# Computing Sum of Sources over a Classical-Quantum MAC

Touheed Anwar Atif1, Arun Padakandla2 and S. Sandeep Pradhan1
Department of Electrical Engineering and Computer Science,
1University of Michigan, Ann Arbor, USA.
2University of Tennessee, Knoxville, USA
Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We consider the problem of communicating a general bivariate function of two
classical sources observed at the encoders of a classical-quantum multiple
access channel. Building on the techniques developed for the case of a
classical channel, we propose and analyze a coding scheme based on coset
codes. The proposed technique enables the decoder recover the desired function
without recovering the sources themselves. We derive a new set of sufficient
conditions that are weaker than the current known for identified examples.
This work is based on a new ensemble of coset codes that are proven to achieve
the capacity of a classical-quantum point-to-point channel.

##  I Introduction

Early research in quantum state discrimination led to the investigation of the
information carrying capacity of quantum states. Suppose Alice - a sender -
can prepare any one of the states in the collection
$\\{\rho_{x}\in\mathcal{D}(\mathcal{H}_{Y}):x\in\mathcal{X}\\}$ and Bob - the
receiver - has to rely on a measurement to infer the label $x$ of the state,
then what is the largest sub-collection $\mathcal{C}\subseteq\mathcal{X}$ of
states that Bob can distinguish perfectly? Studying this question in a
Shannon-theoretic sense, Schumacher, Westmoreland [1] and Holevo [2]
characterized the exponential growth of this sub-collection, thereby
characterizing the capacity of a classical-quantum (CQ) point-to-point (PTP)
channel. In the following years, generalizations of this question with
multiple senders and/or receivers have been studied with an aim of
characterizing the corresponding information carrying capacity of quantum
states in network scenarios [3].

In this work, we consider the problem of computing functions of information
sources over a CQ multiple access channel (MAC). Let
$(\rho_{x_{1}x_{2}}\in\mathcal{D}(\mathcal{H}_{Y}):(x_{1},x_{2})\in\mathcal{X}_{1}\times\mathcal{X}_{2})$
model a CQ-MAC. Sender $j$ \- the party having access to the choice of label
$x_{j}\in\mathcal{X}_{j}$ \- observes a classical information stream
$S_{jt}\in\mathcal{S}_{j}:t\geq 1$. The pairs $(S_{1t},S_{2t}):t\geq 1$ are
independent and identically distributed (IID) with a single-letter joint
distribution $\mathbb{W}_{S_{1}S_{2}}$. The receiver, who is provided with the
prepared quantum state, intends to reconstruct a specific function
$f(S_{1},S_{2})$ of the information observed by the senders. The question of
interest is under what conditions, specified in terms of the CQ-MAC,
$\mathbb{W}_{S_{1}S_{2}}$ and $f$, can the receiver reconstruct the desired
function losslessly?

The conventional approach to characterizing sufficient conditions for this
problem relies on enabling the receiver reconstruct the pair of classical
source sequences. Since the receiver is only interested in recovering the
bivariate function $f$, and not the pair, this approach can be strictly sub-
optimal. Can we exploit this and design a more efficient communication
strategy, thereby weakening the set of sufficient conditions? In this work, we
present one such communication strategy for a general CQ-MAC that is more
efficient than the conventional approach. This strategy is based on
asymptotically good random nested coset codes. We analyze its performance and
derive new sufficient conditions for a general problem instance and identify
examples for which the derived conditions are strictly weaker.

Figure 1: CQ-MAC used for computing sum of classical sources.

Our findings here are built on the ideas developed in the classical setting.
Focusing on a source coding formulation, i.e. a noiseless MAC, Körner and
Marton [4] devised an ingenious coding technique that enabled the receiver
recover the sum of the sources without recovering either source. In [5], the
linearity of the Körner-Marton (KM) source coding map was further exploited to
enable the receiver recover the sum of the sources using only the sum of the
KM indices, not even requiring the pair. Leveraging this observation and
focusing on the subclass of additive MACs, specific MAC channel coding
techniques are devised in [5] that enabled the receiver recover the sum of two
channel coding message indices.

The techniques of [4], [5] are instances of a broader framework of coding
strategies. Decoding functions of sources or channel inputs efficiently
require codes endowed with algebraic closure properties. To emphasize, the
conventional approach of deriving inner bounds/achievable rate region by
analyzing expected performance of IID random codes is incapable of yielding
performance limits - capacity or rate-distortion regions as the case may be-
in network communication scenarios. To improve upon this, it is necessary to
analyze the expected performance of random codes endowed with algebraic
closure properties. In a series of works [6], an information theoretic study
of the latter codes has been carried out yielding new inner bounds for
multiple network communication scenarios.

In this work, we embark on developing these ideas in the CQ setup. After
having provided the problem statement in Sec. II, we focus on a simplified CQ
MAC and illustrate the core idea of our coding scheme. The latter relies on
developing a nested coset code (NCC) based communication scheme for a CQ PTP
channel and analyzing its performance (Sec. IV). Leveraging this building
block, we design and analyze the performance of an NCC-based coding scheme for
computing sum over a general CQ-MAC (Sec. VI). Going further we generalize
this idea for computing arbitrary functions over a general CQ-MAC.

##  II Preliminaries and Problem Statement

We supplement the notation in [7] with the following. For positive integer
$n$,
$[n]\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\left\\{1,\cdots,n\right\\}$.
For a Hilbert space $\mathcal{H}$,
$\mathcal{L}(\mathcal{H}),\mathcal{P}(\mathcal{H})$ and
$\mathcal{D}(\mathcal{H})$ denote the collection of linear, positive and
density operators acting on $\mathcal{H}$, respectively. The von Neumann
entropy of a density operator $\rho\in\mathcal{D}(\mathcal{H})$ is denoted by
$S(\rho)$. Given any ensemble $\\{p_{i},\rho_{i}\\}_{i\in[1,m]}$, the Holevo
information [8] is denoted as $\chi\big{(}\\{p_{i};\rho_{i}\\}\big{)}$. A POVM
acting on $\mathcal{H}$ is a collection
$\lambda_{\mathcal{X}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\\{\lambda_{x}\\}_{x\in\mathcal{X}}$
of positive operators that form a resolution of the identity:
$\sum_{x\in\mathcal{X}}\lambda_{x}=I$, where $\mathcal{X}$ is a finite set. We
employ an underline notation to aggregate objects of similar type. For
example, $\underline{s}$ denotes $(s_{1},s_{2})$, $\underline{x}^{n}$ denotes
$(x_{1}^{n},x_{2}^{n})$, $\underline{\mathcal{S}}$ denotes the Cartesian
product $\mathcal{S}_{1}\times\mathcal{S}_{2}$.

Consider a (generic) CQ-MAC
$(\rho_{x_{1}x_{2}}\in\mathcal{D}(\mathcal{H}_{Y}):(x_{1},x_{2})\in\mathcal{X}_{1}\times\mathcal{X}_{2})$
specified through (i) finite sets $\mathcal{X}_{j}:j\in[2]$, (ii) Hilbert
space $\mathcal{H}_{Y}$, and (iii) a collection
$(\rho_{x_{1},x_{2}}\in\mathcal{D}(\mathcal{H}_{Y}):(x_{1},x_{2})\in\mathcal{X}_{1}\times\mathcal{X}_{2})$
of density operators. This CQ-MAC is employed to enable the receiver
reconstruct a bivariate function of the classical information streams observed
by the senders. Let $\mathcal{S}_{1},\mathcal{S}_{2}$ be finite sets and
$(S_{1},S_{2})\in\mathcal{S}_{1}\times\mathcal{S}_{2}$ distributed with PMF
$\mathbb{W}_{S_{1}S_{2}}$ models the pair of information sources observed at
the encoders. Specifically, sender $j$ observes the sequence
$S_{jt}\in\mathcal{S}_{j}:t\geq 1$ and the sequence $(S_{1t},S_{2t}):t\geq 1$
are IID with single-letter PMF $\mathbb{W}_{S_{1}S_{2}}$. The receiver aims to
recover the sequence $f(S_{1t},S_{2t}):t\geq 1$ losslessly, where
$f:\mathcal{S}_{1}\times\mathcal{S}_{2}\rightarrow\mathcal{R}$ is a specified
function.

A CQ-MAC code $c_{f}=(n,e_{1},e_{2},\lambda_{\mathcal{R}^{n}})$ of block-
length $n$ for recovering $f$ consists of two encoders maps
$e_{j}:\mathcal{S}^{n}\rightarrow\mathcal{X}_{j}^{n}:j\in[2]$, and a POVM
$\lambda_{\mathcal{R}^{n}}=\\{\lambda_{r^{n}}\in\mathcal{P}(\mathcal{H}_{Y}^{\otimes
n}):r^{n}\in\mathcal{R}^{n}\\}$. The average error probability of the CQ-MAC
code $c_{f}$ is

$\displaystyle\overline{\xi}(c_{f})$ $\displaystyle=$ $\displaystyle
1-\sum_{\underline{s}^{n}:f(\underline{s}^{n})=r^{n}}\mathbb{W}_{S_{1}S_{2}}^{n}(s_{1}^{n},s_{2}^{n})\tr(\lambda_{r^{n}}\rho^{\otimes
n}_{c,\underline{s}^{n}})$

where $\rho^{\otimes
n}_{c,\underline{s}^{n}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\otimes_{i=1}^{n}\rho_{x_{1i}(s_{1}^{n})x_{2i}(s_{2}^{n})}$,
where
$e_{j}(s_{j}^{n})=x_{j1}(s_{j}^{n}),x_{j2}(s_{j}^{n}),\cdots,x_{jn}(s_{j}^{n})$
for $j\in[2]$.

