# Lossy Quantum Source Coding with Global Error Criterion using Posterior
Reference Map

Touheed Anwar Atif, Mohammad Aamir Sohail, and S. Sandeep Pradhan
Department of Electrical Engineering and Computer Science,
University of Michigan, Ann Arbor, MI 48109, USA.
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Lossy Quantum Source Coding with a Global Error Criterion based on a
Posterior Reference Map

Touheed Anwar Atif, Mohammad Aamir Sohail, and S. Sandeep Pradhan
Department of Electrical Engineering and Computer Science,
University of Michigan, Ann Arbor, MI 48109, USA.
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider the lossy quantum source coding problem where the task is to
compress a given quantum source below its von Neumann entropy. Inspired by the
duality connections between the rate-distortion and channel coding problems in
the classical setting, we propose a new formulation for the lossy quantum
source coding problem. This formulation differs from the existing quantum
rate-distortion theory in two aspects. Firstly, we require that the
reconstruction of the compressed quantum source fulfill a global error
constraint as opposed to the sample-wise local error criterion used in the
standard rate-distortion setting. Secondly, instead of a distortion
observable, we employ the notion of a backward quantum channel, which we refer
to as a “posterior reference map”, to measure the reconstruction error. Using
these, we characterize the asymptotic performance limit of the lossy quantum
source coding problem in terms of single-letter coherent information of the
given posterior reference map. We demonstrate a protocol to encode (at the
specified rate) and decode, with the reconstruction satisfying the provided
global error criterion, and therefore achieving the asymptotic performance
limit. The protocol is constructed by decomposing coherent information as a
difference of two Holevo information quantities, inspired from prior works in
quantum communication problems. To further support the findings, we develop
analogous formulations for the quantum-classical and classical variants and
express the asymptotic performance limit in terms of single-letter mutual
information quantities with respect to appropriately defined channels
analogous to posterior reference maps. We also provide various examples for
the three formulations, and shed light on their connection to the standard
rate-distortion formulation wherever possible.

##  I Introduction

A fundamental problem from an information theoretic perspective is the
asymptotic characterization of the rate required to compress a source that can
be recovered to a certain measurable degree. Such a problem in quantum
information theory is referred to as quantum source coding or a quantum data
compression problem. In the lossless regime, Schumacher [1, 2] proved that a
quantum source could be compressed at a rate given by von Neumann entropy
while incurring a very small error between the reconstruction and the source
state. The error in this model is defined for the entire block, also called as
block error or _global error_. Considering the block error, a strong converse
was also proved in the lossless regime [3, 4], which states that it is
impossible to achieve any rate below von Neumann entropy even when the
asymptotic probability of block error is relaxed from being (almost) zero.

As for the lossy regime, where the objective is to further reduce the rate at
the expense of increased but bounded error, Barnum [5] conjectured minimal
coherent information as a candidate in characterizing the asymptotic
performance limit. Generalizing the formulation from the classical rate-
distortion theory [6], Barnum in [5] introduced a _local_ distortion criterion
as averaged symbol-wise entanglement fidelity based on marginal operations
(partial trace) between the reconstruction and the reference of the original
source. In [7], Datta _et. al_ obtained a regularized expression for the
quantum rate-distortion distortion function in terms of the entanglement of
purification. Further, the authors also formulated the entanglement-assisted
quantum rate-distortion problem and characterized its asymptotic performance
limit using a single-letter expression. Wilde _et. al_ further refined the
characterization of the quantum rate-distortion function in terms of
regularized entanglement of formation, and also generalized the problem setup
to various scenarios, including side information in [8]. Works toward the
asymptotic simulation of a memoryless quantum channel in [9, 10] have shown to
be useful in achieving the above results, in particular, the entanglement-
assisted formulations. Authors in [11] formulated a quantum-to-classical rate-
distortion problem and provided a single-letter formula. A rate-distortion
version of the quantum state redistribution task [12, 13] was considered in
[14]. Investigations on a rate-distortion framework of generic mixed quantum
sources have been the focus of [15, 16]. Other works that addressed related
problems include [17, 18, 19, 20, 21, 22, 23].

In this work, we consider a new formulation of the problem of lossy quantum
source coding, and characterize a rate function, no larger than von Neumann
entropy, while allowing for bounded error in the reconstruction. We use a
global error criterion as opposed to the approach of local symbol-wise error
studied in the literature. The problem we consider is without any shared
entanglement resources between the encoder and the decoder. We motivate this
formulation with the following observations.

The local error criterion in the quantum rate-distortion framework is inspired
by the corresponding additive local single-letter distortion criterion in the
classical source coding formulation of Shannon [6], where a single-letter
characterization is available. The motivation for considering a local
criterion is the strong converse of the lossless source coding theorem which
states that the entropy bound cannot be breached even when the asymptotic
probability of block error is relaxed to any number in $(0,1)$ [24, Theorem
1.1].

In [6, 24], a duality connection between the source coding problem and the
channel coding problem was observed. These problems were interpreted in terms
of a covering versus packing perspective. In both problems, the same
information measure, namely the mutual information, captures the asymptotic
performance limits. A similar duality connection exists between the classical-
quantum communication problem [25, 26] and the quantum-classical source coding
problem [27, 11], with the performance limits of both problems characterized
in terms of single-letter Holevo information quantities [28]. This has been
further explored in [29]. In the fully quantum setting, from this standpoint,
its well known that the quantum channel coding problem has an asymptotic
performance limit characterized using regularized coherent information [30,
31, 32, 33]. Among others, Devetak developed a proof of this result by
employing a coherent approach to covering and packing, and combined them
cohesively, inspired by his work on the private channel capacity problem [32].
Coherent information can be interpreted in terms of packing of subspaces as
elucidated in [30]. Quantum error-correcting codes have been extensively
studied along these lines in the coding theory literature, e.g., quantum
Hamming bound [34]. This leads us to the question: why is such a limit based
on coherent information absent for the lossy quantum source compression
problem?

Toward answering this question, we take a closer look at the classical
discrete memoryless setting. We find that in addition to Shannon’s pioneering
work of characterizing the rate-distortion problem [6, 35], there have been
several works discussing the lossy source compression problem. A concept that
has received particular attention is the notion of a backward channel [24,
Problem 8.3], which characterizes the posterior distribution of the source
given the reconstruction. The structure of this channel has been studied in
[36, 37, 38]. Although the forward channel, relating the reconstruction to the
source, achieving the rate-distortion function need not be unique, the
resulting backward channel is indeed unique. Moreover, the rate-distortion
achievability result in [24, Theorem 2.3] is shown by constructing a channel
code for a backward channel with a large probability of error and by using the
encoder of the latter as a decoder of the former and vice versa. Highlighting
this duality further, inspired by results on the output statistics of good
channel codes [39], the following was shown in [40]. The $n$-letter actual
posterior conditional distribution of the source vector given the
reconstruction vector of any rate-distortion achieving code converges in
normalized divergence to the $n$-product of the unique minimum-mutual-
information backward channel conditional distribution. In other words,
although the encoder and decoder are block operations, the induced posterior
$n$-letter channel becomes discrete memoryless in the asymptotic limit for a
rate-distortion achieving code. For further developments on this concept see
[41, 42, 43, 44]. This channel also plays a fundamental role in Bayesian
estimation and detection theory [45], e.g., maximum a posteriori (MAP)
estimation. Therefore, we ask the question, can we use such a channel to
formulate a lossy source coding problem?

Contributions of this work: In light of this, in this work, we explore a new
formulation of the source compression problem in the memoryless setting. This
formulation is based on the notion of a posterior channel that produces the
reference of the source from that of the reconstruction. Instead of a single-
letter distortion function, now, we are given a single-letter posterior
channel that characterizes the nature of the loss incurred in the encoding and
decoding operations. More precisely, we want to construct an encoder and a
decoder such that the joint effect of encoding and decoding – to produce a
reconstruction sequence from the source sequence – is close to the effect of
the $n$-product posterior channel acting on the non-product reconstruction
sequence.The closeness is measured using the trace distance in the quantum
case and the total variation in the classical case, manifesting as a global
error constraint. A related concept is the Petz recovery map which has found
significant relevance in information-theoretic problems [46, 47, 48]. However,
we take a different approach and consider a quantum channel, i.e., a CPTP map,
acting on the reference of the reconstruction to produce the reference of the
source, whose existence is guaranteed using Uhlmann’s theorem. We refer to
this as a posterior reference map.

As one of the main contributions of our work, we provide a single-letter
characterization of the asymptotic performance limit of this source coding
problem using the minimal coherent information of the posterior reference map,
where the minimization is over all reconstructions (see Theorem 1).
Furthermore, our work establishes a duality connection between quantum lossy
compression and the quantum channel coding problem. Our proof is based on the
coherent application of two fundamental tools of quantum information theory,
namely, packing and covering, implying a duality relationship with Devetak’s
proof for the channel coding problem [32] (also see [31, 33]).

We also provide a correspondingly new formulation for the quantum-classical
(QC) and classical lossy source coding problems. In the quantum-classical
setup, we provide a single-letter characterization of the asymptotic
performance limit using the minimal Holevo information (or the corresponding
quantum mutual information) of the posterior classical-quantum (CQ) channel,
where the minimization is over all reconstruction distributions (see Theorem
2). In the classical setup, the minimal mutual information of the posterior
channel determines the single letter characterization of the asymptotic
performance limit of classical source coding problem (see Theorem 3). The
posterior CQ channel and the posterior channel are defined analogous to the
posterior reference map for the QC and classical settings, respectively.

At one end of the spectrum, when the posterior reference map is specified as
the identity transformation, our rate expression in the quantum case reduces
to the von Neumann entropy of the given quantum source, demonstrating the
connection with the Schumacher’s lossless compression [1]. In fact, the two
formulations can be shown to be equivalent to one another. The same follows in
the classical and quantum-classical formulations where the rate equals
Shannon’s entropy and von Neumann’s entropy of the sources, respectively. On
the other end, when the specified posterior reference map is such that
coherent information is negative for some reference of the reconstruction, we
characterize the asymptotic performance limit of the lossy quantum source
coding problem to be zero.

The techniques employed to prove our results can be summarized as follows. For
the achievability of the Theorem 1, we first construct a posterior reference
isometry $V$ (as in Definition 1) and decompose it as a coherent measurement.
We then make use of Winter’s measurement compression protocol [27], and apply
it in a coherent fashion to compress the output of the above isometry. This
involves using the Uhlmann’s Theorem [49] (or [50, Theorem 9.2.1]) followed by
incorporating additional phases to achieve a coherent faithful simulation of
the posterior reference map. To further decrease the compression rate, we
exploit the fact that a noiseless quantum channel can preserve arbitrary
superpositions. Therefore, we perform additional encoding to embed the
information at the output of $V$ as superpositions within itself. This
requires availing the HSW classical communication result [25, 26] to construct
information decoding POVMs, and Naimark’s extension theorem to construct a
unitary from POVM elements. The method used for expurgation is another
interesting feature of the proof. The protocol as it stands only permits
operations that are unitary or isometric, followed by partial tracing. It can
be challenging to guarantee this when there are repeated codewords in a code.
A similar phenomenon was observed in the Devetak’s proof [32].

As for the achievability of Theorem 2, we make use of Winter’s measurement
compression protocol [27] to construct the encoding POVM. For Theorem 3, we
use the likelihood encoder as discussed in [51, 52] to prove the achievability
of lossy classical source coding.

For the converse of Theorem 1, we use the quantum data processing inequality
for coherent information, the Fannes-Audenart inequality, and monotonicity
results. In the case of the quantum-classical setup, proof of the converse of
Theorem 2 uses inequalities such as the quantum data processing inequality,
the concavity of conditional quantum entropy, and the continuity of quantum
mutual information (AFW inequality). In the classical setup, similar tools are
used to prove a converse to Theorem 3.

The paper is organized as follows. We provide some necessary definitions and
useful lemmas in Section II. In Section III, we formulate the problems and
provide the main results pertaining to quantum lossy compression (Theorem 1),
QC lossy compression (Theorem 2), and classical lossy compression (Theorem 3).
We provide examples corresponding to these three results in section IV. In
Sections V, VI, and VII, we provide proofs of the main results. Within each of
these sections, we provide the achievability proof followed by proof of the
converse. Finally, Section VIII concludes the paper.

##  II Preliminaries and Notations

We supplement the notations in [50] with the following. Let $I_{A}$ denote the
identity operator acting on a Hilbert space $\mathcal{H}_{A}$. The set of
density operators on $\mathcal{H}_{A}$ are denoted by
$\mathcal{D}(\mathcal{H}_{A})$, and linear operators by
$\mathcal{L}(\mathcal{H}_{A})$. We denote $\mathcal{H}_{{A_{R}}}$ as the
Hilbert space associated with the reference space of $\mathcal{H}_{A}$, with
$\dim{\mathcal{H}_{{A_{R}}}}=\dim{\mathcal{H}_{A}}$. In this work, we focus
exclusively on references obtained from canonical purifications of quantum
states [27, Lemma 14 (Pretty Good Purifications)], and define canonical
purification $\ket{\psi_{\rho}}^{{A_{R}}A}$ of $\rho^{A}$ as
$\ket{\psi_{\rho}}^{{A_{R}}A}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(I_{{A_{R}}}\otimes\sqrt{\rho^{A}})\Gamma_{{A_{R}}A}$,
where $\Gamma_{{A_{R}}A}$ is defined as the unnormalized maximally entangled
state. We use $\Psi_{\rho}^{{A_{R}}A}$ to denote the density operator
corresponding to $\ket{\psi_{\rho}}^{{A_{R}}A}$. As is the convention, for two
states acting on the same Hilbert space, we use the same $\Gamma$ when
defining their canonical purifications. We denote the finite alphabet of a
source as $\mathsf{X}$, and the set of probability distributions on the finite
alphabet $\mathsf{X}$ as $\mathcal{P}(\mathsf{X})$. Let
$[\Theta]\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\left\\{{1,2,\cdots,\Theta}\right\\}$.
For a CPTP map $\mathcal{N}:\mathcal{H}_{A}\rightarrow\mathcal{H}_{B}$, and an
input density operator $\rho^{A}\in\mathcal{D}(\mathcal{H}_{A})$, we use
$I_{c}(\mathcal{N},\rho^{A})$ to denote the coherent information of
$\mathcal{N}$ with respect to $\rho^{A}$.

Figure 1: Figure demonstrating the construction of the posterior reference map
$W$ from the isometry $V$ (the Stinespring’s dilation of $\mathcal{N}_{V}$)
and the source state $\rho^{B}$.

######  Definition 1 (Posterior Reference Map).

Given a source $\rho^{B}\in\mathcal{D}(\mathcal{H}_{B})$ and a channel
$\mathcal{N}_{V}:\mathcal{H}_{B}\rightarrow\mathcal{H}_{A}$, let
$\rho^{A}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\mathcal{N}_{V}(\rho^{B})$.
Let $V:\mathcal{H}_{B}\rightarrow\mathcal{H}_{A}\otimes\mathcal{H}_{E}$ be a
Stinespring’s isometry corresponding to the CPTP map $\mathcal{N}_{V}$ with
$\dim(\mathcal{H}_{E})\geq\dim(\mathcal{H}_{A})$, such that
$\mathcal{N}_{V}(\cdot)=\Tr_{E}\\{V(\cdot)V^{\dagger}\\}$. As shown in Figure
1, define the “posterior reference map” of $V$ with respect to $\rho^{A}$ as
the CPTP map
$\mathcal{N}_{W}:\mathcal{H}_{{A_{R}}}\rightarrow\mathcal{H}_{{B_{R}}}$
corresponding to the isometry
$W:\mathcal{H}_{{A_{R}}}\rightarrow\mathcal{H}_{{B_{R}}}\otimes\mathcal{H}_{E}$
satisfying $(W\otimes I_{A})\ket{\psi_{\rho}}^{{A_{R}}A}=(I_{{B_{R}}}\otimes
V)\ket{\psi_{\rho}}^{{B_{R}}B}$ where $\ket{\psi_{\rho}}^{{A_{R}}A}$ and
$\ket{\psi_{\rho}}^{{B_{R}}B}$ are the canonical purifications of $\rho^{A}$
and $\rho^{B}$, respectively.

######  Remark 1 (Existence of a Posterior Reference Map).

Using the equivalence of purifications, one can guarantee the existence of
such a posterior reference isometry
$W:\mathcal{H}_{{A_{R}}}\rightarrow\mathcal{H}_{{B_{R}}}\otimes\mathcal{H}_{E}$.
Since $V$ is an isometry with
$\dim(\mathcal{H}_{E})\geq\dim(\mathcal{H}_{A})$, and since
$\ket{\psi_{\rho}}^{{A_{R}}A}$ and
$\ket{\psi_{\rho}}^{{B_{R}}AE}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(I_{{B_{R}}}\otimes
V)\ket{\psi_{\rho}}^{{B_{R}}B}$ are purifications of $\rho^{A}$ (as
$\Tr_{E}(V\rho^{B}V^{\dagger})=\rho^{A}$), from [50, Theorem 5.1.1], there
exists an isometry
$W:\mathcal{H}_{{A_{R}}}\rightarrow\mathcal{H}_{{B_{R}}}\otimes\mathcal{H}_{E}$
such that $(W\otimes
I_{A})\ket{\psi_{\rho}}^{{A_{R}}A}=\ket{\psi_{\rho}}^{{B_{R}}AE}$.

###  II-A Useful Lemmas

######  Lemma 1 ([53], Theorem 9.3.1 [50]).

Given two states $\rho,\sigma\in\mathcal{D}(\mathcal{H})$, we have

$1-\sqrt{F(\rho,\sigma)}\leq\frac{1}{2}\norm{\rho-\sigma}_{1}\leq\sqrt{1-F(\rho,\sigma)}.$

######  Lemma 2.

For $\rho^{B},\sigma^{B}\in\mathcal{D}(\mathcal{H}_{B})$, the following
inequality holds:

$F(\ket{\psi_{\rho}},\ket{\psi_{\sigma}})\geq\left(1-\frac{1}{2}\norm{{\rho^{B}}-{\sigma^{B}}}_{1}\right)^{2},$
(1)

where $\ket{\psi_{\rho}}$ and $\ket{\psi_{\sigma}}$ are the canonical
purifications of $\rho^{B}$ and $\sigma^{B}$, respectively.

###### Proof.

We provide a proof in Appendix A-A. ∎

The above lemma is a slight tightening of the Lemma 14 (“Pretty good
purifications”) of [27].

######  Lemma 3 (Naimark’s extension theorem [54], [55, Theorem 2.1]).

Given a POVM $\\{\Gamma_{x}\\}_{x\in\mathsf{X}}$ acting on the system
$\mathcal{H}_{A}$, there exists a unitary $U_{AA^{\prime}}$ acting on the
system $\mathcal{H}_{A}$ and auxiliary system $\mathcal{H}_{A^{\prime}}$ and
an orthonormal basis $\\{\ket{x}^{A^{\prime}}\\}_{x\in\mathsf{X}}$ such that

$\Tr\left\\{\overline{\Gamma}_{x}(\rho^{A}\otimes\outerproduct{0}{0}_{A^{\prime}})\right\\}=\Tr(\Gamma_{x}\rho^{A}),$

where
$\\{\overline{\Gamma}_{x}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}U_{AA^{\prime}}^{\dagger}(\mathbbm{1}_{A}\otimes\outerproduct{x}{x}^{A^{\prime}})U_{AA^{\prime}}\\}$
are orthogonal projectors acting on system
$\mathcal{H}_{A}\otimes\mathcal{H}_{A^{\prime}}$. Also, $\ket{0}^{A^{\prime}}$
is some fixed state in $\mathcal{H}_{A^{\prime}}$, and independent of
$\Gamma_{x}$ and $\rho^{A}$.

##  III Main Results

###  III-A Lossy Quantum Source Coding

We first formulate a quantum source coding problem as follows. For any
memoryless quantum information source, characterized by
$\rho^{B}\in\mathcal{D}(\mathcal{H}_{B})$, denote its canonical purification
by $\ket{\psi_{\rho}}^{B{B_{R}}}$. Let
$\rho^{B_{R}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\Tr_{B}[\Psi_{\rho}^{B_{R}B}]$.

