A , H B ) is defined by X p := (Tr[(XX † ) p/2 ]) 1/p (p ∈ [1, ∞]). We particularly use the trace (p = 1) and the Hilbert-Schmidt (p = 2) norms. We define the fidelity between quantum states ρ and σ by The conditional min-entropy is defined by for ρ AB ∈ P(H). This is a variant of the conditional entropy H(A|B is the von Neumann entropy. When dimH B = 1, H min (A|B) ρ reduces to the min-entropy

Hayden-Preskill protocol
We overview the general setting of the HP protocol in Subsec. 4.1 and review the results in Ref. [6] with a slight generalization in Subsec. 4.2. We then explain how symmetry shall change the situation in Subsec. 4.3. The diagram of the protocol is given in Fig. 1, and important quantities are summarized in Table 1.

General setting of the HP protocol
In the HP protocol, we consider the situation depicted in Fig. 1. In the protocol, the BH B in initially consists of N qubits. The radiation emitted earlier, which we call the past radiation, is denoted by a quantum system B rad . We emphasize that this is just a toy "Past" radiation that purifies the initial BH B in . A (k qubits) Quantum information source Alice throws into the BH B in . R (k qubits) Reference system that is maximally entangled with A. S (N + k qubits) The BH after the information source A is thrown in (S = AB in = S in S rad ). S rad ( qubits) New radiation from the BH S. S in (N + k − qubits) Remaining BH after S rad is evaporated.
Interior of the BH Exterior of the BH Reference Figure 1: Diagram of the HP protocol. The blue lines represent trajectories of qubits, and the yellow wavy lines indicate that they may be entangled. The black region in the righthand side represents the BH system, and the blue region in the left-hand side is a reference system that is virtually introduced so as to keep track of the quantum information of A. The green boxes in the black region represent the internal scrambling dynamics of the BH. Each U N +k−i is internal unitary time-evolution acting on N + k − i qubits. The purple lines represent the radiation of the BH. The map D is the recovery operation applied by Bob, aiming at recovering the information originally stored in A from the past radiation B rad and the new radiation S rad . model. We do not intend to claim that they represent actual black holes and the Hawking radiation.
We particularly consider two types of the BH. One is a "pure" BH, where B in is in a pure state |ξ Bin , and the other is a "mixed" BH, where B in is in a mixed state ξ Bin . The former models the quantum black hole that does not emit radiation yet or the one whose radiation has been measured by someone. The latter models a sufficiently old quantum black hole that has already emitted a large amount of radiation. For a mixed BH, the system B in B rad is in a pure state |ξ BinB rad , which is a purification of ξ Bin .
The HP protocol is the following thoughtexperiment. At time T 1 , a person, Alice, throws a quantum information source A of k qubits into the BH B in . The quantum information source is defined by introducing a reference system R, and by setting the initial state between A and R to a maximally entangled state Φ AR . The BH, now a composite system S := AB in , undergoes the internal unitary dynamics and emits new radiation. We model this process by repeatedly applying an internal unitary dynamics followed by a transfer of a randomly chosen qubit in S to the exterior of the BH. Since each radiation, i.e., the transfer of a qubit, shrinks the BH by one qubit, the unitary after i qubits are radiated acts on N +k −i qubits. In Fig. 1, the internal unitary dynamics acting on the N + k − i qubits in the BH is denoted by Suppose that the BH emits qubits of radiation, i.e., |S rad | = , by time T 2 . We call S rad the new radiation. The remaining BH S in is composed of N + k − qubits. Since the dynamics from time T 1 to T 2 is quantum mechanical, the dynamics should be represented by a quantum channel. From the construction, the dynamics L S→SinS rad from S to S in S rad is simply a product of unitary dynamics, (16) where U N +k−i (ρ) := (I i ⊗ U N +k−i )ρ(I i ⊗ U N +k−i ) † and the identity I i acts on the radiated i qubits. The quantum channel from the BH S to the remaining BH S in and that to the new radiation S rad , denoted by L S→Sin and L S→S rad , are given by respectively. Note that they are complementary to each other since L S→SinS rad is a unitary dynamics. We then introduce another person Bob. He collects all the qubits of the new radiation S rad and tries to recover the k-qubit information originally stored in A. He may additionally use the past radiation B rad . We assume that Bob knows the initial state |ξ BinB rad and the whole dynamics L S→SinS rad . In the recovery process, Bob tries to reproduce the state Φ AR maximally entangled with the reference R by applying a quantum channel D to the quantum systems he has, i.e., to the new and past radiations, S rad and B rad . If he succeeds, it implies that he can access the information source A in the sense that, if the initial state of A had been |ψ , Bob would be able to recover |ψ . Due to the no-cloning theorem [39], this implies that the information stored in A is not left in the interior of the BH. Thus, the information in A has been carried away by the radiation S rad of qubits.
To quantitatively address whether Bob succeeds in recovering information of A, we define and analyze a recovery error ∆. LetΦ AR be the final state of the whole process, i.e., Since the goal of Bob is to establish the state maximally entangled with the reference R, we define the recovery error by where the minimum is taken over all possible quantum channels D S rad B rad →A applied by Bob onto the new and the past radiation, S rad and B rad . Note that 0 ≤ ∆(ξ, L) ≤ 1, and that the recovery error generally depends on the initial state ξ of the initial BH B in and its dynamics L.

