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

The HP protocol is a quantum information-theoretic toy model of a quantum black hole. In the model, a quantum black hole is represented by a closed quantum many-body system that undergoes unitary time evolution. After some time, a qubit is displaced from the system, which models an emission of a single qubit as the Hawking radiation from the black hole. The rest of the system again undergoes unitary time evolution, and another qubit is displaced. By repeating the unitary time evolution and displacement of a single qubit, the initial system is eventually split into two subsystems: one is considered to be a remaining black hole and the other is a number of qubits representing the Hawking radiation.

One may regard this model as the AdS/CFT correspondence of a quantum black hole, which predicts that a quantum black hole has a quantum mechanical description without gravity. This paper, however, does not argue the origin of the model and simply follows the original proposal of the protocol [6]. By focusing on the minimal setting, we can investigate the effect of symmetry to the HP protocol from a purely information-theoretic perspective.

Following the original modelling, we refer to the initial many-body system as a BH and to the random subsystems as a radiation and a remaining BH. The size of the radiation increases as time passes, and the size of a remaining BH decreases. Eventually, the whole BH becomes radiation.

A brief description of the HP protocol is as follows [6]. The full details are given in Sec. 4. See also Fig. 1 therein.

1. 1.

   Quantum information source $A$ of $k$ qubits is thrown into an initial BH $B_{\rm in}$ composed of $N$ qubits, which has already emitted some radiation $B_{\rm rad}$. The quantum information in $A$ is kept track of by a reference system $R$.

2. 2.

   The new BH $S:=AB_{\rm in}$ of $N+k$ qubits undergoes scrambling dynamics and gradually emits new radiation, i.e., a random subsystem $S_{\rm rad}$ of $S$. The number of qubits in $S_{\rm rad}$ is denoted by $\ell$. The BH shrinks to the remaining BH $S_{\rm in}$ of $N+k-\ell$ qubits. As time goes on, $\ell$ increases.

3. 3.

   The quantum information originally stored in $A$ is tried to be recovered from the radiation $B_{\rm rad}$ and $S_{\rm rad}$.

It was shown in Ref. [6] that, if the size $\ell$ of the new radiation $S_{\rm rad}$ exceeds a certain threshold, which is roughly given by

$$\ell>k+\frac{N-H(B_{\rm in})}{2},$$

(1)

where $H(B_{\rm in})$ is the entropy of the initial BH $B_{\rm in}$, the remaining BH $S_{\rm in}$ is decoupled from the reference $R$. As a result, the error in recovering the information from the radiations $B_{\rm rad}$ and $S_{\rm rad}$ decreases exponentially quickly as $\ell$ increases. The threshold depends on the properties of the initial BH $B_{\rm in}$, 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 $B_{\rm in}$ and by $\delta L$ 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, one needs to collect more radiation in order to recover quantum information of $A$. The information remnant means 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 $\delta L$ of the initial Kerr BH $B_{\rm in}$, the delay ranges from $O(\sqrt{k})$ to macroscopically large values, and the information remnant varies from infinitesimal to constant.

Our result implies that, 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. See Sec. 5 for details.

To physically understand the delay and the information remnant, we further investigate them from different perspectives. The details are given in Sec. 6.

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 $S_{\rm rad}$,

$$k<H(B_{\rm in})+\log\biggl{[}\frac{\dim{\mathcal{H}}_{n}^{S_{\rm rad}}}{\dim{\mathcal{H}}_{L-n}^{S_{\rm in}}}\biggr{]},$$

(2)

where $H(B_{\rm in})$ is the entropy of the initial Kerr BH $B_{\rm in}$, ${\mathcal{H}}_{n}^{S_{\rm rad}}$ is the subspace of the Hilbert space of $S_{\rm rad}$ with the $Z$-axis AM $n$, and ${\mathcal{H}}^{S_{\rm in}}_{L-n}$ is that of the remaining Kerr BH $S_{\rm in}$ 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: in this case, the condition simply reduces to

$$k<H(B_{\rm in})+\log\biggl{[}\frac{\dim{\mathcal{H}}^{S_{\rm rad}}}{\dim{\mathcal{H}}^{S_{\rm in}}}\biggr{]},$$

(3)

due to the absence of conserved quantities. From this, one obtains $\ell>k+(N-H(B_{\rm in}))/2$, which is the same as Eq. (1).