######  Definition 2 (Quantum Source Coding Setup).

A quantum source coding setup is characterized by a triple
$(\rho^{B},\mathcal{H}_{A},\mathcal{N}_{W})$, where
$\rho^{B}\in\mathcal{D}(\mathcal{H}_{B})$ is a density operator,
$\mathcal{H}_{{A}}$ is a reconstruction Hilbert space, and $\mathcal{N}_{W}$
is a single-letter CPTP map from $\mathcal{H}_{{A_{R}}}$ to
$\mathcal{H}_{{B_{R}}}$, where $\mathcal{H}_{{A_{R}}}$ and
$\mathcal{H}_{{B_{R}}}$ are reference spaces corresponding to
$\mathcal{H}_{A}$ and $\mathcal{H}_{B}$, respectively.

######  Definition 3 (Lossy Quantum Compression Protocol).

For a given input and reconstruction Hilbert spaces
$(\mathcal{H}_{B},\mathcal{H}_{A})$, an $(n,\Theta)$ lossy quantum compression
protocol consists of a encoding CPTP map
$\mathcal{N}^{(n)}_{\mathcal{E}}:\mathcal{H}_{B^{n}}\rightarrow\mathcal{H}_{M}$
and a decoding CPTP map
$\mathcal{N}^{(n)}_{\mathcal{D}}:\mathcal{H}_{M}\rightarrow\mathcal{H}_{A^{n}}$,
such that $\dim(\mathcal{H}_{M})=\Theta$, as shown in Figure 2.

Figure 2: Illustration of Lossy Quantum Compression protocol

######  Definition 4 (Achievability).

For a quantum source coding setup
$(\rho^{B},\mathcal{H}_{A},\mathcal{N}_{W})$, a rate $R$ is said to be
achievable, if for all $\epsilon>0$ and all sufficiently large $n$, there
exists an $(n,\Theta)$ lossy quantum compression protocol satisfying

$\displaystyle\left\|{\omega^{B_{R}^{n}A^{n}}-(\mathcal{N}_{W}^{\otimes
n}\otimes I_{A^{n}})\Psi_{\omega}^{A_{R}^{n}A^{n}}}\right\|_{1}\leq\epsilon,$
(2)

and $\frac{1}{n}\log{\Theta}\leq R+\epsilon$, where
$\omega^{B_{R}^{n}A^{n}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(I\otimes\mathcal{N}_{\mathcal{D}}^{(n)})(I\otimes\mathcal{N}_{\mathcal{E}}^{(n)})(\Psi_{\rho}^{B_{R}^{n}B^{n}})$,
and $\Psi_{\rho}^{B_{R}^{n}B^{n}}$ and $\Psi_{\omega}^{A_{R}^{n}A^{n}}$ are
the canonical purifications of ${\rho^{B}}^{\otimes n}$ and $\omega^{A^{n}}$,
respectively.

In other words, the protocol ensures that the joint state of the
reconstruction on $\mathcal{H}_{A}^{\otimes n}$ and the original reference
$\mathcal{H}_{{B_{R}}}^{\otimes n}$ is close to the effect of the $n$-product
posterior channel acting on the reference of the non-product reconstruction
sequence. Our objective is to characterize the set of all achievable rates
using single-letter quantum information quantities.

######  Theorem 1 (Lossy Quantum Compression Theorem).

For a $(\rho^{B},\mathcal{H}_{A},\mathcal{N}_{W})$ quantum source coding
setup, a rate $R$ is achievable if and only if $S(\rho^{B},\mathcal{N}_{W})$
is non empty, and

$R\geq\min_{\rho^{{A_{R}}}\in\mathcal{S}(\rho^{B},\mathcal{N}_{W})}I^{+}_{c}(\mathcal{N}_{W},\rho^{{A_{R}}}),$

where for any real $x$,
$x^{+}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\max(x,0)$
and

$\mathcal{S}(\rho^{B},\mathcal{N}_{W})\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\\{\rho^{{A_{R}}}\in\mathcal{D}(\mathcal{H}_{{A_{R}}}):\mathcal{N}_{W}(\rho^{{A_{R}}})=\rho^{{B_{R}}}\\}.$

###### Proof.

A proof of the achievability is provided in Sections V-A and V-B, and a proof
of converse is provided in Section V-C. ∎

######  Remark 2 (Covering of Subspaces).

The asymptotic rate obtained in the statement of Theorem 1 can be interpreted
using a subspace covering argument. Let us assume we are given a source
$\rho^{B}$ and a CPTP map $\mathcal{N}_{W}$ whose coherent information is
positive for all $\rho^{{A_{R}}}\in\mathcal{S}(\rho^{B},\mathcal{N}_{W}).$ Let
$W:\mathcal{H}_{{A_{R}}}\rightarrow\mathcal{H}_{{B_{R}}}\otimes\mathcal{H}_{E}$
be a Stinespring’s dilation of $\mathcal{N}_{W}$. This implies
$I_{c}(\mathcal{N}_{W},\rho^{{A_{R}}})=S({B_{R}})_{\sigma}-S(E)_{\sigma}$,
where
$\sigma^{{B_{R}}E}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}W\rho^{{A_{R}}}W^{\dagger},$
for $\rho^{{A_{R}}}\in\mathcal{S}(\rho^{B},\mathcal{N}_{W})$. We know that the
$n$-product source state ${\rho^{B}}^{\otimes n}$ can be compressed using
Schumacher compression to a subspace of normalized logarithmic dimension
$S(B_{R})_{\sigma}$ with high probability. In order to further reduce the
rate, we use the posterior reference map of $W$ with respect to
$\rho^{{B_{R}}}$ such that its action on the source produces the state
$\rho^{A}$. Each basis vector in the reconstruction space can be thought of as
covering a subspace of normalized logarithmic dimension of $S(E)_{\sigma}$ in
the reference space. Therefore, one needs a rate of coherent information
(which is the difference of the two entropies) to cover the entire source
space with high probability. A similar observation was made for the quantum
channel coding problem in [30].

######  Remark 3 (Comparison with Schumacher’s lossless compression).

Schumuacher’s compression [1] requires
$\lim_{n\rightarrow\infty}\|\omega^{B_{R}^{n}A^{n}}-\Psi_{\rho}^{B_{R}^{n}B^{n}}\|=0$.
In the current formulation, if one chooses the identity map as the posterior
reference map, i.e., $\mathcal{N}_{W}=I_{{A_{R}}\rightarrow{B_{R}}}$, we
require the condition
$\lim_{n\rightarrow\infty}\|\omega^{B_{R}^{n}A^{n}}-\Psi_{\omega}^{A_{R}^{n}A^{n}}\|=0$.
Using Lemma 2, monotonicity of the trace norm, and the triangle inequality,
one can show that the two conditions are equivalent. Subsequently, both
formulations yield the same asymptotic performance limit of von Neumann
entropy. Observe that the standard source coding formulation using the average
single-letter distortion criterion at zero distortion level is not equivalent
to Schumacher’s compression.

######  Remark 4 (Comparison with average single-letter rate distortion).

Given any sequence of $(n,\Theta)$ lossy quantum compression protocol for a
quantum source coding setup $(\rho^{B},\mathcal{H}_{A},\mathcal{N}_{W})$ that
achieves the optimality in Theorem 1, we observe that the following is true.
Let
$\omega^{B_{R}^{n}A^{n}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(I\otimes\mathcal{N}_{\mathcal{D}}^{(n)})(I\otimes\mathcal{N}_{\mathcal{E}}^{(n)})(\Psi_{\rho}^{B_{R}^{n}B^{n}})$
be the induced state of the $n$-letter reference and the reconstruction by the
protocol. Since the protocol satisfies (2), by monotonicity of trace distance,
we obtain

$\lim_{n\rightarrow\infty}\|\omega^{{B_{R}}_{i}A_{i}}-(\mathcal{N}_{W}\otimes
I_{A})(\Psi_{\omega}^{{A_{R}}_{i}A_{i}})\|_{1}=0,\ \ \forall\;1\leq i\leq n,$

where
$\Psi_{\omega}^{{A_{R}}_{i}A_{i}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\Tr_{A^{n\backslash
i}A^{n\backslash i}_{R}}[\Psi_{\omega}^{A_{R}^{n}A^{n}}].$ It is worth noting
that $\Psi_{\omega}^{{A_{R}}_{i}A_{i}}$ is not necessarily a pure state.
Moreover, this does not necessarily provide any guarantee on the average
single-letter distortion between the reference and the reconstruction as
considered in the standard formulation of the problem [7, Lemma 1], where a
single-letter purification of the source is taken into account. From this
perspective, the current formulation is more “optimistic” in terms of
measuring the quality of the reconstruction.

######  Remark 5 (Comparison with Entanglement Assistance).

We note that

$I_{c}(\mathcal{N}_{W},\rho^{{A_{R}}})=\frac{1}{2}\left[{I({B_{R}};A)_{\sigma}-I(A;E)_{\sigma}}\right]\leq\frac{1}{2}I({B_{R}};A)_{\sigma},$

where
$\sigma^{{B_{R}}AE}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(I\otimes
V)\Psi_{\rho}^{{B_{R}}B}(I\otimes V)^{\dagger}$, and
$V:\mathcal{H}_{B}\rightarrow\mathcal{H}_{A}\otimes\mathcal{H}_{E}$ is a
posterior reference map of $W$ with respect of $\rho^{{B_{R}}}$. It was shown
in [7] that $\frac{1}{2}I({B_{R}};A)_{\sigma}$ characterizes the asymptotic
performance limit for the rate-distortion problem (with a local single-letter
distortion function) with unlimited entanglement assistance. Hence, this also
provides a lower bound on the asymptotic performance limit for the
corresponding problem in the unassisted case. Fortunately, this does not lead
to any contradiction, as the current formulation differs from the former by
being more optimistic.

###  III-B Lossy Quantum-Classical Source Coding

This section provides the main results regarding the quantum-to-classical (QC)
setup. A memoryless quantum information source is characterized by
${\rho^{B}}\in\mathcal{D}(\mathcal{H}_{B})$.

######  Definition 5 (QC Source Coding Setup).

A QC source coding setup is characterized by a triple
$({\rho^{B}},\mathsf{X},\mathcal{W})$ where ${\rho^{B}}$ is the source density
operator acting on $\mathcal{H}_{B}$, $\mathsf{X}$ is the reconstruction
alphabet, and $\mathcal{W}:\mathsf{X}\rightarrow\mathcal{D}(\mathcal{H}_{B})$
is a single-letter posterior classical-quantum (CQ) channel.

######  Definition 6 (Lossy QC Compression Protocol).

For a given source density operator ${\rho^{B}}$ and the reconstruction
alphabet $\mathsf{X}$, an $(n,\Theta)$ lossy QC compression protocol is
characterized by $(i)$ a POVM
$\Gamma^{(n)}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\\{A_{m}\\}_{m=1}^{\Theta}$
and $(ii)$ a decoding map
$f:\left\\{{1,2,\cdots,\Theta}\right\\}\rightarrow\mathsf{X}^{n}$, as shown in
Figure 3.

Figure 3: Illustration of Quantum-Classical Lossy Source Compression Protocol.

######  Definition 7 (Achievability).

For a given QC source coding setup $({\rho^{B}},\mathsf{X},\mathcal{W})$, a
rate $R$ is said to be achievable if for all $\epsilon>0$ and all sufficiently
large $n$, there exists an $(n,\Theta)$ QC lossy compression protocol such
that $\frac{1}{n}\log\Theta\leq R+\epsilon$, and
$\Xi(\Gamma^{(n)},f)\leq\epsilon$, where

$\Xi(\Gamma^{(n)},f)\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{x^{n}}\norm{\sqrt{{\rho^{B}}^{\otimes
n}}A_{f^{-1}(x^{n})}\sqrt{{\rho^{B}}^{\otimes
n}}-\Tr(A_{f^{-1}(x^{n})}{\rho^{B}}^{\otimes
n})\bigotimes_{i=1}^{n}\mathcal{W}_{x_{i}}}_{1}.$ (3)

In other words, the post-measurement reference state should look like
$n$-tensored posterior CQ channel $\mathcal{W}^{\otimes n}$. Our objective is
to characterize the set of all achievable rates using single-letter quantum
information quantities.

######  Theorem 2 (Lossy QC Source Compression Theorem).

For a $({\rho^{B}},\mathsf{X},\mathcal{W})$ QC source coding setup, a rate $R$
is achievable if and only if $\mathcal{A}({\rho^{B}},\mathcal{W})$ is non-
empty, and

$R\geq\min_{P_{X}\in\mathcal{A}({\rho^{B}},\mathcal{W})}I(X;B_{{R}})_{\sigma},$

where the quantum mutual information is computed with respect to the
classical-quantum state,

$\sigma^{XB_{{R}}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{x}P_{X}(x)\outerproduct{x}{x}^{X}\otimes\mathcal{W}_{x},$

$\mathcal{A}$ is the set of reconstruction distributions defined as

$\mathcal{A}({\rho^{B}},\mathcal{W})\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\\{P_{X}\in\mathcal{P}(\mathsf{X}):\sum_{x}P_{X}(x)\mathcal{W}_{x}={\rho^{B}}\\},$

and $\\{\ket{x}\\}_{\left\\{{x\in\mathsf{X}}\right\\}}$ is an orthonormal
basis for the Hilbert space $\mathcal{H}_{X}$ with
$\dim{(\mathcal{H}_{X})}=|\mathsf{X}|$.

###### Proof.

A proof of the achievability is provided in Section VI-A, and a converse proof
is provided in Section VI-B. ∎

###  III-C Lossy Classical Source Coding

Consider a stationary discrete memoryless source (DMS) $X$ characterized by a
source distribution ${P}_{X}$ over a finite alphabet $\mathsf{X}$.

######  Definition 8 (Source Coding Setup).

A source coding setup is characterized by a triple
$({P}_{X},\hat{\mathsf{X}},{W}_{X|\hat{X}})$ where ${P}_{X}$ is the source
distribution over a finite alphabet $\mathsf{X}$, $\hat{\mathsf{X}}$ is the
reconstruction alphabet, and
${W}_{X|\hat{X}}:\hat{\mathsf{X}}\rightarrow\mathsf{X}$ is the posterior
(backward) channel, i.e., the single-letter conditional distribution of source
given the reconstruction.

We use ${P}_{X}^{n}$ and ${W}_{X|\hat{X}}^{n}$ to denote IID distributions,
i.e.,

${P}_{X}^{n}(x^{n})=\prod_{i=1}^{n}{P}_{X}(x_{i})\mbox{ and
}{W}_{X|\hat{X}}^{n}(x^{n}|\hat{x}^{n})=\prod_{i=1}^{n}{W}_{X|\hat{X}}(x_{i}|\hat{x}_{i}).$

######  Definition 9 (Lossy Source Compression Protocol).

For a given source distribution ${P}_{X}$ and reconstruction alphabet
$\hat{\mathsf{X}}$, an $(n,\Theta)$ lossy source compression protocol consists
of $(i)$ a randomized encoding map
$\mathcal{E}^{(n)}:\mathsf{X}^{n}\longrightarrow[\Theta]$ and $(ii)$ a
randomized decoding map
$\mathcal{D}^{(n)}:[\Theta]\longrightarrow\hat{\mathsf{X}}^{n}$, as shown in
Figure 4.

Figure 4: Illustration of Lossy Classical Source Compression Protocol.

######  Definition 10 (Achievability).

Given a source coding setup $({P}_{X},\hat{\mathsf{X}},{W}_{X|\hat{X}})$, a
rate $R$ is said to be achievable if for all $\epsilon>0$ and all sufficiently
large $n$, there exists an $(n,\Theta)$ lossy source compression protocol such
that $\frac{1}{n}\log\Theta\leq R+\epsilon$, and
$\Xi(\mathcal{E}^{(n)},\mathcal{D}^{(n)})\leq\epsilon$, where

$\Xi(\mathcal{E}^{(n)},\mathcal{D}^{(n)})\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\frac{1}{2}\sum_{x^{n}\hat{x}^{n}}\left|{P_{X^{n}\hat{X}^{n}}(x^{n},\hat{x}^{n})-P_{\hat{X}^{n}}(\hat{x}^{n})\prod_{i=1}^{n}{W}_{X|\hat{X}}(x_{i}|\hat{x}_{i})}\right|,$
(4)

and

$P_{X^{n}\hat{X}^{n}}(x^{n},\hat{x}^{n})={{P}_{X}^{n}(x^{n})}\sum_{m\in[\Theta]}\mathcal{E}^{(n)}(m|x^{n})\mathcal{D}^{(n)}(\hat{x}^{n}|m),\text{
for all }(x^{n},\hat{x}^{n})\in\mathsf{X}^{n}\times\hat{\mathsf{X}}^{n},$

is the system-induced distribution, and $P_{\hat{X}^{n}}{W}_{X|\hat{X}}^{n}$
is the approximating distribution.

In other words, the posterior distribution of the source given the
reconstruction should look like $n$-product posterior channel
${W}_{X|\hat{X}}^{n}$. Our objective is to characterize the set of all
achievable rates using single-letter information quantities.

######  Theorem 3 (Lossy Source Compression Theorem).

For a $({P}_{X},\hat{\mathsf{X}},{W}_{X|\hat{X}})$ source coding setup, a rate
$R$ is said to be achievable if and only if
$\mathcal{A}({P}_{X},{W}_{X|\hat{X}})$ is non-empty, and

$R\geq\min_{{P}_{\hat{X}}\in\mathcal{A}({P}_{X},{W}_{X|\hat{X}})}I(X;\hat{X}),$
(5)

where $\mathcal{A}$ is the set of reconstruction distributions defined as

$\mathcal{A}({P}_{X},{W}_{X|\hat{X}})\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\\{{P}_{\hat{X}}\in\mathcal{P}(\hat{\mathsf{X}}):\sum_{\hat{x}}{P}_{\hat{X}}(\hat{x}){W}_{X|\hat{X}}(x|\hat{x})={P}_{X}(x),\text{
for all }x\in\mathsf{X}\\}.$

###### Proof.

A proof of the achievability is provided in Section VII-A, and a converse
proof is provided in VII-B. ∎

######  Remark 6 (Comparison with Shannon’s noiseless source compression).

Noiseless source compression requires
$\lim_{n\rightarrow\infty}P(X^{n}\neq\hat{X}^{n})=0$. In the current
formulation, if one chooses the identity posterior channel, i.e.,
${W}_{X|\hat{X}}(x|\hat{x})=\mathbbm{1}_{\\{x=\hat{x}\\}}$, for all
$x\in\mathsf{X}$, $\hat{x}\in\hat{\mathsf{X}}$, we require
$\lim_{n\rightarrow\infty}\norm{P_{X^{n}\hat{X}^{n}}-P_{\hat{X}^{n}}{W}_{X|\hat{X}}^{n}}_{\text{TV}}=0$.
One can easily see that the two conditions are equivalent, and both
formulations yield the same asymptotic performance limit of Shannon’s entropy.
However, the standard source coding formulation using the average single-
letter distortion criterion at zero distortion level is not equivalent to
noiseless source compression.

##  IV Illustrative Examples

######  Example 1 (Quantum Source Coding using Bit-Flip Channel).

In this example, we analyze the performance of a lossy quantum compression
protocol corresponding to a quantum source coding setup
$(\rho^{B},\mathcal{H}_{A},\mathcal{N}_{W}),$ where $\rho^{B}$ is chosen as
the maximally mixed state ($\rho^{B}=I_{B}/2$), and
$\mathcal{N}_{W}{:\mathcal{H}_{{A_{R}}}\rightarrow\mathcal{H}_{{B_{R}}}}$ is
specified as a bit-flip channel. An isometry
$W{:\mathcal{H}_{{A_{R}}}\rightarrow\mathcal{H}_{{B_{R}}}\otimes\mathcal{H}_{E}}$
for $\mathcal{N}_{W}$ can be specified as

$W=\sqrt{1-p}I\otimes\ket{0}^{E}+\sqrt{p}X\otimes\ket{1}^{E},$

where $\mathcal{N}_{W}(\rho^{{A_{R}}})=\Tr_{E}(W\rho^{{A_{R}}}W^{\dagger})$
for all $p\in(0,1/2)$. Note that the canonical purification
$\ket{\psi_{\rho}}^{{B_{R}}B}$ of $\rho^{B}$ is given by

$\displaystyle\ket{\psi_{\rho}}^{{B_{R}}B}=\frac{1}{\sqrt{2}}\left({\ket{0}^{{B_{R}}}\ket{0}^{B}+\ket{1}^{{B_{R}}}\ket{1}^{B}}\right),$
(6)

where
$\ket{0}^{{B_{R}}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(I\otimes\bra{0}^{B})\ket{\Gamma}^{{B_{R}}B}$.
This implies, $\rho^{{B_{R}}}=I_{{B_{R}}}/2$. To compute the asymptotic
performance of the protocol for this source coding setup, as characterized by
Theorem 1, we first need to identify a $\rho^{{A_{R}}}$ such that
$\mathcal{N}_{W}(\rho^{{A_{R}}})=\rho^{{B_{R}}}$. A simple computation reveals
$\mathcal{S}(\rho^{B},\mathcal{N}_{W})=\\{{I_{{A_{R}}}/2}\\}$. This gives

$\min_{\rho^{{A_{R}}}\in\mathcal{S}(\rho^{B},\mathcal{N}_{W})}{I^{+}_{c}(\mathcal{N}_{W},\rho^{{A_{R}}})}=I_{c}(\mathcal{N}_{W},{I_{{A_{R}}}}/{2})=S({B_{R}})_{\sigma}-S(E)_{\sigma},$

where $\sigma^{{B_{R}}E}=W\rho^{{A_{R}}}W^{\dagger}$. Note that
$\sigma^{{B_{R}}}=I_{{B_{R}}}/2$ and
$\sigma^{E}=(1-p)\outerproduct{0}{0}^{E}+p\outerproduct{1}{1}^{E},$ which
gives $I_{c}(\mathcal{N}_{W},{I_{{A_{R}}}}/{2})=1-h_{b}(p)$, where
$h_{b}(p)\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}-p\log(p)-(1-p)\log(1-p).$
Therefore, a maximally mixed source can be compressed at a rate $1-h_{b}(p)$
while satisfying the error criterion as defined in (2).

