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

Yoshifumi Nakata1,2,3 Eyuri Wakakuwa4 Masato Koashi2 1Yukawa Institute for
Theoretical Physics, Kyoto university, Kitashirakawa Oiwakecho, Sakyo-ku,
Kyoto, 606-8502, Japan
2Photon Science Center, Graduate School of Engineering, The University of
Tokyo, Bunkyo-ku, Tokyo 113-8656, Japan
3JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan
4Department of Communication Engineering and Informatics, Graduate School of
Informatics and Engineering, The University of Electro-Communications, Tokyo
182-8585, Japan

###### Abstract

The Hayden-Preskill protocol is a quantum information theoretic model of the
black hole information paradox. Based on the protocol, it was revealed that
information scrambling and entanglement lead to an instant leakage of
information. In this paper, we study the information paradox with symmetry in
the framework of the Hayden-Preskill protocol. Symmetry is an important
feature of black holes that induces yet more conceptual puzzles in the regime
of quantum gravity. We especially consider an axial symmetry and clarify its
consequences in the information leakage problem. Using a partial decoupling
approach, we first show that symmetry induces a _delay_ of information leakage
and an _information remnant_ , both of which can be macroscopically large for
certain initial conditions. We then clarify the physics behind the delay and
the information remnant. By introducing the concept of _clipping of
entanglement_ , we show that the delay is characterized by thermodynamic
properties of the black hole associated with the symmetry. We also show that
the information remnant is closely related to the symmetry-breaking of the
black hole. These relations indicate the existence of non-trivial microscopic-
macroscopic correspondences in the information leakage problem.

##  I 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 H1974 ; H1975 : 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 I1967 ;
W1968 ; C1971 , the holographic principle indicates that information should
leak out as a black hole evaporates, indicating a gap between our macroscopic
and microscopic understandings of black holes. This problem 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 HP2007 . Based
on a quantum mechanical model of black holes with scrambling dynamics SS2008 ;
LSHOH2013 , it was shown that information is extremely quickly leaked out from
a quantum black hole. This result spiked a large number of research topics
aiming to understand quantum gravity from the quantum information point of
view SS2008 ; S2011 ; LSHOH2013 ; SS2014 ; SS2015 ; RS2015 ; RY2017 ; Y2019 ;
SY1993 ; K2015 ; Jensen2016 ; MS2016 ; Sachdev2015 ; B2016 ; vKRPS2018 ;
KVH2018 ; HQRY2016 ; PYHP2015 ; PEW2017 ; KC2019 ; HP2019 , and substantially
contributed to the progress not only in fundamental physics but also in a wide
range of theoretical physics.

The problem has, however, yet to be resolved. One of the obstacles is
symmetry. Black holes in reality have symmetry at least in the classical
limit, which prevents the dynamics from being fully scrambling due to the
conservation law. This implies that there are cases where the basic postulate
in the HP protocol fails to hold, highlighting the importance of studying
scrambling with symmetry and its consequence in the information leakage
problem.

Symmetry in quantum gravity has been argued for a long time. It leads to a
couple of conceptual puzzles, such as weak violation of global symmetries
BS2011 ; HO2019  and the weak gravity conjecture AHMNV2007 . The information
leakage problem with symmetry will also provide insights into these problems.
In the early time of the black hole evaporation, the weak violation of global
symmetry shall not be observed, and the scrambling dynamics should effectively
remain symmetric. In contrast, the violation will be gradually amplified in
the long-time limit, and the dynamics eventually approach the full scrambling.
Thus, the weak violation of symmetry shall result in a transition of the
behavior of the information leakage from the symmetric one to the non-
symmetric one. To understand such phenomena related to the violation of
symmetry, it is important to understand both symmetric and non-symmetric
cases.

In this paper, we study the information leakage problem of a quantum black
hole with exact symmetry in the framework of the HP protocol. We first develop
a quantum information-theoretic tool, called a _partial decoupling_ , which is
useful to simultaneously deal with quantum information and symmetry. Based on
the partial decoupling, we show that the symmetry induces two effects into the
information leakage problem: a _delay_ of information leakage and an
_information remnant_ , both of which can be macroscopic depending of the
initial condition of the black hole. The physics behind these phenomena is
also clarified: the delay and the information remnant are, respectively,
characterized by thermodynamic properties and symmetry-breaking of the black
hole. These relations unveil non-trivial connections between microscopic
features, i.e., quantum information, of black holes and its macroscopic
physics associated with the symmetry.

The paper is organized as follows. For clarity of the presentation, we provide
a brief overview of our results in Sec. II. We start the main analysis with
introducing our notation in Sec. III. In Sec. IV, 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. V. The physics behind the delay and the information remnant is discussed
in Sec. VI. We conclude this paper with discussions in Sec. VII. Technical
statements are proven in Appendices.

##  II Overview of the Results

In the HP protocol, the black hole is represented by a quantum many-body
system, and the Hawking radiation by its random subsystem. Throughout the
paper, we refer to the many-body system as a _BH_ and to the random subsystem
as a _radiation_. Roughly speaking, the HP protocol is as follows:

  1. 1.

Quantum information source $A$ of $k$ qubits is thrown into an initial BH
$B_{\rm in}$ that 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 BH $S:=AB_{\rm in}$ undergoes the scrambling dynamics and gradually emits
the new radiation $S_{\rm rad}$. The BH shrinks to the remaining BH $S_{\rm
in}$.

  3. 3.

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

It was shown in Ref. HP2007  that, if the size of the new radiation $S_{\rm
rad}$ exceeds a certain threshold, the phenomena called _decoupling_ occurs in
the BH. As a result, the error in recovering the information decreases
exponentially quickly as the amount of new radiation increases. This implies
that most of the information has been carried away by that threshold amount of
radiation. The threshold depends on the initial condition of the BH $B_{\rm
rad}$, and can be the same order of the size of the quantum information source
$A$.

In this paper, we consider the BH with the $Z$-axial symmetry, which is one of
the symmetries that the black hole is allowed to have in general relativity
I1967 ; W1968 ; C1971 . We call it the Kerr BH and study the HP protocol under
the constraint that the scrambling dynamics respects the axial symmetry. The
conserved quantity is the $Z$-component of the angular momentum, which we
denote by the _$Z$ -axis AM_ for short. We assume that the $Z$-axis AM in the
Kerr BH $B_{\rm in}$ is initially $L$ and its fluctuation is $\delta L$.

We show that the constraint on scrambling dynamics by the symmetry results in
the _partial decoupling_ of the Kerr BH rather than the full decoupling as in
the case of BH without any symmetry. Due to this difference, two substantial
changes are brought into the HP protocol, the _delay_ of information leakage
and the _information remnant_. The delay means that, compared to the BH
without any symmetry, more radiation is needed to recover the 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 $\delta L$ 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. Thus, with certain initial conditions, the symmetry radically
changes the behavior of how radiation carries information away. In particular,
when the Kerr BHs has extremely large angular momentum 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 the two 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 to be possible: for most of possible values of the $Z$-axis AM $n$ in
the new radiation $S_{\rm rad}$,

$k>H(B_{\rm in})+\log\biggl{[}\frac{\dim{\mathcal{H}}_{n}^{S_{\rm
rad}}}{\dim{\mathcal{H}}_{L-n}^{S_{\rm in}}}\biggr{]},$ (1)

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 HP2007  can be also obtained
from the same idea. We further argue that the clipping argument should hold
for any BH with extensive conserved quantities, such as energy and electric
charges. In those cases, the angular momentum in Eq. (1) should be replaced by
the corresponding conserved quantities.

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 BH
and the dimension of the Hilbert spaces with respect to the conserved
quantity. Hence, for any BH with 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: it unveils a non-trivial relation between the delay of
information leakage and thermodynamic properties of the BH. Let
$\omega(T,\lambda)$ be the state function of $B_{\rm in}$ conjugate to the
conserved quantity, and $\alpha(T,\lambda)$ be thermal sensitivity of the
conserved quantity. We show that, when the initial Kerr BH $B_{\rm in}$ has a
small $Z$-axis AM,

${\rm delay}\propto\sqrt{k_{B}|\omega(T,\lambda)\alpha(T,\lambda)|},$ (2)

where $k_{B}$ is the Boltzman constant. 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 Kerr BH. Importantly, this relation is
independent of the details of the HP protocol. Hence, we conjecture that this
relation should hold beyond the framework of the HP protocol. Note also that a
similar relation shall hold for the black holes with any abelian symmetries.

The information remnant can be also intuitively understood. Since the
remaining Kerr BH $S_{\rm in}$ keeps some correlation of the conserved
quantity with the reference $R$ even after the partial decoupling sets, the
decoherence in the quantum information in $A$ should be necessarily induced.
This is the same mechanism as quantum eraser. By investigating how much
classical information about the reference $R$ can be inferred from the
remaining Kerr BH $S_{\rm in}$, we quantitatively show that

${\rm information}\ {\rm remnant}\gtrsim\frac{1}{\sqrt{\zeta(S_{\rm in})}},$
(3)

where $\zeta(S_{\rm in})$ represents the degree of symmetry-breaking in
$S_{\rm in}$. This indicates that the symmetry-breaking in the Kerr BH is the
dominant factor of the information remnant.

The micro-macro relations, Eqs. (2) and (3), make it possible to investigate
the information leakage problem in terms of macroscopic physics without
directly referring to quantum information itself. Hence, they should provide
useful insights into the further analysis of quantum black holes.

##  III Preliminaries

Throughout the paper, we denote the sequence $\\{i,i+1,\dots,j-1,j\\}$ of
integers between two natural numbers $i$ and $j$ ($i\leq j$) by $[i,j]$. The
logarithm is always taken in base two.

The basic unit of quantum information is a qubit represented by a spin-1/2
particle. 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
$\mathcal{H}^{S}$ of a system $S$, an operator $X^{SR}$ on $SR$, and a
superoperator $\mathcal{E}^{S\rightarrow B}$ from $S$ to $B$. For
superoperators from $S$ to itself, we denote it by $\mathcal{E}^{S}$.

The partial trace over a system $X$ is denoted by $\operatorname{Tr}_{X}$. A
reduced operator on $S$ of $\rho^{SR}$ is denoted simply by $\rho^{S}$, that
is, $\rho^{S}=\operatorname{Tr}_{R}[\rho^{SR}]$. Furthermore, we denote
$(M^{S}\otimes I^{R})\rho^{SR}(M^{S\dagger}\otimes I^{R})$, where $I$ is the
identity operator, by $M^{S}\rho^{SR}M^{S\dagger}$. The identity superoperator
on $S$ is denoted by ${\rm id}^{S}$.

###  III.1 Linear Operators and Superoperators

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.

$\displaystyle\mathcal{P}(\mathcal{H})=\\{X\in\mathcal{L}(\mathcal{H}):X\geq
0\\},$ (4)
$\displaystyle\mathcal{S}(\mathcal{H})=\\{\rho\in\mathcal{P}(\mathcal{H}):\operatorname{Tr}[\rho]=1\\},$
(5)
$\displaystyle\mathcal{S}_{\leq}(\mathcal{H})=\\{\rho\in\mathcal{P}(\mathcal{H}):\operatorname{Tr}[\rho]\leq
1\\}.$ (6)

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

$(V^{{\mathcal{H}}\rightarrow{\cal
K}})^{\dagger}V^{{\mathcal{H}}\rightarrow{\cal K}}=I^{{\mathcal{H}}}.$ (7)

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 S1955  for a quantum channel $\mathcal{T}^{S\rightarrow B}$, i.e.,
$V^{S\rightarrow BE}_{{\cal T}}$ is an isometry satisfying

$\operatorname{Tr}_{E}[V_{{\cal T}}^{S\rightarrow BE}\rho^{S}(V_{{\cal
T}}^{S\rightarrow BE})^{\dagger}]=\mathcal{T}^{S\rightarrow B}(\rho^{S}),$ (8)

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

$\bar{\mathcal{T}}^{S\rightarrow E}(\rho^{S}):=\operatorname{Tr}_{B}[V_{{\cal
T}}^{S\rightarrow BE}\rho^{S}(V^{S\rightarrow BE}_{{\cal T}})^{\dagger}],$ (9)

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 J1972 ; C1975  defined by

$\mathfrak{J}(\mathcal{T}^{S\rightarrow B}):=({\rm
id}^{S^{\prime}}\otimes\mathcal{T}^{S\rightarrow B})(\Phi^{SS^{\prime}}).$
(10)

This is sometimes called the channel-state duality.

###  III.2 Norm and Entropy

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

$F(\rho,\sigma):=\|\sqrt{\rho}\sqrt{\sigma}\|_{1}^{2}.$ (11)

The conditional min-entropy is defined by

$H_{\rm min}(A|B)_{\rho}\\\
=\sup_{\sigma^{B}\in\mathcal{S}(\mathcal{H}^{B})}\sup\\{\lambda\in\mathbb{R}|2^{-\lambda}I^{A}\otimes\sigma^{B}-\rho^{AB}\geq
0\\},$ (12)

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

$H_{\rm
min}(A)_{\rho}=\sup\\{\lambda\in\mathbb{R}|2^{-\lambda}I^{A}-\rho^{A}\geq
0\\}.$ (13)

###  III.3 Haar scrambling

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}).$ (14)

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
$\mathbb{E}_{U\sim\mu}[f(U)]$.

##  IV Hayden-Preskill protocol

The Hayden-Preskill (HP) protocol was proposed to analyze the black hole
information paradox by the information-theoretic manner. We overview the
general setting of the HP protocol in Subsec. IV.1 and review the results in
Ref. HP2007  with a slight generalization in Subsec. IV.2. We then explain how
symmetry shall change the situation in Subsec. IV.3. The diagram of the
protocol is given in Fig. 1, and important quantities are summarized in Table
1.

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 right-hand 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 ${\mathcal{D}}$ is the recovery operation applied by Bob, aiming at recovering the information originally stored in $A$ from the past radiation $B_{\rm rad}$ and the new radiation $S_{\rm rad}$.  Table 1: Notation in the investigation of the BH information problem $B_{\rm in}$ ($N$ qubits) | Initial BH before quantum information source is thrown in.
---|---
$B_{\rm rad}$ ($N^{\prime}$ qubits) | “Past” radiation that purifies the initial BH $B_{\rm in}$.
$A$ ($k$ qubits) | Quantum information source Alice throws into the BH $B_{\rm in}$.
$R$ ($k$ qubits) | Reference system that is maximally entangled with $A$.
$S$ ($N+k$ qubits) | The BH after the information source $A$ is thrown in ($S=AB_{\rm in}=S_{\rm in}S_{\rm rad}$).
$S_{\rm rad}$ ($\ell$ qubits) | New radiation from the BH $S$.
$S_{\rm in}$ ($N+k-\ell$ qubits) | Remaining BH after $S_{\rm rad}$ is evaporated.

###  IV.1 General setting of the HP protocol

In the HP protocol, we consider the situation depicted in Fig. 1. In the
protocol, the BH $B_{\rm in}$ initially consists of $N$ qubits, where $N$ is
supposed to be the Bekenstein-Hawking entropy. The Hawking 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 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 $U_{N+k-i}$.

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,

$\mathcal{L}^{S\rightarrow S_{\rm in}S_{\rm
rad}}=\mathcal{U}_{N+k-\ell+1}\circ\mathcal{U}_{N+k-\ell+2}\dots\mathcal{U}_{N+k-1}\circ\mathcal{U}_{N+k},$
(15)

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

$\displaystyle\mathcal{L}^{S\rightarrow S_{\rm in}}:=\operatorname{Tr}_{S_{\rm
rad}}\circ\mathcal{L}^{S\rightarrow S_{\rm in}S_{\rm rad}},$ (16)
$\displaystyle\mathcal{L}^{S\rightarrow S_{\rm
rad}}:=\operatorname{Tr}_{S_{\rm in}}\circ\mathcal{L}^{S\rightarrow S_{\rm
in}S_{\rm rad}},$ (17)

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 WZ1982 , 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.,

$\hat{\Phi}^{AR}_{{\mathcal{D}}}:={\mathcal{D}}^{S_{\rm rad}B_{\rm
rad}\rightarrow A}\circ\mathcal{L}^{AB_{\rm in}\rightarrow S_{\rm
rad}}(\Phi^{AR}\otimes\xi^{B_{\rm in}B_{\rm rad}}).$ (18)

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

$\Delta(\xi,\mathcal{L}):=\min_{{\mathcal{D}}}\bigl{[}1-F\bigl{(}\Phi^{AR},\hat{\Phi}^{AR}_{{\mathcal{D}}}\bigr{)}\bigr{]},$
(19)

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 $\mathcal{L}$.