A merit of the entanglement clipping is that the condition for the information leakage, Eq. (2), 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 $B_{\rm in}$ in terms of the $Z$-axis AM. The derivatives are then connected to thermodynamic properties of $B_{\rm in}$ as follows. Let $\omega(T,\lambda)$ be the state function of $B_{\rm in}$ conjugate to the $Z$-axis AM, which equals to the angular velocity of the Kerr BH, and $\alpha(T,\lambda)$ be thermal sensitivity of the $Z$-axis AM. We can show that, when the initial Kerr BH $B_{\rm in}$ has a small $Z$-axis AM,

$${\rm delay}\propto\frac{L}{S}\sqrt{k_{B}|\omega(T,\lambda)\alpha(T,\lambda)|},$$

(4)

where $k_{B}$ is the Boltzmann constant, and $S$ is the entropy of the initial Kerr BH. Note that both $\omega(T,\lambda)$ and $\alpha(T,\lambda)$ 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 $Z$-axis AM of the remaining Kerr BH $S_{\rm in}$ remains correlated to that with the reference $R$. In other words, the information originally stored in $A$ can be partially obtained by measuring the $Z$-axis AM in $S_{\rm in}$. From the viewpoint of the radiation $S_{\rm rad}$, this implies that the information about the coherence of the $Z$-axis AM in $A$ cannot be fully accessed, leading to the information remnant. The amount of the information remnant is closely related to the fluctuation of the $Z$-axis AM in the remaining Kerr BH $S_{\rm in}$: if the fluctuation is small, the change in the $Z$-axis AM caused by throwing $A$ into the Kerr BH is easy to detect. We quantitatively confirm this expectation and show that

$\displaystyle{\rm information}\ {\rm remnant}$

$\displaystyle\gtrsim\frac{1}{\sqrt{\langle\delta\nu^{2}\rangle}},$

(5)

where $\sqrt{\langle\delta\nu^{2}\rangle}$ is the standard deviation of the $Z$-axis AM in the remaining Kerr BH $S_{\rm in}$. We further show that the fluctuation $\sqrt{\langle\delta\nu^{2}\rangle}$ is characterized by the degree $\zeta(S_{\rm in})$ of symmetry-breaking in $S_{\rm in}$, leading to

$\displaystyle{\rm information}\ {\rm remnant}$

$\displaystyle\gtrsim\frac{1}{\sqrt{2\zeta(S_{\rm in})}}.$

(6)

This reveals that the information remnant is characterized by the degree of symmetry-breaking of the Kerr BH.

The relations, Eqs. (4) and (6), 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. (4) and (6) should be replaced by the corresponding conserved quantities.

A canonical instance is energy conservation, in which the relations similar to Eqs. (4) and (6) can be obtained. The delay of information leakage when energy is conserved should be given in terms of the heat capacity $C_{V}$ of the quantum system as

$${\rm delay}\propto\frac{E}{S}\sqrt{k_{B}|C_{V}|}$$

(7)

where $E$ and $S$ are the energy and the entropy of the initial BH, respectively, and the amount of information remnant is determined by the energy fluctuation of the BH.

Our results reveal, both in quantitative and physically intuitive manners, the presence of symmetry leads to non-negligible changes in the process of information leakage in the HP protocol. The changes were quantified by using the method based on partial decoupling as in Subsec. 2.2, and were qualitatively estimated by physical quantities of the system as explained in Subsec. 2.3. We are mainly concerned with rotational symmetry in our analysis and show that both the delay of information leakage and the information remnant can be large for certain initial conditions. Our approach, methods, and results offer a solid basis for studying quantum information in the many-body systems with symmetry.

In the context of quantum gravity, one of our main contributions is to show that more careful treatments of the symmetry, including the energy conservation, will be needed to fully understand the information recovery from the Hawking radiation. In fact, the presence of symmetry can result in the situation where information thrown into a quantum black hole cannot be recovered until it is completely evaporated. This conclusion is in sharp contrast to that of the original analysis of the HP protocol without symmetry and tells us that taking the black hole’s symmetry, such as energy conservation, into account is crucial for understanding the quantum information perspective of quantum gravity through the HP protocol.

A couple of analyses were already carried out about the energy conservation in the HP protocol [14, 15]. Therein, the physical modes of the radiation responsible for the instant information recovery and the relations between the HP protocol and other puzzles in quantum gravity were discussed [33, 34, 35, 36]. Despite the fact that these studies shed novel light on quantum gravity, the arguments highly depend on information-theoretic assumptions about the details of the model, which are sometimes unimportant from the viewpoint of physics or even unphysical. Hence, the conclusions are strongly depending on information-theoretic details of the model: one may arrive at entirely different results depending of the assumptions. To avoid such undesired situations, understanding the information perspectives of quantum gravity without referring to too much details about quantum information will be helpful.

Our approach provides a sophisticated method to this end. The simple formulas obtained in this paper can be used to understand information recovery only in terms of physical quantities. For instance, Eq. (2) is a formula to investigate the information recovery only from the degeneracy of eigenspaces. This implies, in the case of energy conservation, that the information recovery can be assessed only from the energy spectrum of the Hamiltonian of a quantum black hole. If one accepts a controversial assumption that a quantum black hole has usual thermodynamic features, which may be justified by the AdS/CFT correspondence, Eq. (7) provides a more direct way of evaluating the information recovery only from macroscopic physical properties of the black hole. The information remnant induced by the energy conservation can be also discussed based on Eq. (5), which may however be negligible in the thermodynamic limit if a black hole has a typical amount of energy fluctuation.