Review of the HP protocol without symmetry
When the BH has no symmetry, the dynamics L S→SinS rad is considered to be fully scrambling. This is formulated by choosing each internal unitary dynamics U N +k−i in U N +k−i of the BH (see Eq. (16)) at Haar random. When this is the case, we denote the dynamics by L Haar . The recovery error ∆ in this case is characterized by the min-entropy H min (B in ) ξ of the initial state of the BH B in : for the scrambling dynamics L Haar of the BH, the recovery error ∆ satisfies [6,40] with high probability. This clearly shows that, when , the recovery error ∆(ξ, L Haar ) decreases exponentially quickly. Note that the minentropy H min (B in ) ξ is large if and only if the initial BH B in and the past radiation B rad are strongly entangled. Hence, in this framework, it is the entangle-ment between the initial BH B in and the past radiation B rad that determines the time scale for the information recovery to be possible.
For the pure BH, H min (B in ) ξ = 0, so that ∆(ξ, L Haar ) ≤ 2 N/2+k− . In contrast, for the mixed BH after the Page time [41], H min (B in ) ξ = N , resulting in ∆ tot (ξ : L Haar ) ≤ 2 k− . The latter case is particularly interesting since the recovery error does not depend on N . Thus, no matter how large the initial BH is, Bob can recover the k-qubit information when a little more than k qubits are radiated.
The mechanism behind Eq. (21) is decoupling [42,43,44,40] in the sense that L S→Sin which is induced by the fully scrambling dynamics of the BH. Decoupling is a basic concept in quantum information theory, which guarantees that information is encoded into good codewords, enabling Bob to retrieve information from the new radiation S rad easily.

The HP protocol with symmetry
Here, we describe how situation should be changed in the case of the Kerr BH with the Z-axial symmetry.

Symmetry, Hilbert spaces, and Quantum Information
When a quantum system has symmetry, the associated Hilbert space is accordingly decomposed. Since the axial symmetry is abelian, the Hilbert spaces H A , H Bin , and H S are decomposed into the subspaces invariant under the Z-axial symmetry: respectively. Note that the labels of the subspaces are based on the number of up-spins, but it can be readily transformed to the Z-axis AM. We denote the projection onto each subspace by Π S κ , Π Bin µ , and Π S m . The symmetry also affects the quantum information stored in A. As mentioned, quantum information in A is represented by the maximally entangled state Φ AR with the reference R. The Hilbert space H R is also decomposed according to the decomposi- with p κ being Tr[( is invariant under the axial rotation of A, this state contains the information of A that is symmetry-invariant, leading us to call it symmetry-invariant information.
Accordingly, we define two errors in recovering information when the BH has symmetry. One is ∆ inv for the symmetry-invariant information in Φ AR diag , and the other is ∆ tot for the total information in Φ AR . They are, respectively, defined by The recovery error ∆ tot for the total information is the same as ∆ in Eq. (20), but we below denote it by ∆ tot to clearly distinguish it from ∆ inv . Using a diagram is useful to represent the statesΦ AR diag,D and Φ AR D in Eqs. (24) and (25): . This is also represented aŝ

Symmetry-preserving scrambling
Since the Kerr BH has a Z-axial symmetry, its internal unitary dynamics, namely, each U N +k−i in Eq. (16), should be in the form of where U (m) N +k−i acts on the subspace of N + k − i spins spanned by the states with m up-spins. These unitaries form a group. We assume that each U (m) N +k−i is Haar random on each subspace, namely, U (m) N +k−i scrambles all quantum states with exactly m upspins. However, unlike the Haar scrambling, U N +k−i does not change the value of conserved quantity. We call such U N +k−i symmetry-preserving scrambling.
We denote the corresponding dynamics of the Kerr BH by L Kerr . Using the unitarily invariant property of the Haar measure, it is straightforward to observe that the statistics induced by the dynamics L S→SinS rad The fact that the symmetry-preserving scrambling cannot change the conserved quantity immediately implies the absence of full decoupling in the Kerr BH, which was also pointed out in Ref. [14]. Since decoupling is the key technique used to derive the recovery error, Eq. (21), it is unclear at all whether and how information retrieval can be done in the presence of symmetry. The main goal of this paper is to clarify the recovery errors ∆ inv and ∆ tot for the symmetryinvariant and total information when the dynamics is L Kerr , or equivalently U Kerr .

Information recovery -partial decoupling approach-
To investigate the recovery errors ∆ inv and ∆ tot , we use the partial decoupling approach [45], which we briefly summarize in Subsec. 5.1. We also provide a slight generalization. We then apply partial decoupling to the Kerr BH in Subsec. 5.2. We derive formulas for the upper bounds on the recover errors in Subsec. 5.4. Based on the formulas, we numerically investigate the recovery errors in Subsec. 5.5.

Partial decoupling theorem
The partial decoupling theorem is proven by two of the authors in Ref. [45]. We here explain its simplest form. In this subsection, the notation and the systems' labeling, such as S, R, and E, do not correspond to those in the HP protocol given in Sec. 4.
The situation of partial decoupling is as follows. Let S and R be quantum systems. We assume that the Hilbert space H S has a direct-sum structure, such that The dimensions are denoted as dim H S = D S and dim H S j = d j . Let SR be an initial state on SR. A random unitary U S in the form of j∈J U S j is first applied to S, and then a given CPTP map M S→E is applied. This leads to the state Despite the fact that this state is dependent on U S , the partial decoupling theorem states that, when where H j is the Haar measure on the unitary group U(d j ) on H S j , the state is typically close to a fixed state independent of U S as far as a certain entropic condition is satisfied. For a later purpose, we slightly extend the situation, where the initial state SR is subnormalized, i.e., SR ∈ S ≤ (H SR ), and the map M S→E is a trace-nonincreasing CP map.
The fixed (subnormalized) state is explicitly given by Γ ER ∈ S ≤ (H S * RE ), which is constructed as where with ζ S E = J(M S→E ) being the Choi-Jamiołkowski representation of M S→E (see Eq. (11)). Note that the state Γ ER has correlation between E and R through the classical value j and has no quantum correlation. Hence, we call the state partially decoupled.
The aforementioned entropic condition is given in terms of the conditional min-entropy of an extension Γ S * RE ∈ S ≤ (H S * RE ) of Γ ER by a system S * = S S , where S and S are replicas of the system S: where Here, Using these subnormalized states, the partial decoupling theorem is stated as follows.