######  Example 2 (Quantum Source Coding using Depolarizing Channel).

In this example, we study the performance of another candidate channel, namely
a depolarising channel. We again proceed with the objective of compressing a
maximally mixed state $\rho^{{B_{R}}}=\frac{I_{{B_{R}}}}{2}$, with
$\mathcal{N}_{W}$ defined as

$\displaystyle\mathcal{N}_{W}(\rho^{{A_{R}}})=\left(1-\frac{3p}{4}\right)\rho^{{A_{R}}}+\frac{p}{4}(X\rho^{{A_{R}}}X^{\dagger}+Y\rho^{{A_{R}}}Y^{\dagger}+Z\rho^{{A_{R}}}Z^{\dagger}).$

for some $p\in[0,1].$ A simple calculation to satisfy
$\mathcal{N}_{W}(\rho^{{A_{R}}})=\rho^{{B_{R}}}=\frac{I_{{B_{R}}}}{2}$ reveals
$\mathcal{S}(\rho^{B},\mathcal{N}_{W})=\\{{I_{{A_{R}}}/2}\\}$, for all
$p\in(0,1)$. Analogous to the above example, finding an isometric extension of
$\mathcal{N}_{W}$ gives

$\displaystyle\min_{\rho^{{A_{R}}}\in\mathcal{S}(\rho^{B},\mathcal{N}_{W})}I^{+}_{c}(\mathcal{N}_{W},\rho^{{A_{R}}})=I^{+}_{c}(\mathcal{N}_{W},{I_{{A_{R}}}}/{2})=\max\big{\\{}0,1-h_{b}({3p}/{4})-\frac{3p}{4}\log(3)\big{\\}}.$

######  Example 3 (Hamming codes for quantum source compression).

In this example, we look at how Hamming codes perform when evaluated using the
standard single-letter (local) entanglement fidelity criterion. Hamming codes
are perfect codes, and achieve the Delsarte upper bound on the covering radius
[56]. Again, let $\rho^{B}=\frac{I_{B}}{2}$. Let a maximally entangled
bipartite state $\ket{\psi_{m}}^{{B_{R}}B}$, defined as

$\displaystyle\ket{\psi_{m}}^{{B_{R}}B}=\frac{1}{\sqrt{2}}\left(\ket{00}^{{B_{R}}B}+\ket{11}^{{B_{R}}B}\right),$
(7)

be the purification of $\rho^{B}$. Let $\mathbb{F}_{2}$ denote a binary finite
field, and let $G\in\mathbb{F}_{2}^{k\times n}$ be the generator matrix of a
Hamming code. To encode $\rho^{B}$, we appeal to the duality perspective, and
use the decoder of a Hamming code. Then the encoding is defined as
${\mathcal{E}(x^{n})}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\mbox{argmin}_{u^{k}}\\{w_{H}({u^{k}\;G\oplus
x^{n}})\\},$ for all $x^{n}\in\mathbb{F}_{2}^{n},$ where $w_{H}$ denotes the
Hamming weight. Similarly, the decoder can be described as mapping
$\mathcal{D}\circ\mathcal{E}((x^{n}))=\mathcal{E}(x^{n})G$. We describe this
encoding as an isometric action $V_{H}:\mathcal{H}_{B}^{\otimes
n}\rightarrow\mathcal{H}_{A}^{\otimes n}\otimes\mathcal{H}_{E}^{\otimes n}$
taking the basis $\ket{x^{n}}^{B^{n}}$ to a vector
$\ket{\mathcal{E}(x^{n})}^{A^{n}}\otimes\ket{x^{n}\oplus\mathcal{E}(x^{n})}^{E^{n}}\in\mathcal{H}_{A}^{\otimes
n}\otimes\mathcal{H}_{E}^{\otimes n}$, where the subsystem
$\mathcal{H}_{A}^{\otimes n}$ stores the reconstruction and
$\mathcal{H}_{E}^{\otimes n}$ is eventually traced out, and $\mathcal{H}_{A}$
is assumed to be an isomorphic copy of $\mathcal{H}_{B}$. This implies that
the encoded state can be characterized as

$\displaystyle\rho^{B_{R}^{n}A^{n}}$
$\displaystyle=\Tr_{E^{n}}\left\\{{V_{H}\outerproduct{\psi_{m}^{\otimes
n}}{\psi_{m}^{\otimes n}}^{B_{R}^{n}B^{n}}V_{H}^{\dagger}}\right\\}.$

Using

$\displaystyle V_{H}\ket{\psi_{m}^{\otimes n}}^{B_{R}^{n}B^{n}}$
$\displaystyle=\frac{1}{\sqrt{2^{n}}}\sum_{x^{n}}\ket{x^{n}}_{B_{R}^{n}}\ket{\mathcal{E}(x^{n})}_{A^{n}}\ket{x^{n}\oplus\mathcal{E}(x^{n})}_{E^{n}}$
$\displaystyle=\frac{1}{\sqrt{2^{n}}}\sum_{c^{n}\in\mathcal{C}}\;\sum_{e^{n}\in\mathbb{F}_{2}^{n}:w_{H}(e^{n})\leq
1}\ket{c^{n}\oplus e^{n}}_{B_{R}^{n}}\ket{c^{n}}_{A^{n}}\ket{e^{n}}_{E^{n}},$

we can simplify $\rho^{B_{R}^{n}A^{n}}$ as

$\displaystyle\rho^{B_{R}^{n}A^{n}}=\frac{1}{2^{n}}\sum_{c^{n},c^{\prime
n},e^{n}}\ket{c^{n}\oplus e^{n}}\langle{c^{\prime n}\oplus
e^{n}}|\otimes\ket{c^{n}}\langle{c^{\prime n}}|,$ (8)

where $\mathcal{C}$ denotes the set of codewords of the Hamming code. To
compute the single-letter entanglement fidelity, we compute

$\displaystyle\rho^{B_{R_{i}}A_{i}}=\Tr_{B_{R}^{n\backslash i}A^{n\backslash
i}}\left\\{{\rho^{B_{R}^{n}A^{n}}}\right\\}=\frac{1}{2^{n}}\sum_{c^{n},e^{n}}\ket{c_{i}\oplus
e_{i}}\bra{c_{i}\oplus e_{i}}\otimes\ket{c_{i}}\bra{c_{i}},$ (9)

where tracing is performed on all the subsystems except corresponding to
${B_{R}^{n\backslash i}A^{n\backslash i}}$, and the second equality follows
from using the fact that minimum Hamming distance of any Hamming code is
three. This gives,

$\displaystyle\langle{\psi}_{m}^{{B_{R}}B}|\rho^{B_{R_{i}}A_{i}}|{\psi}_{m}^{{B_{R}}B}\rangle=\frac{1}{2}\frac{1}{2^{n}}\sum_{c^{n}e^{n}}\left[{\mathbbm{1}_{\left\\{{c_{i}\oplus
e_{i}=0,c_{i}=0}\right\\}}+\mathbbm{1}_{\left\\{{c_{i}\oplus
e_{i}=1,c_{i}=1}\right\\}}}\right]=\frac{1}{2^{n+1}}\sum_{c^{n}e^{n}}\mathbbm{1}_{\left\\{{e_{i}=0}\right\\}}.$
(10)

Therefore,

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\langle{\psi_{m}}^{{B_{R}}B}|\rho^{B_{R_{i}}A_{i}}|{\psi_{m}}^{{B_{R}}B}\rangle$
$\displaystyle=\frac{1}{2^{n+1}n}\sum_{c^{n}e^{n}}\sum_{i=1}^{n}\mathbbm{1}_{\left\\{{e_{i}=0}\right\\}}=\frac{|\mathcal{C}|n^{2}}{2^{n+1}n}=\frac{n}{2\cdot
2^{n-k}}.$ (11)

We know that for Hamming codes $k=2^{r}-r-1$ and $n=2^{r}-1$, which simplifies
as

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\langle{\psi_{m}}^{{B_{R}}B}|\rho^{B_{R_{i}}A_{i}}|{\psi_{m}}^{{B_{R}}B}\rangle$
$\displaystyle=\frac{2^{r}-1}{2\cdot 2^{r}},$ (12)

and goes to half as $r$ goes to infinity. Note that $r\rightarrow\infty$
serves as both a demonstration of the code’s asymptotic performance and the
condition for the rate $k/n$ to reach unity. This results in a discontinuous
asymptotic performance, since at rate exactly one, trivial identity encoding
can be used to achieve the average single-letter fidelity of unity. Further,
note that $S(E^{n})=\log(n+1)=r$. Hence the normalized amount of qubits that
is dissipated, given by $\frac{S(E^{n})}{n}$, approaches zero as
$r\rightarrow\infty$, indicating that there is significant entanglement
between the reconstruction and the reference.

As was demonstrated in Example 1, it is possible to compress a maximally mixed
source in a continuous fashion, when the error is measured in accordance with
the suggested definition in (2),

######  Example 4.

(Lossy QC Source Coding for Binary Quantum Source with Binary Symmetric
Posterior CQ Channel) We develop an example similar to that studied in [11].
Here we analyze the performance of the lossy QC source compression protocol
corresponding to a lossy source coding setup
$({\rho^{B}},{\mathsf{X}},\mathcal{W})$. The quantum source ${\rho^{B}}$
generates the state $\ket{+}$ and $\ket{0}$ with probability $p$ and $(1-p)$,
respectively, where $p\in[0,1/2]$, so the source density operator can be
written as

${\rho^{B}}=p\outerproduct{+}{+}+(1-p)\outerproduct{0}{0},$

the reconstruction set ${\mathsf{X}}=\left\\{{0,1}\right\\}$, and the
posterior CQ channel $\mathcal{W}_{x}=(1-q)\ \omega_{x}+q\ \omega_{\bar{x}}$,
where

$\omega_{0}=\frac{1}{4}\outerproduct{+}{+}+\frac{3}{4}\outerproduct{0}{0},\quad\omega_{1}=\frac{3}{4}\outerproduct{+}{+}+\frac{1}{4}\outerproduct{0}{0},$

$q\in[0,1/2]$, and
$\bar{x}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}x\oplus
1$. Toward identifying the set $\mathcal{A}$, we assume ${P}_{X}(0)=r$, which
characterizes the set $\mathcal{A}$, and solve the following

${\rho^{B}}=r\mathcal{W}_{0}+(1-r)\mathcal{W}_{1},\quad 0\leq r\leq 1.$ (13)

This gives,

$\mathcal{A}({\rho^{B}},\mathcal{W})=\begin{cases}\left\\{{\frac{1}{2}+\frac{1-2p}{1-2q}}\right\\}&\mbox{
if }0\leq q\leq
2\min\left\\{{\left({\frac{3}{4}-p}\right),\left({p-\frac{1}{4}}\right)}\right\\},q<\frac{1}{2}\\\
{[0,1]}&\mbox{ if }q=p=0.5\\\ \phi&\mbox{ otherwise,}\end{cases}$

where $\phi$ denotes the empty set. We now compute the asymptotic performance
described in Theorem 2. For the above source coding setup, we have

$\displaystyle
I(X;B_{{R}})_{\sigma}=S({\rho^{B}})-rS(\mathcal{W}_{0})-(1-r)S(\mathcal{W}_{1}).$
(14)

where

$\sigma^{XB_{{R}}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}r\outerproduct{0}{0}\otimes\mathcal{W}_{0}+(1-r)\outerproduct{1}{1}\otimes\mathcal{W}_{1}.$

Figure 5 shows the QC lossy source compression rate curve for the range of
values of the parameter $q$ and source ${\rho^{B}}$ with $p=0.4$ and $0.5$.
Note that the curve decreases monotonically with $q$, as expected.

Figure 5: Example for Lossy QC Source Coding for Binary Quantum Source with
Binary Symmetric Posterior CQ Channel.

######  Example 5.

(Lossy Classical Source Coding for Binary Source with Binary Symmetric Channel
(BSC) as Posterior Channel) In this example, we analyze the performance of the
lossy source compression protocol corresponding to a lossy source coding setup
$({P}_{X},\hat{\mathsf{X}},{W}_{X|\hat{X}})$, where
${P}_{X}\sim\text{Bernoulli}(p)$, $\hat{\mathsf{X}}=\left\\{{0,1}\right\\}$,
${W}_{X|\hat{X}}\sim\text{BSC}(q)$, and $p,q\in[0,1/2]$. Toward identifying
the set $\mathcal{A}$, we assume ${P}_{\hat{X}}(0)=r$, which characterizes the
set $\mathcal{A}$, and solve the following system of linear equations:

$p=r(1-q)+(1-r)q\quad\mbox{ and }\quad(1-p)=rq+(1-r)(1-q).$ (15)

This gives,

$\mathcal{A}({P}_{X},{W}_{X|\hat{X}})=\begin{cases}\left\\{{\frac{p-q}{1-2q}}\right\\}&\mbox{
if }0\leq q\leq\min\\{p,(1-p)\\},q<\frac{1}{2}\\\ [0,1]&\mbox{ if }q=p=0.5\\\
\phi&\mbox{ otherwise,}\end{cases}$

where $\phi$ denotes the empty set. We now compute the asymptotic performance
described in Theorem 3. For the above source coding setup, we have

$\displaystyle I(X;\hat{X})=H(X)-H(X|\hat{X})=h_{b}(p)-h_{b}(q).$ (16)

Moreover, observe that the rate in (16) is identical to the rate-distortion
function for a Bernoulli($p$) source with Hamming distortion criterion for
$D<p$ [57, Theorem 10.3.1].

##  V Proof of Theorem 1

###  V-A Achievability Proof Overview

We provide a brief overview of the achievability proof before formally
presenting one. The proof we present here is inspired by Devetak’s work in
[32] for the quantum channel communication problem (also detailed in [50,
Chapter 24]). An integral component of that work is the decomposition of
coherent information as the difference of two Holevo information quantities.
We intend to perform a similar decomposition, but from the perspective of the
given map $\mathcal{N}_{W}$. Toward this, for the given source $\rho^{B}$, we
first search for a $\rho^{{A_{R}}}\in\mathcal{D}(\mathcal{H}_{{A_{R}}})$,
satisfying $\mathcal{N}_{W}(\rho^{{A_{R}}})=\rho^{{B_{R}}}.$ Once found, using
the spectral decomposition, we expand $\rho^{{A_{R}}}$ as
$\rho^{{A_{R}}}=\sum_{a\in\mathcal{A}}\lambda_{a}^{A}\outerproduct{a}{a}^{{A_{R}}}$,
for some finite set $\mathcal{A}$. Observe that since
$\outerproduct{a}{a}^{{A_{R}}}$ is pure,
$S(\mathcal{N}_{W}(\outerproduct{a}{a}^{{A_{R}}}))=S(\mathcal{N}^{c}_{W}(\outerproduct{a}{a}^{{A_{R}}}))$,
where $\mathcal{N}^{c}_{W}:\mathcal{H}_{{A_{R}}}\rightarrow\mathcal{H}_{E}$ is
a complementary CPTP map of $\mathcal{N}_{W}$, defined using the Stinespring’s
dilation
$W:\mathcal{H}_{{A_{R}}}\rightarrow\mathcal{H}_{{B_{R}}}\otimes\mathcal{H}_{E}$
corresponding to $\mathcal{N}_{W}$. This also means that

$\sum_{a\in\mathcal{A}}\lambda_{a}^{A}S(\mathcal{N}_{W}(\outerproduct{a}{a}^{{A_{R}}}))=\sum_{a\in\mathcal{A}}\lambda_{a}^{A}S(\mathcal{N}^{c}_{W}(\outerproduct{a}{a}^{{A_{R}}})).$

Furthermore, from the linearity of CPTP maps, we see

$\sum_{a\in\mathcal{A}}\lambda_{a}^{A}\mathcal{N}_{W}(\outerproduct{a}{a}^{{A_{R}}})=\mathcal{N}_{W}(\rho^{{A_{R}}})\mbox{\quad
and\quad}\sum_{a\in\mathcal{A}}\lambda_{a}^{A}\mathcal{N}^{c}_{W}(\outerproduct{a}{a}^{{A_{R}}})=\mathcal{N}^{c}_{W}(\rho^{{A_{R}}}).$

This implies, we can rewrite $I_{c}(\mathcal{N}_{W},\rho^{{A_{R}}})$ as

$\displaystyle I_{c}$
$\displaystyle(\mathcal{N},\rho^{{A_{R}}})=S(\mathcal{N}_{W}(\rho^{{A_{R}}}))-S(\mathcal{N}^{c}_{W}(\rho^{{A_{R}}}))$
$\displaystyle=\left[{S(\mathcal{N}_{W}(\rho^{{A_{R}}}))-\sum_{a\in\mathcal{A}}S(\lambda_{a}^{A}\mathcal{N}_{W}(\outerproduct{a}{a}^{{A_{R}}}))}\right]-\left[{S(\mathcal{N}^{c}_{W}(\rho^{{A_{R}}}))-\sum_{a\in\mathcal{A}}\lambda_{a}^{A}S(\mathcal{N}^{c}_{W}(\outerproduct{a}{a}^{{A_{R}}}))}\right]$
$\displaystyle=\chi\left({\left\\{{\lambda_{a}^{A},\mathcal{N}_{W}(\outerproduct{a}{a}^{{A_{R}}})}\right\\}}\right)-\chi\left({\left\\{{\lambda_{a}^{A},\mathcal{N}^{c}_{W}(\outerproduct{a}{a}^{{A_{R}}})}\right\\}}\right).$
(17)

Now our aim is to show the achievability of a rate equal to the above
difference. After obtaining a similar decomposition, Devetak achieved the
performance limit by applying a coherent version of the CQ packing lemma [50,
Chapter 16] followed by an application of the QC covering lemma [50, Chapter
17]. Inspired by this, and the duality connections between the two problems,
we achieve the difference obtained in (17). In particular, we start with the
objective of applying a coherent version of the QC covering lemma (or the
measurement compression result [27]). Toward this, as shown in Figure 1, we
first obtain a posterior reference map $V$ corresponding to the isometry $W$.
Then we identify the action of $V$ on the state $\rho^{B}$ as a coherent
quantum measurement. Now, using the approximating POVMs constructed in [27],
we perform a coherent covering that allows us to compress the obtained
measurement, and in turn the output of $V$, at rate given by the first Holevo
information. The compression is performed while _faithfully simulating_ the
action of $V$, giving a reconstruction satisfying the error criterion (as in
(2)). This procedure is delineated in Step 1.1 where an encoder is constructed
to perform coherent covering and in Step 2.1 where the effect of covering is
analyzed, and a rate corresponding to the first Holevo information is
achieved.

To get the needed coherent information, the rate corresponding to the second
Holevo information must be further decreased. This entails diffusing more data
or qubits into the environment (partial tracing). However, as will be
demonstrated in the proof below, such an action would destroy quantum
correlations present in the source, possibly turning it into a classical
mixture. Therefore, before such partial tracing operation, in Step 1.2
(Section V-B) we construct a unitary operation that can condense the
information into fewer qubits in the form of entanglement, and thus allowing
for further decrease in the rate. This includes using the coherent post-
measurement state of the subsystem $E$ as side information available at the
encoder. The Step 2.2 of Section V-B details this procedure and achieves the
desired rate. Finally, an additional step (Step 2.3) is required to show the
intended closeness as required in (2).