To summarize, our analysis provides 1. a caution not to literally accept the instant recovery of quantum information from the Hawking radiation in a realistic situation, and 2. convenient tools of investigating the information recovery only in terms of the quantities often studied in physics. Although our result itself does not provide conclusive answers to the most intriguing questions, such as whether the process of information leakage is changed by energy conservation in the leading order, our approach will be a stepping stone toward the full understanding of the information recovery from the Hawking radiation in a realistic situation with energy conservation. To this end, a couple of physical properties of a quantum black hole, such as the energy spectrum, its initial state, and the radiation process with energy conservation, should be clarified.

We denote a set of linear operators from ${\mathcal{H}}^{A}$ to ${\mathcal{H}}^{B}$ by $\mathcal{L}({\mathcal{H}}^{A},{\mathcal{H}}^{B})$, and $\mathcal{L}({\mathcal{H}}^{A},{\mathcal{H}}^{A})$ by $\mathcal{L}({\mathcal{H}}^{A})$. We also use the following notation for the sets of positive semi-definite operators, quantum states, and sub-normalized states.

	$\displaystyle\mathcal{P}(\mathcal{H})=\{X\in\mathcal{L}(\mathcal{H}):X\geq 0\},$		(8)
	$\displaystyle\mathcal{S}(\mathcal{H})=\{\rho\in\mathcal{P}(\mathcal{H}):\operatorname{Tr}[\rho]=1\},$		(9)
	$\displaystyle\mathcal{S}_{\leq}(\mathcal{H})=\{\rho\in\mathcal{P}(\mathcal{H}):\operatorname{Tr}[\rho]\leq 1\}.$		(10)

A maximally entangled state between $S$ and $S^{\prime}$, where ${\mathcal{H}}^{S}\cong{\mathcal{H}}^{S^{\prime}}$, is denoted by $\Phi^{SS^{\prime}}$. The completely mixed state on ${\mathcal{H}}^{S}$ is denoted by $\pi^{S}=I^{S}/d_{S}$ ($d_{S}={\rm dim}{\mathcal{H}}^{S}$).

A purification of $\rho^{S}\in{\mathcal{S}}({\mathcal{H}}^{S})$ by $R$ (${\mathcal{H}}^{R}\cong{\mathcal{H}}^{S}$) is denoted by $|\rho\rangle^{SR}$, namely, $\operatorname{Tr}_{R}[|\rho\rangle\langle\rho|^{SR}]=\rho^{S}$. For instance, since the marginal state of a maximally entangled state is the completely mixed state, we have $\Phi^{S}=\pi^{S}$.

The fundamental superoperators are the conjugations by a unitary, an isometry, and a partial isometry. An isometry $V^{{\mathcal{H}}\rightarrow{\cal K}}$ is the linear operator such that

$$(V^{{\mathcal{H}}\rightarrow{\cal K}})^{\dagger}V^{{\mathcal{H}}\rightarrow{\cal K}}=I^{{\mathcal{H}}}.$$

(11)

When $\dim{\mathcal{H}}=\dim{\cal K}$, the isometry is called a unitary and also satisfies $V^{{\mathcal{H}}\rightarrow{\cal K}}(V^{{\mathcal{H}}\rightarrow{\cal K}})^{\dagger}=I^{{\cal K}}$. A partial isometry is a linear operator from ${\mathcal{H}}$ to ${\cal K}$ such that it is an isometry on its support. Projections, isometries, and unitaries are special classes of a partial isometry.

A quantum channel $\mathcal{T}^{S\rightarrow B}$ is a completely-positive (CP) and trace-preserving (TP) map. A map is called CP if $({\rm id}^{S^{\prime}}\otimes\mathcal{T}^{S\rightarrow B})(\rho^{S^{\prime}S})\geq 0$ for any $\rho^{S^{\prime}S}\geq 0$ and is TP if $\operatorname{Tr}[\mathcal{T}^{S\rightarrow B}(\rho^{S})]=\operatorname{Tr}[\rho^{S}]$. We also say a superoperator $\mathcal{T}^{S\rightarrow B}$ is sub-unital and unital if $\mathcal{T}^{S\rightarrow B}(I^{S})\leq I^{B}$ and $\mathcal{T}^{S\rightarrow B}(I^{S})=I^{B}$, respectively.