Theorem 1. For any δ > 0, it holds that
The sub-normalized states Γ ER and Γ S * RE have relatively simple expressions if we use a diagram. Let C S→SS * J be a CPTP map corresponding to a "Jcorrelator" given by where C S→SS * is the isometry given by Here, |Φ j SS is the state maximally entangled only in the subspace H S j ⊗ H S j . In other words, the Jcorrelator generates the maximally entangled state in the subspace H S j ⊗ H S j depending on the value of j ∈ J in the system S, and swaps the system S with S . Using this CPTP map, the subnormalized states Γ ER and Γ S * ER are expressed as follows: In the above diagram, the black horizontal double bars represent the trace over the corresponding system, namely, S * in this case. Theorem 1 implies that, if H min (S * |ER) Γ is sufficiently large, the state M S→E U S SR U S † is close to Γ ER = Tr S * [Γ S * ER ] with high probability. In the di-agram representation, this can be rewritten as up to the approximation 2 −Hmin(S * |ER) Γ /2 in the trace norm with high probability.
When d min is small, such as d min = 1, Theorem 1 fails to provide a strong concentration since the probability in the statement becomes tiny for small δ. When this is the case, we can set a "threshold" dimension d th .

Corollary 2.
Consider the same setting as in Theorem 1. Let d th be a positive integer, and Π S ≥ be given by The proofs of Theorem 1 and Corollary 2 are given in Appendices A and B, respectively.

Partial decoupling in the Kerr BH
We now apply the partial decoupling to the Kerr BH by identifying the dynamics L S→SinS rad Kerr with the symmetry-preserving scrambling U S Kerr ∼ H Kerr , which is in the form of Since this is the same form as the random unitary used in the partial decoupling theorem, we can directly apply Theorem 1 as well as Corollary 2.
We set and M in Theorem 1 to Φ AR ⊗ ξ Bin and Tr S rad , respectively. The system E in Theorem 1 corresponds to S in in this case.
Using the doubled replica S * = S S of the BH S, with high probability, where C AM is the AMcorrelator that generates the state maximally entangled in a subspace. More explicitly, C AM is the conjugation map by the following isometry where |Φ m SS is the state maximally entangled only in the subspace H S m ⊗ H S m . According to Theorem 1, Eq. (47) holds up to ≈ 2 −Hmin(S * |SinR) Γ /2 in the trace norm, where Hence, if H min (S * |S in R) Γ 1, it is highly likely that the L.H.S. of Eq. (47) is partially decoupled, namely, the Kerr BH is partially decoupled from the reference R.

Empirical smoothing of the conditional min-entropy
From Theorem 1, the probability for Eq. (47) to hold up to the approximation 2 − 1 2 Hmin(S * |ER) Γ + δ is at Since d min = 1 for the axial symmetry, Theorem 1 fails to assure a high probability if δ < 2 −k/2 . This problem can be circumvented by using Corollary 2 instead of Theorem 1 and by setting a threshold dimension. However, the degree of approximation 2 − 1 2 Hmin(S * |ER) Γ turns out not to be tight in general, which typically happens when the conditional min-entropy is used. A smoothing is the technique that is exploited to obtain a better bound [46]. We here explain how the bound can be improved by the smoothing technique.
The most general way of smoothing is to use the smooth conditional min-entropy instead of the conditional min-entropy [47]. We here exploit a limited smoothing by ignoring "less probable" subspaces H S rad n of the radiation S rad . This is because the smooth conditional min-entropy is computationally intractable.
To make the idea more precise, we consider the subnormalized state in which the Z-axis AM in the new radiation S rad is n. We can explicitly compute the average weight p n , namely, the trace of the subnormalized state, by taking the average over U S Kerr : where χ µ = Tr[ξ Bin Π Bin µ ] (see Appendix C). Using p n , we define a probable set I by for ≥ 0. We call the subspaces H S rad n for n / ∈ I rare events. Here, the parameter defines the range of the probable set and rare events. We approximate, by using Theorem 1, only the subnormalized state in the subspace corresponding to the probable events, and count the part corresponding to the rare events as an additional error.
This procedure can be formulated by introducing Γ( ) corresponding to the probable events: where Π S rad := n∈I Π S rad n . The error from the rare events is then given by Using these, we have the following Proposition (see Appendix D for the proof).

Proposition 3.
Let ≥ 0. In the above setting, the probability that holds up to approximation Θ δ, ξ in the trace norm is at least In the definition of Θ δ, ξ , there is a term 2 1− 1 2 Hmin(S * |SinR)Γ ( ) . This term does not recover the non-smoothing bound 2 − 1 2 Hmin(S * |SinR) Γ in the limit of → 0. This may imply that the term could be further improved by factor 2.

Upper bounds on recovery errors
Based on the partial decoupling of the Kerr BH, i.e., Proposition 3 and using the standard technique in quantum information theory [42,43,44,40], we now provide upper bounds on the recovery errors ∆ inv (ξ, L Kerr ) and ∆ tot (ξ, L Kerr ).
The key technique is the Uhlmann's theorem [48], which states that, if any state ρ AB and a pure state We apply this to Eq. (54).
with approximation at least 1 − Θ δ, ξ in terms of the fidelity, where D is a CPTP map from the radiations S rad B rad to S * ∼ = S S = S A B in . Taking the partial trace over S B in as well as the remaining BH S in , we obtain ≈ , where C is the pinching map with respect to the Zaxis AM defined by C S (ρ S ) = m Π S m ρ S Π S m . Note that S = AB in . The last line simply follows from the fact that the AM-correlator C AM generates a normalized state depending on the value of the Z-axis AM in S (see Eq. (48)), but it is traced out in Eq. (58).
To obtain the recovery error ∆ inv for the symmetry-invariant information, we need to compute the fidelity to Recalling that the approximation is at least 1 − Θ δ, ξ in the fidelity, we arrive at For the recovery error ∆ tot for the total information, we can similarly show that where η ξ is given by Here, and χ µ = Tr[ξ Bin Π Bin µ ]. See Appendix E for the derivation.
In summary, we have derived upper bounds on the recovery errors as follows.