###  IV.2 Review of the HP protocol without symmetry

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. (15)) 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
HP2007 ; DBWR2010

$\log_{2}[\Delta(\xi,\mathcal{L}_{\rm
Haar})]\leq\min\bigl{\\{}0,k+\frac{N-H_{\rm min}(B_{\rm
in})_{\xi}}{2}-\ell\bigr{\\}},$ (20)

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 Page1993 , $H_{\rm min}(B_{\rm in})_{\xi}=N$,
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. (20) is _decoupling_ HOW2005 ; HOW2007 ; ADHW2009 ;
DBWR2010  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.

###  IV.3 The HP protocol with symmetry

Let us now describe how situation should change in the case of the Kerr BH
with the $Z$-axial symmetry.

####  IV.3.1 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 ${\mathcal{H}}^{A}$, ${\mathcal{H}}^{B_{\rm in}}$, and
${\mathcal{H}}^{S}$ are decomposed into the subspaces invariant under the
$Z$-axial symmetry:

${\mathcal{H}}^{A}=\bigoplus_{\kappa=0}^{k}\mathcal{H}_{\kappa}^{A},\
{\mathcal{H}}^{B_{\rm in}}=\bigoplus_{\mu=0}^{N}\mathcal{H}_{\mu}^{B_{\rm
in}}\ \text{and}\ {\mathcal{H}}^{S}=\bigoplus_{m=0}^{N+k}\mathcal{H}_{m}^{S},$
(21)

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 $\Pi_{\kappa}^{S}$, $\Pi_{\mu}^{B_{\rm in}}$,
and $\Pi_{m}^{S}$.

The symmetry also affects the quantum information stored in $A$. As mentioned,
quantum information in $A$ is represented by the maximally entangled state
$\Phi^{AR}$ with the reference $R$. The Hilbert space ${\mathcal{H}}^{R}$ is
also decomposed according to the decomposition of ${\mathcal{H}}^{A}$, namely,
${\mathcal{H}}^{R}=\bigoplus_{\kappa}\mathcal{H}^{R}_{\kappa}$, where
$\mathcal{H}^{R}_{\kappa}$ is the range of
$\operatorname{Tr}_{A}[\Pi_{\kappa}^{A}\Phi^{AR}\Pi_{\kappa}^{A}]$. Using the
projection $\Pi^{R}_{\kappa}$ onto $\mathcal{H}^{R}_{\kappa}$, we define

$\Phi^{AR}_{\rm diag}=\sum_{\kappa}p_{\kappa}\Phi_{\kappa}^{AR},$ (22)

with $p_{\kappa}$ being
$\operatorname{Tr}[(I^{A}\otimes\Pi^{R}_{\kappa})\Phi^{AR}]$ and
$\Phi_{\kappa}^{AR}$ being
$(I^{A}\otimes\Pi^{R}_{\kappa})\Phi^{AR}(I^{A}\otimes\Pi^{R}_{\kappa})/p_{\kappa}$.
Since $\Phi_{\rm diag}^{AR}$ 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 $\Delta_{inv}$ for the symmetry-invariant information in
$\Phi^{AR}_{\rm diag}$, and the other is $\Delta_{tot}$ for the total
information in $\Phi^{AR}$. They are, respectively, defined by

$\displaystyle\Delta_{inv}(\xi,\mathcal{L})=\min_{{\mathcal{D}}}\bigl{[}1-F\bigl{(}\Phi^{AR}_{\rm
diag},\hat{\Phi}^{AR}_{{\rm diag},{\mathcal{D}}}\bigr{)}\bigr{]},$ (23)
$\displaystyle\Delta_{tot}(\xi,\mathcal{L})=\min_{{\mathcal{D}}}\bigl{[}1-F\bigl{(}\Phi^{AR},\hat{\Phi}^{AR}_{{\mathcal{D}}}\bigr{)}\bigr{]}.$
(24)

The recovery error $\Delta_{tot}$ for the total information is the same as
$\Delta$ in Eq. (19), but we below denote it by $\Delta_{tot}$ to clearly
distinguish it from $\Delta_{inv}$. Using a diagram is useful to represent the
states $\hat{\Phi}^{AR}_{{\rm diag},{\mathcal{D}}}$ and
$\hat{\Phi}^{AR}_{{\mathcal{D}}}$ in Eqs. (23) and (24):

$\displaystyle\hat{\Phi}^{AR}_{{\rm diag},{\mathcal{D}}}=\ \ \
{\begin{array}[]{c}\vbox{\vskip 0.75pt\hbox{\hskip
0.5pt\includegraphics[scale={0.25}]{TensorNetworkKerrBH0diag}}}\end{array}},$
(26) $\displaystyle\hat{\Phi}^{AR}_{{\mathcal{D}}}=\ \ \
{\begin{array}[]{c}\vbox{\vskip 0.75pt\hbox{\hskip
0.5pt\includegraphics[scale={0.25}]{TensorNetworkKerrBH0}}}\end{array}},$ (28)

respectively. Here, Eq. (28) represents the state given by Eq. (18). The state
defined by Eq. (26) is defined similarly by replacing $\Phi$ in Eq. (18) with
$\Phi_{\rm diag}$.

The states $\hat{\Phi}^{AR}_{{\rm diag},{\mathcal{D}}}$ and
$\hat{\Phi}^{AR}_{{\mathcal{D}}}$ are related to each other by the pinching
map ${\cal C}^{R}$ on $R$, defined by ${\cal
C}^{R}(\rho):=\sum_{\kappa}\Pi_{\kappa}^{R}\rho\Pi_{\kappa}^{R}$ such as
$\hat{\Phi}^{AR}_{{\rm diag},{\mathcal{D}}}={\cal
C}^{R}(\hat{\Phi}^{AR}_{{\mathcal{D}}})$. This is also represented as

$\displaystyle\hat{\Phi}^{AR}_{{\rm diag},{\mathcal{D}}}=\ \ \
{\begin{array}[]{c}\vbox{\vskip 0.75pt\hbox{\hskip
0.5pt\includegraphics[scale={0.25}]{TensorNetworkKerrBH0anddiag}}}\end{array}}.$
(30)

####  IV.3.2 Symmetry-preserving scrambling

Since the Kerr BH has a $Z$-axial symmetry, its internal unitary dynamics,
namely, each $U_{N+k-i}$ in Eq. (15), should be in the form of

$U_{N+k-i}=\bigoplus_{m=0}^{N+k-i}U_{N+k-i}^{(m)},$ (31)

where $U_{N+k-i}^{(m)}$ 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
$U_{N+k-i}^{(m)}$ is Haar random on each subspace, namely, $U_{N+k-i}^{(m)}$
scrambles all quantum states with exactly $m$ up-spins. 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 $\mathcal{L}_{\rm
Kerr}$. Using the unitarily invariant property of the Haar measure, it is
straightforward to observe that the statistics induced by the dynamics
$\mathcal{L}_{\rm Kerr}^{S\rightarrow S_{\rm in}S_{\rm rad}}$ is the same as
that induced by a single application of a symmetry-preserving scrambling
$U^{S}_{\rm Kerr}$ on the Kerr BH $S$. Thus, in the following analysis, we
consider $U^{S}_{\rm Kerr}$ instead of $\mathcal{L}_{\rm Kerr}^{S\rightarrow
S_{\rm in}S_{\rm rad}}$.

The symmetry-preserving scrambling unitary $U_{\rm Kerr}^{S}$ is formally
defined by introducing a product of Haar measures: let ${\sf H}_{m}$ be the
Haar measure on the unitary group acting on the subspace with $m$ up-spins.
Then, we define the symmetry-preserving scrambling $U_{\rm Kerr}^{S}$ by

$U_{\rm Kerr}^{S}\sim{\sf H}_{\rm Kerr}:={\sf H}_{0}\times{\sf
H}_{1}\times\dots\times{\sf H}_{N+k}.$ (32)

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. Y2019 . Since decoupling is the key
technique used to derive the recovery error, Eq. (20), 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 $\Delta_{inv}$
and $\Delta_{tot}$ for the symmetry-invariant and total information when the
dynamics is $\mathcal{L}_{\rm Kerr}$, or equivalently $U_{\rm Kerr}$.

##  V Information recovery –partial decoupling approach–

To investigate the recovery errors $\Delta_{inv}$ and $\Delta_{tot}$, we use
the partial decoupling approach WN2019 , which we briefly summarize in Subsec.
V.1. We also provide a slight generalization. We then apply partial decoupling
to the Kerr BH in Subsec. V.2. We derive formulas for the upper bounds on the
recover errors in Subsec. V.4. Based on the formulas, we numerically
investigate the recovery errors in Subsec. V.5.

###  V.1 Partial decoupling theorem

The partial decoupling theorem is proven by two of the authors in Ref. WN2019
. 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. IV.

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

$\mathcal{H}^{S}=\bigoplus_{j\in\mathcal{J}}\mathcal{H}^{S}_{j}.$ (33)

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

${\mathcal{M}}^{S\rightarrow E}(U^{S}\varrho^{SR}U^{S\dagger}).$ (34)

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

$\Gamma^{ER}=\sum_{j\in\mathcal{J}}\frac{D_{S}}{d_{j}}\zeta^{E}_{jj}\otimes\varrho^{R}_{jj},$
(35)

where

$\displaystyle\zeta^{E}_{jj}=\operatorname{Tr}_{S^{\prime}}\bigl{[}\Pi^{S^{\prime}}_{j}\zeta^{S^{\prime}E}\bigr{]},$
(36)
$\displaystyle\varrho^{R}_{jj}=\operatorname{Tr}_{S}\bigl{[}\Pi^{S}_{j}\varrho^{SR}\bigr{]},$
(37)

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. (10)). 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 $S$:

$\Gamma^{S^{*}ER}:=\sum_{j,j^{\prime}\in\mathcal{J}}\frac{D_{S}}{\sqrt{d_{j}d_{j^{\prime}}}}\zeta^{S^{\prime}E}_{jj^{\prime}}\otimes\varrho^{S^{\prime\prime}R}_{jj^{\prime}},$
(38)

where

$\displaystyle\zeta^{S^{\prime}E}_{jj^{\prime}}=\Pi^{S^{\prime}}_{j}\zeta^{S^{\prime}E}\Pi^{S^{\prime}}_{j^{\prime}},$
(39) $\displaystyle\varrho^{S^{\prime\prime}R}_{jj^{\prime}}=\Pi^{S\rightarrow
S^{\prime\prime}}_{j}\varrho^{SR}(\Pi^{S\rightarrow
S^{\prime\prime}}_{j^{\prime}})^{\dagger}.$ (40)

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.

######  Theorem 1.

For any $\delta>0$, it holds that

$\bigl{\|}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\varrho^{SR}U^{S\dagger}\bigr{)}-\Gamma^{ER}\bigr{\|}_{1}\leq
2^{-\frac{1}{2}H_{\rm min}(S^{*}|ER)_{\Gamma}}+\delta$ (41)

with probability at least $1-\exp\bigl{[}-\frac{\delta^{2}d_{\rm
min}}{48|\\!|\varrho^{S}|\\!|_{\infty}}\bigr{]}$ in terms of the choice of the
unitary $U^{S}\sim{\sf H}_{\times}$, where $d_{\rm
min}=\min_{j\in{\mathcal{J}}}\\{d_{j}\\}$.

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

${\cal C}^{S\rightarrow SS^{*}}_{{\mathcal{J}}}(\rho^{S})=C^{S\rightarrow
SS^{*}}\rho^{S}(C^{S\rightarrow SS^{*}})^{\dagger},$ (42)

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

$C^{S\rightarrow SS^{*}}=\sum_{j\in{\mathcal{J}}}\Pi_{j}^{S\rightarrow
S^{\prime\prime}}\otimes|\Phi_{j}\rangle^{SS^{\prime}}.$ (43)

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:

$\displaystyle\Gamma^{ER}=\ \ \ {\begin{array}[]{c}\vbox{\vskip
0.72002pt\hbox{\hskip
0.48001pt\includegraphics[scale={0.24}]{TensorNetwork3}}}\end{array}}\ ,$ (45)
$\displaystyle\Gamma^{S^{*}ER}=\ \ \ {\begin{array}[]{c}\vbox{\vskip
0.72002pt\hbox{\hskip
0.48001pt\includegraphics[scale={0.24}]{TensorNetwork0}}}\end{array}}\ .$ (47)

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

$\displaystyle{\begin{array}[]{c}\vbox{\vskip 0.81001pt\hbox{\hskip
0.54001pt\includegraphics[scale={0.27}]{TensorNetwork2}}}\end{array}}\ \ \
\approx\ \ \ {\begin{array}[]{c}\vbox{\vskip 0.72002pt\hbox{\hskip
0.48001pt\includegraphics[scale={0.24}]{TensorNetwork3}}}\end{array}}\ ,$ (50)

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 $d_{\rm th}$.

######  Corollary 2.

Consider the same setting as in Theorem 1. Let $d_{\rm th}$ be a positive
integer, and $\Pi_{\geq}^{S}$ be given by
$\Pi_{\geq}^{S}=\sum_{j\in{\mathcal{J}}_{\geq}}\Pi_{j}^{S}$, where
${\mathcal{J}}_{\geq}=\\{j\in{\mathcal{J}}:d_{j}\geq d_{\rm th}\\}$. If
$\operatorname{Tr}[\varrho^{SR}\Pi_{\geq}^{S}]\geq 1-\delta$, then it holds
for any $\delta>0$ that

$\bigl{\|}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\varrho^{SR}U^{S\dagger}\bigr{)}-\Gamma^{ER}\bigr{\|}_{1}\\\
\leq\frac{2^{-\frac{1}{2}H_{\rm
min}(S^{*}|ER)_{\Gamma}}}{\sqrt{1-\delta}}+\delta+f(\delta),$ (51)

with probability at least $1-\exp\bigl{[}-\frac{\delta^{2}d_{\rm
th}}{48C}\bigr{]},$. where
$C=\min\\{1,\frac{|\\!|\varrho^{S}|\\!|_{\infty}}{1-\delta}\\}$, and
$f(\delta)=2\sqrt{\delta}+\delta+\frac{\delta}{1-\delta}$.

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

###  V.2 Partial decoupling in the Kerr BH

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

$U^{S}_{\rm Kerr}=\bigoplus_{m=0}^{N+k}U_{m}^{S}.$ (52)

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

$\displaystyle{\begin{array}[]{c}\vbox{\vskip 0.86998pt\hbox{\hskip
0.57999pt\includegraphics[scale={0.29}]{TensorNetworkBH1}}}\end{array}}\ \
\approx\ \ {\begin{array}[]{c}\vbox{\vskip 0.72002pt\hbox{\hskip
0.48001pt\includegraphics[scale={0.24}]{TensorNetworkBH2-2}}}\end{array}},$
(55)

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

$C_{\rm AM}^{S\rightarrow SS^{*}}=\sum_{m=0}^{N+k}\Pi_{m}^{S\rightarrow
S^{\prime\prime}}\otimes|\Phi_{m}\rangle^{SS^{\prime}},$ (56)

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. (55) holds up to $\approx 2^{-H_{\rm
min}(S^{*}|S_{\rm in}R)_{\Gamma}/2}$ in the trace norm, where

$\displaystyle\Gamma^{S^{*}S_{\rm in}R}$
$\displaystyle={\begin{array}[]{c}\vbox{\vskip 0.75pt\hbox{\hskip
0.5pt\includegraphics[scale={0.25}]{TensorNetworkBH2full}}}\end{array}}\ .$
(58)

Hence, if $H_{\rm min}(S^{*}|S_{\rm in}R)_{\Gamma}\gg 1$, it is highly likely
that the L.H.S. of Eq. (55) is partially decoupled, namely, the Kerr BH is
partially decoupled from the reference $R$.