A function $f$ of the sources $\mathbb{W}_{S_{1}S_{2}}$ is said to be
reconstructible over a CQ-MAC if for $\epsilon>0$, $\exists$ a sequence
$c_{f}^{(n)}=(n,e_{1}^{(n)},e_{2}^{(n)},\lambda_{\mathcal{R}^{n}}):n\geq 1$
such that $\lim_{n\rightarrow\infty}\overline{\xi}(c_{f}^{(n)})=0$.

In this article, we are concerned with the problem of characterizing
sufficient conditions under which a function $f$ of the sources
$\mathbb{W}_{S_{1}S_{2}}$ is reconstructible over a generic MAC
$(\rho_{x_{1}x_{2}}\in\mathcal{D}(\mathcal{H}_{Y}):(x_{1},x_{2})\in\mathcal{X}_{1}\times\mathcal{X}_{2})$.
One of our findings - Proposition 2 \- provides a characterization of
sufficient conditions in terms of a computable function of the associated
objects- density operators that characterize the CQ-MAC, function $f$ and the
source distribution $\mathbb{W}_{S_{1}S_{2}}$.

As we shall see, the specific problem of computing sum of sources will play an
important role in our work. In this case,
$\mathcal{S}=\mathcal{S}_{1}=\mathcal{S}_{2}=\mathcal{F}_{q}$ is a finite
field with $q$ elements and the receiver aims to reconstruct
$f(S_{1},S_{2})=S_{1}\oplus_{q}S_{2}$ where $\oplus_{q}$ denotes addition in
$\mathcal{F}_{q}$. A CQ-MAC code
$c_{\oplus}=(n,e_{1},e_{2},\lambda_{\mathcal{S}^{n}})$ of block-length $n$ for
recovering the sum consists of two encoders maps
$e_{j}:\mathcal{S}^{n}\rightarrow\mathcal{X}_{j}^{n}:j\in[2]$, and a POVM
$\lambda_{\mathcal{S}^{n}}=\\{\lambda_{s^{n}}\in\mathcal{P}(\mathcal{H}_{Y}^{\otimes
n}):s^{n}\in\mathcal{S}^{n}\\}$.

Restricting $f$ to a sum, we say the sum of sources $\mathbb{W}_{S_{1}S_{2}}$
over field $\mathcal{F}_{q}$ is reconstructible over a CQ-MAC if
$\mathcal{S}_{1}=\mathcal{S}_{2}=\mathcal{F}_{q}$ and the function
$f(S_{1},S_{2})=S_{1}\oplus_{q}S_{2}$ is reconstructible over the CQ-MAC. The
problem of characterizing sufficient conditions under which a sum of sources
is reconstructible over a CQ-MAC plays an important role in this work. One of
our findings - Theorem 2 \- provides a computable characterization of a set of
sufficient conditions under which a sum of sources is reconstructible over a
CQ-MAC. As the reader will note, this encapsulates the central element of our
characterization in Proposition 2.

We also formalize the notions of a CQ-PTP and CQ-MAC codes for communicating
uniform messages. A CQ-MAC code
$c_{\underline{m}}=(n,\mathcal{I}_{1},\mathcal{I}_{2},e_{1},e_{2},\lambda_{\underline{\mathcal{I}}})$
for a CQ-MAC
$(\rho_{\underline{x}}\in\mathcal{D}(\mathcal{H}_{Y}):\underline{x}\in\underline{\mathcal{X}})$
consists of (i) index sets $\mathcal{I}_{j}:j\in[2]$, (ii) encoder maps
$e_{j}:\mathcal{I}_{j}\rightarrow\mathcal{X}_{j}^{n}:j\in[2]$ and a decoding
POVM
$\lambda_{\underline{\mathcal{I}}}=\\{\lambda_{\underline{m}}\in\mathcal{P}(\mathcal{H}_{Y}^{\otimes
n}):\underline{m}\in\mathcal{I}_{1}\times\mathcal{I}_{2}\\}$. For
$\underline{m}\in\mathcal{I}_{1}\times\mathcal{I}_{2}$, we let $\rho^{\otimes
n}_{c,\underline{m}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\otimes_{i=1}^{n}\rho_{x_{1i},x_{2i}}$
where $e_{j}(m_{j})=x_{j1}\cdots x_{jn}$ for $j\in[2]$.

A CQ-PTP code $c_{m}=(n,\mathcal{I},e,\lambda_{\mathcal{I}})$ for a CQ-PTP
$(\rho_{x}\in\mathcal{D}(\mathcal{H}_{Y}):x\in\mathcal{X})$ consists of (i) an
index set $\mathcal{I}$, (ii) and encoder map
$e:\mathcal{I}\rightarrow\mathcal{X}^{n}$ and a decoding POVM
$\lambda_{\mathcal{I}}=\\{\lambda_{m}\in\mathcal{P}(\mathcal{H}_{Y}^{\otimes
n}):m\in\mathcal{I}\\}$. For $m\in\mathcal{I}$, we let $\rho_{c,m}^{\otimes
n}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\otimes_{i=1}^{n}\rho_{x_{i}}$
where $e(m)=x_{1}\cdots x_{n}$.

##  III The Central Idea

Let us consider the specific problem of reconstructing the sum of sources each
taking values in $\mathcal{S}=\mathcal{F}_{q}$. We begin by reviewing the KM
coding scheme for the case of a noiseless classical MAC. It was shown in [4]
the existence of linear code with a parity matrix $H\in\mathcal{S}^{l\times
n}$ and decoder map $d:\mathcal{F}_{q}^{l}\rightarrow\mathcal{S}^{n}$ such
that
$\sum_{\underline{s}^{n}\in\underline{\mathcal{S}}^{n}}\mathbb{W}_{\underline{S}}^{n}(\underline{s}^{n})\mathds{1}_{\\{d(Hs_{1}^{n}\oplus_{q}Hs_{2}^{n})\neq
s_{1}^{n}\oplus_{q}s_{2}^{n}\\}}\leq\epsilon$, for any $\epsilon>0$, and
sufficiently large $n$, so long as
$\frac{l\log_{2}q}{n}>H(S_{1}\oplus_{q}S_{2})$. This implies that a receiver
equipped with the decoding map $d$ can recover the sum if it possesses the sum
$M_{1}^{l}\oplus_{q}M_{2}^{l}$ of the Körner-Marton indices
$M_{j}^{l}=HS_{j}^{l}:j\in[2]$.

We are therefore led to building an efficient CQ-MAC coding scheme that
enables the receiver only reconstruct the sum of the two message indices.
Indeed, if the two senders send the KM indices to such a CQ-MAC channel code
and the receiver employs the above source decoder $d$ on the decoded sum of
the KM indices, it can recover the sum of sources. To illustrate the design of
the desired CQ-MAC channel code, let us consider a CQ-MAC
$(\rho_{x_{1}x_{2}}\in\mathcal{D}(\mathcal{H}_{Y}):(x_{1},x_{2})\in\mathcal{X}_{1}\times\mathcal{X}_{2})$
wherein $\mathcal{X}_{1}=\mathcal{X}_{2}=\mathcal{F}_{q}$ and the collection
$\rho_{\underline{x}}:\underline{x}\in\underline{\mathcal{X}}$ satisfies
$\rho_{x_{1}x_{2}}=\rho_{\hat{x}_{1}\hat{x}_{2}}$ whenever
$x_{1}\oplus_{q}x_{2}=\hat{x}_{1}\oplus_{q}\hat{x}_{2}$. Consider a CQ-PTP
$(\mathcal{X}=\mathcal{F}_{q},\sigma_{u}:u\in\mathcal{X})$ where
$\sigma_{u}=\rho_{x_{1}\oplus x_{2}}$ for any $(x_{1},x_{2})$ satisfying
$x_{1}\oplus_{q}x_{2}=u$. Suppose we are able to communicate over this CQ-PTP
via a linear CQ-PTP code $\mathcal{C}\subseteq\mathcal{X}^{n}$. Specifically,
suppose there exists a generator matrix $G\in\mathcal{X}^{l\times n}$ and a
POVM $\\{\lambda_{m^{l}}:m^{l}\in\mathcal{F}_{q}^{l}\\}$ so that
$1-q^{-l}\sum_{m^{l}}\tr(\lambda_{m^{l}}\sigma^{\otimes
n}_{m^{l}G})\leq\epsilon$. for any $\epsilon>0$ and sufficiently large $n$,
where
$\sigma^{\otimes}_{m^{l}G}=\sigma_{x_{1}}\otimes\cdots\otimes\sigma_{x_{n}}$
where $m^{l}G=x^{n}$. We can then use this linear CQ-PTP code as our desired
CQ-MAC channel code. Indeed, observe that, suppose both senders employ this
same linear CQ-PTP code, then sender $j$ maps its KM index
$m_{j}^{l}=Hs_{j}^{n}$ to the channel codeword $x_{j}^{n}=m_{j}^{l}G$. Observe
that the structure of the CQ-MAC implies $\rho^{\otimes
n}_{x_{1}^{n},x_{2}^{n}}=\sigma^{\otimes n}_{x_{1}^{n}\oplus
x_{2}^{n}}=\sigma^{\otimes}_{(m_{1}^{l}\oplus_{q}m_{2}^{l})G}$. If the
receiver employs the POVM $\\{\lambda_{m^{l}}:m^{l}\in\mathcal{F}_{q}^{l}\\}$
designed for the CQ-PTP, it ends up decoding the sum of the KM indices
$m_{1}^{l}\oplus_{q}m_{2}^{l}$, and consequently, recover the sum of the
sources.