Proposition 4.
Let , δ ≥ 0. The recovery errors ∆ inv and ∆ tot for symmetry-invariant and total information satisfy and d min ( ) = min N +k m+n : m ∈ [0, N +k− ], n ∈ I . Since Proposition 4 holds for any ≥ 0, we define and investigate it in the following analysis. By definition, Θ δ ξ (N, k, ) is the best possible upper bound of the recovery errors that can be obtained by the smoothing procedure described in Subsec. 5.3. We have numerically checked that this generally provides better bounds than those directly obtained from Theorem 1, namely, those without smoothing the entropy.

Numerical evaluation
The parameter δ in Proposition 4 controls the tradeoff between the upper bounds on the recovery errors and the probability for the symmetry-preserving scrambling L Kerr to result in that errors. In the following numerical analysis, we consider only the term independent of δ, that is, This suffices since, in the large-N limit, we can typically choose δ such that Θ ξ (N, k, ) δ and, at the same time, that the probability is arbitrarily close to one. To see this, suppose that the Z-axis AM in B in is a constant fraction of N . In this case, the dimension d min ( ) scales as N c0 (c 0 is some constant). Hence, we can choose δ to be values with a slightly greater scaling than N −c0 /2 , such as N −c1 /2 with c 1 < c 0 , and make the probability close to one in the large-N limit. On the other hand, as will be turned out later, Θ ξ (N, k, ) typically scales as 2 −c2 (c 2 is some constant). Since Θ ξ (N, k, ) ≈ 2 −c2 N −c1 /2 = δ for sufficiently large N , δ is negligible and the probability is arbitrarily close to one.
To numerically compute Θ ξ (N, k, ), we need to simplify H min (S * |S in R)Γ ( ) that appears in Θ ξ . This can be done by using the property of the conditional min-entropy for the special type of states.
Let {H A j } be mutually orthogonal subspaces of H A , and π A j be the completely mixed state on H A j . For any state Λ ABC in the form of J j=0 p j π A j ⊗ ρ BC j , where ρ BC j ∈ S(H BC ) and {p j } is a probability distribution, it holds that where d j = dim H A j . This immediately follows from the definition of the conditional min-entropy. See Appendix F for the derivation.
We also use the fact that, for a pure state |ψ AB , the conditional min-entropy is given by (74) Below, we provide upper bounds on the recovery errors for the pure and mixed Kerr BH. In both cases, we characterize the state ξ of the initial Kerr BH B in by the expectation value L of the Z-axis AM and the standard deviation δL, defined by where L Z = 2 N i=1 Z i with being the Plank constant and Z i being the Pauli-Z operator acting on the ith qubits. In the remaining of this paper, we set = 1 except the last part of Subsec. 6.1, where we discuss a physical origin of information leakage.

The pure Kerr BH
Let us first consider the pure Kerr BH B in with the initial state in the form of where µ stands for the number of up-spins in B in . We assume that {χ µ (L, δL)} µ are given by with a proper normalization such that N µ=0 χ µ (L, δL) = 1, and |ϕ µ Bin is an arbitrary pure state in H Bin µ . Due to the symmetry-preserving scrambling, the recovery errors do not depend on the choice of |ϕ µ Bin . Note that the expectation value of L Z and its standard deviation for the state |ξ(L, δL) Bin deviates from L and δL, respectively, due to the discretization, but the deviations are negligibly small.
In this case, the subnormalized stateΓ S * SinR ( ) can be explicitly written as where {p n } is the probability distribution given by Eq. (50), and with a proper normalization constant. Here, we have used the notation that Π a is the zero operator if a < 0. Using Eqs. (73) and (74), we obtain Since w( ) = n / ∈I p n , we can numerically evaluate this bound, from which we obtain upper bounds on ∆ inv and ∆ tot .
In Fig. 2, upper bounds on ∆ inv and ∆ tot , i.e., are provided as functions of the number of qubits in the new radiation S rad for various L and δL. We observe that ∆ inv starts decaying exponentially after a certain number of . The number, as well as the decaying speed, strongly depends on both |L| and The size N of the initial Kerr BH and the size k of the quantum information source are set to 500 and 5, respectively. We observe that the error ∆inv for symmetry-invariant information decays exponentially quickly after = |Sin| exceeds a certain number, while the error ∆tot for the whole information typically has a plateau. The dependence of these features on L and δL, as well as their physical mechanism, are discussed in the main text. Note however that, in any case, ∆inv, ∆tot ≈ 1 unless > (N + k)/2, which is a consequence of the no-cloning property of quantum information [39].
δL. To characterize this, we introduce ∆ (L, δL) for ∆ > 0 by Then, it is observed from Fig. 2 that ∆ (L, δL) ≥ ∆ (0, 0) for any ∆ ≥ 0. This implies that the Kerr BH has a delay in the onset of releasing information compared to the trivial case (L = δL = 0), which is given by the yellow dotted lines in Fig. 2. The delay is especially large when L is large (see figure (iv)) or when δL = Θ(N ) (see brown plots), indicating that the symmetry-invariant information cannot be recovered from the radiation if either L or δL is extremely large.
We also observe that ∆ tot behaves differently de- , ∆ tot first behaves similarly to ∆ inv . However, soon after that, the decreasing of the error becomes slow, and ∆ tot remains at a non-negligible value until ≈ N . This indicates that a part of the information remains in the Kerr BH until the last moment. We call such residual information information remnant. From the plots for different L but same δL, we also observe that the amount of information remnant is independent of L. When δL = O(N ), ∆ tot ≈ ∆ inv for any , implying that there is no information remnant.