Another intriguing aspect of the proof is the technique used for expurgation.
As clear from the definition of the protocol, it only allows unitary or
isometric operations, followed by partial tracing. When a code contains
repeated codewords, it can be difficult to guarantee this. An approach to
removing all repetitions is to perform expurgations. This is achieved by
finding a good code (satisfying all its constraints) while allowing a small
fraction of repeats and then expurgating just this fraction of the code.
However, if there are exponentially many constraints, it becomes challenging
to finding a good code. The exponentially many covering constraints in
Devetak’s problem have a doubly exponential decreasing probability of error,
which Devetak was able to take advantage of. In the current problem we instead
have exponentially many packing constraints which only have an exponential
decay. In order to combat this, we construct our proof to just require one
packing constraint: the average of all exponentially many packing constraints.
This enables us to find a good code and successfully expurgate it. We now
formally construct the arguments toward proving the statement of the theorem.

###  V-B Proof of Achievability

The proof is mainly composed of four parts. In the first part, we develop the
necessary single-letter ensembles required in the proof. In the next part, we
provide the random coding setup and the distributions on the ensembles with
which the codewords are generated. We also state here the constraints that a
good code must satisfy and argue the existence of one code with non-zero
probability. We further use an expurgation strategy to make all the codewords
distinct. In the third part, we construct a protocol by developing all the
actions of the encoder and the decoder and describing them as unitary (or
isometry) evolutions. Note that the only actions allowed by the protocol
(Definition 3) are quantum channels which can be described as unitary or
isometric evolutions followed by partial trace operations. In parallel, we
also provide the necessary lemmas needed for the next part. The last part
deals with analyzing the action of encoding and decoding operations on the
source $\rho^{B}$, and then bounding the trace distance as in Definition 3.

Toward this, fix two positive integers $M$ and $K$, and $\epsilon\in(0,1)$.
Let $\mathcal{M}$ and $\mathcal{K}$ denote the sets $[0,M-1]$ and $[0,K-1]$,
respectively. Given a quantum source coding setup
$(\rho^{B},\mathcal{H}_{A},\mathcal{N}_{W})$, let
$\ket{\psi_{\rho}}^{{B_{R}}B}$ be the canonical purification of $\rho^{B}$ and
$\rho^{{B_{R}}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\Tr_{B}\\{{\Psi_{\rho}}^{{B_{R}}B}\\}$.
Moreover, let $\mathcal{H}_{{A_{R}}}$ be the reference space associated with
$\mathcal{H}_{A}$. Now choose $\rho^{{A_{R}}}\in S(\rho^{B},\mathcal{N}_{W})$.
Let $\mathcal{H}_{E}$ denote the Hilbert space such that
$W:\mathcal{H}_{{A_{R}}}\rightarrow\mathcal{H}_{{B_{R}}}\otimes\mathcal{H}_{E}$
forms an isometric extension (or Stinespring’s dilation) of $\mathcal{N}_{W}$
according to [50, Definition 5.2.1] with
$\dim(\mathcal{H}_{E})\geq\dim(\mathcal{H}_{{B_{R}}})$. As shown in Figure 1,
define a posterior reference isometry of $W$ with respect to $\rho^{{B_{R}}}$
(according to Definition 1) as the isometry
$V:\mathcal{H}_{B}\rightarrow\mathcal{H}_{A}\otimes\mathcal{H}_{E}$ satisfying
$(W\otimes I_{A})\ket{\psi_{\rho}}^{{A_{R}}A}=(I_{{B_{R}}}\otimes
V)\ket{\psi_{\rho}}^{{B_{R}}B}$ where $\ket{\psi}^{{A_{R}}A}$ is the canonical
purification of $\rho^{{A_{R}}}$. Let
$\rho^{A}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\Tr_{{B_{R}}E}\\{(I\otimes
V){\Psi_{\rho}}^{{B_{R}}B}(I\otimes V)^{\dagger}\\}$.

####  V-B1 Defining the ensembles

In this section, we construct the single-letter ensembles corresponding to two
Holevo information quantities used in the decomposition of coherent
information discussed in Section V-A. We begin by using the definition of $W$
to obtain,

$\displaystyle(I_{{B_{R}}}\otimes V)\ket{\psi_{\rho}}^{{B_{R}}B}=(W\otimes
I_{A})\ket{\psi}^{{A_{R}}A}=\sum_{a\in\mathcal{A}}\sqrt{\lambda_{a}^{A}}W\ket{a}^{{A_{R}}}\otimes\ket{a}^{A},$
(18)

where we use
$\rho^{A}=\sum_{a\in\mathcal{A}}\lambda_{a}^{A}\outerproduct{a}{a}^{A}$ as its
spectral decomposition, and define
$\ket{a}^{{A_{R}}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(I_{{A_{R}}}\otimes\bra{a}^{A})\ket{\Gamma}^{{A_{R}}A}$
for $a\in\mathcal{A}$, for some finite set $\mathcal{A}$. This also gives,

$\displaystyle W\ket{a}^{{A_{R}}}=\frac{(\bra{a}^{A}\otimes
I_{{B_{R}}E})(I_{{B_{R}}}\otimes
V)\ket{\psi_{\rho}}^{{B_{R}}B}}{\sqrt{\lambda_{a}^{A}}}.$ (19)

Using the spectral decomposition of $\rho^{B}$ as
$\rho^{B}=\sum_{b\in\mathcal{B}}\lambda_{b}^{B}\outerproduct{b}{b}^{B}$, for
$b\in\mathcal{B}$ for some finite set $\mathcal{B}$, we can rewrite the action
of $V$ on $\rho^{B}$ as

$\displaystyle(I_{{B_{R}}}\otimes V)\ket{\psi_{\rho}}^{{B_{R}}B}$
$\displaystyle=\sum_{b\in\mathcal{B}}\sqrt{\lambda_{b}^{B}}\ket{b}^{{B_{R}}}\otimes
V\ket{b}^{B}$ $\displaystyle=\left(I_{{B_{R}}}\otimes
I_{E}\otimes\sum_{a\in\mathcal{A}}\outerproduct{a}{a}^{A}\right)\sum_{b\in\mathcal{B}}\sqrt{\lambda_{b}^{B}}\ket{b}^{{B_{R}}}\otimes
V\ket{b}^{B}$
$\displaystyle=\sum_{a\in\mathcal{A}}\sum_{b\in\mathcal{B}}\sqrt{\lambda_{b}^{B}}\ket{b}^{{B_{R}}}M_{a}\ket{b}^{B}\otimes\ket{a}^{A},$
(20)

where we define
$\ket{b}^{{B_{R}}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(I_{{B_{R}}}\otimes\bra{b}^{B})\ket{\Gamma}^{{B_{R}}B}$,
and $M_{a}:\mathcal{H}_{B}\rightarrow\mathcal{H}_{E}$ as

$\displaystyle
M_{a}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\big{(}I_{E}\otimes\bra{a}^{A}\big{)}V.$
(21)

By defining a POVM
$\Lambda\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\\{M_{a}^{\dagger}M_{a}\\}_{a\in\mathcal{A}}$,
we can identify a coherent measurement (isometry) $U_{\Lambda}$ corresponding
to $\Lambda$ with
$U_{\Lambda}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{a\in\mathcal{A}}M_{a}\otimes\ket{a}^{A}$,
and therefore express the action of $V$ as

$\displaystyle(I_{{B_{R}}}\otimes
V)\ket{\psi_{\rho}}^{{B_{R}}B}=(I_{{B_{R}}}\otimes
U_{\Lambda})\ket{\psi_{\rho}}^{{B_{R}}B}.$ (22)

Now our objective is to _faithfully simulate_ the action of the isometry (or
the coherent measurement) $U_{\Lambda}$ while using an exponentially smaller
subspace in $\mathcal{H}_{A^{n}}$. Equivalently, we intend to minimize the
amount of qubits needed to represent the quantum state in the Hilbert space
$\mathcal{H}_{A^{n}}$. Employing Schumacher’s compression [1], one can only
achieve a rate of Von-Neumann entropy while faithfully simulating
$U_{\Lambda}$. However, since $U_{\Lambda}$ is a coherent measurement, we
employ a coherent version of the measurement compression protocol [27] and
demonstrate a faithful simulation of the isometry while further decreasing the
resource requirement. In particular, an approximating coherent measurement
(henceforth referred to as the covering isometry) $U_{\mathcal{M}}$ is
constructed to faithfully simulate the action of $U_{\Lambda}$ while requiring
the rate equal to Holevo quantity corresponding to the canonical ensemble
$\\{\lambda_{a}^{A},\hat{\rho}_{a}^{{B_{R}}}\\}$, where

$\displaystyle\hat{\rho}_{a}^{{B_{R}}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\frac{\sqrt{\rho^{{B_{R}}}}(M_{a}^{\dagger}M_{a})^{T}\sqrt{\rho^{{B_{R}}}}}{\lambda_{a}^{A}}\mbox{\quad
and\quad}(M_{a}^{\dagger}M_{a})^{T}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{b,b^{\prime}}\ket{b}\langle{b^{\prime}}|^{{B_{R}}}\langle{b^{\prime}}|(M_{a}^{\dagger}M_{a})\ket{b}^{B}.$
(23)

Observe that using the definition of $M_{a}$ from (21), it follows

$\displaystyle\Tr{M^{\dagger}_{a^{\prime}}M_{a}\rho^{B}}$
$\displaystyle=\Tr{(I_{{B_{R}}E}\otimes\bra{a})V\outerproduct{\psi_{\rho}}{\psi_{\rho}}^{{B_{R}}B}V^{\dagger}(|{a^{\prime}}\rangle\otimes
I_{{B_{R}}E})}$
$\displaystyle=\sum_{b}\sqrt{\lambda_{a}^{A}\lambda_{a^{\prime}}^{A}}\Tr{\bra{b}W|{a^{\prime}}\rangle\bra{a}W^{\dagger}\ket{b}}$
$\displaystyle=\sum_{b}\sqrt{\lambda_{a}^{A}\lambda_{a^{\prime}}^{A}}\bra{a}W^{\dagger}\outerproduct{b}{b}W|{a^{\prime}}\rangle=\lambda_{a}^{A}\cdot\mathbbm{1}_{\\{a=a^{\prime}\\}},$
(24)

for all $a,a^{\prime}\in\mathcal{A}$, where the first equality uses the
definition of $M_{a}$, and the second follows from using the relation (18).
Using the simplification from (19), it is useful to note

$\displaystyle W\ket{a}^{{A_{R}}}=\frac{(I_{{B_{R}}}\otimes
M_{a})\ket{\psi_{\rho}}^{{B_{R}}B}}{\sqrt{\lambda_{a}^{A}}}.$ (25)

For the second Holevo information, we define the packing ensemble
$\\{\lambda_{a}^{E},\tau_{a}^{E}\\}$ as

$\displaystyle\tau_{a}^{E}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\frac{\Tr_{{B_{R}}}{(I_{{B_{R}}}\otimes
M_{a})\Psi_{\rho}^{{B_{R}}B}}(I_{{B_{R}}}\otimes
M_{a})^{\dagger}}{\lambda_{a}^{E}}=\frac{M_{a}\rho^{B}M_{a}^{\dagger}}{\lambda_{a}^{E}},\;\;\mbox{
and
}\;\;\lambda_{a}^{E}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\lambda_{a}^{A}.$
(26)

The discussion on how this ensemble is employed to reduce the rate follows in
the sequel.

####  V-B2 Random Coding and Expurgation

In this section, we construct the random coding argument, and simultaneously,
define all the conditions that pertain to the construction of a good random
code. Subsequently, we randomly generate one code that satisfies these
constraints. We then expurgate this code to ensure no repetitions are present.
Toward constructing an approximating coherent measurement $U_{\mathcal{M}}$,
randomly and independently select $|\mathcal{M}|\times|\mathcal{K}|$ sequences
$A^{n}(m,k)$ according to the following pruned distribution

$\displaystyle\mathbb{P}\left(A^{n}(m,k)=a^{n}\right)=\left\\{\begin{array}[]{cc}\dfrac{\lambda_{a^{n}}^{A}}{(1-\varepsilon)}&\mbox{for}\quad
a^{n}\in\mathcal{T}_{\delta}^{(n)}(A)\\\
0&\quad\mbox{otherwise},\end{array}\right.\\!\\!$ (29)

where
$\varepsilon=\sum_{a^{n}\notin\mathcal{T}_{\delta}^{(n)}(A)}\lambda_{a^{n}}^{A}$,
$\mathcal{T}_{\delta}^{(n)}(A)$ is the $\delta$-typical set corresponding to
the distribution $\lambda_{a}^{A}$ on the set $\mathcal{A}$, and
$\lambda_{a^{n}}^{A}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\Pi_{i=1}^{n}\lambda_{a_{i}}^{A}$.
Let $\mathcal{C}^{(m)}$ denote the codebook
$\\{A^{n}(m,k)\\}_{k\in\mathcal{K}}$ for a given $m$, and $\mathcal{C}$ denote
the collection of all codebooks $\\{\mathcal{C}^{(m)}\\}_{m\in\mathcal{M}}$.
Further, for each $a^{n}\in\mathcal{T}_{\delta}^{(n)}(A)$ define

$\displaystyle\tilde{\rho}_{a^{n}}^{{B_{R}}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\hat{\pi}\pi_{\rho^{{B_{R}}}}\pi_{a^{n}}\hat{\rho}_{a}^{{B_{R}}}\pi_{a^{n}}\pi_{\rho^{{B_{R}}}}\hat{\pi},$
(30)

and $\tilde{\rho}_{a^{n}}^{{B_{R}}}=0,$ for
$a^{n}\notin\mathcal{T}_{\delta}^{(n)}(A)$, where
$\hat{\rho}_{a}^{{B_{R}}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\bigotimes_{i}\hat{\rho}^{{B_{R}}}_{a_{i}}$,
${\pi}_{\rho^{{B_{R}}}}$ and $\pi_{a^{n}}$ are the $\delta-$typical and
conditionally typical projectors defined as in [50, Def. 15.1.3] and [50, Def.
15.2.4], with respect to
$\rho^{{B_{R}}}=\sum_{a\in\mathcal{A}}\lambda_{a}^{A}\hat{\rho}_{a}^{{B_{R}}}$
and $\hat{\rho}_{a}^{{B_{R}}}$, respectively, and $\hat{\pi}$ is the cut-off
projector as defined in [27]. Using the Average Gentle Measurement Lemma [50,
Lemma 9.4.3], for any given $\epsilon\in(0,1)$, and all sufficiently large $n$
and all sufficiently small $\delta$, we have

$\displaystyle\sum_{a^{n}\in\mathcal{A}^{n}}\lambda_{a^{n}}^{A}\|\hat{\rho}_{a^{n}}^{{B_{R}}}-\tilde{\rho}_{a^{n}}^{{B_{R}}}\|_{1}\leq\epsilon.$
(31)

A detailed proof of the above statement can be found in [58, Eq. 35]. Using
these definitions, construct operators

$\displaystyle A_{a^{n}}^{B_{R}^{n}}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\gamma_{a^{n}}\bigg{(}\sqrt{{\rho^{{B_{R}}}}^{\otimes
n}}^{-1}\tilde{\rho}_{a}^{{B_{R}}}\sqrt{{\rho^{{B_{R}}}}^{\otimes
n}}^{-1}\bigg{)},\;\;\gamma_{a^{n}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\frac{1-\varepsilon}{1+\eta}\frac{1}{|\mathcal{M}||\mathcal{K}|}|\\{(m,k):A^{n}(m,k)=a^{n}\\}|,$
(32)

and $\eta\in(0,1)$ is a parameter that determines the probability of not
obtaining a sub-POVM. Note that in the definition of $A_{a^{n}}^{B_{R}^{n}}$
the right hand side operates on $\mathcal{H}_{B_{R}^{n}}$, however, we define
$A_{a^{n}}$ belonging to $\mathcal{L}(\mathcal{H}_{A^{n}})$. To obtain this,
we transform $A_{a^{n}}^{B_{R}^{n}}$ as

$A_{a^{n}}=\sum_{b^{n},\bar{b}^{n}}\langle
b^{n}|A_{a^{n}}^{B_{R}^{n}}|\bar{b}^{n}\rangle_{{B_{R}}}\ket{b^{n}}\langle\bar{b}^{n}|_{B}.$

Then construct a sub-POVM $\Gamma^{(n)}$ as

$\displaystyle\Gamma^{(n)}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\\{A_{a^{n}}\colon
a^{n}\in\mathcal{T}_{\delta}^{(n)}(A)\\}.$ (33)

Let $\mathbbm{1}_{\\{\mbox{sP}\\}}$ denote the indicator random variable
corresponding to the event that $\Gamma^{(n)}$ forms a sub-POVM. We have the
following result.

######  Proposition 1.

For any $\epsilon\in(0,1)$, any $\eta\in(0,1)$, any $\delta\in(0,1)$
sufficiently small, and any $n$ sufficiently large, we have
$\mathbb{E}\left[\mathbbm{1}_{\\{\mbox{sP}\\}}\right]>1-\epsilon,$ if
$\frac{1}{n}\left(\log M+\log
K\right)>\chi({\lambda_{a}^{{B_{R}}},\hat{\rho}_{a}^{{B_{R}}}})$.

###### Proof.

The result follows from [27]. ∎

Define the code dependent random variables $\mathit{E}_{1}$ and
$\mathit{E}_{2}$ as

$\displaystyle\mathit{E}_{1}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{m\in\mathcal{M}}\sum_{k\in\mathcal{K}}(|\mathcal{M}||\mathcal{K}|)^{-{1}}\Tr{\tilde{\rho}^{{B_{R}}}_{m,k}},\mbox{\quad
and\quad}\mathit{E}_{2}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{m\in\mathcal{M}}\sum_{k\in\mathcal{K}}(|\mathcal{M}||\mathcal{K}|)^{-{1}}\left\|\tilde{\rho}^{{B_{R}}}_{m,k}-\hat{\rho}^{{B_{R}}}_{m,k}\right\|_{1},$

where $\hat{\rho}^{{B_{R}}}_{m,k}$, and $\tilde{\rho}^{{B_{R}}}_{m,k}$ are
used as shorthand notations to denote $\tilde{\rho}^{{B_{R}}}_{a^{n}(m,k)}$
and $\tilde{\rho}^{{B_{R}}}_{a^{n}(m,k)}$, respectively. Further, using the
results [58, Eq. (28) and Eq. (35)], for all $\epsilon\in(0,1)$, we have
$\mathbb{E}[\mathit{E}_{1}]\geq 1-\epsilon,$ and
$\mathbb{E}[\mathit{E}_{2}]\leq\epsilon,$ for all sufficiently large $n$ and
all sufficiently small $\delta>0$.

Now, considering the ensemble $\\{\lambda_{a}^{E},\tau_{a}^{E}\\}$, we
construct the operators $\\{\tau_{a^{n}(m,k)}^{E}\\}$ using the codebook
$\mathcal{C}$ and the distribution defined in (29), where
$\tau_{a^{n}}^{E}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\bigotimes_{i}\tau_{a_{i}}^{E}$.
For this ensemble, we construct a collection of $n$-letter POVMs, one for each
$m\in\mathcal{M}$, capable of decoding the message $k\in\mathcal{K}$. In
particular, we employ the Holevo POVMs [28] defined as

$\displaystyle\xi_{k}^{(m)}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\pi^{\tau}\pi^{(m)}_{k}\pi^{\tau}\quad\mbox{
and
}\quad\Xi^{(m)}_{k}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\left(\sum_{k^{\prime}\in\mathcal{K}}\xi^{(m)}_{k^{\prime}}\right)^{-1/2}\xi^{(m)}_{k}\left(\sum_{k^{\prime}\in\mathcal{K}}\xi^{(m)}_{k^{\prime}}\right)^{-1/2},$
(34)

where $\pi^{\tau}$ is the $\delta-$typical projector (as in [50, Def. 15.1.3])
defined for the density operator
$\tau\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{a\in\mathcal{A}}\lambda_{a}^{E}\tau_{a}^{E}$,
and $\pi^{(m)}_{k}$ denotes the strong conditional typical projectors (as in
[50, Def. 15.2.4]) for the operators $\tau_{a^{n}(m,k)}$. For these POVMs, we
know the average probability of error can be made arbitrarily small. More
formally, we have the following.