###  V.3 Empirical smoothing of the conditional min-entropy

From Theorem 1, the probability for Eq. (55) 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 RennerThesis . 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 T2016 . 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 $U_{\rm Kerr}^{S}$:

$p_{n}=\frac{1}{2^{k}}\sum_{m=0}^{N+k}\sum_{\kappa=0}^{k}\chi_{m-\kappa}\frac{\binom{\ell}{n}\binom{N+k-\ell}{m-n}\binom{k}{\kappa}}{\binom{N+k}{m}},$
(59)

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

$I_{\epsilon}=\\{n\in[0,\ell]:p_{n}\geq\epsilon\\},$ (60)

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.

This procedure can be formulated by introducing $\tilde{\Gamma}(\epsilon)$
corresponding to the probable events:

$\displaystyle\tilde{\Gamma}^{S^{*}S_{\rm in}R}(\epsilon):=\Pi^{S_{\rm
rad}}_{\epsilon}\Gamma^{S^{*}S_{\rm in}R}\Pi^{S_{\rm rad}}_{\epsilon},$ (61)

where $\Pi^{S_{\rm rad}}_{\epsilon}:=\sum_{n\in I_{\epsilon}}\Pi_{n}^{S_{\rm
rad}}$. The error from the rare events is then given by

$w(\epsilon)=1-\operatorname{Tr}[\tilde{\Gamma}^{S^{*}S_{\rm
in}R}(\epsilon)]=\sum_{n\notin I_{\epsilon}}p_{n}.$ (62)

Using these, we have the following Proposition (see Appendix D for the proof).

######  Proposition 3.

Let $\epsilon\geq 0$. In the above setting, the probability that

$\displaystyle{\begin{array}[]{c}\vbox{\vskip 0.86998pt\hbox{\hskip
0.57999pt\includegraphics[scale={0.29}]{TensorNetworkBH1}}}\end{array}}\ \
\approx\ \ {\begin{array}[]{c}\vbox{\vskip 0.72002pt\hbox{\hskip
0.48001pt\includegraphics[scale={0.24}]{TensorNetworkBH2-2}}}\end{array}},$
(65)

holds up to approximation $\Theta_{\xi}^{\delta,\epsilon}$ in the trace norm
is at least $1-\exp[-\frac{\delta^{2}d_{\rm min}(\epsilon)}{48}]$, where

$\Theta^{\delta,\epsilon}_{\xi}(N,k,\ell)=2^{1-\frac{1}{2}H_{\rm
min}(S^{*}|S_{\rm in}R)_{\tilde{\Gamma}(\epsilon)}}+2w(\epsilon)+2\delta,$
(66)

and $d_{\rm
min}(\epsilon)=\min\bigl{\\{}\binom{N+k}{m+n}:m\in[0,N+k-\ell],n\in
I_{\epsilon}\bigr{\\}}$.

In the definition of $\Theta^{\delta,\epsilon}_{\xi}$, there is a term
$2^{1-\frac{1}{2}H_{\rm min}(S^{*}|S_{\rm in}R)_{\tilde{\Gamma}(\epsilon)}}$.
This term does not recover the non-smoothing bound $2^{-\frac{1}{2}H_{\rm
min}(S^{*}|S_{\rm in}R)_{\Gamma}}$ in the limit of $\epsilon\rightarrow 0$.
This may imply that the term could be further improved by factor $2$.

###  V.4 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 HOW2005 ; HOW2007 ;
ADHW2009 ; DBWR2010 , we now provide upper bounds on the recovery errors
$\Delta_{inv}(\xi,\mathcal{L}_{\rm Kerr})$ and
$\Delta_{tot}(\xi,\mathcal{L}_{\rm Kerr})$.

The key technique is the Uhlmann’s theorem U1976 , which states that, if any
state $\rho^{AB}$ and a pure state $|\sigma\rangle^{AC}$ (both normalized)
satisfy $\|\rho^{A}-\sigma^{A}\|_{1}\leq\varepsilon$, there exists a CPTP map
${\mathcal{D}}^{C\rightarrow B}$ such that

$1-\varepsilon\leq F\bigl{(}{\mathcal{D}}^{C\rightarrow
B}(|\sigma\rangle\langle\sigma|^{AC}),\rho^{AB}\bigr{)}.$ (67)

We apply this to Eq. (65). Since $\xi^{B_{\rm in}}=\operatorname{Tr}_{B_{\rm
rad}}[|\xi\rangle\langle\xi|^{B_{\rm in}B_{\rm rad}}]$, we have

$\displaystyle{\begin{array}[]{c}\vbox{\vskip 0.81001pt\hbox{\hskip
0.54001pt\includegraphics[scale={0.27}]{TensorNetworkBH1-1}}}\end{array}}\approx\
\ {\begin{array}[]{c}\vbox{\vskip 0.72002pt\hbox{\hskip
0.48001pt\includegraphics[scale={0.24}]{TensorNetworkBH2full}}}\end{array}}\
,$ (70)

with approximation at least $1-\Theta_{\xi}^{\delta,\epsilon}$ in terms of the
fidelity, where ${\mathcal{D}}$ is a CPTP map from the radiations $S_{\rm
rad}B_{\rm rad}$ to $S^{*}\cong
S^{\prime}S^{\prime\prime}=S^{\prime}A^{\prime\prime}B^{\prime\prime}_{\rm
in}$. Taking the partial trace over $S^{\prime}B^{\prime\prime}_{\rm in}$ as
well as the remaining BH $S_{\rm in}$, we obtain

$\displaystyle{\begin{array}[]{c}\vbox{\vskip 0.81001pt\hbox{\hskip
0.54001pt\includegraphics[scale={0.27}]{TensorNetworkBH1-2}}}\end{array}}$
$\displaystyle\approx\ \ {\begin{array}[]{c}\vbox{\vskip 0.72002pt\hbox{\hskip
0.48001pt\includegraphics[scale={0.24}]{TensorNetworkBH4}}}\end{array}},$ (73)
$\displaystyle=\ \ {\begin{array}[]{c}\vbox{\vskip 0.81001pt\hbox{\hskip
0.54001pt\includegraphics[scale={0.27}]{TensorNetworkBH5}}}\end{array}},$ (75)

where ${\cal C}$ is the pinching map with respect to the $Z$-axis AM defined
by ${\cal C}^{S}(\rho^{S})=\sum_{m}\Pi_{m}^{S}\rho^{S}\Pi_{m}^{S}$. Note that
$S=AB_{\rm in}$. The last line simply follows from the fact that the AM-
correlator ${\cal C}_{\rm AM}$ generates a normalized state depending on the
value of the $Z$-axis AM in $S$ (see Eq. (56)), but it is traced out in Eq.
(73).

To obtain the recovery error $\Delta_{inv}$ for the symmetry-invariant
information, we need to compute the fidelity to $\Phi_{\rm diag}={\cal
C}^{R}(\Phi^{AR})=\sum_{\kappa=0}^{k}\Pi_{\kappa}^{R}\Phi^{AR}\Pi_{\kappa}^{R}$.
Using Eq. (75), we have

${\begin{array}[]{c}\vbox{\vskip 0.75pt\hbox{\hskip
0.5pt\includegraphics[scale={0.25}]{TensorNetworkBH1-2d}}}\end{array}}=\
{\begin{array}[]{c}\vbox{\vskip 0.75pt\hbox{\hskip
0.5pt\includegraphics[scale={0.25}]{TensorNetworkBH1-2dc}}}\end{array}},\\\
\approx\ \ {\begin{array}[]{c}\vbox{\vskip 0.81001pt\hbox{\hskip
0.54001pt\includegraphics[scale={0.27}]{TensorNetworkBH5-2}}}\end{array}}=\Phi_{\rm
diag}^{AR}.$ (76)

Recalling that the approximation is at least
$1-\Theta_{\xi}^{\delta,\epsilon}$ in the fidelity, we arrive at

$\Delta_{inv}(\xi,\mathcal{L}_{\rm Kerr})\leq 1-F\bigl{(}\Phi^{AR}_{\rm
diag},\hat{\Phi}^{AR}_{{\rm
diag},{\mathcal{D}}}\bigr{)}\leq\Theta_{\xi}^{\delta,\epsilon}.$ (77)

For the recovery error $\Delta_{tot}$ for the total information, we can
similarly show that

$\displaystyle\Delta_{tot}(\xi,\mathcal{L}_{\rm Kerr})$
$\displaystyle\leq\bigl{\|}\mathcal{L}_{\rm Kerr}^{S\rightarrow S_{\rm
in}}(\Psi^{SR})-\Gamma^{S_{\rm in}}\otimes\pi^{R}\bigr{\|}_{1},$ (78)
$\displaystyle\leq\bigl{\|}\mathcal{L}_{\rm Kerr}^{S\rightarrow S_{\rm
in}}(\Psi^{SR})-\Gamma^{S_{\rm in}R}\bigr{\|}_{1}$ $\displaystyle\hskip
42.67912pt+\bigl{\|}\Gamma^{S_{\rm in}R}-\Gamma^{S_{\rm
in}}\otimes\pi^{R}\bigr{\|}_{1},$ (79)
$\displaystyle\leq\Theta_{\xi}^{\delta,\epsilon}+\eta_{\xi},$ (80)

where $\eta_{\xi}$ is given by

$\displaystyle\eta_{\xi}$ $\displaystyle=\|\Gamma^{S_{\rm in}R}-\Gamma^{S_{\rm
in}}\otimes\pi^{R}\|_{1}$ (81)
$\displaystyle=\frac{1}{2^{k}}\sum_{\nu=0}^{N+k-\ell}\sum_{\kappa=0}^{k}F_{\kappa,\nu}(\xi)\binom{N+k-\ell}{\nu}\binom{k}{\kappa}.$
(82)

Here,

$\displaystyle F_{\kappa,\nu}(\xi)$
$\displaystyle=\biggl{|}\sum_{m=0}^{N+k}\frac{\binom{\ell}{m-\nu}}{\binom{N+k}{m}}\biggl{(}\chi_{m-\kappa}-\frac{1}{2^{k}}\sum_{\kappa^{\prime}=0}^{k}\binom{k}{\kappa^{\prime}}\chi_{m-\kappa^{\prime}}\biggr{)}\biggr{|},$
(83)

and $\chi_{\mu}=\operatorname{Tr}[\xi^{B_{\rm in}}\Pi^{B_{\rm in}}_{\mu}]$.
See Appendix E for the derivation.

In summary, we have derived upper bounds on the recovery errors as follows.

######  Proposition 4.

Let $\epsilon,\delta\geq 0$. The recovery errors $\Delta_{inv}$ and
$\Delta_{tot}$ for symmetry-invariant and total information satisfy

$\displaystyle\Delta_{inv}(\xi,\mathcal{L}_{\rm
Kerr})\leq\Theta_{\xi}^{\delta,\epsilon},$ (84)
$\displaystyle\Delta_{tot}(\xi,\mathcal{L}_{\rm
Kerr})\leq\Theta_{\xi}^{\delta,\epsilon}+\eta_{\xi}$ (85)

with probability at least $1-\exp[-\frac{\delta^{2}d_{\rm
min}(\epsilon)}{48}]$, where

$\Theta^{\delta,\epsilon}_{\xi}(N,k,\ell)=2^{1-\frac{1}{2}H_{\rm
min}(S^{*}|S_{\rm in}R)_{\tilde{\Gamma}(\epsilon)}}+2w(\epsilon)+2\delta,$
(86)

and $d_{\rm
min}(\epsilon)=\min\bigl{\\{}\binom{N+k}{m+n}:m\in[0,N+k-\ell],n\in
I_{\epsilon}\bigr{\\}}$.

Since Proposition 4 holds for any $\epsilon\geq 0$, we define
$\Theta^{\delta}_{\xi}(N,k,\ell)$ by

$\Theta^{\delta}_{\xi}(N,k,\ell):=\min_{\epsilon\geq
0}\Theta^{\delta,\epsilon}_{\xi}(N,k,\ell),$ (87)

and investigate it in the following analysis. By definition,
$\Theta^{\delta}_{\xi}(N,k,\ell)$ is the best possible upper bound of the
recovery errors that can be obtained by the smoothing procedure described in
Subsec. V.3. We have numerically checked that this generally provides better
bounds than those directly obtained from Theorem 1, namely, those without
smoothing the entropy.

###  V.5 Numerical evaluation

The parameter $\delta$ in Proposition 4 controls the trade-off between the
upper bounds on the recovery errors and the probability for the symmetry-
preserving scrambling $\mathcal{L}_{\rm Kerr}$ to result in that errors. In
the following numerical analysis, we consider only the term independent of
$\delta$, that is,

$\displaystyle\Theta_{\xi}(N,k,\ell)=\min_{\epsilon\geq
0}\bigl{\\{}2^{1-\frac{1}{2}H_{\rm min}(S^{*}|S_{\rm
in}R)_{\tilde{\Gamma}(\epsilon)}}+2w(\epsilon)\bigr{\\}}.$ (88)

This suffices since, in the large-$N$ limit, we can typically choose $\delta$
such that $\Theta_{\xi}(N,k,\ell)\gg\delta$ and, at the same time, that the
probability is arbitrarily close to one. To see this, suppose that the
$Z$-axis AM in $B_{\rm in}$ is a constant fraction of $N$. In this case, the
dimension $d_{\rm min}(\epsilon)$ scales as $N^{c_{0}\ell}$ ($c_{0}$ is some
constant). Hence, we can choose $\delta$ to be values with a slightly greater
scaling than $N^{-c_{0}\ell/2}$, such as $N^{-c_{1}\ell/2}$ 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, $\Theta_{\xi}(N,k,\ell)$
typically scales as $2^{-c_{2}\ell}$ ($c_{2}$ is some constant). Since
$\Theta_{\xi}(N,k,\ell)\approx 2^{-c_{2}\ell}\gg N^{-c_{1}\ell/2}=\delta$ for
sufficiently large $N$, $\delta$ is negligible and the probability is
arbitrarily close to one.

To numerically compute $\Theta_{\xi}(N,k,\ell)$, we need to simplify $H_{\rm
min}(S^{*}|S_{\rm in}R)_{\tilde{\Gamma}(\epsilon)}$ that appears in
$\Theta_{\xi}$. This can be done by using the property of the conditional min-
entropy for the special type of states.

Let $\\{{\mathcal{H}}^{A}_{j}\\}$ be mutually orthogonal subspaces of
${\mathcal{H}}^{A}$, and $\pi_{j}^{A}$ be the completely mixed state on
${\mathcal{H}}^{A}_{j}$. For any state $\Lambda^{ABC}$ in the form of
$\sum_{j=0}^{J}p_{j}\pi_{j}^{A}\otimes\rho_{j}^{BC}$, where
$\rho_{j}^{BC}\in\mathcal{S}(\mathcal{H}^{BC})$ and $\\{p_{j}\\}$ is a
probability distribution, it holds that

$2^{-H_{\rm
min}(AB|C)_{\Lambda}}\leq\sum_{j=0}^{J}\frac{p_{j}}{d_{j}}2^{-H_{\rm
min}(B|C)_{\rho_{j}}},$ (89)

where $d_{j}=\dim{\mathcal{H}}^{A}_{j}$. 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 $|\psi\rangle^{AB}$, the
conditional min-entropy is given by

$H_{\rm
min}(A|B)_{\psi}=-2\log\bigl{[}\operatorname{Tr}[\sqrt{\psi^{A}}]\bigr{]}=-2\log\bigl{[}\operatorname{Tr}[\sqrt{\psi^{B}}]\bigr{]}.$
(90)

Below, we provide upper bounds on the recovery errors for the pure and mixed
Kerr BH. In both cases, we characterize the state $\xi$ of the initial Kerr BH
$B_{\rm in}$ by the expectation value $L$ of the $Z$-axis AM and the standard
deviation $\delta L$, defined by

$\displaystyle L=\operatorname{Tr}[L_{Z}\xi]$ (91) $\displaystyle\delta
L=\bigl{(}\operatorname{Tr}[(L_{Z}-L)^{2}\xi]\bigr{)}^{1/2},$ (92)

where $L_{Z}=\frac{\hbar}{2}\sum_{i=1}^{N}Z_{i}$ with $\hbar$ being the Plank
constant and $Z_{i}$ being the Pauli-$Z$ operator acting on the $i$th qubits.
In the remaining of this paper, we set $\hbar=1$ except the last part of
Subsec. VI.1, where we discuss a physical origin of information leakage.