A careful analysis of the above idea reveals that two MAC channel codes
employed by the encoders do not ‘blow up’ when added, is crucial to the
efficiency of the above scheme. A linear code being algebraically closed
enables this. However, the codewords of a random linear code are uniformly
distributed and cannot achieve the capacity of an arbitrary classical PTP
channel, let alone a CQ-PTP channel. We are therefore forced to enlarge a
linear code to identify sufficiently many codewords of the desired empirical
distribution. We are thus led to a nested coset code (NCC)[9]. A NCC comprises
of cosets of a coarse linear code within a fine code. Within each coset, we
can identify a codeword of the desired empirical distribution. We choose as
many cosets as the number of messages. Analogous to our illustration above
where we chose a linear code that achieves the capacity of the CQ-PTP
$(\mathcal{X}=\mathcal{F}_{q},\sigma_{u}:u\in\mathcal{X})$, our first step
(Sec. IV) is to design a NCC with its POVM that can achieve capacity of an
arbitrary CQ PTP. Our second step is to endow both senders with this same NCC
and analyze decoding the sum of the messages. This gets us to our next
challenge - How do we analyze decoding their message sum, for a general CQ-MAC
$\rho_{\underline{x}}:\underline{x}\in\underline{\mathcal{X}}$ for which
$x_{1}\oplus_{q}x_{2}=\hat{x}_{1}\oplus_{q}\hat{x}_{2}$ does not necessarily
imply $\rho_{x_{1}x_{2}}=\rho_{\hat{x}_{1}\hat{x}_{2}}$. In Sec. V, we address
this challenge, leverage our findings in Sec. IV and generalize the idea for a
general CQ-MAC.

##  IV Nested Coset Codes Achieve Capacity of CQ-PTP

We begin by formalizing the structure of an NCC.

######  Definition 1.

An $(n,k,l,g_{I},g_{O/I},b^{n})$ NCC built over a finite field
$\mathcal{V}=\mathcal{F}_{q}$ comprises of (i) generator matrices
$g_{I}\in\mathcal{V}^{k\times n}$, $g_{O/I}\in\mathcal{V}^{l\times n}$ (ii) a
bias vector $b^{n}$, an encoder map
$e:\mathcal{V}^{l}\rightarrow\mathcal{V}^{k}$. We let
$v^{n}(a,m)=ag_{I}\oplus_{q}mg_{O/I}\oplus_{q}b^{n}:(a,m)\in\mathcal{V}^{k}\times\mathcal{V}^{l}$
denote elements in the range space of the generator matrix
$[g_{I}^{t}~{}g_{O/I}^{t}]^{t}$.

######  Definition 2.

A CQ-PTP code $(n,\mathcal{I}=\mathcal{F}_{q}^{l},e,\lambda_{\mathcal{I}})$ is
an NCC CQ-PTP if there exists an $(n,k,g_{I},g_{O/I},b^{n})$ NCC such that
$e(m)\in\\{u^{n}(a,m):a\in\mathcal{F}_{q}^{k}\\}$ for all
$m\in\mathcal{F}_{q}^{l}$.

######  Theorem 1.

Given a CQ-PTP $(\rho_{v}\in\mathcal{D}(\mathcal{H}_{Y}):v\in\mathcal{F}_{q})$
and a PMF $p_{V}$ on $\mathcal{F}_{q}$, $\epsilon>0$ there exists a CQ-PTP
code $c=(n,\mathcal{I}=\mathcal{F}_{q}^{l},e,\lambda_{\mathcal{I}})$ such that
(i)
$q^{-l}\sum_{m\in[\mathcal{I}]}\sum_{\hat{m}\neq[\mathcal{I}]\setminus\\{m\\}}\tr(\lambda_{\hat{m}}\rho^{\otimes
n}_{c,m})\leq\epsilon$, (ii)
$c=(n,\mathcal{I}=\mathcal{F}_{q}^{l},e,\lambda_{\mathcal{I}})$ is a NCC CQ-
PTP, (iii) $\frac{k\log_{2}q}{n}>\log_{2}q-H(V)$ and
$\frac{(k+l)\log_{2}q}{n}<\log_{2}{q}-H(V)+\chi(\\{p_{v},\rho_{v}\\})$ for all
$n$ sufficiently large.

###### Proof.

In order to achieve a rate $R=\chi(\\{p_{v},\rho_{v}\\})$, the standard
approach is to pick $2^{nR}$ codewords uniformly and independently from
$T_{\delta}^{n}(p_{V})$. However, the resulting code is not algebraically
closed. On the other hand, if we pick a random generator matrix
$G\in\mathcal{F}_{q}^{l\times n}$, with $l=\frac{nR}{\log_{2}q}$, whose
entries from $\mathcal{F}_{q}$ are IID uniform, then its range space - the
resulting collection of $2^{nR}$ codewords - are uniformly distributed and
pairwise independent but not $p_{V}-$typical.

To satisfy the dual requirements of algebraically closure and
$p_{V}-$typicality, we observe the following. If a collection of $q^{k}$
codewords are uniformly distributed in $\mathcal{F}_{q}^{n}$ and pairwise
independent, as we found the range space of $G$ to be, then the expected
number of codewords that are $p_{V}-$typical is
$\frac{q^{k}}{q^{n}}|T_{\delta}^{n}(p_{V})|=\exp\\{n\log_{2}q\left(\frac{k}{n}-\left[1-\frac{H(V)}{\log_{2}q}\right]\right)\\}$.
This indicates that if we pick a generator matrix
$G_{I}\in\mathcal{F}^{k\times n}$ with entries uniformly distributed and IID,
such that $\frac{k}{n}>1-\frac{H(V)}{\log_{2}q}$, then its range space will
contain codewords that are $p_{V}$-typical. The latter codewords can be used
for communication.

Each coset of $G_{I}\in\mathcal{F}^{k\times n}$ where
$\frac{k}{n}>1-\frac{H(V)}{\log_{2}q}$ will play an analogous role as a single
codeword in a conventional IID random code. Just as we pick $2^{nR}$ of the
latter, we consider $2^{nR}$ cosets of $G_{I}$ within a larger linear code
with generator matrix $G=\left[\begin{array}[]{c}G_{I}\\\
G_{O/I}\end{array}\right]\in\mathcal{F}_{q}^{(k+l)\times n}$ with
$l=\frac{nR}{\log_{2}q}$. The messages index the $2^{nR}$ cosets of $G_{I}$. A
predetermined element in each coset that is $p_{V}-$typical is the assigned
codeword for the message and chosen for communication.111The reader is
encouraged to relate to the bounds stated in theorem statement and induced
bounds on the rate of communication $\frac{l\log_{2}q}{n}$. A formal proof we
provide below has two parts - error probability analysis for a generic fixed
code followed by an upper bound on the latter via code randomization.

Upper bound on Error Prob. for a generic fixed code : Consider a generic NCC
$(n,k,l,g_{I},g_{O/I},b^{n})$ with its range space
$v^{n}(a,m)=ag_{I}\oplus_{q}mg_{O/I}\oplus_{q}b^{n}:(a,m)\in\mathcal{V}^{k}\times\mathcal{V}^{l}$.
We shall use this and define a CQ-PTP code
$(n,\mathcal{I}=\mathcal{F}_{q}^{l},e,\lambda_{\mathcal{I}})$ that is an NCC
CQ-PTP. Towards that end, let
$\theta(m)\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{a\in\mathcal{V}^{k}}\mathds{1}_{\left\\{v^{n}(a,m)\in
T_{\delta}^{n}(p_{V})\right\\}}$ and

$\displaystyle
s(m)\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\begin{cases}\\{a\in\mathcal{V}^{K}:v^{n}(a,m)\in
T_{\delta}^{n}(p_{V})\\}&\mbox{if }\theta(m)\geq 1\\\ \\{0^{k}\\}&\mbox{if
}\theta(m)=0,\end{cases}$

for each $m\in\mathcal{V}^{l}$. For $m\in\mathcal{V}^{l}$, a predetermined
element $a_{m}\in s(m)$ is chosen. On receiving message $m\in\mathcal{V}^{l}$,
the encoder prepares the quantum state $\rho_{m}^{\otimes
n}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\rho^{\otimes
n}_{v^{n}(a_{m},m)}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\otimes_{i=1}^{n}\rho_{v_{i}(a_{m},m)}$
and is communicated. The encoding map $e$ is therefore determined via the
collection $(\\!a_{m}\\!\in s(m)\\!:\\!m\in\\!\mathcal{V}^{l})$.

Towards specifying the decoding POVM let
$\rho_{v}=\sum_{y\in\mathcal{Y}}p_{Y|V}(y|v)\ket{e_{y|v}}\bra{e_{y|v}}$ be a
spectral decomposition for $v\in\mathcal{V}$. We let
$p_{VY}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}p_{V}p_{Y|V}$.
For any $v^{n}\in\mathcal{V}^{n}$, let $\pi_{v^{n}}$ be the conditional
typical projector as in [7, Defn. 15.2.4] with respect to the ensemble
$\\{\rho_{v}:v\in\mathcal{V}\\}$ and distribution $p_{V}$. Similarly, let
$\pi_{\rho}$ be the (unconditional) typical projector of the state
$\rho\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{v\in\mathcal{V}}p_{V}(v)\rho_{v}$
as defined in [7, Defn. 15.1.3]. For
$(a,m)\in\mathcal{V}^{k}\times\mathcal{V}^{l}$, we let
$\pi_{a,m}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\pi_{v^{n}(a,m)}\mathds{1}_{\\{v^{n}(a,m)\in
T_{\delta}^{n}(p_{V})\\}}$. We let
$\lambda_{\mathcal{I}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\\{\sum_{a\in\mathcal{V}^{k}}\lambda_{a,m}:m\in\mathcal{I}=\mathcal{V}^{l},\lambda_{-1}\\}$,
where