######  Proposition 2.

Given the ensemble $\\{\lambda_{a}^{E},\tau_{a}^{E}\\}$ and the collection of
POVMs $\\{\Xi^{(m)}_{k}\\}_{k}$, for any $\epsilon\in(0,1)$,

$\displaystyle\mathbb{E}\left[\frac{1}{|\mathcal{K}|}\sum_{k\in\mathcal{K}}\Tr{\Xi^{(m)}_{k}\tau_{k}^{(m)}}\right]\geq
1-\epsilon,$ (35)

for sufficiently small $\delta>0$ and for all sufficiently large $n$, and for
all $m\in\mathcal{M}$, if
$\frac{1}{n}\log{K}<\chi(\\{\lambda_{a}^{E},\tau_{a}^{E}\\})$, where
$\tau_{k}^{(m)}$ is used as a shorthand for $\tau_{a^{n}(m,k)}$.

###### Proof.

The proof follows from the result of classical communication over quantum
channels [28] or the packing lemma of [50, Lemma 16.3.1] while making the
following identification. For each $m\in\mathcal{M}$, identify $\mathcal{M}$
with $\mathcal{K}$, $\mathsf{X}$ with
$\mathcal{T}_{\delta}^{(n)}(\mathcal{A})$, $\\{\sigma_{C_{m}}\\}_{m}$ with
$\\{\tau_{k}^{(m)}\\}_{k}$, $\Pi$ with $\pi^{\tau}$, $\Pi_{x}$ with
$\pi_{k}^{(m)}$, $d$ with $2^{n(S(E|A)_{\bar{\tau}}+\bar{\delta})}$, $D$ with
$2^{n(S(E)_{\bar{\tau}}-\bar{\delta})}$, and $\Lambda_{m}$ with
$\Xi^{(m)}_{k}$, where
$\bar{\tau}^{AE}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{a}\lambda_{a}^{E}\outerproduct{a}{a}_{A}\otimes\tau_{a}^{E}$
and $\bar{\delta}(\delta)\searrow 0$ as $\delta\searrow 0$. ∎

The above result also implies a weaker average result which suffices here.
This can be stated as $\mathbb{E}[\mathit{E}_{3}]\geq 1-\epsilon$, for
sufficiently small $\delta>0$ and for all sufficiently large $n$, if
$\frac{1}{n}\log{K}<\chi(\\{\lambda_{a}^{E},\tau_{a}^{E}\\})$, where

$\displaystyle\mathit{E}_{3}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\frac{1}{MK}\sum_{m\in\mathcal{M}}\sum_{k\in\mathcal{K}}\Tr{\Xi^{(m)}_{k}\tau_{k}^{(m)}}.$
(36)

Finally, toward finding a good code, we need one last property which is that
all its codewords are distinct. In the dual, the quantum channel communication
problem [32], Devetak used the double exponential decay of the covering error
to argue the existence of an expurgated code for exponentially many covering
constraints. However, in the current problem, we have exponentially many
packing constraints, with each having only an exponential decay in the error.
To resolve this issue, we develop a proof that only requires the average of
the packing constraints. However, in such a case, it becomes unclear as to
what should be the expurgation strategy. For this, we introduce another event
that captures the non-distinctness of the codebook, and expurgate with respect
to this event. Precisely, we define a codeword $A^{n}(m,k)$ is bad if there
exists $(m^{\prime},k^{\prime})\neq(m,k)$ such that
$A^{n}(m,k)=A^{n}(m^{\prime},k^{\prime}).$ Let

$E_{4}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\frac{1}{MK}\sum_{m\in\mathcal{M}}\sum_{k\in\mathcal{K}}\mathbbm{1}_{\\{A^{n}(m,k)\mbox{
is bad}\\}}.$

Computing its expectation, we get

$\displaystyle\mathbb{E}[E_{4}]=\mathbb{E}\left[{\frac{1}{MK}\sum_{m\in\mathcal{M}}\sum_{k\in\mathcal{K}}\mathbbm{1}_{\\{{\exists(m^{\prime},k^{\prime})\neq(m,k)\mbox{
such that }A^{n}(m,k)=A^{n}(m^{\prime},k^{\prime})}\\}}}\right]$
$\displaystyle\overset{a}{\leq}\frac{1}{MK}\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}m,m^{\prime}\in\mathcal{M}\\\
k,k^{\prime}\in\mathcal{K}\\\
(m,k)\neq(m^{\prime},k^{\prime})\end{subarray}}\sum_{a^{n}\in\mathcal{T}_{\delta}^{(n)}(A)}\mathbb{E}\left[{\mathbbm{1}_{\\{{A^{n}(m,k)=a^{n}}\\}}}\right]\mathbb{E}\left[{\mathbbm{1}_{\\{{A^{n}(m^{\prime},k^{\prime})={a}^{n}}\\}}}\right]\overset{b}{\leq}MK2^{-n(S(\lambda_{a}^{A})-\delta_{1})}\overset{}{\leq}\epsilon,$
(37)

for all sufficiently large $n$ and sufficiently small $\delta>0$ if
$\frac{1}{n}\left(\log M+\log K\right)<S(\lambda_{a}^{A}),$ where (a) uses the
mutual independence of the codewords, and (b) define $\delta_{1}$ as
$\delta_{1}(\delta,\varepsilon)\searrow 0$ as $\delta,\varepsilon\searrow 0$.
Using the Markov inequality and the union bound, we have

$\displaystyle\mathbb{P}\bigg{(}\\{\mathbbm{1}_{\\{\mbox{sP}\\}}=1\\}$
$\displaystyle\cap\left\\{{E_{1}\geq
1-\sqrt{\epsilon}}\right\\}\cap\left\\{{E_{2}\leq\sqrt{\epsilon}}\right\\}\cap\left\\{{E_{3}\geq
1-\sqrt{\epsilon}}\right\\}\cap\left\\{{E_{4}\leq\sqrt{\epsilon}}\right\\}\bigg{)}\geq
1-5\sqrt{\epsilon}.$

Therefore, for all $\epsilon\in(0,1/25),$ and for all sufficiently small
$\delta>0$, for all sufficiently large $n$ there exists a code $\mathcal{C}$
that satisfies the conditions $\\{\mathbbm{1}_{\\{\mbox{sP}\\}}=1\\}$,
$\left\\{{E_{1}\geq 1-\sqrt{\epsilon}}\right\\}$,
$\left\\{{E_{2}\leq\sqrt{\epsilon}}\right\\}$, $\left\\{{E_{3}\geq
1-\sqrt{\epsilon}}\right\\}$, and $\left\\{{E_{4}\leq{\epsilon}}\right\\}$,
simultaneously if

$\displaystyle\frac{1}{n}\left(\log M+\log
K\right)>\chi({\lambda_{a}^{{B_{R}}},\hat{\rho}_{a}^{{B_{R}}}}),\;\;\;\frac{1}{n}\log{K}<\chi(\\{\lambda_{a}^{E},\tau_{a}^{E}\\}),\;\;\;\frac{1}{n}\left(\log
M+\log K\right)<S(\lambda_{a}^{A}).$ (S-0)

At this point, we choose one such code $\mathcal{C}$ satisfying all the above
conditions, and fix it for the rest of the analysis.

Toward showing that this chosen code achieves the asymptotic performance
stated in the theorem statement, we expurgate the code $\mathcal{C}$ with
respect to the random variable $E_{4}$, ensuring that the code has all
distinct codewords. The assumption of codebook being distinct becomes crucial
at multiple places in the proof and will be highlighted as necessary. Since
$\left\\{{E_{4}\leq\sqrt{\epsilon}}\right\\}$ ensures at most
$\sqrt{\epsilon}MK$ codewords in $\mathcal{C}$ are not distinct, we remove
$\sqrt{\epsilon}MK$ codewords from $\mathcal{C}$. This is performed by first
removing all the non-distinct codewords, and then further removing some more
from the distinct ones arbitrarily (if needed) until we remain with a total of
$(1-\sqrt{\epsilon})MK$ codewords. Let the expurgated set (the remainder of
the codewords) be denoted by $\mathcal{C}_{\mathcal{E}}$, and define the sets
$\mathcal{C}_{\mathcal{E}}^{(m)}$ as
$\mathcal{C}_{\mathcal{E}}^{(m)}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\mathcal{C}_{\mathcal{E}}\cap\mathcal{C}^{(m)}$.
Observe that, all the codewords in $\mathcal{C}_{\mathcal{E}}$ are distinct.
However, as opposed to $\mathcal{C}$ which was consistent with regards to the
size of $\mathcal{C}^{(m)}$ (equal to $K$ for all $m\in\mathcal{M}$),
$\mathcal{C}_{\mathcal{E}}$ has varying sizes. Therefore, we define
$K^{\prime}_{m}$ to denote the size of $\mathcal{C}_{\mathcal{E}}^{(m)}$ and
$M^{\prime}$ to denote number of non-empty sets in the collection
$\\{\mathcal{C}_{\mathcal{E}}^{(m)}\\}_{m\in\mathcal{M}}$. Note that for some
$m\in\mathcal{M}$, $K^{\prime}_{m}$ may be zero. Let $\mathcal{M}^{\prime}$
denote the subset of $\mathcal{M}$ for which $K^{\prime}_{m}>0$, and let
$\mathcal{H}_{M}^{\prime}$ denote the corresponding Hilbert space with
$\dim(\mathcal{H}_{M}^{\prime})=M^{\prime}+1$. As is evident,
$\sum_{m\in\mathcal{M}^{\prime}}K^{\prime}_{m}=(1-\sqrt{\epsilon})MK$. In
addition, define the set of indices corresponding to the expurgated codebook
as
$\mathcal{I}^{(m)}_{E}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\\{k:a^{n}(m,k)\in\mathcal{C}_{\mathcal{E}}^{(m)}\\}$
and
$\mathcal{I}_{E}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\\{(m,k):a^{n}(m,k)\in\mathcal{C}_{\mathcal{E}}\\}$.
Further, for the expurgated code, we have

$\displaystyle{E}^{\prime}_{1}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\frac{1}{(1-\sqrt{\epsilon})|\mathcal{M}||\mathcal{K}|}\sum_{m\in\mathcal{M}^{\prime}}\sum_{k\in\mathcal{I}^{(m)}_{E}}\Tr{\tilde{\rho}^{{B_{R}}}_{m,k}}\geq
1-2\sqrt{\epsilon},$ (38) $\displaystyle{E}^{\prime}_{2}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\frac{1}{(1-\sqrt{\epsilon})|\mathcal{M}||\mathcal{K}|}\sum_{m\in\mathcal{M}^{\prime}}\sum_{k\in\mathcal{I}^{(m)}_{E}}\left\|\tilde{\rho}^{{B_{R}}}_{m,k}-\hat{\rho}^{{B_{R}}}_{m,k}\right\|_{1}\leq\frac{\sqrt{\epsilon}}{1-\sqrt{\epsilon}}\leq
2\sqrt{\epsilon},$ (39) $\displaystyle E^{\prime}_{3}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\frac{1}{(1-\sqrt{\epsilon})|\mathcal{M}||\mathcal{K}|}\sum_{m\in\mathcal{M}^{\prime}}\sum_{k\in\mathcal{I}^{(m)}_{E}}\Tr{\Xi^{(m)}_{k}\tau_{k}^{(m)}}\geq
1-2\sqrt{\epsilon},$ (40)

where the inequalities above follow from the fact that codebook $\mathcal{C}$
satisfies $\left\\{{E_{1}\geq
1-\sqrt{\epsilon}}\right\\},\left\\{{E_{2}\leq\sqrt{\epsilon}}\right\\}$ and
$\left\\{{E_{3}\geq 1-\sqrt{\epsilon}}\right\\}$ and that only
$\sqrt{{\epsilon}}$ fraction of the code is expurgated. Observe that the event
$\\{{\mathbbm{1}_{\\{\mbox{sP}\\}}=1}\\}$ remains true for the expurgated
$\mathcal{C}_{\mathcal{E}}$. Define the collection

$\Gamma_{\mathcal{E}}^{(n)}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\\{A_{a^{n}(m,k)}\\}_{m\in\mathcal{M}^{\prime},k\in\mathcal{I}_{\mathcal{E}}^{(m)}}.$

The collection $\Gamma_{\mathcal{E}}^{(n)}$ is completed using the operator
$I-\sum_{m\in\mathcal{M}^{\prime}}\sum_{k\in\mathcal{I}_{\mathcal{E}}^{(m)}}A_{a^{n}(m,k)}$,
and the operator is associated with sequence $a^{n}_{0}$ chosen arbitrarily
from $\mathcal{A}^{n}\backslash\mathcal{T}_{\delta}^{(n)}(A)$, i.e.,

$A_{a^{n}_{0}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}I-\sum_{m\in\mathcal{M}^{\prime}}\sum_{k\in\mathcal{I}_{\mathcal{E}}^{(m)}}A_{a^{n}(m,k)}.$

Corresponding to this expurgated code, we now construct our encoding and
decoding operations.

####  V-B3 Encoding and Decoding Isometries

The encoding isometry $U_{\mathcal{E}}$ is constructed by concatenating three
isometries: (i) the covering isometry
$U_{\mathcal{M}}:\mathcal{H}_{B^{n}}\rightarrow\mathcal{H}_{B^{n}}\otimes\mathcal{H}_{M}^{\prime}\otimes\mathcal{H}_{K}$,
(ii) the rotation isometry
$U_{\mathcal{R}}:\mathcal{H}_{B^{n}}\otimes\mathcal{H}_{M}^{\prime}\otimes\mathcal{H}_{K}\rightarrow\mathcal{H}_{E^{n}}\otimes\mathcal{H}_{M}^{\prime}\otimes\mathcal{H}_{K}$,
and (iii) the packing isometry
$U_{\mathcal{P}}:\mathcal{H}_{E^{n}}\otimes\mathcal{H}_{{\bar{E}}}\otimes\mathcal{H}_{M}^{\prime}\otimes\mathcal{H}_{K}\rightarrow\mathcal{H}_{E^{n}}\otimes\mathcal{H}_{{\bar{E}}}\otimes\mathcal{H}_{M}^{\prime}\otimes\mathcal{H}_{K}$,
where $\mathcal{H}_{M}^{\prime},\mathcal{H}_{K}$ and $\mathcal{H}_{{\bar{E}}}$
are auxiliary Hilbert spaces with dimensions $M^{\prime}+1,K+1$, and $K+1$,
respectively.

Step 1.1: Covering Isometry
To define the covering isometry $U_{\mathcal{M}}$, we use the completion
$[\Gamma_{\mathcal{E}}^{(n)}]$ as

$\displaystyle U_{\mathcal{M}}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{m\in\mathcal{M}^{\prime}}\sum_{k\in\mathcal{I}_{\mathcal{E}}^{(m)}}\sqrt{A_{a^{n}(m,k)}}\otimes\ket{m}\otimes\ket{k}+\sqrt{A_{a_{0}^{n}}}\otimes\ket{M^{\prime}}_{M}\otimes\ket{K}_{K}.$
(41)

Note that, for the chosen code, the event
$\\{\mathbbm{1}_{\\{\mbox{sP}\\}}=1\\}$ makes $U_{\mathcal{M}}$ a valid
isometry. From now on, for the ease of notation, we use
$M_{m,k},\lambda_{m,k}^{{B_{R}}},$ and $A_{m,k}$ to denote the corresponding
$n-$letter objects constructed for the codewords $A^{n}(m,k)$.

Step 1.2: Rotation Isometry
Although the above covering unitary aims to cover the source, it only does so
for the reference system. To be able to apply the next step of packing, we
wish to use the post-measured state as side information. This could be
possible if the post-measured state also looks close to being product. For
this, we employ a rotation unitary. A similar operation is discussed in [59,
Fact 6] with regards to classical-quantum states obtained post measurement.
The construction below generalizes this to a coherent application of a
measurement. More formally, for the expurgated code
$\mathcal{C}_{\mathcal{E}}$, we construct the states

$\displaystyle\ket{\hat{\sigma}}^{{B_{R}^{n}}E^{n}MK}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{m\in\mathcal{M}^{\prime}}\sum_{k\in\mathcal{I}^{(m)}_{\mathcal{E}}}\frac{1}{\sqrt{(1-\sqrt{\epsilon})|\mathcal{M}||\mathcal{K}|}}\frac{(I\otimes{M_{m,k})}}{\sqrt{\lambda_{m,k}}}\ket{\psi_{\rho}^{\otimes
n}}^{B_{R}^{n}B^{n}}\otimes\ket{m,k}\mbox{ and }$
$\displaystyle\ket{\tilde{\sigma}}^{B_{R}^{n}B^{n}MK}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{m\in\mathcal{M}^{\prime}}\sum_{k\in\mathcal{I}^{(m)}_{\mathcal{E}}}\frac{1}{\sqrt{(1-\sqrt{\epsilon})|\mathcal{M}||\mathcal{K}|}}\frac{(I\otimes\sqrt{A_{m,k})}}{\sqrt{\delta_{m,k}}}\ket{\psi_{\rho}^{\otimes
n}}^{B_{R}^{n}B^{n}}\otimes\ket{m,k},$ (42)

where
$\delta_{m,k}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\Tr{A_{m,k}{\rho^{B}}^{\otimes
n}}=\gamma\Tr{\tilde{\rho}^{{B_{R}}}_{m,k}}$, and
$\gamma\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\frac{1-\varepsilon}{1+\eta}\frac{1}{|\mathcal{M}||\mathcal{K}|}.$
For brevity in notation, we skip the sets in the summations over $m$ or $k$
when summations are performed over the codewords belonging to the set
$\mathcal{I}_{\mathcal{E}}$ corresponding to the expurgated codebook
$\mathcal{C}_{\mathcal{E}}$. Clearly,
$\ket{\hat{\sigma}}^{{B_{R}^{n}}E^{n}MK}$ and
$\ket{\tilde{\sigma}}^{B_{R}^{n}B^{n}MK}$ are valid states. Now to construct
$U_{\mathcal{R}}$, consider the following lemma which upper bounds the
fidelity.

######  Lemma 4.

For any $\epsilon,\eta\in(0,1)$, there exists a collection of isometries
$\\{U_{r}(m,k):\mathcal{H}_{B^{n}}\rightarrow\mathcal{H}_{E^{n}}\\}$ and a
collection of phases $\theta_{m,k}$ such that
$F(\ket{\hat{\sigma}}^{{B_{R}^{n}}E^{n}MK},(I_{{B_{R}}}\otimes
U_{\mathcal{R}})\ket{\tilde{\sigma}}^{B_{R}^{n}B^{n}MK})\geq
1-4\sqrt{\epsilon}$, for all sufficiently large $n$ and all sufficiently small
$\delta>0$, where

$\displaystyle
U_{\mathcal{R}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{m\in\mathcal{M}^{\prime}\cup\left\\{{M^{\prime}}\right\\}}\sum_{k\in\mathcal{K}\cup\left\\{{K}\right\\}}e^{-i\theta_{m,k}}U_{r}(m,k)\otimes\outerproduct{m}{m}_{M}\otimes\outerproduct{k}{k}_{K}.$
(43)

and $U_{r}(m,k)=I$ and $\theta_{m,k}=0$ for all
$(m,k)\in(\mathcal{M}\times\mathcal{K})$ such that
$a^{n}(m,k)\notin\mathcal{C}_{\mathcal{E}}$.

###### Proof.

The proof of the lemma follows using (38), (39) and from the Lemma 7. For
completeness, we detail the proof in Appendix A-B. ∎

Step 1.3: Packing Isometry
Observe that by coherently performing the covering and rotation operations,
one can show that the source $\rho_{B}$ can be successfully recovered by the
quantum registers $\\{\ket{m,k}\\}$. This implies that the quantum states
$\\{\ket{m,k}\\}$ can be used by the decoder to faithfully reconstruct the
source as per Definition 3. As a result, we would require a rate of
$\frac{1}{n}\log M+\frac{1}{n}\log K$ which has to be greater than the Holevo
information $\chi({\lambda_{a}^{{B_{R}}},\hat{\rho}_{a}^{{B_{R}}}})$, as
constrained by Proposition 1. However, we intend to further reduce the rate
from this Holevo information to the coherent information provided in the
statement of the theorem.