The mixed Kerr BH
We next consider the mixed Kerr BH, which is initially entangled with the past radiation B rad . For simplicity, we only consider the following entangled states: where {χ µ } µ are given by Eq. (78), and |Φ µ BinB rad is the maximally entangled state in H Bin µ ⊗ H B rad µ . In this case, the state Γ S * SinR reduces to where p n,µ and the normalized pure stateΨ SinS in A R n,µ are given by respectively. Note that µ p n,µ = p n , where p n is given by Eq. (50).
and, we obtain upper bounds on ∆ inv and ∆ tot . We again note that ±L provide the same bounds.
The upper bounds on the recovery errors are numerically plotted in Fig. 3 as functions of = |S rad |. Similarly to the pure Kerr BH, the plots are for various L and δL. We observe that ∆ inv decays in a manner similar to the pure Kerr BH. That is, the recovery error starts decreasing after a certain number of , as well as the onset of decreasing has a delay compared to the trivial case (L = δL = 0, yellow dotted lines in Fig. 3). While the delay is not so large for small L and δL, it becomes substantially large when |L| is large or δL = Θ(N ).
The recovery error ∆ tot for the total information also behaves similarly to the pure Kerr BH, and it remains at a non-zero value until ≈ N . This implies that, even for the mixed Kerr BH, the information remnant exists. It turns out that the amount of information remnant depends only on δL and is the same as that for the pure Kerr BH for the same δL, which will be elaborated on later.
The significant difference from the pure Kerr BH is that the timing at which the errors start decreasing can be much earlier. It indeed ranges widely from O(k) to O(N ) depending on the Z-axis AM L and on its fluctuation δL of the initial Kerr BH.

Scaling of the delay with respect to N
We also provide numerical results about how the delay ∆ (L, δL) scales with respect to the size N of the initial Kerr BH B in for various Z-axis AMs L and their fluctuations δL. The result is summarized in Tab. 2 for the pure and mixed Kerr BHs. See Appendix G for the details.
6 Information recovery -macroscopic physics approach-In Sec. 5, we have numerically shown that the delay and the information remnant are the consequences of the symmetry of the Kerr BH. In this section, we analyze them from different perspectives. This lead to great insights into their physical origins.   We consider the delay in Subsec. 6.1 and the information remnant in Subsec. 6.2.

Delay of information leakage and themodynamics
To interpret the delay of information leakage, we propose a new concept that we call clipping of entanglement and show that it reproduces both the result for the BH without symmetry and the delay of information leakage for the Kerr BH. As we will see, the clipping argument provides a concise condition for the information leakage to occur only in terms of dimensions of the Hilbert spaces.
Before we start, we emphasize that the argument based on the entanglement clipping is applicable not only for the axial symmetry but for any symmetry with extensive conserved quantity.

Clipping of Entanglement
One of the significant features of the scrambling dynamics is that it generates nearly maximal entanglement between subsystems [41,49,50]. We argue that such a maximal entanglement is responsible also for the quick information leakage.
For clarity, we first consider the BH without symmetry. Moreover, instead of directly considering the quantum information source Φ AR , i.e., the maximally entangled state between A and the reference R, we consider the situation where the reference R is measured in an arbitrary basis to obtain a k-bit outcome. It is known that if the k-bit outcome can be inferred from the radiations B rad and S rad , then the maximally entangled state Φ AR can be reproduced [51], that is, the quantum information in A is recovered.
When the BH has no symmetry, its dynamics is fully scrambling and generates nearly maximal entanglement between S rad and S in . This implies that the marginal state in the smaller subsystem should be nearly completely mixed. In contrast, the support of the marginal state in the larger subsystem needs to be "clipped" at the dimension of the smaller one, be-cause any bipartite pure state should have an equal size, known as the Schmidt rank, in the two subsystems [52].
The information recovery can be understood from the entanglement clipping in the larger subsystem. In the case of the pure BH, no information is obtained from the radiation when |S rad | ≤ |S in | since no clipping occurs in the new radiation S rad . Note that all the eigenvalues of the marginal state in S rad are nearly the same, from which no information would be retrieved. When |S rad | ≥ |S in |, the clipping takes place in S rad , namely, the marginal state in S rad is released from the full rank state. In the case of the mixed BH, the entanglement clipping in S rad sets in much earlier since the BH is initially already on the verge of clipping due to the entanglement with the past radiation B rad , which is under control of the decoding person Bob.
To capture the information-theoretic consequence of the entanglement clipping in the new radiation S rad , we introduce a degree of clipping C in H S rad : Note that the degree of clipping C(H S rad ) can be negative. For instance, C(H S rad ) = −(N + k) when |S rad | = 0 and H(B in ) ξ = 0. When C(H S rad ) > k, the marginal state in the new radiation S rad is sufficiently clipped so that the 2 k states, corresponding to all the possible 2 k outcomes in R, can be fit in the marginal state without significant overlaps each other. Thus, we propose that the condition for the information recovery is given by By explicitly writing down the dimensions, we obtain as a condition for the information recovery to be possible. Note that this agrees well with the result of the HP protocol without symmetry, namely, Eq. (21).
We can apply the same argument to the Kerr BH. In this case, it is of crucial importance that the symmetry-preserving scrambling in the form of m U S m , where U S m is the Haar scrambling in the subspace with m up-spins, generates entanglement between S in and S rad only in the subspaces. This introduces a special structure of entanglement between them. To be more precise, we consider the initial Kerr BH B in with a fixed Z-axis AM L = λN and δL = 0, where λ ∈ [−1/2, 1/2]. The number of upspins in B in is λN + N/2. Due to the symmetrypreserving scrambling, the subspace H S rad n with n up-spins in S rad is entangled only with H Sin n satisfying n + n ≈ λN + (N + k)/2.
Taking this constraint into account, the entanglement clipping should be considered in each subspace H S rad n separately. We thus define the degree of clipping in H S rad and require that for most n.
We here emphasize that we have not used any property of the Z-axial symmetry of the Kerr BH except that it is additive. Thus, the above clipping condition should be valid for any BH with extensive conserved quantity.
If Eq. (98) is not satisfied for certain n, the information in the corresponding subspace should be counted as recovery error. This point can be elaborated by considering the probability distribution W (n) over the number n of spins in S rad induced by the random choice of S rad from S. The probability is given by With respect to this probability distribution, we define probable values of n and require for Eq. (98) to hold for any probable n. We count the total probability over non-probable values of n as recovery errors. This is similar to the empirical smoothing we exploited in Subsec. 5.3.