For a quantum channel ${\cal T}^{S\rightarrow B}$, a complementary channel is defined as follows. Let $V_{{\cal T}}^{S\rightarrow BE}$ be a Steinspring dilation for a quantum channel $\mathcal{T}^{S\rightarrow B}$, i.e., $V^{S\rightarrow BE}_{{\cal T}}$ is an isometry satisfying

$$\operatorname{Tr}_{E}[V_{{\cal T}}^{S\rightarrow BE}\rho^{S}(V_{{\cal T}}^{S\rightarrow BE})^{\dagger}]=\mathcal{T}^{S\rightarrow B}(\rho^{S}),$$

(12)

for any $\rho^{S}\in\mathcal{L}(\mathcal{H}^{S})$. The map $\bar{\mathcal{T}}^{S\rightarrow E}$ defined by

$$\bar{\mathcal{T}}^{S\rightarrow E}(\rho^{S}):=\operatorname{Tr}_{B}[V_{{\cal T}}^{S\rightarrow BE}\rho^{S}(V^{S\rightarrow BE}_{{\cal T}})^{\dagger}],$$

(13)

for any $\rho^{S}\in\mathcal{L}(\mathcal{H}^{S})$ is called a complementary channel.

Any superoperator $\mathcal{T}^{S\rightarrow B}$ from $S$ to $B$ has a Choi-Jamiołkowski representation defined by

$$\mathfrak{J}(\mathcal{T}^{S\rightarrow B}):=({\rm id}^{S^{\prime}}\otimes\mathcal{T}^{S\rightarrow B})(\Phi^{SS^{\prime}}).$$

(14)

This is sometimes called the channel-state duality.

The Schatten $p$-norm for a linear operator $X\in\mathcal{L}({\mathcal{H}}^{A},{\mathcal{H}}^{B})$ is defined by $\|X\|_{p}:=(\operatorname{Tr}[(XX^{\dagger})^{p/2}])^{1/p}$ ($p\in[1,\infty]$). We particularly use the trace ($p=1$) and the Hilbert-Schmidt ($p=2$) norms. We define the fidelity between quantum states $\rho$ and $\sigma$ by

$$F(\rho,\sigma):=\|\sqrt{\rho}\sqrt{\sigma}\|_{1}^{2}.$$

(15)

The conditional min-entropy is defined by

$$H_{\rm min}(A|B)_{\rho}\\ =\sup_{\sigma^{B}\in\mathcal{S}(\mathcal{H}^{B})}\sup\{\lambda\in\mathbb{R}|2^{-\lambda}I^{A}\otimes\sigma^{B}-\rho^{AB}\geq 0\},$$

(16)

for $\rho^{AB}\in\mathcal{P}({\mathcal{H}})$. This is a variant of the conditional entropy $H(A|B)_{\rho}:=H(AB)-H(B)$, where $H(A)=-\operatorname{Tr}[\rho^{A}\log\rho^{A}]$ is the von Neumann entropy. When ${\rm dim}{\mathcal{H}}^{B}=1$, $H_{\rm min}(A|B)_{\rho}$ reduces to the min-entropy

$$H_{\rm min}(A)_{\rho}=\sup\{\lambda\in\mathbb{R}|2^{-\lambda}I^{A}-\rho^{A}\geq 0\}.$$

(17)

The Haar measure is often used to formulate the scrambling dynamics. The Haar measure ${\sf H}$ on a unitary group ${\sf U}(d)$ of finite degree $d$ is the unique left- and right- unitarily invariant probability measure. Namely, it satisfies,

$$\text{for any $\mathcal{W}\subset{\sf U}(d)$ and $V\in{\sf U}(d)$},\\ {\sf H}(V\mathcal{W})={\sf H}(\mathcal{W}V)={\sf H}(\mathcal{W}).$$

(18)

When a unitary $U$ is chosen with respect to the Haar measure ${\sf H}$, denoted by $U\sim{\sf H}$, it is called a Haar random unitary. Throughout the paper, the average of a function $f(U)$ over a random unitary $U\sim\mu$, where $\mu$ is a probability measure, is denoted by $\mathbb{E}_{U\sim\mu}[f(U)]$.

In the HP protocol, we consider the situation depicted in Fig. 1. In the protocol, the BH $B_{\rm in}$ initially consists of $N$ qubits. The radiation emitted earlier, which we call the past radiation, is denoted by a quantum system $B_{\rm rad}$.

We particularly consider two types of the BH. One is a “pure” BH, where $B_{\rm in}$ is in a pure state $|\xi\rangle^{B_{\rm in}}$, and the other is a “mixed” BH, where $B_{\rm in}$ is in a mixed state $\xi^{B_{\rm in}}$. 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 $B_{\rm in}B_{\rm rad}$ is in a pure state $|\xi\rangle^{B_{\rm in}B_{\rm rad}}$, which is a purification of $\xi^{B_{\rm in}}$.

The HP protocol is the following thought-experiment. At time $T_{1}$, a person, Alice, throws a quantum information source $A$ of $k$ qubits into the BH $B_{\rm in}$. 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 $\Phi^{AR}$. The BH, now a composite system $S:=AB_{\rm in}$, undergoes the internal unitary dynamics and emits new radiation. We model this process by repeatedly applying an internal unitary dynamics followed by a displacement of a qubit in $S$ to the exterior of the BH. Since each radiation, i.e., the displacement 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 $U_{N+k-i}$.