Figure 2: Recovery Errors from the radiation emitted by the pure Kerr BH.
Upper bounds on $\Delta_{inv}(\xi:\mathcal{L}_{\rm Kerr})$ (dashed lines) and
those on $\Delta_{tot}(\xi:\mathcal{L}_{\rm Kerr})$ (filled markers) for the
pure Kerr BH are plotted. We particularly consider the initial pure states
$|\xi\rangle^{B_{\rm in}}$ with various $Z$-axis AM $L$ and fluctuation
$\delta L$. The figures (i) - (iv) show the cases for $|L|=0,N/8,N/4$, and
$3N/8$, respectively, while $\delta L$ is chosen to be $0.1\sqrt{N}$ (red),
$0.5\sqrt{N}$ (blue), $0.9\sqrt{N}$ (green), and $0.3N$ (brown) for each $L$.
For comparison, $\Delta_{inv}$ for $(L,\delta L)=(0,0)$ is plotted by a yellow
dash-dotted line in each figure. 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 $\Delta_{inv}$ for symmetry-invariant
information decays exponentially quickly after $\ell=|S_{\rm in}|$ exceeds a
certain number, while the error $\Delta_{tot}$ for the whole information
typically has a plateau. The dependence of these features on $L$ and $\delta
L$, as well as their physical mechanism, are discussed in the main text. Note
however that, in any case, $\Delta_{inv},\Delta_{tot}\approx 1$ unless
$\ell>(N+k)/2$, which is a consequence of the no-cloning property of quantum
information WZ1982 .

####  V.5.1 The pure Kerr BH

Let us first consider the pure Kerr BH $B_{\rm in}$ with the initial state in
the form of

$|\xi(L,\delta L)\rangle^{B_{\rm
in}}=\sum_{\mu=0}^{N}\sqrt{\chi_{\mu}(L,\delta
L)}|\varphi_{\mu}\rangle^{B_{\rm in}},$ (93)

where $\mu$ stands for the number of up-spins in $B_{\rm in}$. We assume that
$\\{\chi_{\mu}(L,\delta L)\\}_{\mu}$ are given by

$\chi_{\mu}(L,\delta L)\propto\exp\biggl{[}-\frac{(\mu-L-N/2)^{2}}{\delta
L^{2}}\biggr{]},$ (94)

with a proper normalization such that $\sum_{\mu=0}^{N}\chi_{\mu}(L,\delta
L)=1$, and $|\varphi_{\mu}\rangle^{B_{\rm in}}$ is an arbitrary pure state in
${\mathcal{H}}^{B_{\rm in}}_{\mu}$. Due to the symmetry-preserving scrambling,
the recovery errors do not depend on the choice of
$|\varphi_{\mu}\rangle^{B_{\rm in}}$. Note that the expectation value of
$L_{Z}$ and its standard deviation for the state $|\xi(L,\delta
L)\rangle^{B_{\rm in}}$ deviates from $L$ and $\delta L$, respectively, due to
the discretization, but the deviations are negligibly small.

In this case, the subnormalized state $\tilde{\Gamma}^{S^{*}S_{\rm
in}R}(\epsilon)$ can be explicitly written as

$\tilde{\Gamma}^{S^{*}S_{\rm in}R}_{L,\delta L}(\epsilon)=\sum_{n\in
I_{\epsilon}}p_{n}\pi_{n}^{S^{\prime}_{\rm rad}}\otimes\tilde{\Phi}^{S_{\rm
in}S_{\rm in}^{\prime}S^{\prime\prime}R}_{n}(L,\delta L),$ (95)

where $\\{p_{n}\\}$ is the probability distribution given by Eq. (59), and

$|\tilde{\Phi}_{n}(L,\delta L)\rangle^{S_{\rm in}S^{\prime}_{\rm
in}S^{\prime\prime}R}\propto\\\
\sum_{m=0}^{N+k}\sqrt{\frac{1}{\binom{N+k}{m}}}\bigl{(}\Pi_{m-n}^{S_{\rm
in}}|\Phi\rangle^{S_{\rm in}S_{\rm
in}^{\prime}}\bigr{)}\otimes\bigl{(}\Pi_{m}^{S\rightarrow
S^{\prime\prime}}(|\Phi\rangle^{AR}\otimes|\xi\rangle^{B_{\rm in}})\bigr{)},$
(96)

with a proper normalization constant. Here, we have used the notation that
$\Pi_{a}$ is the zero operator if $a<0$.

Using Eqs. (89) and (90), we obtain

$H_{\rm min}(S^{*}|ER)_{\tilde{\Gamma}(\epsilon)}\geq k-\log[\gamma_{\rm
pure}(N,k,\ell|\xi)],$ (97)

where

$\gamma_{\rm pure}(N,k,\ell|\xi)\\\ =\sum_{n\in
I_{\epsilon}}\biggl{(}\sum_{m=0}^{N+k}\sum_{\kappa=0}^{k}\sqrt{\frac{\chi_{m-\kappa}}{\binom{N+k}{m}}}\binom{N+k-\ell}{m-n}\binom{k}{\kappa}\biggr{)}^{2},$
(98)

with $\chi_{m-\kappa}=\operatorname{Tr}[\xi^{B_{\rm in}}\Pi^{B_{\rm
in}}_{m-\kappa}]$. This bound does not depend on the sign of $L$. Namely,
$|\xi(\pm L,\delta L)\rangle$ result in the same bound. This is naturally
expected since the rotation direction of the Kerr BH should not affect any
features of the protocol.

From Eqs. (88) and (97), we obtain

$\displaystyle\Theta_{\xi}(N,k,\ell)$ $\displaystyle\leq\min_{\epsilon\geq
0}\bigl{\\{}2^{1-k/2}\sqrt{\gamma_{\rm
pure}(N,k,\ell|\xi)}+2w(\epsilon)\bigr{\\}}.$ (99)

Since $w(\epsilon)=\sum_{n\notin I_{\epsilon}}p_{n}$, we can numerically
evaluate this bound, from which we obtain upper bounds on $\Delta_{inv}$ and
$\Delta_{tot}$.

In Fig. 2, upper bounds on $\Delta_{inv}$ and $\Delta_{tot}$, i.e.,

$\displaystyle\Delta_{inv}(\xi,\mathcal{L}_{\rm Kerr})\leq\Theta_{\xi},$ (100)
$\displaystyle\Delta_{tot}(\xi,\mathcal{L}_{\rm
Kerr})\leq\Theta_{\xi}+\eta_{\xi}$ (101)

are provided as functions of the number $\ell$ of qubits in the new radiation
$S_{\rm rad}$ for various $L$ and $\delta L$. We observe that $\Delta_{inv}$
starts decaying exponentially after a certain number of $\ell$. The number, as
well as the decaying speed, strongly depends on both $|L|$ and $\delta L$. To
characterize this, we introduce $\ell_{\Delta}(L,\delta L)$ for $\Delta>0$ by

$\ell_{\Delta}(L,\delta
L)=\min\bigl{\\{}\ell\in[0,N+k]|\Theta_{\xi}\leq\Delta\bigr{\\}}.$ (102)

Then, it is observed from Fig. 2 that $\ell_{\Delta}(L,\delta
L)\geq\ell_{\Delta}(0,0)$ for any $\Delta\geq 0$. This implies that the Kerr
BH has a _delay_ in the onset of releasing information compared to the trivial
case ($L=\delta L=0$), 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
$\delta L=\Theta(N)$ (see brown plots), indicating that the symmetry-invariant
information cannot be recovered from the radiation if either $L$ or $\delta L$
is extremely large.

We also observe that $\Delta_{tot}$ behaves differently depending on whether
$\delta L=O(\sqrt{N})$ or $O(N)$. When $\delta L=O(\sqrt{N})$, $\Delta_{tot}$
first behaves similarly to $\Delta_{inv}$. However, soon after that, the
decreasing of the error becomes slow, and $\Delta_{tot}$ remains at a non-
negligible value until $\ell\approx N$. 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 $\delta L$, we also observe that the amount of information remnant is
independent of $L$. When $\delta L=O(N)$, $\Delta_{tot}\approx\Delta_{inv}$
for any $\ell$, implying that there is no information remnant.

####  V.5.2 The mixed Kerr BH

Figure 3: Recovery Errors from the radiation emitted by the mixed Kerr BH.
Upper bounds on $\Delta_{inv}(\xi:\mathcal{L}_{\rm Kerr})$ (dashed lines) and
those on $\Delta_{tot}(\xi:\mathcal{L}_{\rm Kerr})$ (filled markers) for the
mixed Kerr BH are plotted. The mixed Kerr BH $B_{\rm in}$ is entangled with
the past radiation $B_{\rm rad}$. We consider a family of specific entangled
states with various $Z$-axis AM $L$ and fluctuation $\delta L$. The figures
(i) - (iv) show the bounds for $|L|=0,N/8,N/4$, and $3N/8$, respectively,
where $\pm L$ leads to the same result as explained in the caption of Fig. 2,
while $\delta L$ is chosen to be $0.1\sqrt{N}$ (red), $0.5\sqrt{N}$ (blue),
$0.9\sqrt{N}$ (green), and $0.3N$ (brown). The recovery error $\Delta_{inv}$
for $(L,\delta L)=(0,0)$ is also plotted by a yellow dash-dotted line for
comparison. The size $N$ of the initial Kerr BH is set to $500$, and the size
$k$ of the quantum information source to $5$. Similarly to the pure Kerr BH,
we observe that both $\Delta_{inv}(\xi:\mathcal{L}_{\rm Kerr})$ and
$\Delta_{tot}(\xi:\mathcal{L}_{\rm Kerr})$ start decreasing at almost same
timing, and that only $\Delta_{tot}$ stops decreasing soon after, except the
one for $\delta L=0.3N$.

We next consider the mixed Kerr BH, which is initially entangled with the past
radiation $B_{\rm rad}$. For simplicity, we only consider the following
entangled states:

$|\xi(L,\delta L)\rangle^{B_{\rm in}B_{\rm
rad}}=\sum_{\mu=0}^{N}\sqrt{\chi_{\mu}(L,\delta L)}|\Phi_{\mu}\rangle^{B_{\rm
in}B_{\rm rad}},$ (103)

where $\\{\chi_{\mu}\\}_{\mu}$ are given by Eq. (94), and
$|\Phi_{\mu}\rangle^{B_{\rm in}B_{\rm rad}}$ is the maximally entangled state
in ${\mathcal{H}}^{B_{\rm in}}_{\mu}\otimes{\mathcal{H}}^{B_{\rm rad}}_{\mu}$.

In this case, the state $\Gamma^{S^{*}S_{\rm in}R}$ reduces to

$\Gamma^{S^{*}S_{\rm
in}R}=\sum_{n=0}^{\ell}\sum_{\mu=0}^{N}p_{n,\mu}\pi_{n}^{S^{\prime}_{\rm
rad}}\otimes\pi_{\mu}^{B^{\prime\prime}_{\rm
in}}\otimes\tilde{\Psi}_{n,\mu}^{S_{\rm in}S^{\prime}_{\rm
in}A^{\prime\prime}R},$ (104)

where $p_{n,\mu}$ and the normalized pure state $\tilde{\Psi}_{n,\mu}^{S_{\rm
in}S^{\prime}_{\rm in}A^{\prime\prime}R}$ are given by

$\displaystyle p_{n,\mu}=\frac{\chi_{\mu}(L,\delta
L)\binom{\ell}{n}}{2^{k}}\sum_{m=0}^{N+k}\frac{\binom{N+k-\ell}{m-n}\binom{k}{m-\mu}}{\binom{N+k}{m}}$
(105) $\displaystyle|\tilde{\Psi}_{n,\mu}\rangle^{S_{\rm in}S^{\prime}_{\rm
in}A^{\prime\prime}R}\propto$ $\displaystyle\ \ \ \ \ \
\sum_{m=0}^{N+k}\frac{1}{\sqrt{\binom{N+k}{m}}}\bigl{(}\Pi^{S_{\rm
in}}_{m-n}|\Phi\rangle^{S_{\rm in}S^{\prime}_{\rm
in}}\bigr{)}\otimes\bigl{(}\Pi^{A\rightarrow
A^{\prime\prime}}_{m-\mu}|\Phi\rangle^{AR}\bigr{)},$ (106)

respectively. Note that $\sum_{\mu}p_{n,\mu}=p_{n}$, where $p_{n}$ is given by
Eq. (59).

Using Eqs. (89) and (90), we obtain

$H_{\rm min}(S^{*}|ER)_{\tilde{\Gamma}(\epsilon)}\geq
k-\log\bigl{[}\gamma_{\rm mixed}(N,k,\ell|\xi)\bigr{]}$ (107)

where

$\gamma_{\rm mixed}(N,k,\ell|\xi):=\sum_{\mu=0}^{N}\frac{\chi_{\mu}(L,\delta
L)}{\binom{N}{\mu}}\sum_{n\in I_{\epsilon}}\\\
\biggl{(}\sum_{m=0}^{N+k}\sqrt{\frac{1}{\binom{N+k}{m}}}\binom{N+k-\ell}{m-n}\binom{k}{m-\mu}\biggr{)}^{2}.$
(108)

This leads to

$\displaystyle\Theta_{\xi}(N,k,\ell)$ $\displaystyle\leq\min_{\epsilon\geq
0}\bigl{\\{}2^{1-k/2}\sqrt{\gamma_{\rm
mixed}(N,k,\ell|\xi)}+2w(\epsilon)\bigr{\\}},$ (109)

and, we obtain upper bounds on $\Delta_{inv}$ and $\Delta_{tot}$. We again
note that $\pm L$ provide the same bounds.

|  $\delta L=O(\sqrt{N})$ |  $\delta L=O(N)$
---|---|---
Delay for the pure Kerr BH | for any $L$ |  $O(\sqrt{N})$ |  $O(N)$
Delay for the mixed Kerr BH |  for small $L$ |  $O(1)$ |  $O(\sqrt{N})$?
for large $L$ |  $O(1)$ |  $O(N)$?
Table 2: A summary of the delay of information leakage for the pure and mixed
Kerr BHs, which initially have the $Z$-axis AM $L$ and the fluctuation $\delta
L$. The results are obtained from the numerical analysis based on partial
decoupling, where we varies $N$ up to $500$. The order is only in terms of
$N=|B_{\rm in}|$. In the case of the mixed Kerr BH with small $L$ and $\delta
L=O(N)$, it was hard to decide the scaling from our numerics.

The upper bounds on the recovery errors are numerically plotted in Fig. 3 as
functions of $\ell=|S_{\rm rad}|$. Similarly to the pure Kerr BH, the plots
are for various $L$ and $\delta L$. We observe that $\Delta_{inv}$ decays in a
manner similar to the pure Kerr BH. That is, the recovery error starts
decreasing after a certain number of $\ell$, as well as the onset of
decreasing has a delay compared to the trivial case ($L=\delta L=0$, yellow
dotted lines in Fig. 3). While the delay is not so large for small $L$ and
$\delta L$, it becomes substantially large when $|L|$ is large or $\delta
L=\Theta(N)$.

The recovery error $\Delta_{tot}$ for the total information also behaves
similarly to the pure Kerr BH, and it remains at a non-zero value until
$\ell\approx N$. 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 $\delta L$ and is the same as that for the pure Kerr
BH for the same $\delta L$, 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
$\delta L$ of the initial Kerr BH.

####  V.5.3 Scaling of the delay with respect to $N$

We also provide numerical results about how the delay $\ell_{\Delta}(L,\delta
L)$ scales with respect to the size $N$ of the initial Kerr BH $B_{\rm in}$
for various $Z$-axis AMs $L$ and their fluctuations $\delta L$. The result is
summarized in Tab. 2 for the pure and mixed Kerr BHs. See Appendix G for the
details.

##  VI Information recovery –macroscopic physics approach–

In Sec. V, 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. VI.1 and the information remnant in Subsec.
VI.2.

###  VI.1 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.

####  VI.1.1 Clipping of Entanglement

One of the significant features of the scrambling dynamics is that it
generates nearly maximal entanglement between subsystems Page1993 ; L1978 ;
HLW2006 . 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 $\Phi^{AR}$, 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 $B_{\rm rad}$ and $S_{\rm rad}$, then the maximally
entangled state $\Phi^{AR}$ can be reproduced K2007 , 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 $S_{\rm rad}$ and $S_{\rm in}$. 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, because
any bipartite pure state should have an equal size, known as the Schmidt rank,
in the two subsystems NC2000 .