One can perhaps argue why we cannot simply release the information in the
$\mathcal{H}_{K}$ system into the environment (partial tracing)? But as
expected for a purely quantum setup, this would lead to the protocol becoming
incoherent. More precisely, the subsystem $\mathcal{H}_{E^{n}}$ that the
encoder has in its possession is entangled with the subsystem
$\mathcal{H}_{K}$, and tracing out the latter without decoupling the two
systems would render the former in a mixed state. Once this entanglement is
lost, the decoder would not be able to faithfully reconstruct the source by
using such a (mixed) state.

Therefore, a major task here is to successfully decouple the system
$\mathcal{H}_{K}$ before releasing it to the environment. To achieve this, we
introduce the notion of coherent packing or coherent binning. This notion is
built on the idea that the post-measured system present in
$\mathcal{H}_{E^{n}}$ contains information about the quantum state
$\ket{k}_{K}$, and hence, conditioned on the state $\ket{m}$, a copy of the
state $\ket{k}$ can be recovered from the state present in subsystem
$\mathcal{H}_{E^{n}}$, albeit with a small probability of error. Using this
copy, we intend to decouple the existing copy of $\ket{k}$ from the latter.
However, this new copy can erase (decouple) the original, but will itself
still remain. Therefore, as will become evident in the sequel, we perform the
process of erasing the information in $\mathcal{H}_{K}$ intrinsically without
producing any additional copies.

Toward this, we employ the packing code consisting of the sub-POVMs
$\\{\Xi_{k}^{(m)}\\}_{k\in\mathcal{I}_{\mathcal{E}}^{(m)}}$, generated for the
ensemble $\\{\lambda_{a}^{E},\tau_{a}^{E}\\}$. We complete this sub-POVM for
each $m\in\mathcal{M}^{\prime}$ as

$\Xi_{K}^{(m)}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}I-\sum_{k\in\mathcal{I}_{\mathcal{E}}^{(m)}}\Xi_{k}^{(m)}.$

In addition, we also make use of Naimark’s extension theorem (also provided in
Lemma 3). This lemma gives us a collection of orthogonal projectors
$\\{\Pi_{k}^{(m)}\\}$ each acting on
$\mathcal{H}_{E^{n}}\otimes\mathcal{H}_{\bar{E}}$, corresponding to the
collection $\\{\Xi^{(m)}_{k}\\}_{k}$, such that

$\displaystyle\Tr{\Pi_{k}^{(m)}(\tau_{k}^{(m)}\otimes\outerproduct{0}{0}_{\bar{E}})}=\Tr{\Xi^{(m)}_{k}\tau_{k}^{(m)}},$
(44)

for all $m\in\mathcal{M}^{\prime}$ and
$k\in\mathcal{I}_{\mathcal{E}}^{(m)}\cup\\{K\\}$, and
$\dim{\mathcal{H}_{\bar{E}}}=K+1$. Finally, we define the packing unitary
$U_{\mathcal{P}}$ as

$\displaystyle
U_{\mathcal{P}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{m\in\mathcal{M}^{\prime}}$
$\displaystyle\left[\sum_{k\in\mathcal{I}_{\mathcal{E}}^{(m)}\cup\\{K\\}}\\!\\!\\!\\!\\!\\!\\!\Pi_{k}^{(m)}\\!\otimes\\!\bigg{(}\sum_{k^{\prime}\in\mathcal{K}\cup\\{K\\}}e^{i\alpha_{k^{\prime}}^{(m)}}\ket{(k-k^{\prime})\\!\\!\\!\\!\mod(K+1)}\bra{k^{\prime}}\bigg{)}\right]\otimes\\!\outerproduct{m}{m}$
$\displaystyle\hskip
267.39877pt+I_{E^{n}\bar{E}K}\otimes\outerproduct{M^{\prime}}{M^{\prime}},$
(45)

where the phases $\\{\alpha_{k}^{(m)}\\}$ are introduced for later
convenience, and will be specified in the sequel111Moving forward, we
implicitly assume the modulus operation rather than explicitly mentioning it
for the purpose of brevity.. Note that by using $\Pi_{k}^{(m)}$ in the above
definition, instead of $\Xi_{k}^{(m)}$ ensures that
$U_{\mathcal{P}}U_{\mathcal{P}}^{\dagger}=I$, implying $U_{\mathcal{P}}$ is a
valid unitary.

As a result, we can express the encoding CPTP map
$\mathcal{N}_{\mathcal{E}}^{(n)}$ as

$\displaystyle\left(I_{B_{R}^{n}}\otimes\mathcal{N}_{\mathcal{E}}^{(n)}\right)$
$\displaystyle\left(\outerproduct{\psi_{\rho}^{\otimes
n}}{\psi_{\rho}^{\otimes n}}^{B_{R}^{n}B^{n}}\right)$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\Tr_{\bar{E}E^{n}K}\left((I\otimes
U_{\mathcal{P}}U_{\mathcal{R}}U_{\mathcal{M}})\outerproduct{\psi_{\rho}^{\otimes
n}}{\psi_{\rho}^{\otimes
n}}^{B_{R}^{n}B^{n}}\otimes\outerproduct{0}{0}_{\bar{E}}(I\otimes
U_{\mathcal{P}}U_{\mathcal{R}}U_{\mathcal{M}})^{\dagger}\right).$ (46)

The quantum state in $\mathcal{H}_{M}^{\prime}$ is now sent to the decoder.

Step 1.4: Decoding Isometry:
The following decoding isometry is applied on the state in
$\mathcal{H}_{M}^{\prime}$:

$\displaystyle
U_{\mathcal{D}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{m\in\mathcal{M}^{\prime}}\bigg{(}\frac{1}{\sqrt{K^{\prime}_{m}}}\sum_{k\in\mathcal{I}_{\mathcal{E}}^{(m)}}e^{-i\beta_{k}^{(m)}}\ket{a^{n}(m,k)}\bigg{)}\bra{m}+\ket{a^{n}_{0}}\langle{M^{\prime}}|,$
(47)

where the phases $\\{\beta_{k}^{(m)}\\}$ will be identified in the
continuation. Observe that, to argue $U_{\mathcal{D}}$ is a valid isometric
operation, we need the vectors $\\{\ket{a^{n}(m,k)}\\}$ to be distinct. By
expurgating the codebook to generate $\mathcal{C}_{\mathcal{E}}$, and only
using the codewords from $\mathcal{C}_{\mathcal{E}}$ ensures this
distinctness. With the definitions of encoder and decoder, we move on to
bounding the error incurred by the protocol (as defined in Definition 2).

####  V-B4 Trace Distance

We begin by defining the following terms

$\displaystyle\ket{\omega}^{B_{R}^{n}E^{n}\bar{E}A^{n}K}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(I\otimes
U_{\mathcal{D}})(I\otimes U_{\mathcal{P}})(I\otimes
U_{\mathcal{R}}U_{\mathcal{M}})\ket{\psi_{\rho}^{\otimes
n}}^{B_{R}^{n}B^{n}}\ket{0}_{\bar{E}},$
$\displaystyle\ket{\zeta}^{B_{R}^{n}E^{n}A^{n}}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(W^{\otimes
n}\otimes I)\ket{\psi_{\omega}}^{A_{R}^{n}A^{n}},$ (48)

where222For conciseness, we drop the $\otimes n$ from
$\ket{\psi_{\rho}^{\otimes n}}^{B_{R}^{n}B^{n}}$ when understood from context.
$\ket{\psi_{\omega}}^{A_{R}^{n}A^{n}}$ is the canonical purification of
$\omega^{A^{n}}$. Let

$G\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\|\omega^{B_{R}^{n}A^{n}}-\zeta^{B_{R}^{n}A^{n}}\|_{1}.$

Following Definition 2, our objective now is to show $G$ can be made
arbitrarily small for all sufficiently large $n$ for the code
$\mathcal{C}_{\mathcal{E}}$.

Step 2.1: Closeness of $\ket{\omega}\mbox{ and }(I\otimes
U_{\mathcal{D}})(I\otimes U_{\mathcal{P}})\ket{\hat{\sigma}}:$
Recall the definitions of $\ket{\hat{\sigma}}\mbox{ and }\ket{\tilde{\sigma}}$
from (V-B3), and let
$\ket{\omega_{1}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(I\otimes
U_{\mathcal{R}}U_{\mathcal{M}})\ket{\psi_{\rho}}^{B_{R}^{n}B^{n}}$ and
$\varepsilon_{1}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}{(1-\varepsilon)/(1+\eta)}$.
Consider

$\displaystyle\sqrt{F\left(\ket{\omega_{1}}^{B_{R}^{n}E^{n}MK},(I\otimes
U_{\mathcal{R}})\ket{\tilde{\sigma}}^{B_{R}^{n}E^{n}MK}\right)}$
$\displaystyle=\sum_{m,k}\frac{1}{\sqrt{(1-\sqrt{\epsilon})|\mathcal{M}||\mathcal{K}|}}\frac{\bra{\psi_{\rho}}(I\otimes
U_{r}(m,k)\sqrt{A_{m,k}})^{\dagger}((I\otimes
U_{r}(m,k)\sqrt{A_{m,k}})\ket{\psi_{\rho}}}{\sqrt{\delta_{m,k}}}$
$\displaystyle=\frac{1}{\sqrt{1-\sqrt{\epsilon}}}\sum_{m,k}\frac{\sqrt{\varepsilon_{1}}}{{|\mathcal{M}||\mathcal{K}|}}\sqrt{\Tr{\tilde{\rho}_{m,k}}}\geq\frac{1}{\sqrt{1-\sqrt{\epsilon}}}{\sum_{m,k}\frac{\sqrt{\varepsilon_{1}}}{{|\mathcal{M}||\mathcal{K}|}}\Tr{\tilde{\rho}_{m,k}}}\geq\sqrt{\varepsilon_{1}}\sqrt{1-\sqrt{\epsilon}}(1-2\sqrt{\epsilon}),$
(49)

where we note that there is no overlap between the term corresponding to
$\sqrt{A_{a^{n}_{0}}}\otimes\ket{M^{\prime}}_{M}\otimes\ket{K}_{K}$ of
$\ket{\omega_{1}}^{B_{R}^{n}E^{n}MK}$ and the state $(U_{\mathcal{R}}\otimes
I)\ket{\tilde{\sigma}}^{B_{R}^{n}E^{n}MK}$, and the last inequality follows
from (38).

Using Lemma 1, and the inequality (49) , we get 333At times, the subspace
notation is omitted when it is clear from the context.

$\displaystyle\left\|{\omega_{1}}^{B_{R}^{n}E^{n}MK}-(I\otimes
U_{\mathcal{R}}){\tilde{\sigma}}^{B_{R}^{n}E^{n}MK}(I\otimes
U_{\mathcal{R}})^{\dagger}\right\|_{1}$ $\displaystyle\hskip 72.26999pt\leq
2\sqrt{1-(1-\sqrt{\epsilon})(1-2\sqrt{\epsilon})^{2}\left(1-\frac{\eta+\varepsilon}{1+\eta}\right)}\leq
2\sqrt{\frac{\eta+\varepsilon}{1+\eta}+5\sqrt{\epsilon}}\leq
6\sqrt[4]{\epsilon},$ (50)

for all sufficiently large $n$ and sufficiently small $\eta,\delta>0$.
Further, using the unitary invariance of trace distance, we get the closeness
of the states:

$\displaystyle\left\|(I\otimes
U_{\mathcal{D}}U_{\mathcal{P}}){\omega_{1}}^{B_{R}^{n}E^{n}MK}\otimes\outerproduct{0}{0}_{\bar{E}}(I\otimes
U_{\mathcal{D}}U_{\mathcal{P}})^{\dagger}\right.$ $\displaystyle\hskip
72.26999pt\left.-(I\otimes
U_{\mathcal{D}}U_{\mathcal{P}}U_{\mathcal{R}}){\tilde{\sigma}}^{B_{R}^{n}E^{n}MK}\otimes\outerproduct{0}{0}_{\bar{E}}(I\otimes
U_{\mathcal{D}}U_{\mathcal{P}}U_{\mathcal{R}})^{\dagger}\right\|_{1}\leq
6\sqrt[4]{\epsilon}.$ (51)

Using Lemma 4 and the fact that trace norm is invariant under isometric
transformations, we have

$\displaystyle\left\|(I\otimes
U_{\mathcal{D}}U_{\mathcal{P}}){\hat{\sigma}^{B_{R}^{n}E^{n}MK}}\otimes\outerproduct{0}{0}_{\bar{E}}(I\otimes
U_{\mathcal{D}}U_{\mathcal{P}})^{\dagger}\right.$ $\displaystyle\hskip
36.135pt\left.-(I\otimes
U_{\mathcal{D}}U_{\mathcal{P}}U_{\mathcal{R}}){\tilde{\sigma}}^{B_{R}^{n}E^{n}MK}\otimes\outerproduct{0}{0}_{\bar{E}}(I\otimes
U_{\mathcal{D}}U_{\mathcal{P}}U_{\mathcal{R}})^{\dagger}\right\|_{1}$
$\displaystyle\hskip 28.90755pt=\left\|\hat{\sigma}-(I\otimes
U_{\mathcal{R}}){\tilde{\sigma}}(I\otimes
U_{\mathcal{R}})^{\dagger}\right\|_{1}\leq
2\sqrt{1-F\left(\ket{\hat{\sigma}},(I\otimes
U_{\mathcal{R}})\ket{\tilde{\sigma}}\right)}\leq 4\sqrt[4]{\epsilon}.$ (52)

Using triangle inequality and inequalities (51) and (V-B4), we obtain

$\displaystyle\left\|{\omega}^{B_{R}^{n}E^{n}\bar{E}A^{n}K}-(I\otimes
U_{\mathcal{D}}U_{\mathcal{P}})({\hat{\sigma}}^{B_{R}^{n}E^{n}MK}\otimes\outerproduct{0}{0}_{\bar{E}})(I\otimes
U_{\mathcal{D}}U_{\mathcal{P}})^{\dagger}\right\|_{1}\leq
10\sqrt[4]{\epsilon},$ (S-1)

for all sufficiently large $n$ and sufficiently small $\eta,\delta>0$, which
concludes Step 2.1.

For the next step, define $|{\hat{\zeta}}\rangle^{B_{R}^{n}E^{n}\bar{E}MK}$ as

$\displaystyle|{\hat{\zeta}}\rangle^{B_{R}^{n}E^{n}\bar{E}MK}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{m\in\mathcal{M}^{\prime}}\sum_{k\in\mathcal{I}_{\mathcal{E}}^{(m)}}\frac{1}{\sqrt{(1-\sqrt{\epsilon})|\mathcal{M}||\mathcal{K}|}}e^{i\beta_{k}^{(m)}}\frac{(I\otimes
M_{m,k})\ket{\psi_{\rho}}}{\sqrt{\lambda_{m,k}}}\otimes\ket{m}_{M}\otimes\ket{0}_{K}\otimes\ket{0}_{\bar{E}},$
(53)

where the phases $\beta_{k}^{(m)}$ will be specified shortly. Observe that
$|{\hat{\zeta}}\rangle^{B_{R}^{n}E^{n}\bar{E}MK}$ is a valid pure state due to
(i) the distinctness of codewords in $\mathcal{C}_{\mathcal{E}}$ and (ii) the
identity (V-B1). Furthermore, in its definition, the information in the
subsystem $\mathcal{H}_{K}$ is decoupled from the remaining subsystems. Since
$U_{\mathcal{P}}$ acts on $\mathcal{H}_{E^{n}}\otimes\mathcal{H}_{\bar{E}}$,
an additional pure ancilla is attached for appropriate comparisons. We aim to
show that this state is close to the action of $(I\otimes U_{\mathcal{P}})$ on
the state $\ket{\hat{\sigma}}\ket{0}_{\bar{E}}$.

Step 2.2: Closeness of $(I\otimes
U_{\mathcal{P}})\ket{\hat{\sigma}}\ket{0}_{\bar{E}}$ and
$|{\hat{\zeta}}\rangle:$
We begin by simplifying $(I\otimes
U_{\mathcal{P}})\ket{\hat{\sigma}}\ket{0}_{\bar{E}}$ as

$\displaystyle(I\otimes
U_{\mathcal{P}})\ket{\hat{\sigma}}\ket{0}_{\bar{E}}=\sum_{m\in\mathcal{M}^{\prime}}\sum_{k\in\mathcal{I}_{\mathcal{E}}^{(m)}}\frac{1}{\sqrt{(1-\sqrt{\epsilon})|\mathcal{M}||\mathcal{K}|}}e^{i\alpha_{k}^{(m)}}|{\phi_{k}^{(m)}}\rangle\otimes\ket{m}_{M},$

where

$\displaystyle|{\phi_{k}^{(m)}}\rangle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{k^{\prime}\in\mathcal{I}_{\mathcal{E}}^{(m)}\cup\\{K\\}}\frac{(I\otimes\Pi_{k^{\prime}}^{(m)}M_{m,k})\ket{\psi_{\rho}}\ket{0}_{\bar{E}}}{\sqrt{\lambda_{m,k}}}\otimes\ket{k^{\prime}-k}_{K},\mbox{
for all }k\in\mathcal{I}_{\mathcal{E}}^{(m)}\mbox{ and
}m\in\mathcal{M}^{\prime}.$ (54)

Similarly, let

$\displaystyle|{\hat{\zeta}}\rangle=\sum_{m,k}\frac{1}{\sqrt{(1-\sqrt{\epsilon})|\mathcal{M}||\mathcal{K}|}}e^{i\beta^{(m)}_{k}}|{\chi_{k}^{(m)}}\rangle\otimes\ket{m}_{M},\;\;\;\;|{\chi_{k}^{(m)}}\rangle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\frac{(I\otimes
M_{m,k})\ket{\psi_{\rho}}\ket{0}_{\bar{E}}}{\sqrt{\lambda_{m,k}}}\otimes\ket{0}_{K},$
(55)

for all $m\in\mathcal{M}^{\prime}$ and $k\in\mathcal{I}_{\mathcal{E}}^{(m)}$,
and the phases $\\{\beta_{k}^{(m)}\\}$ are the same phases incorporated in the
construction of the decoding isometry $U_{\mathcal{D}}$. Further, from (44),
we know for all $m\in\mathcal{M}^{\prime}$,

$\displaystyle\frac{1}{K^{\prime}_{m}}\sum_{k\in\mathcal{I}_{\mathcal{E}}^{(m)}}\langle{\phi_{k}^{(m)}}|{\chi_{k}^{(m)}}\rangle=\frac{1}{K^{\prime}_{m}}\sum_{k\in\mathcal{I}_{\mathcal{E}}^{(m)}}\Tr{\Xi^{(m)}_{k}\tau_{k}^{(m)}}.$
(56)

Now the fidelity between $|{\hat{\zeta}}\rangle\mbox{ and }(I\otimes
U_{\mathcal{P}})\ket{\hat{\sigma}}\ket{0}_{\bar{E}}$ can be written as

$\displaystyle\sqrt{F\left((I\otimes
U_{\mathcal{P}})\ket{\hat{\sigma}}\ket{0}_{\bar{E}},|{\hat{\zeta}}\rangle\right)}=\frac{1}{M^{\prime}}\left|\sum_{m\in\mathcal{M}^{\prime}}\bra{\phi_{m}}\ket{\chi_{m}}\right|,$
(57)

where, for all $m\in\mathcal{M}^{\prime}$,

$\displaystyle\ket{\phi_{m}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}c\sum_{k\in\mathcal{I}_{\mathcal{E}}^{(m)}}e^{i\alpha_{k}^{(m)}}|{\phi_{k}^{(m)}}\rangle\mbox{\quad
and\quad}\ket{\chi_{m}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}c\sum_{k\in\mathcal{I}_{\mathcal{E}}^{(m)}}e^{i\beta_{k}^{(m)}}|{\chi_{k}^{(m)}}\rangle,$

and
$c\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sqrt{\frac{M^{\prime}}{(1-\sqrt{\epsilon})|\mathcal{M}||\mathcal{K}|}}$.
Toward a lower bound on the fidelity, we provide the following proposition.

######  Proposition 3.

For any $\epsilon\in(0,1)$, there exists phases $\\{\alpha_{k}^{(m)}\\},$ and
$\\{{\beta_{k}^{(m)}}\\}$ such that

$\displaystyle\left|\frac{1}{M^{\prime}}\sum_{m\in\mathcal{M}^{\prime}}\bra{\phi_{m}}\ket{\chi_{m}}\right|\geq
1-2\sqrt{\epsilon},$ (58)

for all sufficiently small $\delta>0$ and all sufficiently large $n$.