Suppose that the BH emits $\ell$ qubits of radiation, i.e., $|S_{\rm rad}|=\ell$, by time $T_{2}$. We call $S_{\rm rad}$ the new radiation. The remaining BH $S_{\rm in}$ is composed of $N+k-\ell$ 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 $\mathcal{L}^{S\rightarrow S_{\rm in}S_{\rm rad}}$ from $S$ to $S_{\rm in}S_{\rm rad}$ is simply a product of unitary dynamics,

$$\mathcal{L}^{S\rightarrow S_{\rm in}S_{\rm rad}}=\mathcal{U}_{N+k-\ell+1}\circ\mathcal{U}_{N+k-\ell+2}\circ\dots\circ\mathcal{U}_{N+k},$$

(19)

where $\mathcal{U}_{N+k-i}(\rho):=(I_{i}\otimes U_{N+k-i})\rho(I_{i}\otimes U_{N+k-i})^{\dagger}$ and the identity $I_{i}$ acts on the radiated $i$ qubits. The quantum channel from the BH $S$ to the remaining BH $S_{\rm in}$ and that to the new radiation $S_{\rm rad}$, denoted by $\mathcal{L}^{S\rightarrow S_{\rm in}}$ and $\mathcal{L}^{S\rightarrow S_{\rm rad}}$, are given by

	$\displaystyle\mathcal{L}^{S\rightarrow S_{\rm in}}:=\operatorname{Tr}_{S_{\rm rad}}\circ\mathcal{L}^{S\rightarrow S_{\rm in}S_{\rm rad}},$		(20)
	$\displaystyle\mathcal{L}^{S\rightarrow S_{\rm rad}}:=\operatorname{Tr}_{S_{\rm in}}\circ\mathcal{L}^{S\rightarrow S_{\rm in}S_{\rm rad}},$		(21)

respectively. Note that they are complementary to each other since $\mathcal{L}^{S\rightarrow S_{\rm in}S_{\rm rad}}$ is a unitary dynamics.

We then introduce another person Bob. He collects all the $\ell$ qubits of the new radiation $S_{\rm rad}$ and tries to recover the $k$-qubit information originally stored in $A$. He may additionally use the past radiation $B_{\rm rad}$. We assume that Bob knows the initial state $|\xi\rangle^{B_{\rm in}B_{\rm rad}}$ and the whole dynamics $\mathcal{L}^{S\rightarrow S_{\rm in}S_{\rm rad}}$. In the recovery process, Bob tries to reproduce the state $\Phi^{AR}$ maximally entangled with the reference $R$ by applying a quantum channel ${\mathcal{D}}$ to the quantum systems he has, i.e., to the new and past radiations, $S_{\rm rad}$ and $B_{\rm rad}$. 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 $|\psi\rangle$, Bob would be able to recover $|\psi\rangle$. Due to the no-cloning theorem [41], 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 $S_{\rm rad}$ of $\ell$ qubits.

To quantitatively address whether Bob succeeds in recovering information of $A$, we define and analyze a recovery error $\Delta$. Let $\hat{\Phi}^{AR}$ be the final state of the whole process, i.e.,

$$\hat{\Phi}^{AR}_{{\mathcal{D}}}:={\mathcal{D}}^{S_{\rm rad}B_{\rm rad}\rightarrow A}\circ\mathcal{L}^{AB_{\rm in}\rightarrow S_{\rm rad}}(\Phi^{AR}\otimes\xi^{B_{\rm in}B_{\rm rad}}).$$

(22)

Since the goal of Bob is to establish the state maximally entangled with the reference $R$, we define the recovery error by

$$\Delta(\xi,\mathcal{L}):=\min_{{\mathcal{D}}}\bigl{[}1-F\bigl{(}\Phi^{AR},\hat{\Phi}^{AR}_{{\mathcal{D}}}\bigr{)}\bigr{]},$$

(23)

where the minimum is taken over all possible quantum channels ${\mathcal{D}}^{S_{\rm rad}B_{\rm rad}\rightarrow A}$ applied by Bob onto the new and the past radiation, $S_{\rm rad}$ and $B_{\rm rad}$. Note that $0\leq\Delta(\xi,\mathcal{L})\leq 1$, and that the recovery error generally depends on the initial state $\xi$ of the initial BH $B_{\rm in}$ and its dynamics $\mathcal{L}$.