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 $|S_{\rm rad}|\leq|S_{\rm in}|$ since no clipping
occurs in the new radiation $S_{\rm rad}$. Note that all the eigenvalues of
the marginal state in $S_{\rm rad}$ are nearly the same, from which no
information would be retrieved. When $|S_{\rm rad}|\geq|S_{\rm in}|$, the
clipping takes place in $S_{\rm rad}$, namely, the marginal state in $S_{\rm
rad}$ is released from the full rank state. In the case of the mixed BH, the
entanglement clipping in $S_{\rm rad}$ sets in much earlier since the BH is
initially already on the verge of clipping due to the entanglement with the
past radiation $B_{\rm rad}$, which is under control of the decoding person
Bob.

To capture the information-theoretic consequence of the entanglement clipping
in the new radiation $S_{\rm rad}$, we introduce a degree of clipping $C$ in
${\mathcal{H}}^{S_{\rm rad}}$:

$C({\mathcal{H}}^{S_{\rm rad}}):=H(B_{\rm
in})_{\xi}+\log\biggl{[}\frac{\dim{\mathcal{H}}^{S_{\rm
rad}}}{\dim{\mathcal{H}}^{S_{\rm in}}}\biggr{]}.$ (110)

Note that the degree of clipping $C({\mathcal{H}}^{S_{\rm rad}})$ can be
negative. For instance, $C({\mathcal{H}}^{S_{\rm rad}})=-(N+k)$ when $|S_{\rm
rad}|=0$ and $H(B_{\rm in})_{\xi}=0$. When $C({\mathcal{H}}^{S_{\rm rad}})>k$,
the marginal state in the new radiation $S_{\rm rad}$ 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

$C({\mathcal{H}}^{S_{\rm rad}})>k.$ (111)

By explicitly writing down the dimensions, we obtain

$\ell>k+\frac{N-H(B_{\rm in})_{\xi}}{2},$ (112)

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.
(20).

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
$\bigoplus_{m}U^{S}_{m}$, where $U^{S}_{m}$ is the Haar scrambling in the
subspace with $m$ up-spins, generates entanglement between $S_{\rm in}$ and
$S_{\rm rad}$ only in the subspaces. This introduces a special structure of
entanglement between them. To be more precise, we consider the initial Kerr BH
$B_{\rm in}$ with a fixed $Z$-axis AM $L=\lambda N$ and $\delta L=0$, where
$\lambda\in[-1/2,1/2]$. The number of up-spins in $B_{\rm in}$ is $\lambda
N+N/2$. Due to the symmetry-preserving scrambling, the subspace
${\mathcal{H}}_{n}^{S_{\rm rad}}$ with $n$ up-spins in $S_{\rm rad}$ is
entangled only with ${\mathcal{H}}_{n^{\prime}}^{S_{\rm in}}$ satisfying
$n+n^{\prime}\approx\lambda N+(N+k)/2$.

Taking this constraint into account, the entanglement clipping should be
considered in each subspace ${\mathcal{H}}_{n}^{S_{\rm rad}}$ separately. We
thus define the degree of clipping in ${\mathcal{H}}_{n}^{S_{\rm rad}}$ by

$C({\mathcal{H}}_{n}^{S_{\rm rad}}):=H(B_{\rm
in})_{\xi}+\log\biggl{[}\frac{\dim{\mathcal{H}}^{S_{\rm
rad}}_{n}}{\dim{\mathcal{H}}^{S_{\rm in}}_{\lambda N+(N+k)/2-n}}\biggr{]},$
(113)

and require that

$C({\mathcal{H}}_{n}^{S_{\rm rad}})>k,$ (114)

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. (114) 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 $S_{\rm rad}$ induced by the random choice of $S_{\rm rad}$
from $S$. The probability is given by

$W(n)=\frac{\dim{\mathcal{H}}_{\lambda N+(N+k)/2-n}^{S_{\rm
in}}\dim{\mathcal{H}}_{n}^{S_{\rm rad}}}{\dim{\mathcal{H}}_{\lambda
N+(N+k)/2}^{S}}.$ (115)

With respect to this probability distribution, we define probable values of
$n$ and require for Eq. (114) 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. V.3.

####  VI.1.2 Comparison between entanglement clipping and partial decoupling

To check the validity of the argument of entanglement clipping, let us compare
the condition (114) 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 $\langle n\rangle$ under the probability distribution
of Eq. (115). The average is explicitly given by

$\langle n\rangle=\biggl{(}\frac{L}{N+k}+1/2\biggr{)}\ell.$ (116)

More precisely, we require that Eq. (114) holds for all $n$ such that

$|n-\langle n\rangle|\leq c\sqrt{\langle\delta n^{2}\rangle},$ (117)

where $\delta n:=n-\langle n\rangle$. Here, $c>0$ is a parameter related to
the recovery error since taking a larger $c$ means that we require Eq. (114)
for a wider range of $n$, resulting in less error. We then define the number
$\hat{\ell}_{c}(L)$ by the minimum of $\ell=|S_{\rm rad}|$ for the clipping
condition $C({\mathcal{H}}_{n}^{S_{\rm rad}})>k$ to hold for any $n$
satisfying Eq. (117).

The quantity $\hat{\ell}_{c}(L)$ should correspond to the
$\ell_{\Delta}(L,0)$, defined based on the partial decoupling by Eq. (102). In
the following, we show that $\hat{\ell}_{c}(L)\approx\ell_{\Delta}(L,0)$,
which indicates that the clipping argument results in nearly the same
prediction as the partial decoupling.

To this end, we provide an approximate expression of $\hat{\ell}_{c}(L)$. The
dimensions in Eq. (113) are given by

$\displaystyle\dim{\mathcal{H}}_{n}^{S_{\rm rad}}=\binom{\ell}{n},$ (118)
$\displaystyle\dim{\mathcal{H}}_{\lambda N+(N+k)/2-n}^{S_{\rm
in}}=\binom{N+k-\ell}{\lambda N+(N+k)/2-n}.$ (119)

These are well-approximated by a function
$s(\lambda):=-(1/2-\lambda)\log[1/2-\lambda]-(1/2+\lambda)\log[1/2+\lambda]$.
We then obtain

$\log\biggl{[}\frac{\dim{\mathcal{H}}_{n}^{S_{\rm
rad}}}{\dim{\mathcal{H}}_{\lambda N+(N+k)/2-n}^{S_{\rm in}}}\biggr{]}\\\
=\ell\ \\!s\biggl{(}\frac{n}{\ell}-\frac{1}{2}\biggr{)}-(N+k-\ell)\
\\!s\biggl{(}\frac{\lambda N+(N+k)/2-n}{N+k-\ell}-\frac{1}{2}\biggr{)}.$ (120)

We can further simplify the expression by assuming $k\ll N$ and expanding $n$
around its average $\langle n\rangle\approx(\lambda+1/2)\ell$ as

$n\approx(\lambda+1/2)\ell+\delta n.$ (121)

It is straightforward to show

$\log\biggl{[}\frac{\dim{\mathcal{H}}_{n}^{S_{\rm
rad}}}{\dim{\mathcal{H}}_{\lambda N+(N+k)/2-n}^{S_{\rm in}}}\biggr{]}\\\
\approx-(N+k-2\ell)\ \\!s(\lambda)+2\delta n|s^{\prime}(\lambda)|.$ (122)

We also introduce the initial degree of clipping $C_{\rm ini}$, i.e., the
degree of clipping when $|S_{\rm rad}|=0$. It is approximately given by

$C_{\rm ini}\approx H(B_{\rm in})_{\xi}-(N+k)s(\lambda)<0.$ (123)

Using these, we arrive at

$C({\mathcal{H}}^{S_{\rm rad}}_{n})\approx C_{\rm ini}+2\ell
s(\lambda)+2\delta n|s^{\prime}(\lambda)|.$ (124)

As mentioned, we require $C({\mathcal{H}}^{S_{\rm rad}}_{n})>k$ for all
$\delta n$ such that $|\delta n|\leq c\sqrt{\langle\delta n^{2}\rangle}$. The
standard deviation $\sqrt{\langle\delta n^{2}\rangle}$ can be also rephrased
in terms of $s(\lambda)$. From Eq. (115), we have

$\displaystyle\log[W(n)]$
$\displaystyle=-\frac{1}{2}\frac{(N+k)}{\ell(N+k-\ell)}|s^{\prime\prime}(\lambda)|\delta
n^{2}+O(\delta n^{3}).$ (125)

Since $W(n)$ can be approximated by a Gaussian distribution when $1\ll k\ll
N$, the variance $\langle\delta n^{2}\rangle$ is approximately given by

$\langle\delta
n^{2}\rangle\approx\biggl{(}1-\frac{\ell}{N+k}\biggr{)}\frac{\ell}{|s^{\prime\prime}(\lambda)|}.$
(126)

Altogether, the number $\hat{\ell}_{c}(L)$ is given by the minimum $\ell$ that
satisfies

$C_{\rm ini}+2\ell
s(\lambda)+2c\sqrt{\biggl{(}1-\frac{\ell}{N+k}\biggr{)}\frac{\ell}{|s^{\prime\prime}(\lambda)|}}|s^{\prime}(\lambda)|>k.$
(127)

By solving this, we obtain an explicit form of $\hat{\ell}_{c}(L)$, which is

$\hat{\ell}_{c}(L)\approx\ell_{0}(L)+c\ \\!\ell_{\rm fl}(L),$ (128)

where

$\displaystyle\ell_{0}(L)=-\frac{C_{\rm
ini}}{2s(\lambda)}+\frac{k}{2s(\lambda)},$ (129) $\displaystyle\ell_{\rm
fl}(L)=\frac{|s’(\lambda)|}{s(\lambda)}\sqrt{\frac{\ell_{0}(L)}{|s^{\prime\prime}(\lambda)|}\biggl{(}1-\frac{\ell_{0}(L)}{N+k}\biggr{)}}.$
(130)

Figure 4: Numerical comparison of the delays computed in two ways. One is
$\delta\ell_{\Delta}(\lambda):=\ell_{\Delta}(\lambda N,0)-\ell_{\Delta}(0,0)$,
based on the partial decoupling (Eq. (102)), and the other is
$\delta\hat{\ell}_{c}(\lambda):=\hat{\ell}_{c}(\lambda N)-\hat{\ell}_{c}(0)$
obtained from the clipping of entanglement (Eq. (128)). They are both plotted
as functions of the $Z$-axis AM ratio $\lambda=L/(\hbar N)$ of the initial
Kerr BH $B_{\rm in}$. The upper and lower figures show the delays for the pure
and mixed Kerr BHs, respectively. The size $N$ of the initial Kerr BH $B_{\rm
in}$ is set to $300$, and the size $k$ of the quantum information source $A$
is fixed to $3$. In both figures, $\delta\ell_{\Delta}(\lambda)$ is computed
for $\Delta=0.005$ (red plots), $0.05$ (blue plots), and $0.5$ (green plots),
while $\delta\hat{\ell}_{c}(\lambda)$ is computed for $c=3.4$ (red dashed
line), $2.6$ (blue dashed line), and $1.6$ (green dashed line) for the pure
BH, and $10.8$ (red dashed line), $8.7$ (blue dashed line), and $6.2$ (green
dashed line) for the mixed BH. In both pure and mixed Kerr BHs, they nearly
coincide, indicating that the argument of entanglement clipping provides good
estimations.

In Fig. 4, we numerically compare $\hat{\ell}_{c}(L)$ and
$\ell_{\Delta}(L,0)$. We observe that, by choosing proper $c$ and $\Delta$,
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 $S_{\rm rad}$ for the
information to become recoverable from the radiations.

In Eq. (128), we have decomposed $\hat{\ell}_{c}(L)$ into two terms. The first
term, $\ell_{0}(L)$, represents a delay originated from an information
theoretic reason: when the number of qubits in $S_{\rm rad}$ is increased by
one, the degree of clipping is increased by $2s(\lambda)$ since a qubit has
entropy $\approx s(\lambda)$. Thus, $|S_{\rm rad}|$ should be at least
$\ell_{0}(L)$ so as to cancel the initial negative degree of clipping $C_{\rm
ini}$, and more for $S_{\rm rad}$ to typically have a sufficiently large space
to store $2^{k}$ states without significant overlaps. That is why we refer to
$\ell_{0}(L)$ as an _information-theoretic_ delay.

It is the second term $\ell_{\rm fl}(L)$ 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 $\ell_{\rm fl}(L)$ a _fluctuational_ delay. Note also that, when
the $Z$-axis AM $L$ of the initial Kerr BH is small, the delay of information
leakage, $\delta\hat{\ell}_{c}(L):=\hat{\ell}_{c}(L)-\hat{\ell}_{c}(0)$,
satisfies

$\delta\hat{\ell}_{c}(L)\propto\ell_{\rm fl}(L),$ (131)

since $\ell_{0}(L)\approx\ell_{0}(0)$ and $\ell_{\rm fl}(0)=0$ for small $L$.

####  VI.1.3 Delay and BH Thermodynamics

The fluctuational delay $\ell_{\rm fl}(L)$ 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 $L_{\rm fl}$ by

$L_{\rm fl}:=\hbar\lambda\ell_{\rm fl}(L),$ (132)

and investigate it. In this Subsection, we explicitly write the Plank constant
$\hbar$. Correspondingly, we also introduce $L_{0}:=\hbar\lambda\ell_{0}(L)$,
which is the $Z$-axis AM corresponding to the information-theoretic delay
$\ell_{0}(L)$.

Figure 5: Figure (i) explains the $Z$-axial symmetry of a state and the Q
function. The Q function is defined on a sphere. For simplicity, we expand the
sphere surface to a 2-dimensional plain as in the right side of the figure.
Figure (ii) shows the semi-log plot of $\eta_{\xi}(N,k,\ell)$, which
characterizes the information remnant, for various states $\xi$ of the initial
Kerr BH $B_{\rm in}$ with different fluctuations of the $Z$-axis AM: $\delta
L=0$ (black), $0.1\sqrt{N}$ (red), $0.5\sqrt{N}$ (blue), $0.9\sqrt{N}$
(green), and $0.3N$ (brown). The size $N$ of the initial Kerr BH is fixed to
$1000$, and $k$ is set to $1$. The average $Z$-axis AM $L$ can be arbitrary
since $\eta_{\xi}(N,k,\ell)$ does not depend on $L$. Comparing this with the Q
functions for each $\xi(L,\delta L)$, which visualize the symmetry-breaking of
$\xi$, we observe that $\eta_{\xi}(N,k,\ell)$ is small when the degree of
symmetry-breaking is large.

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 symmetry-preserving 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
$S(\lambda,L,T)$ of the Kerr BH is independent of $T$, and the free energy
$F(T,\lambda,L)$ is approximately given by $-TS(\lambda,L)$.

Let $\omega:=-\bigl{(}\frac{\partial F}{\partial L}\bigl{)}_{T,N}$ be an
intensive state function conjugate to the $Z$-axis AM $L$, and
$\alpha:=\bigl{(}\frac{1}{L}\frac{\partial L}{\partial T}\bigl{)}_{\omega,N}$
be the sensitivity of the $Z$-axis AM to the temperature. Note that the state
function $\omega(T,\lambda)$ corresponds to the angular velocity of the Kerr
BH. Since the thermodynamic entropy $S(\lambda,L)$ is given by

$S(\lambda,L)=k_{B}s(\lambda)N,$ (133)

where $k_{B}$ is the Boltzmann constant, in our particular model of the Kerr
BH, we have

$\displaystyle\omega(T,\lambda)=\frac{k_{B}Ts^{\prime}(\lambda)}{\hbar},$
(134)
$\displaystyle\alpha(T,\lambda)=-\frac{s^{\prime}(\lambda)}{s^{\prime\prime}(\lambda)\lambda
T}.$ (135)

Hence, it follows that

$\biggl{|}\frac{s^{\prime}(\lambda)^{2}}{s^{\prime\prime}(\lambda)}\biggr{|}=\frac{\lambda\hbar}{k_{B}}|\omega(T,L)\alpha(T,L)|.$
(136)

Using these and assuming $k\ll N$, we obtain from Eqs. (132) and (130) that

$\displaystyle L_{\rm
fl}\approx\frac{L}{S(\lambda,L)}\sqrt{k_{B}\bigl{|}\omega(T,\lambda)\alpha(T,\lambda)\bigr{|}\biggl{(}1-\frac{L_{0}}{L}\biggr{)}L_{0}}.$
(137)

Note that $\omega(T,\lambda)\alpha(T,\lambda)$ is independent of the
temperature $T$, and so is $L_{\rm fl}$.