$\displaystyle\lambda_{a,m}\\!\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\\!\Big{(}\\!\sum_{\hat{a}\in\mathcal{V}^{k}}\\!\sum_{\hat{m}\in\mathcal{V}^{l}}\\!\\!\gamma_{\hat{a},\hat{m}}\Big{)}^{-{1}/{2}}\\!\\!\\!\gamma_{a,m}\Big{(}\\!\sum_{\tilde{a}\in\mathcal{V}^{k}}\sum_{\tilde{m}\in\mathcal{V}^{l}}\\!\\!\gamma_{\tilde{a},\tilde{m}}\Big{)}^{-{1}/{2}},$
(1)

$\lambda_{-1}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}I-\sum_{m\in\mathcal{V}^{l}}\sum_{a\in\mathcal{V}^{k}}\lambda_{a,m}$
and
$\gamma_{a,m}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\pi_{\rho}\pi_{a,m}\pi_{\rho}$.
Since $0\leq\gamma_{a,m}\leq I$, we have $0\leq\lambda_{a,m}\leq I$. The
latter lower bound implies
$\lambda_{\mathcal{I}}\subseteq\mathcal{P}(\mathcal{H})$. The same lower bound
coupled with the definition of the generalized inverse implies
$I\geq\sum_{a\in\mathcal{V}^{k}}\sum_{m\in\mathcal{V}^{l}}\lambda_{a,m}\geq
0$. We thus have $0\leq\lambda_{-1}\leq I$. It can now be verified that
$\lambda_{\mathcal{I}}$ is a POVM. In essence, the elements of this POVM is
identical to the standard POVMs except the POVM elements corresponding to a
coset have been added together. Indeed, since each coset corresponds to one
message, there is no need to disambiguate within the coset.

We have thus associated an NCC $(n,k,l,g_{I},g_{O/I},b^{n})$ and a collection
$(a_{m}\in s(m):m\in\mathcal{V}^{l})$ with a CQ-PTP code. The error
probability of this code is

$\displaystyle
q^{-l}\\!\\!\sum_{m\in\mathcal{I}}\\!\mbox{tr}((I-\sum_{a\in\mathcal{V}^{k}}\\!\\!\lambda_{a,m})\rho_{m}^{\otimes
n})\leq
q^{-l}\\!\\!\sum_{m\in\mathcal{I}}\\!\mbox{tr}((I-\lambda_{a_{m},m})\rho_{m}^{\otimes
n}).$ (2)

Denoting event $\mathscr{E}=\\{\theta(m)<1\\}$, its complement
$\mathscr{E}^{c}$ and the associated indicator functions
$\mathds{1}_{\mathscr{E}},\mathds{1}_{\mathscr{E}^{c}}$ respectively, a
generic term in the RHS of the above sum satisfies

$\displaystyle\mbox{tr}((I-\lambda_{a_{m},m})\rho_{m}^{\otimes
n})\mathds{1}_{\mathscr{E}^{c}}+\mbox{tr}((I-\lambda_{a_{m},m})\rho_{m}^{\otimes
n})\mathds{1}_{\mathscr{E}}\leq\mathds{1}_{\mathscr{E}^{c}}+\sum_{i=1}^{3}T_{2i},$

where

$\displaystyle T_{21}=2\tr((I-\gamma_{a_{m},m})\rho_{m}^{\otimes
n})\mathds{1}_{\mathscr{E}},\quad T_{22}=4\sum_{\hat{a}\neq
a_{m}}\tr(\gamma_{\hat{a},m}\rho_{m}^{\otimes
n})\mathds{1}_{\mathscr{E}},\quad\text{and}\quad T_{23}=4\sum_{\hat{m}\neq
m}\sum_{\tilde{a}}\tr(\gamma_{\tilde{a},\hat{m}}\rho_{m}^{\otimes
n})\mathds{1}_{\mathscr{E}},$

where we have used Hayashi-Nagaoka inequality [10].

Distribution of the Random Code : The objects $g_{I}\in\mathcal{V}^{k\times
n},g_{O/I}\in\mathcal{V}^{l\times n},b^{n}\in\mathcal{V}^{n}$ and the
collection $(a_{m}\in s(m):m\in\mathcal{V}^{l})$ specify an NCC CQ-PTP code
unambiguously. A distribution for a random code is therefore specified through
a distribution of these objects. We let upper case letters denote the
associated random objects, and obtain

$\displaystyle\mathcal{P}\\!\left(\begin{array}[]{c}G_{I}=g_{I},G_{O/I}=g_{O/I}\\\
B^{n}=b^{n},A_{m}=a_{m}:m\in
S(m)\end{array}\right)=q^{-(k+l+1)n}\prod_{m\in\mathcal{V}^{l}}\frac{1}{\Theta(m)},$
(5)

and analyze the expectation of $\mathscr{E}$ and the terms $T_{2i};i\in[1,3]$
in regards to the above random code. We begin by
$\mathbb{E}_{\mathcal{P}}[\mathscr{E}]=\mathcal{P}(\sum_{a\in\mathcal{V}^{k}}\mathds{1}_{\\{V^{n}(a,m)\in
T_{\delta}^{n}(p_{V})\\}}<1).$ For this, we provide the following proposition.

######  Proposition 1.

There exist $\epsilon_{T_{1}}(\delta),\delta_{T_{1}}(\delta),$ such that for
all sufficiently small $\delta$ and sufficiently large $n$, we have
$\mathbb{E}_{\mathcal{P}}\left[\mathscr{E}\right]\leq\epsilon_{{T_{1}}}(\delta)$,
if $\frac{k}{n}\geq\log{q}-H(V)+\delta_{S}$, where
$\epsilon_{{S}},\delta_{S}\searrow 0$ as $\delta\searrow 0$.

###### Proof.

The proof follows from Appendix B of [11] with the identification of
$\mathcal{S}=\phi$. ∎

We now consider $T_{21}$. Deriving an upper bound on $T_{21}$ is by deriving a
lower bound $\tr(\lambda_{a_{m},m}\rho_{m}^{\otimes n})$. This follows by an
argument that is colloquially referred to as ‘pinching’. Lemma 2 in Appendix A
proves the existence of $\lambda>0$ such that
$\mathbb{E}_{\mathcal{P}}\\{T_{21}\\}\leq\exp\\{-n\lambda\delta^{2}\\}$ for
sufficiently large $n$. We now analyze $\mathbb{E}_{\mathcal{P}}[T_{22}]$.
Denoting the event

$\displaystyle\mathcal{J}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\left\\{\begin{array}[]{c}\Theta(m)\geq
1,\\!V^{n}(\hat{a},\hat{m})=\hat{x}^{n}\\\
A_{m}=d,V^{n}(d,m)=x^{n}\end{array}\right\\}\subseteq\\!\mathcal{K}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\left\\{\begin{array}[]{c}V^{n}(\hat{a},\hat{m})=\hat{x}^{n}\\\
V^{n}(d,m)=x^{n}\end{array}\right\\}$ (10)

we perform the following steps.

$\displaystyle\mathbb{E}_{\mathcal{P}}[T_{22}]=\sum_{\hat{a}\in\mathcal{V}^{k}}\mathbb{E}_{\mathcal{P}}[\mbox{tr}(\Gamma_{\hat{a},m}\rho_{m}^{\otimes
n})\mathds{1}_{\\{\theta(m)\geq 1\\}}\mathds{1}_{\\{\hat{a}\neq A_{m}\\}}]$
$\displaystyle=\sum_{d\in\mathcal{V}^{k}}\sum_{\hat{a}\in\mathcal{V}^{k}}\sum_{x^{n}\in
T_{\delta}^{n}(p_{V})}\sum_{\hat{x}^{n}\in\mathcal{V}^{n}}\mathbb{E}\left[\mbox{tr}(\Gamma_{\hat{a},m}\rho_{m}^{\otimes
n})\mathbbm{1}_{\hat{a}\neq d}\mathbbm{1}_{\mathcal{J}}\right]$
$\displaystyle=\sum_{d\in\mathcal{V}^{k}}\sum_{\hat{a}\neq d}\sum_{x^{n}\in
T_{\delta}^{n}(p_{V})}\sum_{\hat{x}^{n}\in\mathcal{V}^{n}}\mathbb{E}\left[\mbox{tr}(\Gamma_{\hat{a},m}\rho_{m}^{\otimes
n})\mathbbm{1}_{\mathcal{J}}\right]$

where the restriction of the summation $x^{n}$ to $T_{\delta}^{n}(p_{V})$ is
valid since $S(m)>1$ forces the choice $A_{m}\in S(m)$ such that
$V^{n}(A_{m},m)\in T_{\delta}^{n}(p_{V})$. Going further, we have