When the BH has no symmetry, the dynamics $\mathcal{L}^{S\rightarrow S_{\rm in}S_{\rm rad}}$ is considered to be fully scrambling. This is formulated by choosing each internal unitary dynamics $U_{N+k-i}$ in $\mathcal{U}_{N+k-i}$ of the BH (see Eq. (19)) at Haar random. When this is the case, we denote the dynamics by $\mathcal{L}_{\rm Haar}$. The recovery error $\Delta$ in this case is characterized by the min-entropy $H_{\rm min}(B_{\rm in})_{\xi}$ of the initial state of the BH $B_{\rm in}$: for the scrambling dynamics $\mathcal{L}_{\rm Haar}$ of the BH, the recovery error $\Delta$ satisfies [6, 42]

$$\log_{2}[\Delta(\xi,\mathcal{L}_{\rm Haar})]\\ \leq\min\bigl{\{}0,k+\frac{N-H_{\rm min}(B_{\rm in})_{\xi}}{2}-\ell\bigr{\}},$$

(24)

with high probability. This clearly shows that, when $\ell\geq k+\frac{N-H_{\rm min}(B_{\rm in})_{\xi}}{2}$, the recovery error $\Delta(\xi,\mathcal{L}_{\rm Haar})$ decreases exponentially quickly. Note that the min-entropy $H_{\rm min}(B_{\rm in})_{\xi}$ is large if and only if the initial BH $B_{\rm in}$ and the past radiation $B_{\rm rad}$ are strongly entangled. Hence, in this framework, it is the entanglement between the initial BH $B_{\rm in}$ and the past radiation $B_{\rm rad}$ that determines the time scale for the information recovery to be possible.

For the pure BH, $H_{\rm min}(B_{\rm in})_{\xi}=0$, so that $\Delta(\xi,\mathcal{L}_{\rm Haar})\leq 2^{N/2+k-\ell}$. In contrast, for the mixed BH after the Page time [43], $H_{\rm min}(B_{\rm in})_{\xi}=N$, resulting in $\Delta_{tot}(\xi:\mathcal{L}_{\rm Haar})\leq 2^{k-\ell}$. 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. (24) is decoupling [44, 45, 46, 42] in the sense that $\mathcal{L}_{\rm Haar}^{S\rightarrow S_{\rm in}}(\Phi^{AR}\otimes\xi^{B_{\rm in}})\approx\pi^{S_{\rm in}}\otimes\pi^{R}$, 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 $S_{\rm rad}$ easily.

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

The partial decoupling theorem is proven by two of the authors in Ref. [47]. 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 ${\mathcal{H}}^{S}$ has a direct-sum structure, such that

$$\mathcal{H}^{S}=\bigoplus_{j\in\mathcal{J}}\mathcal{H}^{S}_{j}.$$

(37)

The dimensions are denoted as $\dim{\mathcal{H}}^{S}=D_{S}$ and $\dim{\mathcal{H}}^{S}_{j}=d_{j}$. Let $\varrho^{SR}$ be an initial state on $SR$. A random unitary $U^{S}$ in the form of $\bigoplus_{j\in\mathcal{J}}U^{S}_{j}$ is first applied to $S$, and then a given CPTP map ${\mathcal{M}}^{S\rightarrow E}$ is applied. This leads to the state

$${\mathcal{M}}^{S\rightarrow E}(U^{S}\varrho^{SR}U^{S\dagger}).$$

(38)

Despite the fact that this state is dependent on $U^{S}$, the partial decoupling theorem states that, when $U^{S}\sim{\sf H}_{\times}:={\sf H}_{1}\times\dots\times{\sf H}_{J}$, where ${\sf H}_{j}$ is the Haar measure on the unitary group ${\sf U}(d_{j})$ on ${\mathcal{H}}_{j}^{S}$, 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 $\varrho^{SR}$ is subnormalized, i.e., $\varrho^{SR}\in{\mathcal{S}}_{\leq}({\mathcal{H}}^{SR})$, and the map ${\mathcal{M}}^{S\rightarrow E}$ is a trace-non-increasing CP map.

The fixed (subnormalized) state is explicitly given by $\Gamma^{ER}\in\mathcal{S}_{\leq}(\mathcal{H}^{S^{*}\!RE})$, which is constructed as

$$\Gamma^{ER}=\sum_{j\in\mathcal{J}}\frac{D_{S}}{d_{j}}\zeta^{E}_{jj}\otimes\varrho^{R}_{jj},$$

(39)

where

	$\displaystyle\zeta^{E}_{jj}=\operatorname{Tr}_{S^{\prime}}\bigl{[}\Pi^{S^{\prime}}_{j}\zeta^{S^{\prime}E}\bigr{]},$		(40)
	$\displaystyle\varrho^{R}_{jj}=\operatorname{Tr}_{S}\bigl{[}\Pi^{S}_{j}\varrho^{SR}\bigr{]},$		(41)

with $\zeta^{S^{\prime}E}=\mathfrak{J}({\mathcal{M}}^{S\rightarrow E})$ being the Choi-Jamiołkowski representation of ${\mathcal{M}}^{S\rightarrow E}$ (see Eq. (14)). Note that the state $\Gamma^{ER}$ 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 $\Gamma^{S^{*}\!RE}\in\mathcal{S}_{\leq}(\mathcal{H}^{S^{*}\!RE})$ of $\Gamma^{ER}$ by a system $S^{*}=S^{\prime}S^{\prime\prime}$, where $S^{\prime}$ and $S^{\prime\prime}$ are replicas of the system $S$:

$$\Gamma^{S^{*}ER}:=\sum_{j,j^{\prime}\in\mathcal{J}}\frac{D_{S}}{\sqrt{d_{j}d_{j^{\prime}}}}\zeta^{S^{\prime}E}_{jj^{\prime}}\otimes\varrho^{S^{\prime\prime}R}_{jj^{\prime}},$$