###### Proof.

The proof is provided in Appendix A-D. ∎

Observe that, using the relation in Lemma 1, and the result of Proposition 3
and (57), we get

$\displaystyle\left\|(I\otimes
U_{\mathcal{P}}){\hat{\sigma}}\otimes\outerproduct{0}{0}_{\bar{E}}(I\otimes
U_{\mathcal{P}})^{\dagger}-\hat{\zeta}\right\|_{1}\leq
2\sqrt{1-F\left((U_{\mathcal{P}}\otimes
I)\ket{\hat{\sigma}}\ket{0}_{\bar{E}},|{\hat{\zeta}}\rangle\right)}\leq
4\sqrt[4]{\epsilon},$ (S-2)

for all sufficiently large $n$, and sufficiently small $\eta,\delta>0$.
Observe that
$|{\hat{\zeta}}\rangle^{B_{R}^{n}E^{n}MK}=|{\hat{\zeta}}\rangle^{B_{R}^{n}E^{n}M}\otimes\ket{0}_{K\bar{E}}$,
and hence $|{\hat{\zeta}}\rangle^{B_{R}^{n}E^{n}M}$ remains pure after partial
tracing over the subsystem $\mathcal{H}_{K}\otimes\mathcal{H}_{\bar{E}}$.
Finally, we are left with showing the closeness of the state $(I\otimes
U_{\mathcal{D}})|{\hat{\zeta}}\rangle^{B_{R}^{n}E^{n}M}$ with the state
$\ket{\zeta}^{B_{R}^{n}E^{n}A^{n}}$.

Step 2.3: Closeness of $(I\otimes
U_{\mathcal{D}})|{\hat{\zeta}}\rangle^{B_{R}^{n}E^{n}M}$ and
$\ket{\zeta}^{B_{R}^{n}E^{n}A^{n}}:$
We begin by defining $\sigma^{A^{n}}$ as

$\displaystyle\sigma^{A^{n}}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\Tr_{B_{R}^{n}E^{n}}\\{(I\otimes
U_{\mathcal{D}})\hat{\zeta}^{B_{R}^{n}E^{n}M}(I\otimes
U_{\mathcal{D}})^{\dagger}\\},$

and perform the simplification

$\displaystyle\sigma^{A^{n}}$
$\displaystyle=U_{\mathcal{D}}\bigg{(}\sum_{m,m^{\prime}}\sum_{k,k^{\prime}}\frac{1}{{(1-\sqrt{\epsilon})|\mathcal{M}||\mathcal{K}|}}e^{-i(\beta_{k}^{(m)}-\beta_{k^{\prime}}^{(m^{\prime})})}\frac{\Tr\big(M_{mk}\rho^{B}M_{m^{\prime}k^{\prime}}^{\dagger}\big{missing})}{\sqrt{\lambda_{mk}\lambda_{m^{\prime}k^{\prime}}}}|{m}\rangle\langle{m^{\prime}}|\bigg{)}U_{\mathcal{D}}^{\dagger}$
$\displaystyle=U_{\mathcal{D}}\left(\sum_{m}\frac{K^{\prime}_{m}}{{{(1-\sqrt{\epsilon})|\mathcal{M}||\mathcal{K}|}}}\outerproduct{m}{m}\right)U_{\mathcal{D}}^{\dagger}=\sum_{m}\frac{K^{\prime}_{m}}{{{(1-\sqrt{\epsilon})|\mathcal{M}||\mathcal{K}|}}}\outerproduct{b^{n}(m)}{b^{n}(m)}^{A^{n}},$
(59)

where the second equality uses (V-B1) and the crucial condition that the
codebook $\mathcal{C}_{\mathcal{E}}$ obtained after expurgation is distinct,
and last equality defines $\ket{b^{n}(m)}^{A^{n}}$ as

$\ket{b^{n}(m)}^{A^{n}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\frac{1}{\sqrt{K^{\prime}_{m}}}\sum_{k\in\mathcal{I}_{\mathcal{E}}^{(m)}}e^{-i\beta_{k}^{(m)}}\ket{a^{n}(m,k)}^{A^{n}},$
(60)

for all $m\in\mathcal{M}^{\prime}$. This implies, we can write the canonical
purification of $\sigma^{A^{n}}$ as

$\displaystyle\ket{\psi_{\sigma}}^{A_{R}^{n}A^{n}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(I_{A_{R}^{n}}\otimes\sqrt{\sigma^{A^{n}}})$
$\displaystyle\ket{\Gamma^{\otimes
n}}^{A_{R}^{n}A^{n}}=\sum_{m}\sqrt{\frac{K^{\prime}_{m}}{{{(1-\sqrt{\epsilon})|\mathcal{M}||\mathcal{K}|}}}}\ket{b^{n}(m)}^{A_{R}^{n}}\otimes\ket{b^{n}(m)}^{A^{n}}$
$\displaystyle=(I\otimes
U_{\mathcal{D}}^{A^{n}})\sum_{m}\sqrt{\frac{K^{\prime}_{m}}{{{(1-\sqrt{\epsilon})|\mathcal{M}||\mathcal{K}|}}}}\ket{b^{n}(m)}^{A_{R}^{n}}\otimes\ket{m}^{M},$
(61)

where the first equality follows by defining
$\ket{b^{n}(m)}^{A_{R}^{n}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(I_{A_{R}^{n}}\otimes\bra{b^{n}(m)}^{A^{n}})\ket{\Gamma^{\otimes
n}}^{A_{R}^{n}A^{n}}$. Using the relation from (25) and definition (60), we
can write

$\displaystyle W^{\otimes n}\ket{b^{n}(m)}^{A_{R}^{n}}$
$\displaystyle=\\!\frac{1}{\sqrt{K^{\prime}_{m}}}\\!\sum_{k\in\mathcal{I}_{\mathcal{E}}^{(m)}}\\!\\!\\!e^{i\beta_{k}^{(m)}}W^{\otimes
n}\ket{a^{n}(m,k)}^{A_{R}^{n}}=\\!\frac{1}{\sqrt{K^{\prime}_{m}}}\\!\sum_{k\in\mathcal{I}_{\mathcal{E}}^{(m)}}\\!\\!\\!e^{i\beta_{k}^{(m)}}\frac{(I_{{B_{R}}}\\!\otimes\\!M_{m,k})\ket{\psi_{\rho}^{\otimes
n}}^{B_{R}^{n}B^{n}}}{\sqrt{\lambda_{m,k}^{A}}},$

for all $m\in\mathcal{M}^{\prime}$, which gives

$\displaystyle(W^{\otimes n}\otimes
I_{A^{n}})\ket{\psi_{\sigma}}^{A_{R}^{n}A^{n}}$ $\displaystyle=(I\otimes
U_{\mathcal{D}}^{A^{n}})\sum_{m}\sqrt{\frac{K^{\prime}_{m}}{{{(1-\sqrt{\epsilon})|\mathcal{M}||\mathcal{K}|}}}}W^{\otimes
n}\ket{b^{n}(m)}^{A_{R}^{n}}\otimes\ket{m}^{M}$ $\displaystyle=(I\otimes
U_{\mathcal{D}}^{A^{n}})\sum_{m,k}\frac{1}{\sqrt{(1-\sqrt{\epsilon})|\mathcal{M}||\mathcal{K}|}}e^{i\beta_{k}^{(m)}}\frac{(I_{{B_{R}}}\otimes
M_{m,k})\ket{\psi_{\rho}^{\otimes
n}}^{B_{R}^{n}B^{n}}}{\sqrt{\lambda_{m,k}^{A}}}\otimes\ket{m}^{M}$
$\displaystyle=(I\otimes
U_{\mathcal{D}}^{A^{n}})|{\hat{\zeta}}\rangle^{B_{R}^{n}E^{n}M},$ (62)

where the last equality follows from the definition of
$|{\hat{\zeta}}\rangle^{B_{R}^{n}E^{n}M}$ in (53). Observing that

$\ket{\zeta}^{B_{R}^{n}E^{n}A^{n}}=(W^{\otimes n}\otimes
I_{A^{n}})\ket{\psi_{\omega}}^{A_{R}^{n}A^{n}},$

we are left with showing the closeness of
$\ket{\psi_{\sigma}}^{A_{R}^{n}A^{n}}$ and
$\ket{\psi_{\omega}}^{A_{R}^{n}A^{n}}$. Using (S-1) and (S-2), the triangle
inequality, the monotonicity of trace distance, and identification of
appropriate purifications, we obtain for all sufficiently large $n$ and
sufficiently small $\eta,\delta>0$,

$\displaystyle\|\omega^{A^{n}}-\sigma^{A^{n}}\|_{1}$
$\displaystyle\leq\|\omega^{B_{R}^{n}E^{n}A^{n}}-(I\otimes
U_{\mathcal{D}})\hat{\zeta}^{B_{R}^{n}E^{n}M}(I\otimes
U_{\mathcal{D}})^{\dagger}\|_{1}\leq 14\sqrt[4]{\epsilon}.$

This implies,

$\displaystyle\left\|(I\otimes
U_{\mathcal{D}}^{A^{n}})\hat{\zeta}^{B_{R}^{n}E^{n}M}(I\otimes
U_{\mathcal{D}}^{A^{n}})^{\dagger}-\zeta^{B_{R}^{n}E^{n}A^{n}}\right\|_{1}$
$\displaystyle\overset{a}{=}\left\|{\psi_{\sigma}}^{A^{n}A_{R}^{n}}-{\psi_{\omega}}^{A^{n}A_{R}^{n}}\right\|_{1}$
$\displaystyle\overset{b}{\leq}2{\sqrt{1-F(|{\psi_{\sigma}}\rangle^{A^{n}A_{R}^{n}},|{\psi_{\omega}}\rangle^{A^{n}A_{R}^{n}})}}$
$\displaystyle{\overset{c}{\leq}2\sqrt{\|\omega^{A^{n}}-\sigma^{A^{n}}\|_{1}}\leq
2\sqrt{{14\sqrt[4]{\epsilon}}}\leq 8\sqrt[8]{\epsilon},}$ (S-3)

for all sufficiently large $n$ and sufficiently small $\eta,\delta>0$, where
(a) follows from the isometric invariance of trace distance, (b) uses Lemma 1,
and (c) uses Lemma 2, which concludes this step.

In summary, combining results of (S-0), (S-1), (S-2) and (S-3), we have showed
that there exist a code $\mathcal{C}$ satisfying
$G\leq{14\sqrt[4]{\epsilon}+8\sqrt[8]{\epsilon}}$ with the following rate
constraints:

$\displaystyle S(\lambda_{a}^{A})>\frac{1}{n}\left(\log M+\log K\right)$
$\displaystyle>\chi({\lambda_{a}^{{B_{R}}},\hat{\rho}_{a}^{{B_{R}}}}),\;\;\frac{1}{n}\log
K<\chi(\\{\lambda_{a}^{E},\tau_{a}^{E}\\}),\;\;\frac{1}{n}\log{M}\geq
0,\quad\frac{1}{n}\log K\geq 0,$

for all sufficiently large $n$ and sufficiently small $\eta,\delta>0$, where
we have also included the necessary non-negativity constraints. Eliminating
$\frac{1}{n}\log{K}$ using Fourier-Motzkin elimination [60] gives

$\frac{1}{n}\log{M}>\chi({\lambda_{a}^{{B_{R}}},\hat{\rho}_{a}^{{B_{R}}}})-\chi(\\{\lambda_{a}^{E},\tau_{a}^{E}\\}),\mbox{\quad
and\quad}\frac{1}{n}\log{M}\geq 0,$

where we remove the redundant constraints. This completes the proof.

######  Remark 7 (Zero performance rate).

The coherent information $I_{c}(\mathcal{N}_{W},\rho^{A_{R}})$ is negative
when the Holevo information quantities are such that
$\chi({\lambda_{a}^{{B_{R}}},\hat{\rho}_{a}^{{B_{R}}}})<\chi(\\{\lambda_{a}^{E},\tau_{a}^{E}\\})$.
The asymptotic performance limit for these situations is zero, according to
the statement of the theorem. We must therefore demonstrate that a rate of
zero is feasible. To put it another way, we must construct a protocol (see
Definition (3)) that satisfies (2) with a rate that can be made arbitrarily
close to zero. The constraints imposed by the preceding proof are still met if
we select $M=1$ and $\frac{1}{n}\log
K=\chi({\lambda_{a}^{{B_{R}}},\hat{\rho}_{a}^{{B_{R}}}})+\delta_{0}<\chi(\\{\lambda_{a}^{E},\tau_{a}^{E}\\})$,
while achieving a rate of $log(2)/n$, for a sufficiently small $\delta_{0}$.
This rate $(1/n)$ can be made arbitrarily close to zero (i.e., smaller than
the provided $\epsilon$) for any given $\epsilon$, for all sufficiently large
$n$ and sufficiently small $\eta,\delta>0.$ Similarly, when the coherent
information $I_{c}(\mathcal{N}_{W},\rho^{A_{R}})$ is exactly zero, i.e.,
$\chi({\lambda_{a}^{{B_{R}}},\hat{\rho}_{a}^{{B_{R}}}})=\chi(\\{\lambda_{a}^{E},\tau_{a}^{E}\\})$,
we choose $M$ and $K$ such that $\frac{1}{n}\log M=2\delta_{0}$, and
$\frac{1}{n}\log K=\chi(\\{\lambda_{a}^{E},\tau_{a}^{E}\\})-\delta_{0}$. This
gives a rate of $2\delta_{0}$ which can be again made arbitrarily close to
zero. Therefore, even though the coherent information is not necessarily
positive, the rate in the theorem can still be achieved.

###  V-C Proof of Converse

Let $R$ be an achievable rate. Then from Definition 4, given a triple
($\rho_{B}$, $\mathcal{H}_{A}$, $\mathcal{N}_{W}$), for all $\epsilon>0$, and
all sufficiently large $n$, there exists $(n,\Theta)$ lossy compression
protocol with an encoding CPTP map $\mathcal{N}_{\mathcal{E}}^{(n)}$ and a
decoding CPTP map $\mathcal{N}_{\mathcal{D}}^{(n)}$ that satisfies the
following constraints:

$\displaystyle c_{0}:\frac{1}{n}\log{\Theta}\leq R+\epsilon,\mbox{\quad
and\quad}c_{1}:\|\omega^{B_{R}^{n}A^{n}}-\upsilon^{B_{R}^{n}A^{n}}\|_{1}\leq\epsilon,$
(63)

where
$\omega^{B_{R}^{n}A^{n}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(I\otimes\mathcal{N}_{\mathcal{D}}^{(n)})(I\otimes\mathcal{N}_{\mathcal{E}}^{(n)})(\ket{\psi_{\rho}}^{B_{R}^{n}B^{n}})$,

$\upsilon^{B_{R}^{n}A^{n}}=\Tr_{E^{n}}\left\\{\upsilon^{{B}^{n}_{R}A^{n}E^{n}}\right\\}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\Tr_{E^{n}}\left\\{(W^{\otimes
n}\otimes I)\Psi_{\omega}^{A_{R}^{n}A^{n}}(W^{\otimes n}\otimes
I)^{\dagger}\right\\},$

and $\ket{\psi_{\omega}}^{A^{n}{A_{R}}^{n}}$ is the canonical purification of
$\omega^{A^{n}}$, and $W$ is the Stinespring’s dilation of the CPTP map
$\mathcal{N}_{W}$. Let
$\omega^{A_{R}^{n}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\Tr_{A^{n}}(\Psi_{\omega}^{A^{n}A_{R}^{n}})$.

Step 1: Quantum Data Processing Inequality: Let $M$ denote the quantum state
at the output of the encoder. Let
$V_{\mathcal{E}}^{(n)}:\mathcal{H}_{B^{n}}\rightarrow\mathcal{H}_{M}\otimes\mathcal{H}_{\tilde{E}_{1}}$
and
$V_{\mathcal{D}}^{(n)}:\mathcal{H}_{M}\rightarrow\mathcal{H}_{A^{n}}\otimes\mathcal{H}_{\tilde{E}_{2}}$
be Stinesping dilations of encoding and decoding maps
$\mathcal{N}_{\mathcal{E}}^{(n)}$ and $\mathcal{N}_{\mathcal{D}}^{(n)}$,
respectively, such that
$\dim(\mathcal{H}_{\tilde{E}_{1}})\geq\dim(\mathcal{H}_{M})$ and
$\dim(\mathcal{H}_{\tilde{E}_{2}})\geq\dim(\mathcal{H}_{A^{n}})$, as shown in
Figure 6(a). Let

$\displaystyle{\omega_{1}}^{B_{R}^{n}M\tilde{E}_{1}}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(I_{B_{R}^{n}}\otimes
V_{\mathcal{E}}^{(n)})(\Psi_{\rho}^{B_{R}^{n}B^{n}})(I_{B_{R}^{n}}\otimes
V_{\mathcal{E}}^{(n)})^{\dagger}$
$\displaystyle\quad{\omega}^{B_{R}^{n}\tilde{E}_{1}\tilde{E}_{2}A^{n}}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(I_{B_{R}^{n}\tilde{E}_{1}}\otimes
V_{\mathcal{D}}^{(n)})({\omega_{1}}^{B_{R}^{n}\tilde{E}_{1}M})(I_{B_{R}^{n}\tilde{E}_{1}}\otimes
V_{\mathcal{D}}^{(n)})^{\dagger}.$ (64)

Let $|\psi_{\omega_{1}}\rangle^{M_{R}M}$ denote the canonical purification of
the quantum state $\omega_{1}^{M}$. Let
$W_{\mathcal{E}}^{(n)}:\mathcal{H}_{M_{R}}\rightarrow\mathcal{H}_{B_{R}^{n}}\otimes\mathcal{H}_{\tilde{E}_{1}}$
denote the posterior reference isometry (see Definition 1) of
$V_{\mathcal{E}}^{(n)}$ with respect to $\omega_{1}^{M}$, as shown in Figure
6(b). Moreover, let
$W_{\mathcal{D}}^{(n)}:\mathcal{H}_{A_{R}^{n}}\rightarrow\mathcal{H}_{M_{R}}\otimes\mathcal{H}_{\tilde{E}_{2}}$
denote the posterior reference isometry of $V_{\mathcal{D}}^{(n)}$ with
respect to $w^{A^{n}}$, as shown in Figure 6(c). Let
$\mathcal{N}_{W_{\mathcal{E}}}(\cdot)=\Tr_{\tilde{E_{1}}}(W_{\mathcal{E}}^{(n)}\cdot(W_{\mathcal{E}}^{(n)})^{\dagger})$
and
$\mathcal{N}_{W_{\mathcal{D}}}(\cdot)=\Tr_{\tilde{E_{2}}}(W_{\mathcal{D}}^{(n)}\cdot(W_{\mathcal{D}}^{(n)})^{\dagger})$
be the induced CPTP maps. Let

$\tilde{\omega}_{1}^{M_{R}\tilde{E}_{2}A^{n}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(W_{\mathcal{D}}^{(n)}\otimes
I_{A^{n}})(\Psi_{\omega}^{A_{R}^{n}A^{n}})(W_{\mathcal{D}}^{(n)}\otimes
I_{A^{n}})^{\dagger}.$

Using the quantum data processing inequality for coherent information [50,
Theorem 11.3.2], we obtain

$I_{c}(\mathcal{N}_{W_{\mathcal{D}}},\omega^{A_{R}^{n}})\geq
I_{c}(\mathcal{N}_{W_{\mathcal{E}}}\circ\mathcal{N}_{W_{\mathcal{D}}},\omega^{A_{R}^{n}}).$

Expanding the coherent information in terms of Von Neuman entropy, we get

$S(M_{R})_{\tilde{\omega}_{1}}-S(\tilde{E}_{2})_{\tilde{\omega}_{1}}\geq
S(B_{R}^{n})_{\omega}-S(\tilde{E}_{1}\tilde{E}_{2})_{\omega},$

which implies that

$S(M)_{\omega_{1}}\geq S(B_{R}^{n})_{\omega}-S(B_{R}^{n}A^{n})_{\omega}.$ (65)

Step 2: Implication of the constraints $c_{0}$ and $c_{1}$: Consider the
following sequence of inequalities:

Figure 6: Lossy quantum source coding protocol and the associated CPTP maps
and their Stinespring dilations.