$\displaystyle\mathbb{E}_{\mathcal{P}}[T_{22}]$
$\displaystyle=\sum_{\begin{subarray}{c}d,\hat{a}\in\mathcal{V}^{k}\\\
\hat{a}\neq d\end{subarray}}\sum_{x^{n}\in
T_{\delta}^{n}(p_{V})}\sum_{\hat{x}^{n}\in
T_{\delta}^{n}(p_{V})}\\!\\!\\!\\!\\!\\!\mathbb{E}\left[\mbox{tr}(\pi_{\rho}\pi_{\hat{x}^{n}}\pi_{\rho}\rho_{x^{n}}^{\otimes
n})\mathbbm{1}_{\mathcal{J}}\right]$
$\displaystyle=\sum_{\begin{subarray}{c}d,\hat{a}:\hat{a}\neq
d\end{subarray}}\sum_{x^{n}\in T_{\delta}^{n}(p_{V})}\sum_{\hat{x}^{n}\in
T_{\delta}^{n}(p_{V})}\mbox{tr}(\pi_{\rho}\pi_{\hat{x}^{n}}\pi_{\rho}\rho_{x^{n}}^{\otimes
n})\mathcal{P}(\mathcal{J})$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{\leq}}\sum_{\begin{subarray}{c}d,\hat{a}:\hat{a}\neq
d\end{subarray}}\sum_{\hat{x}^{n}\in
T_{\delta}^{n}(p_{V})}\mbox{tr}(\pi_{\hat{x}^{n}}\pi_{\rho})\mathcal{P}(\mathcal{J})2^{-n\left[S(\rho)-H(p_{V})+\epsilon_{V}\right]}$
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\sum_{\begin{subarray}{c}d,\hat{a}:\hat{a}\neq
d\end{subarray}}\sum_{{\hat{x}^{n}\in
T_{\delta}^{n}(p_{V})}}\mbox{tr}(\pi_{\hat{x}^{n}}\pi_{\rho})\mathcal{P}(\mathcal{K})2^{-n\left[S(\rho)-H(p_{V})+\epsilon_{V}\right]}$
$\displaystyle\stackrel{{\scriptstyle(c)}}{{=}}\sum_{\begin{subarray}{c}d,\hat{a}:\hat{a}\neq
d\end{subarray}}\sum_{{\hat{x}^{n}\in
T_{\delta}^{n}(p_{V})}}\mbox{tr}(\pi_{\hat{x}^{n}}\pi_{\rho})\frac{1}{q^{2n}}2^{-n\left[S(\rho)-H(p_{V})+\epsilon_{V}\right]}$
$\displaystyle\stackrel{{\scriptstyle(d)}}{{\leq}}2^{-n\left[\chi(\\{p_{V};\rho_{v}\\})+\epsilon_{V}-2H(p_{V})-\frac{2k}{n}\log{q}+2\log{q}\right]},$
(11)

where the restriction of the summation $\hat{x}^{n}$ to
$T_{\delta}^{n}(p_{V})$ follows from the fact that $\pi_{\hat{x}^{n}}$ is the
zero projector if $\hat{x}^{n}\notin T_{\delta}^{n}(p_{V})$, (a) follows from
the operator inequality $\sum_{x^{n}\in
T_{\delta}(p_{V})}\pi_{\rho}\rho_{x^{n}}\pi_{\rho}\leq
2^{n(H(p_{V})+\epsilon_{V}(\delta))}\pi_{\rho}\rho^{\otimes n}\pi_{\rho}\leq
2^{n(H(p_{V})+\epsilon_{V}(\delta)-S(\rho))}\pi_{\rho}$ found in [12, Eqn.
20.34, 15.20], (b) follows from Def. 10, (c) follows from pairwise
independence of the distinct codewords, and (d) follows from $\pi_{\rho}\leq
I$ and [12, Eqn. 15.77] and $\epsilon_{V}(\delta)\searrow 0$ as
$\delta\searrow 0$. We now derive an upper bound on
$\mathbb{E}_{\mathcal{P}}[T_{23}]$. We have

$\displaystyle\mathbb{E}_{\mathcal{P}}[T_{23}]$
$\displaystyle{=}\sum_{d,\hat{a}\in\mathcal{V}^{k}}\sum_{\hat{m}\neq
m}\sum_{\begin{subarray}{c}x^{n},\hat{x}^{n}\in\\\
T_{\delta}^{n}(p_{V})\end{subarray}}\\!\\!\\!\mathbb{E}\\!\left[\mbox{tr}(\pi_{\rho}\Pi_{\hat{a},\hat{m}}\pi_{\rho}\rho_{A_{m},m}^{\otimes
n})\mathds{1}_{\mathcal{J}}\right]$
$\displaystyle{=}\sum_{d,\hat{a}\in\mathcal{V}^{k}}\sum_{\hat{m}\neq
m}\sum_{\begin{subarray}{c}x^{n},\hat{x}^{n}\in
T_{\delta}^{n}(p_{V})\end{subarray}}\mbox{tr}(\pi_{\hat{x}^{n}}\pi_{\rho}\rho_{x^{n}}^{\otimes
n}\pi_{\rho})\mathcal{P}(\mathcal{J})$
$\displaystyle{\leq}\sum_{d,\hat{a}\in\mathcal{V}^{k}}\sum_{\hat{m}\neq
m}\sum_{\begin{subarray}{c}\hat{x}^{n}\in
T_{\delta}^{n}(p_{V})\end{subarray}}\\!\\!\\!\\!\mbox{tr}(\pi_{\hat{x}^{n}}\pi_{\rho})\mathcal{P}(\mathcal{J})2^{-n\left[S(\rho)-H(p_{V})+\epsilon_{V}\right]}$
$\displaystyle{\leq}\sum_{d,\hat{a}\in\mathcal{V}^{k}}\sum_{\hat{m}\neq
m}\sum_{\begin{subarray}{c}\hat{x}^{n}\in
T_{\delta}^{n}(p_{V})\end{subarray}}\\!\\!\\!\\!\mbox{tr}(\pi_{\hat{x}^{n}}\pi_{\rho})\mathcal{P}(\mathcal{K})2^{-n\left[S(\rho)-H(p_{V})+\epsilon_{V}\right]}$
$\displaystyle{=}\sum_{d,\hat{a}\in\mathcal{V}^{k}}\sum_{\hat{m}\neq
m}\sum_{\begin{subarray}{c}\hat{x}^{n}\in
T_{\delta}^{n}(p_{V})\end{subarray}}\\!\\!\\!\\!\mbox{tr}(\pi_{\hat{x}^{n}}\pi_{\rho})\frac{1}{q^{2n}}2^{-n\left[S(\rho)-H(p_{V})+\epsilon_{V}\right]}$
$\displaystyle{\leq}\;\;2^{-n\left[\chi(\\{p_{V};\rho_{v}\\})+2\log_{2}q-2H(p_{V})-\frac{2k+l}{n}\log_{2}q+\epsilon_{V}\right]},$

where the inequalities above uses similar reasoning as in (IV).

We have therefore obtained three bounds
$\frac{k}{n}>1-\frac{H(p_{V})}{\log_{2}q}$,
$\frac{2k}{n}<2+\frac{\chi(\\{p_{V};\rho_{v}\\})-2H(p_{V})}{\log_{2}q}$,
$\frac{2k+l}{n}<2+\frac{\chi(\\{p_{V};\rho_{v}\\})-2H(p_{V})}{\log_{2}q}$. A
rate of $\chi(\\{p_{V};\rho_{v}\\})-\epsilon$ is achievable by choosing
$\frac{k}{n}=1-\frac{H(p_{V})}{\log_{2}q}+\frac{\epsilon}{2}$,
$\frac{l}{n}=\frac{\chi(\\{p_{V};\rho_{v}\\})-\epsilon\log_{2}\sqrt{q}}{\log_{2}q}$
thus completing the proof. ∎

##  V Decoding Sum over CQ-MAC

Throughout this section, the source alphabets
$\mathcal{S}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\mathcal{S}_{1}=\mathcal{S}_{2}=\mathcal{F}_{q}$
is a finite field with $q$ elements and the receiver intends to reconstruct
the sum $f(S_{1},S_{2})=S_{1}\oplus_{q}S_{2}$ of the sources. As discussed in
Sec. III, we propose a ‘separation based’ coding scheme consisting of a Körner
Marton (KM) source code followed by a CQ MAC channel code designed to
communicate the sum of the message indices input at the channel code encoders.
The focus of this section is to design, analyze and thereby characterize
performance of the latter CQ MAC channel code tasked to communicate the sum of
messages. Towards that end, we begin with a definition.

######  Definition 3.

Let $\mathcal{V}=\mathcal{F}_{q}$ be a finite field and
$(\rho_{x_{1}x_{2}}\in\mathcal{D}(\mathcal{H}_{Y}):(x_{1},x_{2})\in\mathcal{X}_{1}\times\mathcal{X}_{2})$
be a CQ-MAC. A CQ-MAC code
$c_{m\oplus}=(n,\mathcal{I}_{1}=\mathcal{I}_{2}=\mathcal{F}_{q}^{l},e_{1},e_{2},\lambda_{[q^{l}]})$
of block-length $n$ for recovering $\mathcal{F}_{q}-$sum of messages consists
of two encoders maps
$e_{j}:\mathcal{V}^{l}\rightarrow\mathcal{X}_{j}^{n}:j\in[2]$, and a POVM
$\lambda_{q^{l}}=\\{\lambda_{m}\in\mathcal{P}(\mathcal{H}_{Y}^{\otimes
n}):m\in\mathcal{V}^{l}\\}$.

An $\mathcal{F}_{q}-$message-sum rate $R>0$ is achievable if given any
sequence $l(n)\in\mathbb{N}:n\in\mathbb{N}$ such that
$\limsup_{n\rightarrow\infty}\frac{l(n)\log q}{n}<R$, any sequence
$p_{M_{1}M_{2}}^{(n)}$ of PMFs on
$\mathcal{F}_{q}^{l(n)}\times\mathcal{F}_{q}^{l(n)}$, there exists a CQ-MAC
code
$c_{m\oplus}^{(n)}=(n,\mathcal{I}=\mathcal{I}_{1}=\mathcal{I}_{2}=\mathcal{F}_{q}^{l(n)},e_{1}^{(n)},e_{2}^{(n)},\lambda_{\mathcal{I}})$
of block-length $n$ for recovering $\mathcal{F}_{q}^{l(n)}-$sum of messages
such that for every $\delta>0$, have

$\displaystyle\lim_{n\rightarrow\infty}\overline{\xi}(c_{m\oplus}^{(n)})$
$\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}1-\\!\\!\\!\\!\sum_{\begin{subarray}{c}(m_{1},m_{2})\\\
\in\mathcal{I}_{1}\times\mathcal{I}_{2}\end{subarray}}\\!\\!\\!\\!p_{M_{1}}p_{M_{2}}(m_{1},m_{2})\tr(\lambda_{m_{1}\oplus
m_{2}}\rho^{\otimes n}_{c,\underline{m}})=0$

where $\rho^{\otimes
n}_{c,\underline{m}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\otimes_{i=1}^{n}\rho_{x_{1i}(m_{1})x_{2i}(m_{2})}$,
where $e_{j}(m_{j})=x_{j1}(m_{j}),x_{j2}(m_{j}),\cdots,x_{jn}(m_{j})$ for
$j\in[2]$. The closure of the set of all achievable $\mathcal{F}_{q}-$message-
sum rates is the message-sum capacity of the CQ-MAC.