Comparison between entanglement clipping and partial decoupling
To check the validity of the argument of entanglement clipping, let us compare the condition (98) with the partial decoupling result. To this end, we need to specify the probable set of n. We define the probable set by the set of n around its average n under the probability distribution of Eq. (99). The average is explicitly given by More precisely, we require that Eq. (98) holds for all n such that where δn := n − n . Here, c > 0 is a parameter related to the recovery error since taking a larger c means that we require Eq. (98) for a wider range of n, resulting in less error. We then define the number We can further simplify the expression by assuming k N and expanding n around its average n ≈ (λ + 1/2) as n ≈ (λ + 1/2) + δn. We also introduce the initial degree of clipping C ini , i.e., the degree of clipping when |S rad | = 0. It is approximately given by Using these, we arrive at As mentioned, we require C(H S rad n ) > k for all δn such that |δn| ≤ c δn 2 . The standard deviation δn 2 can be also rephrased in terms of s(λ). From Eq. (99), we have Since W (n) can be approximated by a Gaussian distribution when 1 k N , the variance δn 2 is approximately given by Altogether, the numberˆ c (L) is given by the minimum that satisfies (111) By solving this, we obtain an explicit form ofˆ c (L), which isˆ where In Fig. 4, we numerically compareˆ c (L) and ∆ (L, 0). We observe that, by choosing proper c and ∆, which both play the role of a smoothing parameter, they coincide very well. Hence, we conclude that the clipping argument provides a good estimate of the necessary number of qubits in the new radiation S rad for the information to become recoverable from the radiations.
In Eq. (112), we have decomposedˆ c (L) into two terms. The first term, 0 (L), represents a delay originated from an information theoretic reason: when the number of qubits in S rad is increased by one, the degree of clipping is increased by 2s(λ) since a qubit has entropy ≈ s(λ). Thus, |S rad | should be at least 0 (L) so as to cancel the initial negative degree of clipping C ini , and more for S rad to typically have a sufficiently large space to store 2 k states without significant overlaps. That is why we refer to 0 (L) as an information-theoretic delay.
It is the second term fl (L) that is a non-trivial consequence of the symmetry of the Kerr BH. It stems from the fluctuation of the Z-axis AM in the radiation, making the clipping condition harder to be fulfilled. We hence call fl (L) a fluctuational delay. Note also that, when the Z-axis AM L of the initial Kerr BH is small, the delay of information leakage, δˆ c (L) :=ˆ c (L) −ˆ c (0), satisfies since 0 (L) ≈ 0 (0) and fl (0) = 0 for small L.

Delay and BH Thermodynamics
The fluctuational delay fl (L) can be further rewritten in terms of thermodynamic quantities of the Kerr BH. In this context, it is more natural to quantify the delay by the amount of the Z-axis AM rather than the number of qubits. We hence introduce L fl by and investigate it. In this Subsection, we explicitly write the Plank constant . Correspondingly, we also introduce L 0 := λ 0 (L), which is the Z-axis AM corresponding to the information-theoretic delay 0 (L).
In order to consider the thermodynamics of the Kerr BH, it is necessary to take its Hamiltonian into account. The Hamiltonian should be invariant under the rotation around the Z-axis and also sufficiently random for the dynamics to be symmetrypreserving scrambling. Since we are interested in the thermodynamics properties that do not strongly rely on the randomness, we especially consider the Kerr BH with temperature T sufficiently higher than the energy scale of the Hamiltonian, so that the thermodynamic entropy S(λ, L, T ) of the Kerr BH is independent of T , and the free energy F (T, λ, L) is approximately given by −T S(λ, L).
Let ω := − ∂F ∂L T,N be an intensive state function conjugate to the Z-axis AM L, and α := 1 L ∂L ∂T ω,N be the sensitivity of the Z-axis AM to the temperature. Note that the state function ω(T, λ) corresponds to the angular velocity of the Kerr BH. Since the thermodynamic entropy S(λ, L) is given by where k B is the Boltzmann constant, in our particular model of the Kerr BH, we have Hence, it follows that Using these and assuming k N , we obtain from Eqs. (116) and (114) that Note that ω(T, λ)α(T, λ) is independent of the temperature T , and so is L fl .
After radiating L 0 of the Z-axis AM to fulfill the information-theoretic requirement, the Kerr BH must further radiate an extra amount L fl of the Z-axis AM to release the information. It is clear from Eq. (121) that this amount L fl is solely determined by intensive thermodynamic quantities of the initial Kerr BH, that is, the Z-axis AM per entropy L/S(λ, L), the angular velocity ω(T, λ) of the Kerr BH, and the coefficient α(T, λ) of the thermal sensitivity of the Z-axis AM.
The delay of information leakage is, thus, closely related to the thermodynamic properties of the BH. This, in turn, implies that, if we understand the thermodynamics of the BH well, it is possible to predict how quickly the BH releases the quantum information therein.
We emphasize that the above argument is likely to be applicable to the system with any global abelian symmetry. A particularly important application is the case where energy is conserved during the unitary dynamics of the system. Assuming that the dynamics is energy-preserving scrambling in the sense that it scrambles only quantum states with the same energy scale, we can evaluate the delay of information leakage based on Eq. (114). Suppose that the initial quantum many-body system BH is thermodynamical and has energy E with negligibly small fluctuation. The thermodynamic entropy S(E) of the system, where E is the internal energy, and the heat capacity C V satisfy Thus, a similar calculation from Eq. (114) leads to where E 0 and E fl are the energy corresponding to the information-theoretic and fluctuational delays, respectively. As E 0 /E is between 0 and 1, the amount is basically determined by the ratio between the energy E and entropy S(E) of the initial black hole, and the heat capacity C V , connecting the information leakage to thermodynamic quantities of the quantum many-body system.