After radiating $L_{0}$ of the $Z$-axis AM to fulfill the information-
theoretic requirement, the Kerr BH must further radiate an extra amount
$L_{\rm fl}$ of the $Z$-axis AM to release the information. It is clear from
Eq. (137) that this amount $L_{\rm fl}$ is solely determined by intensive
thermodynamic quantities of the initial Kerr BH, that is, the $Z$-axis AM per
entropy $L/S(\lambda,L)$, the angular velocity $\omega(T,\lambda)$ of the Kerr
BH, and the coefficient $\alpha(T,\lambda)$ 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 all the quantities in Eq. (137) are independent of
microscopic details of the BH model we have used. This shall indicate that the
formula should hold beyond the framework of the HP protocol of the Kerr BH.
This leads us to conjecture that Eq. (137) is the new microscopic-macroscopic
correspondence of a quantum black hole with an extensive conserved quantity,
which is useful for understanding the information leakage in terms of the
macroscopic black hole physics.

###  VI.2 Information remnant and symmetry-breaking

We finally study the information remnant characterized by $\eta_{\xi}$ (see
Eq. (81)). As explained in Subsec. V.5, the quantity $\eta_{\xi}$ depends on
$\delta L$ but not on $L$. To intuitively capture the relation between
$\eta_{\xi}$ and $\ell=|S_{\rm rad}|$, we plot it in Fig. 5 (ii) as a function
of $\ell$ for various fluctuations $\delta L$ of the initial Kerr BH $B_{\rm
in}$.

In Fig. 5 (ii), we also provide an unnormalized Q function of the initial
state $\xi^{B_{\rm in}}$ in $B_{\rm in}$, which visualizes how $Z$-axis-
symmetric the state is (see also Fig. 5 (i)). For a pure state
$|\xi\rangle^{B_{\rm in}}$, the Q function is defined with variables $L_{X}$
and $L_{Z}$, which correspond to the $X$\- and $Z$-axis AMs, respectively, by

$Q_{\xi}\biggl{(}\frac{L_{X}}{\sqrt{N/2}},\frac{L_{Z}}{\sqrt{N/2}}\biggr{)}=\bigl{|}\langle\theta,\phi|\xi\rangle^{B_{\rm
in}}\bigr{|}^{2},$ (138)

where
$|\theta,\phi\rangle:=(\cos\frac{\theta}{2}|0\rangle+e^{i\phi}\sin\frac{\theta}{2}|1\rangle)^{\otimes
N}$ and $(\theta,\phi)$ are those satisfying $\sin\theta\cos\phi=2L_{X}/N$ and
$\cos\theta=2L_{Z}/N$. Note that the function is regarded as the Husimi Q
function for $N\rightarrow\infty$ when $\langle L_{Y}\rangle\approx N/2$.

The Q function in Fig. 5 (ii) is for a state $\xi$ in the form of Eq. (93)
with $|\varphi_{n}\rangle^{B_{\rm in}}$ being the state of equal superposition
of all product basis states in ${\mathcal{H}}_{n}^{B_{\rm in}}$. By comparing
the plots of $\eta_{\xi}$ and the Q functions in Fig. 5 (ii), we observe that,
when $\eta_{\xi}$ is small, the $Z$-axial symmetry in $\xi^{B_{\rm in}}$ is
strongly broken. This indicates that the information remnant, characterized by
$\eta_{\xi}$, should be related to the symmetry-breaking of the initial state
$\xi$ in the Kerr BH $B_{\rm in}$.

We can analytically elaborate on this relation. To this end, let us define the
degree $\zeta$ of symmetry-breaking 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 $S_{\rm in}$, $\zeta(S_{\rm in})$,
is defined as follows: Let $|\Psi\rangle^{S_{\rm in}S^{\prime}_{\rm in}}$ be a
purification of the state $\Psi^{S_{\rm in}}$, and $B_{\theta}(\Psi)$ denote

$B_{\theta}(\Psi):=1-\bigl{|}\langle\Psi|e^{i\theta L_{Z}}\otimes
I|\Psi\rangle\bigr{|}^{2},$ (139)

where $L_{Z}$ is the $Z$-axis AM operator in $S_{\rm in}$ and the second
identity acts on $S^{\prime}_{\rm in}$. Then, the degree $\zeta(S_{\rm in})$
of symmetry-breaking is defined by

$\zeta(S_{\rm
in}):=\frac{\partial^{2}}{\partial\theta^{2}}B_{\theta}(\Psi)\biggl{|}_{\theta=0}.$
(140)

This quantifies how sensitive the state $\Psi$ 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 $\langle\delta\nu^{2}\rangle$ of the $Z$-axis AM $\nu$ in
$\Psi^{S_{\rm in}}$. Since we have

$\displaystyle\bigl{|}\langle\Psi|e^{i\theta L_{Z}}\otimes
I|\Psi\rangle\bigr{|}^{2}$ $\displaystyle=\bigl{|}\operatorname{Tr}[e^{i\theta
L_{Z}}\Psi]\bigr{|}^{2}$ (141)
$\displaystyle=\bigl{|}1+i\theta\operatorname{Tr}[L_{Z}\Psi]-\frac{\theta^{2}}{2}\operatorname{Tr}[L_{Z}^{2}\Psi]\bigr{|}^{2}+O(\theta^{3})$
(142)
$\displaystyle=1+\theta^{2}\bigl{(}(\operatorname{Tr}[L_{Z}\Psi])^{2}-\operatorname{Tr}[L_{Z}^{2}\Psi]\bigr{)}+O(\theta^{3})$
(143) $\displaystyle=1-\theta^{2}\langle\delta\nu^{2}\rangle+O(\theta^{3}),$
(144)

we obtain $\zeta(S_{\rm in})=2\langle\delta\nu^{2}\rangle$.

Let us now relate the degree of symmetry-breaking to the information remnant.
The key observation is that, for the whole information in $\Phi^{AR}$ to be
recoverable from the radiation $S_{\rm rad}$, classical information about the
$Z$-axis AM in $R$ should not be inferred from the remaining Kerr BH $S_{\rm
in}$ KoaWin2004 . Since we are interested in the information remnant, we also
assume that the new radiation $S_{\rm rad}$ is sufficiently large so that the
Kerr BH has already get partially decoupled and the state therein and the
reference $R$ is $\Gamma^{S_{\rm in}R}$. The state is explicitly given by (see
Proposition 3)

$\Gamma^{S_{\rm
in}R}=\sum_{m=0}^{N+k}\frac{2^{N+k}}{\binom{N+k}{m}}\operatorname{Tr}_{S^{\prime}}[\Pi_{m}^{S^{\prime}}(\pi^{S^{\prime}_{\rm
rad}}\otimes\Phi^{S_{\rm in}S^{\prime}_{\rm in}})]\\\
\otimes\operatorname{Tr}_{S}[\Pi_{m}^{S}(\Phi^{AR}\otimes\xi^{B_{\rm in}})].$
(145)

When the $Z$-axis AM in $S_{\rm in}$ and that in $R$ are measured on the state
$\Gamma^{S_{\rm in}R}$, the probabilities to obtain $\nu$ in $S_{\rm in}$ and
$\kappa$ in $R$ are given by

$\displaystyle q(\kappa)$
$\displaystyle:=\operatorname{Tr}[\Pi_{\kappa}^{R}\Gamma^{R}]=\operatorname{Tr}[\Pi_{\kappa}^{R}\pi^{R}],$
(146) $\displaystyle P(\nu)$
$\displaystyle:=\operatorname{Tr}[\Pi_{\nu}^{S_{\rm in}}\Gamma^{S_{\rm in}}],$
(147) $\displaystyle P(\nu|\kappa)$
$\displaystyle:=\operatorname{Tr}\bigl{[}(\Pi_{\nu}^{S_{\rm
in}}\otimes\Pi_{\kappa}^{R})\Gamma^{S_{\rm in}R}]/q(\kappa),$ (148)
$\displaystyle=\operatorname{Tr}\bigl{[}\Pi_{\nu}^{S_{\rm in}}\Gamma^{S_{\rm
in}}_{\kappa}],$ (149)

where $\Gamma^{S_{\rm
in}}_{\kappa}:=\operatorname{Tr}_{R}[\Pi_{\kappa}^{R}\Gamma^{S_{\rm
in}R}]/q(\kappa)$ is the post-measured state on $S_{\rm in}$. It is easy to
check that they satisfy

$P(\nu)=\sum_{\kappa}q(\kappa)P(\nu|\kappa).$ (150)

Using these probabilities, $\eta_{\xi}=|\\!|\Gamma^{S_{\rm
in}R}-\Gamma^{S_{\rm in}}\otimes\pi^{R}|\\!|_{1}$ (see Eq. (81)) is bounded
from below by

$\eta_{\xi}\geq\sum_{\kappa}q(\kappa)\sum_{\nu}\bigl{|}P(\nu|\kappa)-P(\nu)\bigr{|}.$
(151)

We now approximate the value of $\nu$ by continuous values and replace the
above probabilities $P(\nu)$ and $P(\nu|\kappa)$ with probability density
functions $p(\nu)$ and $p(\nu|\kappa)$, respectively. We also assume that the
$\kappa$-dependence of $p(\nu|\kappa)$ is approximated by a shift without
changing its common shape. Namely, we assume that

$p(\nu|\kappa)\approx\bar{p}(\nu-\alpha\delta\kappa),$ (152)

where $\bar{p}(\mu):=p(\mu|\kappa=\langle\kappa\rangle)$ with
$\alpha=1-\ell/(N+k)$, $\delta\kappa=\kappa-\langle\kappa\rangle$, and
$\langle\cdot\rangle$ is the expectation over the probability distribution
$q(\kappa)$. Note that the rescaling by $\alpha$ in Eq. (152) is needed since
$S$ is composed of $N+k$ qubits while $S_{\rm in}$ is of $N+k-\ell$ qubits. We
also assume that $|\delta\kappa|$ is sufficiently small so that

$\bar{p}(\nu-\alpha\delta\kappa)=\bar{p}(\nu)-\alpha\delta\kappa\frac{d\bar{p}(\nu)}{d\nu}+O(\delta\kappa^{2}),$
(153)

which further implies that $p(\nu)=\bar{p}(\nu)+O(\delta\kappa^{2})$ due to
Eq. (150) and $\langle\delta\kappa\rangle=0$.

Using these approximations and replacing the summation over $\nu$ in Eq. (151)
by integral, we obtain

$\displaystyle\eta_{\xi}$
$\displaystyle\geq\alpha\sum_{\kappa}q(\kappa)|\delta\kappa|\int
d\nu\bigl{|}\frac{d\bar{p}(\nu)}{d\nu}\bigr{|}+O(\delta\kappa^{2}),$ (154)
$\displaystyle=\alpha\langle|\delta\kappa|\rangle\int
d\nu\biggl{|}\frac{d\bar{p}(\nu)}{d\nu}\biggr{|}+O(\delta\kappa^{2})$ (155)

where $\langle|\delta\kappa|\rangle:=\sum_{\kappa}q(\kappa)|\delta\kappa|$ is
the mean absolute deviation of the $Z$-axis AM in $R$. The integral in Eq.
(155) can be bounded from below by $2\max_{\nu}p(\nu)$. Thus, we have

$\displaystyle\eta_{\xi}$ $\displaystyle\geq
2\alpha\langle|\delta\kappa|\rangle\max_{\nu}p(\nu)+O(\delta\kappa^{2}).$
(156)

The value $\max_{\nu}p(\nu)$ can be further bounded from below in terms of the
variance $\langle\delta\nu^{2}\rangle$ of the $Z$-axis AM in $S_{\rm in}$.
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 $r(x)=r_{\rm max}$ for $x\in[-1/(2r_{\rm
max}),1/(2r_{\rm max})]$ and $0$ otherwise. In that case, the variance $V$ is

$V=\int_{-1/(2r_{\rm max})}^{1/(2r_{\rm max})}x^{2}r_{\rm max}dx=\frac{1}{12\
\\!r_{\rm max}^{2}}.$ (157)

Hence, for any probability density function $p(x)$, it holds that $\max
p(x)\geq(2\sqrt{3}\sqrt{V})^{-1}$. Applying this to our case, we obtain
$\max_{\nu}p(\nu)\geq(2\sqrt{3}\sqrt{\langle\delta\nu^{2}\rangle})^{-1}$. We
thus arrive at

$\eta_{\xi}\geq\frac{1}{\sqrt{3}}\langle|\delta\kappa|\rangle\biggl{(}1-\frac{\ell}{N+k}\biggr{)}\frac{1}{\sqrt{\langle\delta\nu^{2}\rangle}}+O(\delta\kappa^{2}).$
(158)

Recalling the relation between the variance and the degree of symmetry-
breaking, i.e., $\langle\delta\nu^{2}\rangle=\zeta(S_{\rm in})/2$, we obtain

$\eta_{\xi}\gtrsim\sqrt{\frac{2}{3}}\langle|\delta\kappa|\rangle\biggl{(}1-\frac{\ell}{N+k}\biggr{)}\frac{1}{\sqrt{\zeta(S_{\rm
in})}}.$ (159)

This implies that, when information remnant is small, $\zeta(S_{\rm in})$
should be necessarily large, implying that the $Z$-axial symmetry in the
remaining Kerr BH $S_{\rm in}$ should be broken strongly. This clarifies yet
another micro-macro correspondence of a quantum Kerr BH.

The symmetry-breaking in $S_{\rm in}$ has two origins: one is that in the
initial Kerr BH $B_{\rm in}$, and the other is due to the new radiation, where
$\ell$-qubit radiation increases the standard deviation of the $Z$-axis AM in
the remaining Kerr BH $S_{\rm in}$ by $O(\sqrt{\ell})$, and so does the degree
of symmetry breaking. When $\sqrt{\ell}\ll\delta L$ with $\delta L$ being the
standard deviation of the $Z$-axis AM in the initial Kerr BH $B_{\rm in}$, the
former origin is dominant. On the other hand, when $\sqrt{\ell}\gg\delta L$,
the symmetry-breaking in $S_{\rm in}$ is mostly originated from the radiation
process. We thus conclude that

$\eta_{\xi}\gtrsim\begin{cases}\frac{a}{\sqrt{2}}\frac{1}{\delta L}&\text{when
$\sqrt{\ell}\ll\delta L$},\\\ \frac{a}{\sqrt{\ell}}&\text{when
$\sqrt{\ell}\gg\delta L$},\end{cases}$ (160)

where $a=\sqrt{2/3}\langle|\delta\kappa|\rangle(1-\ell/(N+k))$. Here, we have
used the fact that the degree of symmetry breaking in $B_{\rm in}$,
$\zeta(B_{\rm in})$, is given by $2(\delta L)^{2}$.

In the case of the pure Kerr BH, the information remnant is negligible in the
thermodynamic limit ($N\rightarrow\infty$) since the partial decoupling occurs
in the pure Kerr BH only when $\ell\gtrsim N/2$. Thus, Eq. (160) implies that,
regardless of $\delta L$, the information remnant vanishes when
$N\rightarrow\infty$. In contrast, for the mixed Kerr BH, there are cases
where the partial decoupling sets in when $\ell=O(k)$. In this case, the
fluctuation $\delta L$ of the $Z$-axis AM in the initial Kerr BH $B_{\rm in}$
determines the information remnant. In particular, if $\delta L$ is not so
large that the lower case of Eq. (160) holds, a non-negligible amount of
information remains in the Kerr BH even in the thermodynamic limit unless
$\ell$ 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 $\ell=O(k)$ 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 $\delta L$ of the
$Z$-axis AM.

Finally, we refer to the recent result TS2021 , 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.

##  VII Conclusion and Discussions

In this paper, we have investigated the information leakage problem from a
quantum Kerr black hole with $Z$-axial symmetry within the framework of the HP
protocol. Based on the partial decoupling approach, we have first provided
general formulas for upper bounds on the errors in recovering the information
in the Kerr BH from the radiation.

From the numerical evaluations of the recovery errors, we have shown that the
symmetry of the Kerr BH induces two substantial deviations. One is the delay
of information leakage in the sense that more radiation is needed to achieve
the same recovery error as the case without symmetry. The other is the
information remnant: a part of the information remains in the Kerr BH until
the last moment of the complete evaporation. Depending on the initial
condition of the Kerr BH, 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 Kerr BH. The close relation between the
information remnant and the symmetry-breaking of the Kerr BH has been also
revealed. These relations provide new microscopic-macroscopic correspondences
of the Kerr BH.