From our discussion in Sec. III and the above definition, a road map for
characterizing sufficient conditions for computing the sum over a CQ-MAC must
be evident. Referring back to Sec. III, we note that is joint PMF
$\mathbb{W}_{S_{1}S_{2}}$ of the sources is such that
$H(S_{1}\oplus_{q}S_{2})$ is dominated by the message-sum capacity of the CQ-
MAC, then the corresponding sum of sources can be reconstructed over the CQ-
MAC. Therefore, if $R>0$ is an achievable message-sum rate over a CQ-MAC, then
$H(S_{1}\oplus_{q}S_{2})<R$ is a sufficient condition. We now state the main
contribution of this section - a lower bound on the message-sum capacity of a
CQ-MAC. Following its proof, we leverage the above argument in Thm. 2 to
characterize sufficient conditions for reconstructing sum of sources over an
arbitrary CQ-MAC.

######  Definition 4.

Given a CQ-MAC
$\rho_{\underline{\mathcal{X}}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}(\rho_{x_{1}x_{2}}\in\mathcal{D}(\mathcal{H}_{Y}):(x_{1},x_{2})\in\mathcal{X}_{1}\times\mathcal{X}_{2})$
and a prime power $q$, let

$\displaystyle\mathscr{P}(\rho_{\underline{\mathcal{X}}},q)\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\left\\{\begin{array}[]{l}(p_{V_{1}V_{2}U},\rho_{u}:u\in\mathcal{V})\end{array}:\begin{array}[]{l}p_{V_{1}X_{1}}p_{V_{2}X_{2}}\mbox{
is a PMF on
}\mathcal{V}\times\mathcal{X}_{1}\times\mathcal{V}\times\mathcal{X}_{2},\mathcal{V}=\mathcal{F}_{q},\\\
\displaystyle
p_{V_{1}V_{2}U}(v_{1},v_{2},u)=\sum_{x_{1},x_{2}\in\underline{\mathcal{X}}}p_{V_{1}X_{1}}(v_{1},x_{1})p_{V_{2}X_{2}}(v_{2},x_{2})\mathds{1}_{\\{u=v_{1}\oplus_{q}v_{2}\\}}\\\
\displaystyle\rho_{u}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{v_{1}\in\mathcal{F}_{q}}\sum_{v_{2}\in\mathcal{F}_{q}}\\!\\!{p_{V_{1}V_{2}|U}(v_{1},v_{2}|u)}\rho_{v_{1}v_{2}}\mathbbm{1}_{\\{v_{1}\oplus_{q}v_{2}=u\\}},\\\
\displaystyle\rho_{v_{1}v_{2}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{x_{1}\in\mathcal{X}_{1},x_{2}\in\mathcal{X}_{2}}p_{X_{1}|V_{1}}(x_{1}|v_{1})p_{X_{2}|V_{2}}(x_{2}|v_{2})\rho_{x_{1}x_{2}}\end{array}\right\\}$
(17)

For
$(p_{V_{1}V_{2}U},\rho_{u}:u\in\mathcal{F}_{q})\in\mathscr{P}(\rho_{\underline{\mathcal{X}}},q)$,
let

$\displaystyle\mathscr{R}(p_{V_{1}V_{2}U},\rho_{\mathcal{V}})\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\min\\{H(V_{1}),H(V_{2})\\}-H(U)+\chi(\\{p_{U};\rho_{u}\\})\mbox{
and
}\mathscr{R}(\rho_{\underline{\mathcal{X}}},q)\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\displaystyle\sup_{p_{V_{1}V_{2}U},\rho_{\mathcal{V}}\in\mathscr{P}(\rho_{\underline{\mathcal{X}}},q)}\mathscr{R}(p_{V_{1}V_{2}U},\rho_{\mathcal{V}}).$
(18)

######  Lemma 1.

$\mathcal{F}_{q}-$message-sum rate
$\mathscr{R}(\rho_{\underline{\mathcal{X}}},q)$ is achievable over a CQ-MAC
$\rho_{\underline{\mathcal{X}}}=(\rho_{x_{1}x_{2}}\in\mathcal{D}(\mathcal{H}_{Y}):(x_{1},x_{2})\in\mathcal{X}_{1}\times\mathcal{X}_{2})$.

###### Proof.

Let
$(p_{V_{1}V_{2}U},\rho_{u}:u\in\mathcal{V})\in\mathscr{P}(\rho_{\underline{\mathcal{X}}},q)$
with associated collection
$(\rho_{v_{1}v_{2}}:(v_{1},v_{2})\in\mathcal{V}_{1}\times\mathcal{V}_{2})$ of
density operators and PMF $p_{V_{1}X_{1}}p_{V_{2}X_{2}}$ on
$\mathcal{V}_{1}\times\mathcal{X}_{1}\times\mathcal{V}_{2}\times\mathcal{X}_{2}$
where $\mathcal{V}_{1}=\mathcal{V}_{2}=\mathcal{F}_{q}$.

We now describe the coding scheme in terms of a specific code. It is
instructive to revisit Sec. III, wherein we specified the import of both
encoders employing cosets of the the same linear code. In order to choose
codewords of a desired empirical distribution $p_{V_{j}}$, we employ NCCs (as
was done for the same reason in Sec. IV). Following the same notation as in
proof of Thm. 1, we now specify the random coding scheme.

Let $G_{I}\in\mathcal{F}_{q}^{k\times n},G_{O/I}\in\mathcal{F}_{q}^{l\times
n},B_{j}\in\mathcal{F}_{q}^{n}:j\in[2]$ be mutually independent and uniformly
distributed on their respective range spaces. Through out this proof, we let
$\oplus=\oplus_{q}$. Let
$V_{j}^{n}(a,m_{j})\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}a{G_{I}}\oplus
m_{j}G_{O/I}\oplus B_{j}^{n}:(a,m_{j})\in\mathcal{F}_{q}^{k+l}$ for $j\in[2]$
and
$U^{n}(a,m)\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}a{G_{I}}\oplus
mG_{O/I}\oplus B_{1}^{n}\oplus B_{2}^{n}:(a,m)\in\mathcal{F}_{q}^{k+l}$. For
$j\in[2]$, let

$\displaystyle
S_{j}(m_{j})\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\begin{cases}\\{a\in\mathcal{V}^{k}:V^{n}_{j}(a,m_{j})\in
T_{\delta}^{n}(p_{V_{j}})\\}&\mbox{if
}\displaystyle\sum_{a\in\mathcal{V}^{k}}\mathds{1}_{\left\\{V_{j}^{n}(a,m_{j})\in
T_{\delta}^{n}(p_{V_{j}})\right\\}}\geq 1\\\ \\{0^{k}\\}&\mbox{otherwise, i.e
}\displaystyle\sum_{a\in\mathcal{V}^{k}}\mathds{1}_{\left\\{V_{j}^{n}(a,m_{j})\in
T_{\delta}^{n}(p_{V_{j}})\right\\}}=0,\end{cases}$

for each $m_{j}\in\mathcal{V}^{l}$. For $m_{j}\in\mathcal{V}^{l}$, a
predetermined element $A_{j,m_{j}}\in S_{j}(m_{j})$ is chosen. We let
$\Theta_{j}(m_{j})\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}|S_{j}(m_{j})|$.
For $m_{j}\in\mathcal{V}^{l}$, a predetermined
$X_{j}^{n}(m_{j})\in\mathcal{X}_{j}^{n}$ is chosen. As we shall see later, the
choice of $X_{j}^{n}(m_{j})$ is based on $V_{j}^{n}(A_{j,m_{j}},m_{j})$. We
are thus led to the encoding rule.

Encoding Rule: On receiving message
$(m_{1},m_{2})\in\mathcal{V}^{l}\times\mathcal{V}^{l}$, the quantum state
$\rho_{m_{1}m_{2}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\rho_{X_{1}^{n}(m_{1})X_{2}^{n}(m_{2})}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\otimes_{t=1}^{n}\rho_{X_{1t}(m_{1})X_{2t}(m_{2})}$
is (distributively) prepared.