Information remnant and symmetrybreaking
We finally study the information remnant characterized by η ξ (see Eq. (65)). As explained in Subsec. 5.5, the quantity η ξ depends on δL but not on L. To intuitively capture the relation between η ξ and = |S rad |, we plot it in Fig. 5 (ii) as a function of for various fluctuations δL of the initial Kerr BH B in .
In Fig. 5 (ii), we also provide an unnormalized Q function of the initial state ξ Bin in B in , which visualizes how Z-axis-symmetric the state is (see also Fig. 5 (i)). For a pure state |ξ Bin , the Q function is defined with variables L X and L Z , which correspond to the Xand Z-axis AMs, respectively, by where |θ, φ := (cos θ 2 |0 + e iφ sin θ 2 |1 ) ⊗N and (θ, φ) are those satisfying sin θ cos φ = 2L X /N and cos θ = 2L Z /N . Note that the function is regarded as the Husimi Q function for N → ∞ when L Y ≈ N/2.
The Q function in Fig. 5 (ii) is for a state ξ in the form of Eq. (77) with |ϕ n Bin being the state of equal superposition of all product basis states in H Bin n . By comparing the plots of η ξ and the Q functions in Fig. 5 (ii), we observe that, when η ξ is small, the Zaxial symmetry in ξ Bin is strongly broken. This indicates that the information remnant, characterized by η ξ , should be related to the symmetry-breaking of the initial state ξ in the Kerr BH B in .
We can analytically elaborate on this relation. To this end, let us define the degree ζ of symmetrybreaking in a given system by the sensitivity of the state under the symmetric action. For instance, the degree of the symmetry-breaking in the remaining Kerr BH S in , ζ(S in ), is defined as follows: Let |Ψ SinS in be a purification of the state Ψ Sin , and B θ (Ψ) denote where L Z is the Z-axis AM operator in S in and the second identity acts on S in . Then, the degree ζ(S in ) of symmetry-breaking is defined by This quantifies how sensitive the state Ψ is to the symmetric action, which is just a rotation about the Z-axis in this case. The degree of symmetry-breaking defined in this way is closely related to the variance δν 2 of the Zaxis AM ν in Ψ Sin . Since we have we obtain ζ(S in ) = 2 δν 2 .
Let us now relate the degree of symmetry-breaking to the information remnant. The key observation is that, for the whole information in Φ AR to be recoverable from the radiation S rad , classical information about the Z-axis AM in R should not be inferred from the remaining Kerr BH S in [53]. Since we are interested in the information remnant, we also assume that the new radiation S rad is sufficiently large so that the Kerr BH has already get partially decoupled and the state therein and the reference R is Γ SinR . The state is explicitly given by (see Proposition 3) When the Z-axis AM in S in and that in R are measured on the state Γ SinR , the probabilities to obtain ν in S in and κ in R are given by where Γ Sin is the postmeasured state on S in . It is easy to check that they satisfy Using these probabilities, η ξ = ||Γ SinR −Γ Sin ⊗π R || 1 (see Eq. (65)) is bounded from below by We now approximate the value of ν by continuous values and replace the above probabilities P (ν) and P (ν|κ) with probability density functions p(ν) and p(ν|κ), respectively. We also assume that the κ-dependence of p(ν|κ) is approximated by a shift without changing its common shape. Namely, we assume that p(ν|κ) ≈p(ν − αδκ), wherep(µ) := p(µ|κ = κ ) with α = 1 − /(N + k), δκ = κ − κ , and · is the expectation over the probability distribution q(κ). Note that the rescaling by α in Eq. (138) is needed since S is composed of N + k qubits while S in is of N + k − qubits. We also assume that |δκ| is sufficiently small so that which further implies that p(ν) =p(ν) + O(δκ 2 ) due to Eq. (136) and δκ = 0.
Using these approximations and replacing the summation over ν in Eq. (137) by integral, we obtain = α |δκ| dν dp(ν) dν where |δκ| := κ q(κ)|δκ| is the mean absolute deviation of the Z-axis AM in R. The integral in Eq. (141) can be bounded from below by 2 max ν p(ν). Thus, we have The value max ν p(ν) can be further bounded from below in terms of the variance δν 2 of the Z-axis AM in S in . This follows from the fact that the probability distribution that has the least standard deviation under the condition that the maximum probability is given is the rectangle function r(x) = r max for x ∈ [−1/(2r max ), 1/(2r max )] and 0 otherwise. In that case, the variance V is Hence, for any probability density function Applying this to our case, we obtain max ν p(ν) ≥ (2 √ 3 δν 2 ) −1 . We thus arrive at Recalling the relation between the variance and the degree of symmetry-breaking, i.e., δν 2 = ζ(S in )/2, we obtain This implies that, when information remnant is small, ζ(S in ) should be necessarily large, implying that the Z-axial symmetry in the remaining Kerr BH S in should be broken strongly. This clarifies yet another micro-macro correspondence of a quantum Kerr BH.
The symmetry-breaking in S in has two origins: one is that in the initial Kerr BH B in , and the other is due to the new radiation, where -qubit radiation increases the standard deviation of the Z-axis AM in the remaining Kerr BH S in by O( √ ), and so does the degree of symmetry breaking. When √ δL with δL being the standard deviation of the Z-axis AM in the initial Kerr BH B in , the former origin is dominant. On the other hand, when √ δL, the symmetry-breaking in S in is mostly originated from the radiation process. We thus conclude that where a = 2/3 |δκ| (1 − / (N + k)). Here, we have used the fact that the degree of symmetry breaking in B in , ζ(B in ), is given by 2(δL) 2 .
In the case of the pure Kerr BH, the information remnant is negligible in the thermodynamic limit (N → ∞) since the partial decoupling occurs in the pure Kerr BH only when N/2. Thus, Eq. (146) implies that, regardless of δL, the information remnant vanishes when N → ∞. In contrast, for the mixed Kerr BH, there are cases where the partial decoupling sets in when = O(k). In this case, the fluctuation δL of the Z-axis AM in the initial Kerr BH B in determines the information remnant. In particular, if δL is not so large that the lower case of Eq. (146) holds, a non-negligible amount of information remains in the Kerr BH even in the thermodynamic limit unless becomes sufficiently large. This is in sharp contrast to the BH without symmetry. In that case, the information of k qubits can be almost fully recovered from = O(k) radiation, which is independent of N . Thus, the information remnant is a substantial feature induced by the symmetry when the initial Kerr BH is sufficiently mixed with small fluctuation δL of the Z-axis AM.
Finally, we refer to the recent result [54], where a rigorous lower bound on the information remnant for any symmetry was derived in terms of the quantum Fischer information of the initial/remaining BH. Since the quantum Fischer information is another characterization of variance, our and their results are consistent with each other.