The microscopic-macroscopic relations that we have revealed are useful in the
further analyses of quantum black holes. For instance, it may be used to
experimentally verify the information leakage: dealing with quantum
information requires a state-of-the-art technology, but measuring macroscopic
properties is in general much easier. It may also be possible to estimate the
strength of the weak violation of symmetry in quantum gravity, which is an
open question in fundamental physics HO2019 .

The methods and results in this paper can be easily generalized to quantum BHs
with extensive conserved quantity, such as the energy conservation. It is also
possible to apply the partial decoupling approach to the BHs with non-abelian
symmetry. See Ref. WN2019  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 Kerr BH is symmetry-
preserving scrambling, which will not be exactly satisfied since it generally
takes exponentially long time to implement a Haar random unitary. Hence, it is
of crucial importance to consider approximate implementations. Approximations
of random unitary are studied in terms of _unitary $t$-designs_, which mimic
the $t$th order statistical moments of a Haar random unitary on average, and a
couple of results are known about the time needed to achieve $t$-designs
HL2009TPE ; BHH2016 ; NHKW2017 ; HMHEGR2020 . It will be of great interest to
investigate how symmetries affect such results M2020 ; L2020 . More
explicitly, it is interesting to address if we can construct _symmetry-
preserving_ unitary $t$-designs by quantum circuits. By using such quantum
circuits, it will be possible to simulate the Hayden-Preskill protocol with
symmetry and obtain further insights into the black hole information paradox
and symmetry.

##  VIII Acknowledgements

Y.N. is supported by JST CREST, Grant Number JPMJCR1671 and by JST, PRESTO
Grant Number JPMJPR1865, Japan.

## References

  * [1] S. W. Hawking.  Black hole explosions?  Nature, 248:30–31, 1974.
  * [2] S. W. Hawking.  Particle creation by black holes.  Commun. Math. Phys., 43:199–220, 1975.
  * [3] W. Israel.  Event Horizons in Static Vacuum Space-Times.  Phys. Rev., 164:1776–1779, 1967.
  * [4] W. Israel.  Event horizons in static electrovac space-times.  Commun. Math. Phys., 8:245–260, 1968.
  * [5] B. Carter.  Axisymmetric Black Hole Has Only Two Degrees of Freedom.  Phys. Rev. Lett., 26:331–333, 1971.
  * [6] P. Hayden and J. Preskill.  Black holes as mirrors: quantum information in random subsystems.  J. High Energy Phys., 2007:120, 2007.
  * [7] Y. Sekino and L. Susskind.  Fast scramblers.  J. High Energy Phys., 2008:065, 2008.
  * [8] N. Lashkari, D. Stanford, M. Hastings, T. Osborne, and P. Hayden.  Towards the fast scrambling conjecture.  J. High Energy Phys., 2013:22, 2013.
  * [9] L. Susskind.  Addendum to Fast Scramblers.  arXiv:1101.6048, 2011.
  * [10] H. Shenker and D. Stanford.  Black holes and the butterfly effect.  J. High Energy Phys., 2014:67, 2014.
  * [11] S. H. Shenker and D. Stanford.  Stringy effects in scrambling.  J. High Energy Phys., 2015:132, 2015.
  * [12] D. A. Roberts and D. Stanford.  Diagnosing Chaos Using Four-Point Functions in Two-Dimensional Conformal Field Theory.  Phys. Rev. Lett., 115:131603, 2015.
  * [13] D. A. Roberts and B. Yoshida.  Chaos and complexity by design.  J. High Energ. Phys., 2017:121, 2017.
  * [14] B. Yoshida.  Soft mode and interior operator in the hayden-preskill thought experiment.  Phys. Rev. D, 100:086001, 2019.
  * [15] S. Sachdev and J. Ye.  Gapless spin-fluid ground state in a random quantum Heisenberg magnet.  Phys. Rev. Lett., 70:3339–3342, 1993.
  * [16] A. Kitaev.  A simple model of quantum holography.  http://online.kitp.ucsb.edu/online/entangled15/kitaev/, http://online.kitp.ucsb.edu/online/entangled15/kitaev2/, April, May 2015\.  Talks at KITP.
  * [17] K. Jensen.  Chaos in ${\mathrm{ads}}_{2}$ holography.  Phys. Rev. Lett., 117:111601, 2016.
  * [18] J. Maldacena and D. Stanford.  Remarks on the sachdev-ye-kitaev model.  Phys. Rev. D, 94:106002, 2016.
  * [19] S. Sachdev.  Bekenstein-hawking entropy and strange metals.  Phys. Rev. X, 5:041025, 2015.
  * [20] M. Blake.  Universal Charge Diffusion and the Butterfly Effect in Holographic Theories.  Phys. Rev. Lett., 117:091601, 2016.
  * [21] C. W. von Keyserlingk, T Rakovszky, F. Pollmann, and S. L. Sondhi.  Operator Hydrodynamics, OTOCs, and Entanglement Growth in Systems without Conservation Laws.  Phys. Rev. X, 8:021013, 2018.
  * [22] V. Khemani, A. Vishwanath, and D. A. Huse.  Operator Spreading and the Emergence of Dissipative Hydrodynamics under Unitary Evolution with Conservation Laws.  Phys. Rev. X, 8:031057, 2018.
  * [23] P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida.  Chaos in quantum channels.  J. High Energy Phys., 2016:4, 2016.
  * [24] F. Pastawski, B. Yoshida, D Harlow, and J. Preskill.  Holographic quantum error-correcting codes: toy models for the bulk/boundary correspondence.  J. High Energ. Phys., 2015:149, 2015.
  * [25] F. Pastawski, J. Eisert, and H. Wilming.  Towards Holography via Quantum Source-Channel Codes.  Phys. Rev. Lett., 119:020501, 2017.
  * [26] T. Kohler and T. Cubitt.  Toy models of holographic duality between local Hamiltonians.  J. High Energy Phys., 2019:17, 2019.
  * [27] P. Hayden and G. Penington.  Learning the Alpha-bits of black holes.  J. High Energy Phys., 2019:7, 2019.
  * [28] T. Banks and N. Seiberg.  Symmetries and strings in field theory and gravity.  Phys. Rev. D, 83:084019, 2011.
  * [29] D. Harlow and H. Ooguri.  Constraints on Symmetries from Holography.  Phys. Rev. Lett., 122:191601, 2019.
  * [30] N. Arkani-Hamed, L. Motl, A. Nicolis, and C. Vafa.  The string landscape, black holes and gravity as the weakest force.  J. High Energy Phys., 2007:060–060, 2007.
  * [31] W. F. Stinespring.  Positive functions on $c^{*}$-algebras.  Proc. Amer. Math. Soc., 6:211–216, 1955.
  * [32] A. Jamiołkowski.  Linear transformations which preserve trace and positive semidefiniteness of operators.  Rep. Math. Phys., 3:275, 1972.
  * [33] M. D. Choi.  Completely positive linear maps on complex matrices.  Linear Algebra Appl., 10:285, 1975.
  * [34] W. K. Wootters and W. H. Zurek.  A single quantum cannot be cloned.  Nature, 299:802–803, 1982.
  * [35] F. Dupuis, M. Berta, J. Wullschleger, and R. Renner.  One-shot decoupling.  Commun. Math. Phys., 328:251, 2014.
  * [36] D. N. Page.  Average entropy of a subsystem.  Phys. Rev. Lett., 71:1291–1294, 1993.
  * [37] M. Horodecki, J. Oppenheim, and A. Winter.  Partial quantum information.  Nature, 436:673–676, 2005.
  * [38] M. Horodecki, J. Oppenheim, and A. Winter.  Quantum state merging and negative information.  Commun. Math. Phys., 269:107–136, 2007.
  * [39] A. Abeyesinghe, I. Devetak, P. Hayden, and A. Winter.  The mother of all protocols : Restructuring quantum information’s family tree.  Proc. R. Soc. A, 465:2537, 2009.
  * [40] E. Wakakuwa and Y. Nakata.  One-shot randomized and nonrandomized partial decoupling.  arXiv:1903.05796, 2019.
  * [41] R. Renner.  Security of Quantum Key Distribution.  PhD thesis, ETH Zurich, 2005.  arXiv:quant-ph/0512258.
  * [42] M. Tomamichel.  Quantum Information Processing with Finite Resources.  SpringerBriefs in Mathematical Physics, 2016.
  * [43] A. Uhlmann.  The “transition probability” in the state space of a ∗-algebra.  Rep. Math. Phys., 9:273–279, 1976.
  * [44] E. Lubkin.  Entropy of an $n$‐system from its correlation with a $k$‐reservoir.  J. of Math. Phys., 19:1028–1031, 1978.
  * [45] P. Hayden, D. W. Leung, and A. Winter.  Aspects of Generic Entanglement.  Commn. Math. Phys., 265:95–117, 2006.
  * [46] M. Koashi.  Complementarity, distillable secret key, and distillable entanglement.  arXiv:0704.3661, 2007.
  * [47] N. A. Nielsen and I. L. Chuang.  Quantum Computation and Quantum information.  Cambridge University Press, Cambridge, UK, 2000.
  * [48] M. Koashi and A. Winter.  Monogamy of quantum entanglement and other correlations.  Phys. Rev. A, 69:022309, 2004.
  * [49] H. Tajima and K. Saito.  Symmetry hinders quantum information recovery.  arXiv:2103.01876, 2021.
  * [50] A. W. Harrow and R. A. Low.  Efficient Quantum Tensor Product Expanders and k-Designs.  In Proc. RANDOM’09, Lecture Notes in Computer Science, pages 548–561. Springer Berlin Heidelberg, 2009.
  * [51] F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki.  Local Random Quantum Circuits are Approximate Polynomial-Designs.  Commun. Math. Phys., 346:397–434, September 2016.
  * [52] Y. Nakata, C. Hirche, M. Koashi, and A. Winter.  Efficient Quantum Pseudorandomness with Nearly Time-Independent Hamiltonian Dynamics.  Phys. Rev. X, 7:021006, 2017.
  * [53] J. Haferkamp, F. Montealegre-Mora, M. Heinrich, J. Eisert, D. Gross, and I. Roth.  Quantum homeopathy works: Efficient unitary designs with a system-size independent number of non-Clifford gates.  arXiv: 2002.09524, 2020.
  * [54] I. Marvian.  Locality and Conservation Laws: How, in the presence of symmetry, locality restricts realizable unitaries.  arXiv: 2003.05524, 2020.
  * [55] J. Liu.  Scrambling and decoding the charged quantum information.  Phys. Rev. Research, 2:043164, 2020.
  * [56] M. Ledoux.  The Concentration of Measure Phenomenon.  American Mathematical Society Providence, RI, USA, 2001.
  * [57] E. Meckes.  Concentration of measure and the compact classical matrix groups.  https://www.math.ias.edu/files/wam/Haar_notes-revised.pdf, 2014\.  Accessed: 2017-01-10.
  * [58] A. Winter.  Coding theorem and strong converse for quantum channels.  IEEE Trans. Inf. Theory, 45:2481–2485, 1999.

##  Appendix A Proof of Theorem 1

To prove Theorem 1, we use the theorem shown in Ref. [40] and the
concentration of measure phenomena for product measures.

######  Theorem 5 (Non-smoothed partial decoupling theorem [40]).

In the situation described in Subsec. V.1, it holds that

$\mathbb{E}_{U^{S}\sim{\sf
H}_{\times}}\bigl{|}\\!\bigl{|}\mathcal{M}^{S\rightarrow
E}\bigl{(}U^{S}\varrho^{SR}U^{S\dagger}\bigr{)}-\Gamma^{ER}\bigr{|}\\!\bigr{|}_{1}\leq
2^{-\frac{1}{2}H_{\rm min}(S^{*}|ER)_{\Gamma}},$ (161)

where $H_{\rm min}(S^{*}|ER)_{\Gamma}$ is the conditional min-entropy of
$\Gamma^{S^{*}ER}$.

It is well-known that a Haar random unitary has concentration properties when
the degree of the group is large [56], which finds a number of applications in
quantum information science. In most cases, the Haar measure on the whole
unitary group is considered. However, the concentration also happens for the
product measure ${\sf H}_{\times}={\sf H}_{1}\times\cdots\times{\sf H}_{J}$.

To explain the concentration of measure on the product space, let us introduce
the $L_{2}$-sum of the Hilbert-Schmidt norms. Let ${\sf U}_{\times}$ be a
product of unitary groups ${\sf U}(d_{1})\times\dots\times{\sf U}(d_{J})$. For
$U=(U_{1},\cdots,U_{J})\in{\sf U}_{\times}$ and $V=(V_{1},\cdots,V_{J})\in{\sf
U}_{\times}$, the $L_{2}$-sum $D(U,V)$ of the Hilbert-Schmidt norms on ${\sf
U}_{\times}$ is defined by

$D(U,V)=\sqrt{\sum_{j\in{\mathcal{J}}}|\\!|U_{j}-V_{j}|\\!|_{2}^{2}}.$ (162)

We also use Lipschitz functions. A real-valued function $F$ on a metric space
$(X,d)$ is said to be _Lipschitz_ if

$|\\!|F|\\!|_{\rm Lip}:=\sup_{x\neq y\in X}\frac{|F(x)-F(y)|}{d(x,y)}<\infty.$
(163)

The quantity $|\\!|F|\\!|_{\rm Lip}$ 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 [57]).

Let ${\sf U}_{\times}={\sf U}(d_{1})\times\dots\times{\sf U}(d_{J})$ equipped
with the $L_{2}$-sum of Hilbert-Schmidt norms, and ${\sf H}_{\times}={\sf
H}_{1}\times\cdots\times{\sf H}_{J}$ be the product probability measure.
Suppose that a function $F:{\sf U}_{\times}\rightarrow\mathbb{R}$ is
$L$-Lipschitz. Then, for every $\delta>0$,

$F(U_{1},\dots,U_{J})\geq\mathbb{E}_{(U_{1},\cdots,U_{J})\sim{\sf
H}_{\times}}[F]+\delta,$ (164)

with probability at most $\exp[-\frac{\delta^{2}d_{\rm min}}{12L^{2}}]$, where
$d_{\rm min}=\min\\{d_{1},\dots,d_{J}\\}$.

Based on Theorems 5 and 6, we now prove Theorem 1 by identifying the unitary
$U^{S}=\bigoplus_{j\in{\mathcal{J}}}U^{S}_{j}$ as a point
$(U_{1}^{S},\dots,U_{J}^{S})$ on ${\sf U}_{\times}={\sf
U}(d_{1})\times\cdots\times{\sf U}(d_{J})$.

######  Proof (Theorem 1).

For a given $U^{S}=\bigoplus_{j\in{\mathcal{J}}}U^{S}_{j}$, let
$(U_{1}^{S},\dots,U_{J}^{S})$ be the corresponding point on ${\sf U}_{\times}$
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 $F:{\sf U}_{\times}\rightarrow\mathbb{R}$ be a function given by

$F(U_{1}^{S},\dots,U_{J}^{S}):=\bigl{\|}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\varrho^{SR}U^{S\dagger}\bigr{)}-\operatorname{Tr}_{S^{*}}[\Gamma^{S^{*}ER}]\bigr{\|}_{1}.$
(165)

We compute the Lipschitz constant of $F$. Using the triangle inequality, we
obtain

$\displaystyle|F(U_{1}^{S},\dots,U_{J}^{S})-F(V_{1}^{S},\dots,V_{J}^{S})|$
$\displaystyle\leq|\\!|{\mathcal{M}}^{S\rightarrow
E}(U^{S}\Psi^{SR}U^{S\dagger})-{\mathcal{M}}^{S\rightarrow
E}(V^{S}\varrho^{SR}V^{S\dagger})|\\!|_{1}$ (166)
$\displaystyle\leq|\\!|U^{S}\varrho^{SR}U^{S\dagger}-V^{S}\Psi^{SR}V^{S\dagger}|\\!|_{1}$
(167) $\displaystyle\leq 2|\\!|(U^{S}-V^{S})|\varrho\rangle^{SR}|\\!|_{2}$
(168) $\displaystyle\leq
2|\\!|\varrho^{S}|\\!|_{\infty}^{1/2}|\\!|U^{S}-V^{S}|\\!|_{2}.$ (169)

Here, the second inequality follows from the monotonicity of the trace
distance under any trace-non-increasing map, the third one from that
$\Psi^{SR}$ is a pure state and the exact diagonalization with the use of the
inequality $\sqrt{1-x^{2}}\leq\sqrt{2(1-x)}$ for any $x\in[0,1]$, and the last
one from the fact that

$\displaystyle|\\!|M^{X}|\phi\rangle^{XY}|\\!|_{2}^{2}$
$\displaystyle=\operatorname{Tr}[M^{X}\phi^{XY}M^{X\dagger}]$ (170)
$\displaystyle=|\\!|\phi^{X}M^{X\dagger}M^{X}|\\!|_{1}$ (171)
$\displaystyle\leq|\\!|\phi^{X}|\\!|_{\infty}|\\!|M^{X\dagger}M^{X}|\\!|_{1}$
(172) $\displaystyle=|\\!|\phi^{X}|\\!|_{\infty}|\\!|M^{X}|\\!|_{2}^{2},$
(173)

due to the Hölder’s inequality.