Distribution of the Random Code: The distribution of the random code is
completely specified through the distribution $\mathcal{P}(\cdot)$ of
$G_{I},G_{O/I},B_{1}^{n},B_{2}^{n},(A_{1,m_{1}}:m_{1}\in\mathcal{V}^{l}),(A_{2,m_{2}}:m_{2}\in\mathcal{V}^{l})$
and $(X_{j}^{n}(m_{j}):m_{j}\in\mathcal{V}^{l})$. We let

$\displaystyle\mathcal{P}\left(\\!\\!\\!\begin{array}[]{c}(A_{1,m_{1}}=a_{1,m_{1}}:m_{1}\in\mathcal{V}^{l}),(A_{2,m_{2}}=a_{2,m_{2}}:m_{2}\in\mathcal{V}^{l}),\\\
B_{j}^{n}=b_{j}^{n}:j\in[2],(X_{1}(m_{1})=x_{1}^{n}(m_{1}):m_{1}\in\mathcal{V}^{l}),\\\
G_{I}=g_{I},G_{O/I}=g_{O/I},(X_{2}(m_{2})=x_{2}^{n}(m_{2}):m_{2}\in\mathcal{V}^{l})\end{array}\\!\\!\\!\right)=\displaystyle\left[\prod_{m_{1}}\frac{\mathds{1}_{\\{a_{1,m_{1}}\in
s_{1}(m_{1})\\}}}{\Theta(m_{1})}p_{X_{1}|V_{1}}^{n}(x_{1}^{n}(m_{1})|v_{1}^{n}(a_{1,m_{1}},m_{1}))\right]\times$
(22) $\displaystyle\left[\prod_{m_{2}}\frac{\mathds{1}_{\\{a_{2,m_{2}}\in
s_{2}(m_{2})\\}}}{\Theta(m_{2})}p_{X_{2}|V_{2}}^{n}(x_{2}^{n}(m_{2})|v_{2}^{n}(a_{2,m_{2}},m_{2})\right]\times\frac{1}{q^{kn+ln+2n}}.$

Towards specifying a decoding POVM, we state the associated density operators
modeling the quantum systems, their spectral decompositions and projectors.
Let

$\displaystyle\rho\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{y\in\mathcal{Y}}s_{Y}(y)\ket{h_{y}}\bra{h_{y}},~{}~{}\rho_{x_{1}x_{2}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{y\in\mathcal{Y}}p_{Y|X_{1}X_{2}}(y|x_{1},x_{2})\ket{e_{y|x_{1}x_{2}}}\bra{e_{y|x_{1}x_{2}}}:(x_{1},x_{2})\in\underline{\mathcal{X}}$
$\displaystyle\rho_{v_{1}v_{2}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{y\in\mathcal{Y}}q_{Y|V_{1}V_{2}}(y|v_{1},v_{2})\ket{f_{y|v_{1}v_{2}}}\bra{f_{y|v_{1}v_{2}}}:(v_{1},v_{2})\in\underline{\mathcal{V}},~{}~{}\rho_{u}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{y\in\mathcal{Y}}r_{Y|U}(y|u)\ket{g_{y|u}}\bra{g_{y|u}}:u\in\mathcal{U},$

Decoding POVM: Unlike a generic CQ-MAC decoder [3], which aims at decoding
both the classical messages from the quantum state received, the decoder here
is designed to decode only the sum of messages transmitted. For this, the
decoder employs the nested coset code $(n,k,l,G_{I},G_{O/I},B^{n})$, where
$B^{n}=B_{1}^{n}\oplus B_{2}^{n}$. We define
$U^{n}(a,m)\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}aG_{I}+mG_{O/I}+B^{n}$
to represent a generic codeword. We let
$\Pi_{a,m}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\pi_{U^{n}(a,m)}\mathbbm{1}_{\\{U^{n}(a,m)\in\mathcal{T}_{\delta}^{(n)}(p_{U})\\}}$,
where $p_{U}$ is as defined in the theorem statement. The decoder is provided
with a sub-POVM
$\Lambda_{\mathcal{I}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\\{\Lambda_{m}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{a\in\mathcal{F}_{q}^{k}}\Lambda_{a,m}:m\in\mathcal{F}_{q}^{l}\\}$
where

$\displaystyle\Lambda_{a,m}$
$\displaystyle\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\Big{(}\sum_{\hat{a}\in\mathcal{F}_{q}^{k}}\sum_{\hat{m}\in\mathcal{F}_{q}^{l}}\Gamma_{\hat{a},\hat{m}}\Big{)}^{-1/2}\Gamma_{a,m}\Big{(}\sum_{\hat{a}\in\mathcal{F}_{q}^{k}}\sum_{\hat{m}\in\mathcal{F}_{q}^{l}}\Gamma_{\hat{a},\hat{m}}\Big{)}^{-1/2},$

$\Lambda_{-1}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}I-\sum_{{a}\in\mathcal{F}_{q}^{k}}\sum_{{m}\in\mathcal{F}_{q}^{l}}\Lambda_{a,m}$
and
$\Gamma_{a,m}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\pi_{\rho}\Pi_{(a,m)}\pi_{\rho}$.
We note that

$\displaystyle\displaystyle\pi_{\rho}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{y^{n}\in
T_{\delta}^{n}(s_{Y})}\displaystyle\bigotimes_{t=1}^{n}\ket{h_{y_{t}}}\bra{h_{y_{t}}}\mbox{
and
}\pi_{u^{n}}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{y^{n}:(u^{n},y^{n})\in
T_{\delta}^{n}(p_{U}r_{Y|U})}\displaystyle\bigotimes_{t=1}^{n}\ket{g_{y_{t}|u_{t}}}\bra{g_{y_{t}|u_{t}}},$

denote the typical and conditional typical projectors (as stated in Definition
15.2.4 [7]) with respect to
$\rho\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{u\in\mathcal{F}_{q}}p_{U}(u)\rho_{u}$
and $(\rho_{u}:u\in\mathcal{U})$, respectively.

Error Analysis: We derive upper bounds on
$\mathbb{E}_{\mathcal{P}}\\{\overline{\xi}(c_{m\oplus})\\}$. Our derivation
will be similar to those adopted in proof of Thm. 1. Let us define event

$\displaystyle\mathscr{E}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\left\\{\left(\\!\\!\\!\begin{array}[]{c}V_{1}^{n}(A_{1.m_{1}},m_{1}),X_{1}^{n}(m_{1}),\\\
V_{2}^{n}(A_{2.m_{2}},m_{2}),X_{2}^{n}(m_{2}),\\\
V_{1}^{n}(A_{1.m_{1}},m_{1})\oplus
V_{2}^{n}(A_{2.m_{2}},m_{2})\end{array}\\!\\!\\!\right)\in
T_{{8\delta}}(p_{V_{1}X_{1}V_{2}X_{2}U})\right\\}.$ (26)

We have

$\displaystyle\mathbb{E}_{\mathcal{P}}\left\\{\sum_{m_{1}}\sum_{m_{2}}p_{M_{1}M_{2}}(m_{1},m_{2})\tr(\left[I-\Lambda_{m_{1}\oplus
m_{2}}\right])\rho_{m_{1}m_{2}}^{\otimes
n}\right\\}\leq\underbrace{\mathbb{E}_{\mathcal{P}}\left\\{\sum_{m_{1}}\sum_{m_{2}}p_{M_{1}M_{2}}(m_{1},m_{2})\tr(\left[I-\Lambda_{m_{1}\oplus
m_{2}}\right])\rho_{m_{1}m_{2}}^{\otimes
n}\mathds{1}_{\mathscr{E}^{c}}\right\\}}_{T_{1}}$
$\displaystyle+\underbrace{\mathbb{E}_{\mathcal{P}}\left\\{\sum_{m_{1}}\sum_{m_{2}}p_{M_{1}M_{2}}(m_{1},m_{2})\tr(\left[I-\Lambda_{m_{1}\oplus
m_{2}}\right])\rho_{m_{1}m_{2}}^{\otimes
n}\mathds{1}_{\mathscr{E}}\right\\}}_{T_{2}}.$

In regards to $T_{1}$, the sub-POVM nature of $\Lambda_{\mathcal{I}}$ and the
fact that $\rho_{m_{1},m_{2}}^{\otimes n}$ is a density operator enables us
conclude
$T_{1}\leq\mathbb{E}_{\mathcal{P}}\\{\mathds{1}_{\mathscr{E}^{c}}\\}$.
Furthermore, observe that $X_{j}(m_{j)}$ is distributed with PMF
$p_{X_{j}|V_{j}}^{n}$ conditionally on $V_{j}^{n}(A_{j,m_{j},m_{j}})$. (See
(V). In addition, $p_{V_{1}X_{1}V_{2}X_{2}}=p_{V_{1}X_{1}}p_{V_{2}X_{2}}$
implies that, standard conditional typicality arguments yields

$\displaystyle\mathbb{E}_{\mathcal{P}}\\{\mathds{1}_{\mathscr{E}^{c}}\\}\leq\mathbb{E}_{\mathcal{P}}\left\\{\sum_{m_{1}}p_{M_{1}}(m_{1})\mathds{1}_{\\{\Theta_{1}(m_{1})=0\\}}+\sum_{m_{2}}p_{M_{2}}(m_{2})\mathds{1}_{\\{\Theta_{1}(m_{2})=0\\}}\right\\}+\exp\\{-n\delta\\},$
(27)

where $\delta$ is chosen appropriately. In the above inequality, the second
term on the RHS is an upper bound on the probability of the event
$(X_{1}^{n}(m_{1},X_{2}^{n}(m_{2}))\notin
T_{\delta}^{n}(p_{V_{1}X_{1}V_{2}X_{2}U}|v_{1}^{n},v_{2}^{n},v_{1}^{n}\oplus
v_{2}^{n})$ conditioned on
$(V_{1}^{n}(A_{1.m_{1}},m_{1}),V_{2}^{n}(A_{2.m_{2}},m_{2}),V_{1}^{n}(A_{1.m_{1}},m_{1})\oplus
V_{2}^{n}(A_{2.m_{2}},m_{2}))=(v_{1}^{n},v_{2}^{n},v_{1}^{n}\oplus
v_{2}^{n})\in T_{\delta}^{n}(p_{V_{1}V_{2}U})$ and the first term provides an
upper bound on the complement of the latter event. An upper bound on $T_{1}$
therefore reduces to deriving an upper bound on the first term on the RHS of
(27). This task - deriving an upper bound on the first term on the RHS of (27)
- being a classical analysis, has been detailed in several earlier works [13,
14, 15, 16] and in particular [6, Proof of Thm. 2.5] or [13, Appendix B].
Following this, we have