$\displaystyle nR$ $\displaystyle\geq\log\Theta-n\epsilon\geq
S(M)_{\omega_{1}}-n\epsilon$ (66)
$\displaystyle\overset{a}{\geq}S({B}^{n}_{R})_{\omega}-S({B}^{n}_{R},A^{n})_{\omega}-n\epsilon$
(67)
$\displaystyle\overset{b}{\geq}S({B}^{n}_{R})_{\omega}-S({B}^{n}_{R},A^{n})_{\upsilon}-n\epsilon-n\tilde{\epsilon}_{1}$
(68)
$\displaystyle=S({B}^{n}_{R})_{\omega}-S(E^{n})_{\upsilon}-n\epsilon-n\tilde{\epsilon}_{1}$
(69)
$\displaystyle\overset{c}{\geq}S({B}^{n}_{R})_{\omega}-\sum_{i=1}^{n}S(E_{i})_{\upsilon}-n\epsilon-n\tilde{\epsilon}_{1}$
(70)
$\displaystyle\overset{d}{=}\sum_{i=1}^{n}S({B_{R}}_{i})_{\omega}-\sum_{i=1}^{n}S(E_{i})_{\upsilon}-n\epsilon-n\tilde{\epsilon}_{1}$
(71)
$\displaystyle\overset{e}{\geq}\sum_{i=1}^{n}S({B_{R}}_{i})_{\upsilon}-\sum_{i=1}^{n}S(E_{i})_{\upsilon}-n\epsilon-n\tilde{\epsilon}_{1}-n\tilde{\epsilon}_{2}$
(72)
$\displaystyle\overset{f}{=}\sum_{i=1}^{n}I_{c}(\mathcal{N}_{W},\omega^{{A}_{Ri}})-n\epsilon-n\tilde{\epsilon}_{1}-n\tilde{\epsilon}_{2}$
(73)
$\displaystyle\overset{g}{\geq}n\min_{\rho^{{A_{R}}}\in\mathcal{D}(\mathcal{H}_{{A_{R}}}):\|\rho^{{B_{R}}}-\mathcal{N}_{W}(\rho^{{A_{R}}})\|_{1}\leq\epsilon}I_{c}(\mathcal{N}_{W},\rho^{{A}_{R}})-n\epsilon-n\tilde{\epsilon}_{1}-n\tilde{\epsilon}_{2},$
(74)

where the inequalities are argued as follows. (a) follows from (65). (b)
follows from the condition $c_{1}$ and the Fannes-Audenaert inequality [50,
Theorem 11.10.1] by defining
$\tilde{\epsilon}_{1}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\epsilon\log|\mathcal{H}_{A}||\mathcal{H}_{B}|+h_{b}(\epsilon)$.
(c) follows from the subadditivity of entropy. (d) follows from the
memoylessness of the quantum source. (e) follows from condition $c_{2}$ and
the Fannes-Audenaert inequality [50, Theorem 11.10.1], where condition

$c_{2}:\|\omega^{{B_{R}}_{i}}-\upsilon^{{B_{R}}_{i}}\|_{1}\leq\epsilon,\ \
\forall\;1\leq i\leq n,$

is implied by $c_{1}$ using the monotonicity of trace distance with respect to
partial trace. $\tilde{\epsilon}_{2}$ is defined as
$\tilde{\epsilon}_{2}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\epsilon\log|\mathcal{H}_{B}|+h_{b}(\epsilon)$.
(f) follows from the fact that
$\upsilon^{{B_{R}}_{i}E_{i}}=W\omega^{{A}_{Ri}}W^{\dagger}$. (g) follows from
condition $c_{2}$ which can also be stated as

$c_{2}:\|\rho^{{B_{R}}}-\mathcal{N}_{W}(\omega^{{A_{R}}_{i}})\|_{1}\leq\epsilon,\
\ \forall\;1\leq i\leq n,$

and the fact that coherent information is continuous, and the constraint set
is closed and bounded. The continuity follows from the following arguments:
for the fixed CPTP map $\mathcal{N}_{W}$, let a function
$f:\mathcal{D}(\mathcal{H}_{{A_{R}}})\rightarrow\mathbb{R}$ be defined as
$f(\rho^{{A_{R}}})=I_{c}(\mathcal{N}_{W},\rho^{{A}_{R}}).$ One can establish
the continuity of $f$ for a fixed $\mathcal{N}_{W}$ by writing
$I_{c}(\mathcal{N}_{W},\rho^{{A}_{R}})=S({B_{R}})_{W\rho^{{A_{R}}}W^{\dagger}}-S(E)_{W\rho^{{A_{R}}}W^{\dagger}},$
and using the Fannes–Audenaert Inequality [50, Theorem 11.10.2], where $W$ is
the Stinespring’s extension of the given CPTP map $\mathcal{N}_{W}$.

Step 3: Continuity Argument: We have shown that

$R\in\bigcap_{\epsilon>0}\mathcal{I}_{\epsilon},$

where we have defined for all $\epsilon\geq 0$,

$\displaystyle\mathcal{I}_{\epsilon}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\left\\{{R:\exists\;\rho^{{A}_{R}}\in\mathcal{S}_{\epsilon}(\rho^{B},\mathcal{N}_{W})\mbox{
such that }R\geq I_{c}(\mathcal{N}_{W},\rho^{{A}_{R}})-g(\epsilon)}\right\\},$
(75)

and

$\displaystyle\mathcal{S}_{\epsilon}(\rho^{B},\mathcal{N}_{W})$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\left\\{{\rho^{{A}_{R}}\in\mathcal{D}(\mathcal{H}_{{A_{R}}}):\|\mathcal{N}_{W}(\rho^{{A_{R}}})-\rho^{{B_{R}}}\|_{1}\leq\epsilon}\right\\},$
(76)

$g(\epsilon)\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\epsilon+\tilde{\epsilon}_{1}+\tilde{\epsilon}_{2}.$
Condition $c_{2}$ ensures that the set $\mathcal{S}_{\epsilon}$ is non-empty
for all $\epsilon>0$. Now, by arguing continuity of $\mathcal{I}_{\epsilon}$
at $\epsilon=0$, we obtain the desired result.

######  Lemma 5.

For the above definitions of $\mathcal{S}_{\epsilon}$ and
$\mathcal{I}_{\epsilon}$, we have $\mathcal{S}_{0}(\rho^{B},\mathcal{N}_{W})$
non-empty, and

$\mathcal{I}_{0}=\bigcap_{\epsilon>0}\mathcal{I}_{\epsilon}.$

###### Proof.

This is a standard argument used in the literature [61, 62, 51]. A proof is
provided in Appendix A-E for completeness. ∎

This completes the proof.

##  VI Proof of Theorem 2

###  VI-A Proof of Achievability

For a given $({\rho^{B}},\mathsf{X},\mathcal{W})$ QC source coding setup, we
choose a reconstruction distribution
$P_{X}\in\mathcal{A}({\rho^{B}},\mathcal{W})$. Toward specifying the POVM
$\Gamma^{(n)}$ and a decoding map
$f:\left\\{{1,2,\cdots,\Theta}\right\\}\rightarrow\mathsf{X}^{n}$, we
construct a codebook $\mathcal{C}$. From now on, we let $\Theta=2^{nR}$.

####  VI-A1 Codebook Design

We generate a codebook $\mathcal{C}$ consisting of $n$-length codewords by
randomly and independently selecting $2^{nR}$ sequences
$\left\\{{X^{n}(m)}\right\\}_{m\in[2^{nR}]}$ according to the following pruned
distribution:

$\displaystyle\mathbb{P}(X^{n}(m)=x^{n})=\left\\{\begin{array}[]{cc}\dfrac{P_{X}^{n}(x^{n})}{(1-\varepsilon)}&\mbox{for}\quad
x^{n}\in\mathcal{T}_{{\delta}}^{(n)}({X})\\\
0&\quad\mbox{otherwise}\end{array}\right.\\!\\!,$ (79)

where $P_{X}^{n}(x^{n})=\prod_{i=1}^{n}P_{X}(x_{i})$,
$\mathcal{T}_{{\delta}}^{(n)}({X})$ is the $\delta$-typical set corresponding
to the distribution ${P}_{X}$ on the set $\mathsf{X}$, and
$\varepsilon(\delta,n)\triangleq\sum_{x^{n}\not\in\mathcal{T}_{{\delta}}^{(n)}({X})}P_{X}^{n}(x^{n})$.
Note that $\varepsilon(\delta,n)\searrow 0$ as $n\rightarrow\infty$ and for
all sufficiently small $\delta>0$. The generated codebook $\mathcal{C}$ is
revealed to both the encoder and decoder before the QC lossy source
compression protocol begins.

####  VI-A2 Construction of POVM

We use Winter’s POVM construction [27]. Let $\pi_{{\rho^{B}}}$ and
$\pi_{x^{n}}$ denote the $\delta$-typical and conditional $\delta$-typical
projectors defined as in [50, Def. 15.1.3] and [50, Def. 15.2.4], with respect
to ${\rho^{B}}$ and $\mathcal{W}$, respectively. Consider the following
positive operators with a trace of less than one, and we exploit the random
selection of these operators to construct the sub-POVM
$\left\\{{A_{x^{n}}}\right\\}$. For all
$x^{n}\in\mathcal{T}_{{\delta}}^{(n)}({X})$, define:

$\displaystyle\xi_{x^{n}}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\pi_{{\rho^{B}}}\pi_{x^{n}}\mathcal{W}_{x^{n}}\pi_{x^{n}}\pi_{{\rho^{B}}},$
(80)

and $\xi_{x^{n}}=0$ for $x^{n}\not\in\mathcal{T}_{{\delta}}^{(n)}({X})$, where
$\mathcal{W}_{x^{n}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\bigotimes_{i}\mathcal{W}_{x_{i}}$.
We now define $\xi$ as the expectation of $\xi_{x^{n}}$ with respect to the
pruned distribution $\mathbb{P}$ as defined in (79):

$\xi\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\mathbb{E}_{\mathbb{P}}\left[{\xi_{x^{n}}}\right]=\sum_{x^{n}\in\mathcal{T}_{{\delta}}^{(n)}({X})}\frac{P_{X}^{n}(x^{n})}{1-\varepsilon}\xi_{x^{n}}.$

Let $\hat{\pi}$ be the cut-off projector onto the subspaces spanned by the
eigenstates of $\xi$ with eigenvalues greater than $\epsilon d$, where
$d\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}2^{-n(H({\rho^{B}})+\delta_{1})}$
and $\delta_{1}$ will be specified later. With the above notation, we define

$\displaystyle\tilde{\rho}_{x^{n}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\hat{\pi}\xi_{x^{n}}\hat{\pi}\quad\mbox{
and
}\quad\tilde{\rho}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\mathbb{E}_{\mathbb{P}}\left[{\tilde{\rho}_{x^{n}}}\right]=\hat{\pi}\xi\hat{\pi}.$
(81)

Using the Average Gentle Measurement Lemma [50, Lemma 9.4.3], for any given
$\epsilon\in(0,1)$, and all sufficiently large $n$ and all sufficiently small
$\delta$, we have

$\displaystyle\sum_{x^{n}\in\mathsf{X}^{n}}\frac{P_{X}^{n}(x^{n})}{1-\varepsilon}\norm{\tilde{\rho}_{x^{n}}-\mathcal{W}_{x^{n}}}_{1}\leq\epsilon.$
(82)

Detailed proof of the above statement can be found in [58, Eq. 35]. Using the
above definitions, for all $x^{n}\in\mathsf{X}^{n}$, we construct the
operators,

$A_{x^{n}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\gamma_{x^{n}}\
\sqrt{{\rho^{B}}^{\otimes
n}}^{-1}\tilde{\rho}_{x^{n}}\sqrt{{\rho^{B}}^{\otimes n}}^{-1},\text{ where
}\gamma_{x^{n}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\frac{1}{2^{nR}}\frac{(1-\varepsilon)}{(1+\eta)}\sum_{m=1}^{2^{nR}}\mathbbm{1}_{\left\\{{X^{n}(m)=x^{n}}\right\\}},$

and $\eta\in\left({0,1}\right)$ is a parameter that determines the probability
of not obtaining a sub-POVM. Let $\mathbbm{1}_{\left\\{{\mbox{sP}}\right\\}}$
denote the indicator random variable corresponding to the event that
$\\{A_{x^{n}}\colon x^{n}\in\mathcal{T}_{{\delta}}^{(n)}({X})\\}$ forms a sub-
POVM. If $\mathbbm{1}_{\left\\{{\mbox{sP}}\right\\}}=1$, then construct sub-
POVM $\Gamma^{(n)}$ as follows:

$\displaystyle\Gamma^{(n)}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\\{A_{x^{n}}\colon
x^{n}\in\mathcal{T}_{{\delta}}^{(n)}({X})\\}.$ (83)

Since $\Gamma^{(n)}$ is a sub-POVM, we add an extra operator
$A_{x^{n}_{0}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\left({I-\sum_{x^{n}\in\mathcal{T}_{{\delta}}^{(n)}({X})}A_{x^{n}}}\right)$,
associated with an arbitrary sequence
$x^{n}_{0}\in\mathsf{X}^{n}\backslash\mathcal{T}_{{\delta}}^{(n)}({X})$, to
form a valid POVM $\left[{\Gamma^{(n)}}\right]$ with at most $(2^{nR}+1)$
elements. If $\mathbbm{1}_{\left\\{{\mbox{sP}}\right\\}}=0$, then we define
$\Gamma^{(n)}=\left\\{{I}\right\\}$ and associate it with $x^{n}_{0}$. This
defines the POVM and the associated decoder. We now provide a proposition from
[27], which will be helpful later in the analysis.

######  Proposition 4.

For any $\epsilon,\eta\in(0,1)$, for any sufficiently small $\delta>0$, and
sufficiently large $n$, we have
$\mathbb{E}\left[{\mathbbm{1}_{\\{\mbox{sP}\\}}}\right]\geq 1-\epsilon$, if
$R>I(X;B_{{R}})_{\sigma}$, where the quantum mutual information is computed
with respect to the CQ state,

$\sigma^{XB_{{R}}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{x}P_{X}(x)\outerproduct{x}{x}^{X}\otimes\mathcal{W}_{x},$

and $\\{\ket{x}\\}_{\left\\{{x\in\mathsf{X}}\right\\}}$ is an orthonormal
basis for the Hilbert space $\mathcal{H}_{X}$ with
$\dim{(\mathcal{H}_{X})}=|\mathsf{X}|$.

####  VI-A3 Error Analysis

We show that for the above-mentioned POVM and decoder, the sum of unnormalized
post-measurement reference state $\sqrt{{\rho^{B}}^{\otimes
n}}A_{x^{n}}\sqrt{{\rho^{B}}^{\otimes n}}$ is close to the unnormalized
n-product posterior reference state $\Tr(A_{x^{n}}{\rho^{B}}^{\otimes
n})\bigotimes_{i=1}^{n}\mathcal{W}_{x_{i}}$ in the trace distance, averaged
over the random codebook. In other words, we would like to bound the following
error term:

$\mathbb{E}[\Xi(\Gamma^{(n)})]=\mathbb{E}\left[{\sum_{x^{n}}\norm{\sqrt{{\rho^{B}}^{\otimes
n}}A_{x^{n}}\sqrt{{\rho^{B}}^{\otimes n}}-\Tr(A_{x^{n}}{\rho^{B}}^{\otimes
n})\bigotimes_{i=1}^{n}\mathcal{W}_{x_{i}}}_{1}}\right].$

We begin by splitting the error $\Xi(\Gamma^{(n)})$ into two terms using the
indicator function $\mathbbm{1}_{\left\\{{\mbox{sP}}\right\\}}$ as

$\displaystyle\Xi(\Gamma^{(n)})$
$\displaystyle=\mathbbm{1}_{\left\\{{\mbox{sP}}\right\\}}\Xi(\Gamma^{(n)})+\left({1-\mathbbm{1}_{\left\\{{\mbox{sP}}\right\\}}}\right)\Xi(\Gamma^{(n)}),$
$\displaystyle\leq\mathbbm{1}_{\left\\{{\mbox{sP}}\right\\}}\Xi(\Gamma^{(n)})+2\left({1-\mathbbm{1}_{\left\\{{\mbox{sP}}\right\\}}}\right),$
(84)

where (84) follows from upper bounding the trace distance between two density
operators by two, i.e., its maximum value.

Step 1: Isolating the error term induced by not covering

Using the triangle inequality, we now expand the $\Xi(\Gamma^{(n)})$ under the
condition $\mathbbm{1}_{\left\\{{\mbox{sP}}\right\\}}=1$.

$\displaystyle\Xi(\Gamma^{(n)})$ $\displaystyle\leq\\!\\!\\!\\!\\!\\!\\!\\!\
\sum_{x^{n}\in\mathcal{T}_{{\delta}}^{(n)}({X})}\norm{\sqrt{{\rho^{B}}^{\otimes
n}}A_{x^{n}}\sqrt{{\rho^{B}}^{\otimes n}}-\Tr(A_{x^{n}}{\rho^{B}}^{\otimes
n})\mathcal{W}_{x^{n}}}_{1}$ $\displaystyle\hskip
108.405pt+\norm{\sqrt{{\rho^{B}}^{\otimes
n}}A_{x^{n}_{0}}\sqrt{{\rho^{B}}^{\otimes
n}}}_{1}\\!\\!\\!+\Tr(A_{x^{n}_{0}}{\rho^{B}}^{\otimes
n})\norm{\mathcal{W}_{x^{n}_{0}}}_{1},$
$\displaystyle=\\!\\!\\!\\!\\!\\!\\!\\!\
\sum_{x^{n}\in\mathcal{T}_{{\delta}}^{(n)}({X})}\norm{\sqrt{{\rho^{B}}^{\otimes
n}}A_{x^{n}}\sqrt{{\rho^{B}}^{\otimes n}}-\Tr(A_{x^{n}}{\rho^{B}}^{\otimes
n})\mathcal{W}_{x^{n}}}_{1}+2\Tr(A_{x_{0}^{n}}{\rho^{B}}^{\otimes
n})=\zeta+2\tilde{\zeta},$ (85)

where we have defined:

$\displaystyle\zeta$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{x^{n}\in\mathcal{T}_{{\delta}}^{(n)}({X})}\norm{{\sqrt{{\rho^{B}}^{\otimes
n}}A_{x^{n}}\sqrt{{\rho^{B}}^{\otimes n}}}-{\Tr(A_{x^{n}}{\rho^{B}}^{\otimes
n})}\bigotimes_{i=1}^{n}\mathcal{W}_{x_{i}}}_{1},$ $\displaystyle\mbox{ and
}\tilde{\zeta}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\Tr(A_{x^{n}_{0}}{\rho^{B}}^{\otimes
n})=\Tr{\Bigg{(}I-\\!\\!\\!\\!\\!\sum_{x\in\mathcal{T}_{{\delta}}^{(n)}({X})}\\!\\!\\!A_{x^{n}}\Bigg{)}{\rho^{B}}^{\otimes
n}}.$

The error term $\tilde{\zeta}$ captures the error induced by not covering the
$n$-tensored posterior reference state. We provide the following proposition
that bounds this term.

######  Proposition 5.

For all $\epsilon\in(0,1)$, and for all sufficiently small $\eta,\delta>0$,
and sufficiently large $n$, we have
$\mathbb{E}\left[{\mathbbm{1}_{\left\\{{\mbox{sP}}\right\\}}\tilde{\zeta}}\right]\leq\epsilon$.

###### Proof.

The proof is provided in Appendix A-F. ∎

Step 2: Bounding the error induced by covering

We now bound the term $\zeta$, which captures the error induced by covering.
Under the condition $\mathbbm{1}_{\left\\{{\mbox{sP}}\right\\}}=1$, we rewrite
$\zeta$ as

$\displaystyle\zeta$
$\displaystyle=\sum_{x^{n}\in\mathcal{T}_{{\delta}}^{(n)}({X})}\gamma_{x^{n}}\Tr{\tilde{\rho}_{x^{n}}}\norm{\frac{\tilde{\rho}_{x^{n}}}{\Tr{\tilde{\rho}_{x^{n}}}}-\mathcal{W}_{x^{n}}}_{1}.$
(86)

We now provide the following proposition that bounds the error term $\zeta$.

######  Proposition 6.

For all $\epsilon,\eta\in(0,1)$, for all sufficiently small $\delta>0$, and
sufficiently large $n$, we have
$\mathbb{E}\left[{\mathbbm{1}_{\left\\{{\mbox{sP}}\right\\}}\zeta}\right]\leq\epsilon$.

###### Proof.

The proof is provided in Appendix A-G. ∎