(42)

where

	$\displaystyle\zeta^{S^{\prime}E}_{jj^{\prime}}=\Pi^{S^{\prime}}_{j}\zeta^{S^{\prime}E}\Pi^{S^{\prime}}_{j^{\prime}},$		(43)
	$\displaystyle\varrho^{S^{\prime\prime}R}_{jj^{\prime}}=\Pi^{S\rightarrow S^{\prime\prime}}_{j}\varrho^{SR}(\Pi^{S\rightarrow S^{\prime\prime}}_{j^{\prime}})^{\dagger}.$		(44)

Here, $\Pi_{j}^{S\rightarrow S^{\prime\prime}}=V^{S\rightarrow S^{\prime\prime}}\Pi_{j}^{S}$, with $V^{S\rightarrow S^{\prime\prime}}$ is a unitary that maps $S$ to $S^{\prime\prime}$.

Using these subnormalized states, the partial decoupling theorem is stated as follows.

The sub-normalized states $\Gamma^{ER}$ and $\Gamma^{S^{*}\!RE}$ have relatively simple expressions if we use a diagram. Let ${\cal C}^{S\rightarrow SS^{*}}_{{\mathcal{J}}}$ be a CPTP map corresponding to a “${\mathcal{J}}$-correlator” given by

$${\cal C}^{S\rightarrow SS^{*}}_{{\mathcal{J}}}(\rho^{S})=C^{S\rightarrow SS^{*}}\rho^{S}(C^{S\rightarrow SS^{*}})^{\dagger},$$

(46)

where $C^{S\rightarrow SS^{*}}$ is the isometry given by

$$C^{S\rightarrow SS^{*}}=\sum_{j\in{\mathcal{J}}}\Pi_{j}^{S\rightarrow S^{\prime\prime}}\otimes|\Phi_{j}\rangle^{SS^{\prime}}.$$

(47)

Here, $|\Phi_{j}\rangle^{SS^{\prime}}$ is the state maximally entangled only in the subspace ${\mathcal{H}}_{j}^{S}\otimes{\mathcal{H}}_{j}^{S^{\prime}}$. In other words, the ${\mathcal{J}}$-correlator generates the maximally entangled state in the subspace ${\mathcal{H}}_{j}^{S^{\prime\prime}}\otimes{\mathcal{H}}_{j}^{S^{\prime}}$ depending on the value of $j\in{\mathcal{J}}$ in the system $S$, and swaps the system $S$ with $S^{\prime\prime}$. Using this CPTP map, the subnormalized states $\Gamma^{ER}$ and $\Gamma^{S^{*}ER}$ are expressed as follows:

	$\displaystyle\Gamma^{ER}=\ \ \ {\begin{array}[]{c}\vbox{\vskip 0.72002pt\hbox{\hskip 0.48001pt\includegraphics[scale={0.24}]{TensorNetwork3}}}\end{array}}\ ,$		(49)
	$\displaystyle\Gamma^{S^{*}ER}=\ \ \ {\begin{array}[]{c}\vbox{\vskip 0.72002pt\hbox{\hskip 0.48001pt\includegraphics[scale={0.24}]{TensorNetwork0}}}\end{array}}\ .$		(51)

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 $H_{\rm min}(S^{*}|ER)_{\Gamma}$ is sufficiently large, the state ${\mathcal{M}}^{S\rightarrow E}\bigl{(}U^{S}\varrho^{SR}U^{S\dagger}\bigr{)}$ is close to $\Gamma^{ER}=\operatorname{Tr}_{S^{*}}[\Gamma^{S^{*}ER}]$ with high probability. In the diagram representation, this can be rewritten as

$\displaystyle{\begin{array}[]{c}\vbox{\vskip 0.81001pt\hbox{\hskip 0.54001pt\includegraphics[scale={0.27}]{TensorNetwork2}}}\end{array}}\ \ \ \approx\ \ \ {\begin{array}[]{c}\vbox{\vskip 0.72002pt\hbox{\hskip 0.48001pt\includegraphics[scale={0.24}]{TensorNetwork3}}}\end{array}}\ ,$

(54)

up to the approximation $2^{-H_{\rm min}(S^{*}|ER)_{\Gamma}/2}$ in the trace norm with high probability.

When $d_{\rm min}$ is small, such as $d_{\rm min}=1$, Theorem 1 fails to provide a strong concentration since the probability in the statement becomes tiny for small $\delta$. When this is the case, we can set a “threshold” dimension $d_{\rm th}$.

The proofs of Theorem 1 and Corollary 2 are given in Appendices A and B, respectively.