Conclusion and Discussions
In this paper, we have investigated the Hayden-Preskill protocol when the system conserves the number of up-spins, or equivalently the Zcomponent of the angular momentum. Based on the partial decoupling approach, we have first provided general formulas for upper bounds on the errors in recovering the information. From the numerical evaluations of the recovery errors, we have shown that the symmetry induces two substantial deviations from the protocol without symmetry. One is the delay of information leakage and the other is the information remnant. Depending on the initial condition of the system, these phenomena can be negligible or macroscopically large.
We have then investigated the delay and the information remnant from the macroscopic physics point of view. By introducing the clipping of entanglement, we have revealed that the delay is characterized by the thermodynamic properties of the initial system. The close relation between the information remnant and the symmetry-breaking of the system has been also revealed. These relations establish connections between the information leakage problem and macroscopic physics.
The results obtained by partial decoupling can be easily generalized to the systems with any abelian global symmetries, such as energy conservation. It is also possible to apply the partial decoupling approach to the systems with non-abelian global symmetry. See Ref. [45] for the more general form of partial decoupling. We, however, leave the non-abelian case as an interesting future problem. In that case, the information cannot be simply divided into the symmetry-invariant and the other ones, and more careful analysis will be needed.
Another important open problem is about an approximate realization of the symmetry-preserving Haar scrambling. All of our analyses rely on the assumption that the internal unitary dynamics of the system is symmetry-preserving scrambling, which will not be rigorously satisfied since it takes exponentially long time to implement a Haar random unitary. Hence, it is of crucial importance to consider approximate implementations of symmetrypreserving scrambling. Approximations of random unitary have been studied in terms of unitary designs, and a couple of results are known about the time needed to achieve unitary designs [55,56,57,58]. It will be of great interest to investigate how symmetries affect such results [59,15]. More explicitly, it is interesting to address if we can construct symmetrypreserving unitary designs by quantum circuits.
is defined by We also use Lipschitz functions. A real-valued function F on a metric space (X, d) is said to be Lipschitz if The quantity ||F || Lip is called an Lipschitz constant of F . The function F with a Lipschitz constant L is called L-Lipschitz.
The following provides the concentrating property on the product space.
Theorem 6 (Concentration of measure on the product space [61]). Let U × = U(d 1 ) × · · · × U(d J ) equipped with the L 2 -sum of Hilbert-Schmidt norms, and H × = H 1 × · · · × H J be the product probability measure. Suppose that a function F : U × → R is L-Lipschitz. Then, for every δ > 0, with probability at most exp[− δ 2 dmin Based on Theorems 5 and 6, we now prove Theorem 1 by identifying the unitary U S = j∈J U S j as a point (U S 1 , . . . , U S J ) on U × = U(d 1 ) × · · · × U(d J ). Proof (Theorem 1). For a given U S = j∈J U S j , let (U S 1 , . . . , U S J ) be the corresponding point on U × equipped with the L 2 -sum of Hilbert-Schmidt norms D. By a slight abuse of notation, we also denote the point by U S .
Let F : U × → R be a function given by Using these lemmas, Corollary 2 can be shown as follows.
Proof (Corollary 2). We first define the state˜ SR bỹ We also use the fact that Γ ER := Tr S * [Γ S * ER ] = E U S ∼H× M S→E U S SR U S † . Using the triangle inequality, we obtain In the following, we evaluate each term in the righthand side of Eq. (168) separately. For the first term, noting that C S→E is a trace-nonincreasing map, and the trace norm is unitarily invariant, we have where the last inequality follows from the gentle measurement lemma and the assumption that Tr[ SR Π S ≥ ] ≥ 1 − . To evaluate the second term, we use Theorem 1. Recalling thatΨ S does not have support on the subspace H S j with dimension being smaller than d th , it follows that, for any δ > 0, with probability at least 1 − exp[− δ 2 d th 48||˜ S ||∞ ]. Here, Since Π ≥ is commutable with Π S j for any j, ≤ min 1, where we have used the sub-multiplicativity of the operator norm, and the assumption that Tr[ SR Π S ≥ ] ≥ 1 − . Combining all of these together, the second term is bounded as for any δ > 0 with probability at least 1−exp[− δ 2 d th 48C ]. To evaluate the third term in the right-hand side of Eq. (168), we use the explicit form of the averaged operator (179) Further, using the relation Π S ≥ = j∈J ≥ Π S j , we obtain that the third term X in the right-hand side of Eq. (168) satisfy Combining the upper bounds of all three terms in Eq. (168), we obtain the desired result:

C Computation of p n
We here show Eq. (50) by explicitly computing p n , which is defined by We first take the average of U S (π A ⊗ ξ Bin )U S † over U S ∼ H × , which leads to due to the Schur's lemma. By substituting this, we have Tr Π S m (π A ⊗ ξ Bin ) Tr π S m (I Sin ⊗ Π S rad n ) .
(190) It is then straightforward to compute each term.
The first trace is given by In total, we have Eq. (50).

D Empirical smoothing of the conditional entropy
We here show Proposition 3. The statement is that, for any δ > 0, the dynamics L S→Sin  For the pure Kerr BH, we first observe from the case of δL = 0 (black circles) that δ ∆ (L, 0) = O( √ N ) for any L. Note that the delay is in comparison with the case of L = δL = 0. Hence, the delay for δL = 0 is not plotted in Fig. 6  For the mixed BH, the delay δ ∆ (L, 0) is independent of N . This seems to be also the case for δL = O( √ N ). When δL = O(N ), δ ∆ (L, δL) appears to increase as N increases. It also seems that the scaling of δ ∆ (L, δL) with respect to N depends on L: when L = 0, the delay may grow very slowly as N goes to large, while it seems to scales linearly with N when |L| ≥ N/4.