Explicitly writing $U^{S}$ and $V^{S}$ as
$\bigoplus_{j\in\mathcal{J}}U_{j}^{S}$ and
$\bigoplus_{j\in\mathcal{J}}V_{j}^{S}$, respectively,
$|\\!|U^{S}-V^{S}|\\!|_{2}$ is given by
$\sum_{j\in\mathcal{J}}\bigl{\|}U_{j}^{S}-V_{j}^{S}\bigr{|}\\!\bigr{|}_{2}^{2}$,
i.e. $|\\!|U^{S}-V^{S}|\\!|_{2}=D(U^{S},V^{S})$. Hence, for any $U^{S}\in{\sf
U}_{\times}$ and $V^{S}\in{\sf U}_{\times}$, we obtain

$\frac{|F(U^{S}_{\rm rad},\dots,U^{S}_{J})-F(V^{S}_{\rm
rad},\dots,V^{S}_{J})|}{D(U^{S},V^{S})}\leq
2\sqrt{|\\!|\varrho^{S}|\\!|_{\infty}},$ (174)

which implies that the Lipschitz constant of $F$ is bounded from above by
$2\sqrt{|\\!|\varrho^{S}|\\!|_{\infty}}$.

We then apply Theorem 6 and obtain

$\displaystyle{\rm Prob}_{U^{S}\sim{\sf H}_{\times}}\bigl{[}F(U^{S}_{\rm
rad},\dots,U^{S}_{J})\geq 2^{-\frac{1}{2}H_{\rm
min}(S^{*}|ER)_{\Gamma}}+\delta\bigr{]}$ (175) $\displaystyle\leq{\rm
Prob}_{U^{S}\sim{\sf H}_{\times}}\bigl{[}F(U^{S}_{\rm
rad},\dots,U^{S}_{J})\geq\mathbb{E}[F]+\delta\bigr{]}$ (176)
$\displaystyle\leq\exp\bigl{[}-\frac{\delta^{2}d_{\rm
min}}{48|\\!|\varrho^{S}|\\!|_{\infty}}\bigr{]},$ (177)

where $d_{\rm min}=\min_{j{\mathcal{J}}}\\{d_{j}\\}$, and Theorem 5, stating
that $\mathbb{E}[F]\leq 2^{-\frac{1}{2}H_{\rm min}(S^{*}|ER)_{\Gamma}}$, is
used to obtain the first inequality. $\hfill\blacksquare$

##  Appendix B Proof of Corollary 2

To show Corollary 2, we use the gentle measurement lemma:

######  Lemma 7 (Gentle measurement lemma [58]).

Let $\Phi$ be in $\mathcal{S}(\mathcal{H})$ and $\Lambda$ be an Hermitian
operator such that $0\leq\Lambda\leq I$. If they satisfy
$\operatorname{Tr}[\Lambda\Phi]\geq 1-\epsilon$, where $0\leq\epsilon\leq 1$,
then $|\\!|\Phi-\Phi^{\prime}|\\!|_{1}\leq 2\sqrt{\epsilon}$, where

$\Phi^{\prime}=\frac{\sqrt{\Lambda}\Phi\sqrt{\Lambda}}{\operatorname{Tr}[\Lambda\Phi]}.$
(178)

We also use a simple fact about the conditional min-entropy as given in Lemma
8.

######  Lemma 8 (Conditional min-entropy after projective measurement).

Let $\Pi^{A}$ be a projection operator, $\Psi^{AB}$ be a quantum state. A
post-measured state
$\tilde{\Psi}^{AB}:=\Pi^{A}\Psi^{AB}\Pi^{A}/\operatorname{Tr}[\Pi^{A}\Psi^{AB}]$
satisfies

$H_{\rm min}(A|B)_{\tilde{\Psi}}\geq H_{\rm
min}(A|B)_{\Psi}+\log[\operatorname{Tr}[\Pi^{A}\Psi^{AB}]].$ (179)

######  Proof (Lemma 8).

Let $\sigma^{B}\in{\mathcal{S}}({\mathcal{H}}^{B})$ be the state such that
$2^{-H_{\rm min}(A|B)_{\Psi}}I^{A}\otimes\sigma^{B}\geq\Psi^{AB}$. Then, we
have

$2^{-(H_{\rm
min}(A|B)_{\Psi}+\log[\operatorname{Tr}[\Pi^{A}\Psi^{AB}]])}I^{A}\otimes\sigma^{B}\geq\tilde{\Psi}^{AB},$
(180)

which implies the desired result. $\hfill\blacksquare$

Using these lemmas, Corollary 2 can be shown as follows.

######  Proof (Corollary 2).

We first define the state $\tilde{\varrho}^{SR}$ by

$\tilde{\varrho}^{SR}:=\frac{\Pi_{\geq}^{S}\varrho_{SR}\Pi_{\geq}^{S}}{\operatorname{Tr}[\Pi_{\geq}^{S}\varrho_{SR}]}$
(181)

We also use the fact that
$\Gamma^{ER}:=\operatorname{Tr}_{S^{*}}[\Gamma^{S^{*}ER}]=\mathbb{E}_{U^{S}\sim{\sf
H}_{\times}}\bigl{[}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\varrho^{SR}U^{S\dagger}\bigr{)}\bigr{]}$. Using the triangle
inequality, we obtain

$\bigl{\|}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\varrho^{SR}U^{S\dagger}\bigr{)}-\mathbb{E}_{U^{S}\sim{\sf
H}_{\times}}\bigl{[}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\varrho^{SR}U^{S\dagger}\bigr{)}\bigr{]}\bigr{\|}_{1}\\\
\leq\bigl{\|}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\varrho^{SR}U^{S\dagger}-U^{S}\tilde{\varrho}^{SR}U^{S\dagger}\bigr{)}\bigr{\|}_{1}+\bigl{\|}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\tilde{\varrho}^{SR}U^{S\dagger}\bigr{)}-\mathbb{E}_{U^{S}\sim{\sf
H}_{\times}}\bigl{[}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\tilde{\varrho}^{SR}U^{S\dagger}\bigr{)}\bigr{]}\bigr{\|}_{1}\\\
+\bigl{\|}\mathbb{E}_{U^{S}\sim{\sf
H}_{\times}}\bigl{[}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\tilde{\varrho}^{SR}U^{S\dagger}\bigr{)}\bigr{]}-\mathbb{E}_{U^{S}\sim{\sf
H}_{\times}}\bigl{[}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\varrho^{SR}U^{S\dagger}\bigr{)}\bigr{]}\bigr{\|}_{1}.$ (182)

In the following, we evaluate each term in the right-hand side of Eq. (182)
separately.

For the first term, noting that ${\cal C}^{S\rightarrow E}$ is a trace-non-
increasing map, and the trace norm is unitarily invariant, we have

$\displaystyle\bigl{\|}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\varrho^{SR}U^{S\dagger}-U^{S}\tilde{\varrho}^{SR}U^{S\dagger}\bigr{)}\bigr{\|}_{1}$
$\displaystyle\leq\bigl{\|}\varrho^{SR}-\tilde{\varrho}^{SR}\bigr{\|}_{1}$
(183) $\displaystyle\leq 2\sqrt{\epsilon},$ (184)

where the last inequality follows from the gentle measurement lemma and the
assumption that $\operatorname{Tr}[\varrho^{SR}\Pi_{\geq}^{S}]\geq
1-\epsilon$.

To evaluate the second term, we use Theorem 1. Recalling that
$\tilde{\Psi}^{S}$ does not have support on the subspace
${\mathcal{H}}_{j}^{S}$ with dimension being smaller than $d_{\rm th}$, it
follows that, for any $\delta>0$,

$\bigl{\|}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\tilde{\varrho}^{SR}U^{S\dagger}\bigr{)}-\mathbb{E}_{U^{S}\sim{\sf
H}_{\times}}\bigl{[}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\tilde{\varrho}^{SR}U^{S\dagger}\bigr{)}\bigr{]}\bigr{\|}_{1}\\\
\leq 2^{-\frac{1}{2}H_{\rm min}(S^{*}|ER)_{\tilde{\Gamma}}}+\delta$ (185)

with probability at least $1-\exp[-\frac{\delta^{2}d_{\rm
th}}{48|\\!|\tilde{\varrho}^{S}|\\!|_{\infty}}]$. Here,
$\tilde{\Gamma}^{S^{*}ER}=\sum_{j,j^{\prime}\in\mathcal{J}}\frac{D_{S}}{\sqrt{d_{j}d_{j^{\prime}}}}\zeta^{SE}_{jj^{\prime}}\otimes\tilde{\varrho}^{S^{\prime}R}_{jj^{\prime}}$.
Since $\Pi_{\geq}$ is commutable with $\Pi_{j}^{S}$ for any $j$,

$\tilde{\Gamma}^{S^{*}ER}=\frac{(I^{S}\otimes\Pi^{S^{\prime}}_{\geq})\Gamma^{S^{*}ER}(I^{S}\otimes\Pi^{S^{\prime}}_{\geq})}{\operatorname{Tr}[(I^{S}\otimes\Pi^{S^{\prime}}_{\geq})\Gamma^{S^{*}ER}]}.$
(186)

Using Lemma 8, we have

$\displaystyle H_{\rm min}(S^{*}|ER)_{\tilde{\Gamma}}$ $\displaystyle\geq
H_{\rm
min}(S^{*}|ER)_{\Gamma}+\log\bigl{[}\operatorname{Tr}[\Pi^{S}_{\geq}\Gamma^{S^{*}ER}]\bigr{]}$
(187) $\displaystyle=H_{\rm
min}(S^{*}|ER)_{\Gamma}+\log\bigl{[}\operatorname{Tr}[\Pi^{S}_{\geq}\varrho^{SR}]\bigr{]}$
(188) $\displaystyle\geq H_{\rm min}(S^{*}|ER)_{\Gamma}+\log[1-\epsilon],$
(189)

where the second line is obtained since
$\frac{D_{S}}{\sqrt{d_{j}d_{j^{\prime}}}}\operatorname{Tr}[\zeta^{SE}_{jj^{\prime}}]=\delta_{jj^{\prime}}$.
Furthermore, it holds that

$\displaystyle|\\!|\tilde{\varrho}^{S}|\\!|_{\infty}$
$\displaystyle\leq\min\bigl{\\{}1,\frac{|\\!|\Pi_{\geq}^{S}|\\!|_{\infty}|\\!|\varrho^{S}|\\!|_{\infty}}{\operatorname{Tr}[\Pi^{S}_{\geq}\varrho^{S}]}\bigr{\\}}$
(190)
$\displaystyle\leq\min\bigl{\\{}1,\frac{|\\!|\varrho^{S}|\\!|_{\infty}}{1-\epsilon}\bigr{\\}}=:C,$
(191)

where we have used the sub-multiplicativity of the operator norm, and the
assumption that $\operatorname{Tr}[\varrho^{SR}\Pi_{\geq}^{S}]\geq
1-\epsilon$. Combining all of these together, the second term is bounded as

$\bigl{\|}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\tilde{\varrho}^{SR}U^{S\dagger}\bigr{)}-\mathbb{E}_{U^{S}\sim{\sf
H}_{\times}}\bigl{[}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\tilde{\varrho}^{SR}U^{S\dagger}\bigr{)}\bigr{]}\bigr{\|}_{1}\\\
\leq\frac{2^{-\frac{1}{2}H_{\rm
min}(S^{*}|ER)_{\Gamma}}}{\sqrt{1-\epsilon}}+\delta$ (192)

for any $\delta>0$ with probability at least $1-\exp[-\frac{\delta^{2}d_{\rm
th}}{48C}]$.

To evaluate the third term in the right-hand side of Eq. (182), we use the
explicit form of the averaged operator

$\mathbb{E}_{U^{S}\sim{\sf H}_{\times}}\bigl{[}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\Psi^{SR}U^{S\dagger}\bigr{)}\bigr{]}=\sum_{j\notin\mathcal{J}_{\geq}}\frac{D_{S}}{d_{j}}\zeta^{E}_{jj}\otimes\varrho^{R}_{jj}.$
(193)

Further, using the relation
$\Pi^{S}_{\geq}=\sum_{j\in{\mathcal{J}}_{\geq}}\Pi^{S}_{j}$, we obtain that
the third term $X$ in the right-hand side of Eq. (182) satisfy

$\displaystyle X$
$\displaystyle=\bigl{\|}\sum_{j\in\mathcal{J}}\frac{D_{S}}{d_{j}}\zeta^{E}_{jj}\otimes(\tilde{\varrho}^{R}_{jj}-\varrho^{R}_{jj})\bigr{\|}_{1}$
(194)
$\displaystyle\leq\sum_{j\notin\mathcal{J}_{\geq}}\bigl{\|}\frac{D_{S}}{d_{j}}\zeta^{E}_{jj}\otimes\varrho^{R}_{jj}\bigr{\|}_{1}+\sum_{j\in\mathcal{J}_{\geq}}\bigl{\|}\frac{D_{S}}{d_{j}}\zeta^{E}_{jj}\otimes(\tilde{\varrho}^{R}_{jj}-\varrho^{R}_{jj})\bigr{\|}_{1}.$
(195)

Recalling that $\operatorname{Tr}[\frac{D_{S}}{d_{j}}\zeta^{E}_{jj}]\leq 1$
and that
$\tilde{\varrho}^{R}_{j}=\varrho^{R}_{j}/\operatorname{Tr}[\Pi_{\geq}^{S}\varrho^{SR}]$
for $j\in{\mathcal{J}}_{\geq}$, the first term in the right-hand side of Eq.
(195) is bounded from above by

$\sum_{j\notin\mathcal{J}_{\geq}}\operatorname{Tr}[\varrho^{R}_{jj}]=\operatorname{Tr}\bigl{[}\varrho^{SR}(I^{S}-\Pi^{S}_{\geq})\bigr{]}\leq\epsilon.$
(196)

An upper bound of the second term in the right-hand side of Eq. (195) is given
by

$\displaystyle\biggl{|}\frac{1}{\operatorname{Tr}[\Pi^{S}_{\geq}\varrho^{SR}]}-1\biggr{|}\sum_{j\in\mathcal{J}_{\geq}}\operatorname{Tr}[\varrho^{SR}_{jj}]$
$\displaystyle\leq\frac{\epsilon}{1-\epsilon}\sum_{j\in\mathcal{J}}\operatorname{Tr}[\varrho^{SR}_{jj}]$
(197) $\displaystyle=\frac{\epsilon}{1-\epsilon}.$ (198)

Combining the upper bounds of all three terms in Eq. (182), we obtain the
desired result:

$\bigl{\|}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\varrho^{SR}U^{S\dagger}\bigr{)}-\mathbb{E}_{U^{S}\sim{\sf
H}_{\times}}\bigl{[}{\mathcal{M}}^{S\rightarrow
E}\bigl{(}U^{S}\varrho^{SR}U^{S\dagger}\bigr{)}\bigr{]}\bigr{\|}_{1}\\\
\leq\frac{2^{-\frac{1}{2}H_{\rm
min}(S^{*}|ER)_{\Gamma}}}{\sqrt{1-\epsilon}}+\delta+f(\epsilon),$ (199)

with probability at least $1-\exp[-\frac{\delta^{2}d_{\rm th}}{48C}]$, where