We now apply the partial decoupling to the Kerr BH by identifying the dynamics $\mathcal{L}_{\rm Kerr}^{S\rightarrow S_{\rm in}S_{\rm rad}}$ with the symmetry-preserving scrambling $U^{S}_{\rm Kerr}\sim{\sf H}_{\rm Kerr}$, which is in the form of

$$U^{S}_{\rm Kerr}=\bigoplus_{m=0}^{N+k}U_{m}^{S}.$$

(56)

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 $\varrho$ and ${\mathcal{M}}$ in Theorem 1 to $\Phi^{AR}\otimes\xi^{B_{\rm in}}$ and $\operatorname{Tr}_{S_{\rm rad}}$, respectively. The system $E$ in Theorem 1 corresponds to $S_{\rm in}$ in this case.

Using the doubled replica $S^{*}=S^{\prime}S^{\prime\prime}$ of the BH $S$, we have

$\displaystyle{\begin{array}[]{c}\vbox{\vskip 0.86998pt\hbox{\hskip 0.57999pt\includegraphics[scale={0.29}]{TensorNetworkBH1}}}\end{array}}\ \ \approx\ \ {\begin{array}[]{c}\vbox{\vskip 0.72002pt\hbox{\hskip 0.48001pt\includegraphics[scale={0.24}]{TensorNetworkBH2-2}}}\end{array}},$

(59)

with high probability, where ${\cal C}_{\rm AM}$ is the AM-correlator that generates the state maximally entangled in a subspace. More explicitly, ${\cal C}_{\rm AM}$ is the conjugation map by the following isometry

$$C_{\rm AM}^{S\rightarrow SS^{*}}=\sum_{m=0}^{N+k}\Pi_{m}^{S\rightarrow S^{\prime\prime}}\otimes|\Phi_{m}\rangle^{SS^{\prime}},$$

(60)

where $|\Phi_{m}\rangle^{SS^{\prime}}$ is the state maximally entangled only in the subspace ${\mathcal{H}}^{S}_{m}\otimes{\mathcal{H}}^{S^{\prime}}_{m}$. According to Theorem 1, Eq. (59) holds up to $\approx 2^{-H_{\rm min}(S^{*}|S_{\rm in}R)_{\Gamma}/2}$ in the trace norm, where

$\displaystyle\Gamma^{S^{*}S_{\rm in}R}$

$\displaystyle={\begin{array}[]{c}\vbox{\vskip 0.75pt\hbox{\hskip 0.5pt\includegraphics[scale={0.25}]{TensorNetworkBH2full}}}\end{array}}\ .$

(62)

Hence, if $H_{\rm min}(S^{*}|S_{\rm in}R)_{\Gamma}\gg 1$, it is highly likely that the L.H.S. of Eq. (59) is partially decoupled, namely, the Kerr BH is partially decoupled from the reference $R$.

From Theorem 1, the probability for Eq. (59) to hold up to the approximation $2^{-\frac{1}{2}H_{\rm min}(S^{*}|ER)_{\Gamma}}+\delta$ is at least $1-\exp[-\delta^{2}2^{k}d_{\rm min}/48]$, where we have used $\|\pi^{A}\otimes\xi^{B_{\rm in}}\|_{\infty}\leq 2^{-k}$. Since $d_{\rm min}=1$ for the axial symmetry, Theorem 1 fails to assure a high probability if $\delta<2^{-k/2}$. This problem can be circumvented by using Corollary 2 instead of Theorem 1 and by setting a threshold dimension. However, the degree of approximation $2^{-\frac{1}{2}H_{\rm min}(S^{*}|ER)_{\Gamma}}$ 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 [48]. 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 [49]. We here exploit a limited smoothing by ignoring “less probable” subspaces ${\mathcal{H}}_{n}^{S_{\rm rad}}$ of the radiation $S_{\rm rad}$. 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 $S_{\rm rad}$ is $n$. We can explicitly compute the average weight $p_{n}$, namely, the trace of the subnormalized state, by taking the average over $U_{\rm Kerr}^{S}$:

$$p_{n}=\frac{1}{2^{k}}\sum_{m=0}^{N+k}\sum_{\kappa=0}^{k}\chi_{m-\kappa}\frac{\binom{\ell}{n}\binom{N+k-\ell}{m-n}\binom{k}{\kappa}}{\binom{N+k}{m}},$$

(63)

where $\chi_{\mu}=\operatorname{Tr}[\xi^{B_{\rm in}}\Pi^{B_{\rm in}}_{\mu}]$ (see Appendix C). Using $p_{n}$, we define a probable set $I_{\epsilon}$ by

$$I_{\epsilon}=\{n\in[0,\ell]:p_{n}\geq\epsilon\},$$

(64)

for $\epsilon\geq 0$. We call the subspaces ${\mathcal{H}}_{n}^{S_{\rm rad}}$ for $n\notin I_{\epsilon}$ rare events. Here, the parameter $\epsilon$ 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.