$\displaystyle\mathbb{E}_{\mathcal{P}}\left\\{\sum_{m_{j}}p_{M_{j}}(m_{j})\mathds{1}_{\\{\Theta_{j}(m_{j})=0\\}}\right\\}$
$\displaystyle\leq$ $\displaystyle\exp\left\\{-n\left(\frac{k\log q}{n}-[\log
q-H(V_{j})]\right)\right\\}$ (28)

thereby ensurnig $T_{1}\leq 2\exp\\{-n\delta\\}$ if

$\displaystyle\frac{k\log q}{n}$ $\displaystyle\geq\max\left\\{\log
q-H(V_{1}),\log q-H(V_{2})\right\\}=\log q-\min\\{H(V_{1}),H(V_{2})\\}.$ (29)

We now analyze $T_{2}$. Applying the Hayashi-Nagaoka inequality, we
have$T_{2}\stackrel{{\scriptstyle(a)}}{{\leq}}T_{21}+T_{22}+T_{23},$ where

$\displaystyle
T_{21}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\mathbb{E}_{\mathcal{P}}\left\\{2\sum_{m_{1}}\sum_{m_{2}}p_{M_{1}M_{2}}(m_{1},m_{2})\tr([I-\Gamma_{A_{\underline{m}}^{\oplus},\underline{m}^{\oplus}}]\rho_{m_{1}m_{2}}^{\otimes
n}])\mathds{1}_{\mathscr{E}}\right\\}$ (30) $\displaystyle
T_{22}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\mathbb{E}_{\mathcal{P}}\left\\{4\sum_{m_{1}}\sum_{m_{2}}\sum_{\hat{a}\neq
A_{\underline{m}}^{\oplus}}p_{M_{1}M_{2}}(m_{1},m_{2})\tr(\Gamma_{\hat{a},\underline{m}^{\oplus}}\rho_{m_{1}m_{2}}^{\otimes
n})\mathds{1}_{\mathscr{E}}\right\\}$ $\displaystyle
T_{23}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\mathbb{E}_{\mathcal{P}}\left\\{4\sum_{m_{1}}\sum_{m_{2}}\sum_{\hat{a}\neq
A_{\underline{m}}^{\oplus}}\sum_{\hat{m}\neq\underline{m}^{\oplus}}p_{M_{1}M_{2}}(m_{1},m_{2})\tr(\Gamma_{\hat{a},\hat{m}}\rho_{m_{1}m_{2}}^{\otimes
n})\mathds{1}_{\mathscr{E}}\right\\},$

and
$A_{\underline{m}}^{\oplus}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}A_{1,m_{1}}\oplus
A_{2,m_{2}}\in\mathcal{V}^{k},\underline{m}^{\oplus}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}m_{1}\oplus
m_{2}\in\mathcal{V}^{l}$. We note that (30(a)) follows from an argument
analogous to the one in (2). We now analyze $T_{21},T_{22}$ and $T_{23}$. We
begin with $T_{21}$. Deriving an upper bound on $T_{21}$ is by deriving a
lower bound
$\mathbb{E}_{\mathcal{P}}\left\\{\tr(\Gamma_{A_{\underline{m}}^{\oplus},\underline{m}^{\oplus}}\rho_{m_{1}m_{2}}^{\otimes
n}])\mathds{1}_{\mathscr{E}}\right\\}$. This follows by an argument that is
colloquially referred to as ‘pinching’. Refer to Lemma 2 in Appendix A. Set
$\mathcal{A}=\mathcal{V}=\mathcal{F}_{q}$, $\mathcal{B}=\mathcal{X}$,
$p_{AB}=p_{V_{1}\oplus V_{2},X}$ and the density operators correspondingly.
With this choice, Lemma 2 proves the existence of $\lambda>0$ such that
$\mathbb{E}_{\mathcal{P}}\left\\{\tr(\Gamma_{A_{\underline{m}}^{\oplus},\underline{m}^{\oplus}}\rho_{m_{1}m_{2}}^{\otimes
n}])\mathds{1}_{\mathscr{E}}\right\\}\geq 1-\exp\\{-n\lambda\delta^{2}\\}$ for
sufficiently large $n$.

We now analyze $T_{22}$. Denoting the event

$\displaystyle\mathcal{J}\\!\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\\!\left\\{\left(\\!\\!\\!\begin{array}[]{c}V_{1}^{n}(A_{1.m_{1}},m_{1}),X_{1}^{n}(m_{1}),V_{2}^{n}(A_{2.m_{2}},m_{2}),X_{2}^{n}(m_{2})\end{array}\\!\\!\\!\right)=(v_{1}^{n},x_{1}^{n},v_{2},x_{2})\in
T_{\delta_{4}}(p_{V_{1}X_{1}V_{2}X_{2}})\right\\},$ (32)

abbreviating $\underline{v}_{\oplus}^{n}=v_{1}^{n}\oplus v_{2}^{n}$,
$\underline{a}^{\oplus}=a_{1}\oplus a_{2}$, we have

$\displaystyle\mathbb{E}_{\mathcal{P}}[T_{22}]$
$\displaystyle=\mathbb{E}_{\mathcal{P}}\left\\{4\sum_{\underline{m}}\sum_{a_{1},a_{2}}\sum_{\hat{a}\neq\underline{a}^{\oplus}}\sum_{\begin{subarray}{c}(\underline{v}^{n},\underline{x})\in\\\
T_{\delta_{4}}(p_{\underline{V}\underline{X}})\end{subarray}}p_{\underline{M}}(\underline{m})\tr(\Gamma_{\hat{a},\underline{m}^{\oplus}}\rho_{m_{1}m_{2}}^{\otimes
n})\mathds{1}_{\mathcal{J}}\mathds{1}_{\left\\{A_{j,m_{j}}=a_{j}:j\in[2]\right\\}}\right\\}$
(33)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=4\sum_{\underline{m}}\sum_{a_{1},a_{2}}\sum_{\hat{a}\neq\underline{a}^{\oplus}}\sum_{\begin{subarray}{c}(\underline{v}^{n},\underline{x})\in\\\
T_{\delta_{4}}(p_{\underline{V}\underline{X}})\end{subarray}}\sum_{\hat{v}^{n}\in\mathcal{V}^{n}}p_{\underline{M}}(\underline{m})\tr(\pi_{\hat{v}^{n}}\pi_{\rho}\rho_{x_{1}^{n}x_{2}^{n}}^{\otimes
n}\pi_{\rho})\mathbb{E}_{\mathcal{P}}\left\\{\mathds{1}_{\mathcal{J}}\mathds{1}_{\left\\{\\!\\!\\!\begin{array}[]{c}A_{j,m_{j}}=a_{j}:j\in[2]\\\
U^{n}(\hat{a},m_{1}\oplus
m_{2})=\hat{v}^{n}\end{array}\\!\\!\\!\right\\}}\right\\}$ (36)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\leq
4\sum_{\underline{m}}\sum_{a_{1},a_{2}}\sum_{\hat{a}\neq\underline{a}^{\oplus}}\sum_{\begin{subarray}{c}(\underline{v}^{n})\in\\\
T_{\delta_{4}}(p_{\underline{V}})\end{subarray}}\sum_{\underline{x}^{n}\in\underline{\mathcal{X}}^{n}}\sum_{\hat{v}^{n}\in\mathcal{V}^{n}}p_{\underline{M}}(\underline{m})\left[\prod_{j=1}^{2}p_{X_{j}|V_{j}}(x_{j}^{n}|v_{j}^{n})\right]\tr(\pi_{\hat{v}^{n}}\pi_{\rho}\rho_{x_{1}^{n}x_{2}^{n}}^{\otimes
n}\pi_{\rho})\mathcal{P}\left(\\!\\!\\!{\begin{array}[]{c}V_{j}^{n}(a_{j},m_{j})=v_{j}^{n}\\\
A_{j,m_{j}}=a_{j}:j\in[2]\\\ U^{n}(\hat{a},m_{1}\oplus
m_{2})=\hat{v}^{n}\end{array}\\!\\!\\!}\right)$ (40)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=4\sum_{\underline{m}}\sum_{a_{1},a_{2}}\sum_{\hat{a}\neq\underline{a}^{\oplus}}\sum_{\begin{subarray}{c}(\underline{v}^{n})\in\\\
T_{\delta_{4}}(p_{\underline{V}})\end{subarray}}\sum_{\hat{v}^{n}\in\mathcal{V}^{n}}p_{\underline{M}}(\underline{m})\tr(\pi_{\hat{v}^{n}}\pi_{\rho}\rho_{v_{1}^{n}v_{2}^{n}}^{\otimes
n}\pi_{\rho})\mathcal{P}\left(\\!\\!\\!{\begin{array}[]{c}V_{1}^{n}(a_{1},m_{1})=v_{1}^{n},A_{j,m_{j}}=a_{j}:j\in[2]\\\
V_{2}^{n}(a_{2},m_{2})=v_{2}^{n},U^{n}(\hat{a},m_{1}\oplus
m_{2})=\hat{v}^{n}\end{array}\\!\\!\\!}\right)$ (43)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\leq
4\sum_{\underline{m}}\sum_{a_{1},a_{2}}\sum_{\hat{a}\neq\underline{a}^{\oplus}}\sum_{\begin{subarray}{c}(\underline{v}^{n})\in\\\
T_{\delta_{4}}(p_{\underline{V}})\end{subarray}}\sum_{\hat{v}^{n}\in\mathcal{V}^{n}}p_{\underline{M}}(\underline{m})\tr(\pi_{\hat{v}^{n}}\pi_{\rho}\rho_{v_{1}^{n}v_{2}^{n}}^{\otimes
n}\pi_{\rho})\mathcal{P}\left(\\!\\!\\!{\begin{array}[]{c}V_{1}^{n}(a_{1},m_{1})=v_{1}^{n},\\\
V_{2}^{n}(a_{2},m_{2})=v_{2}^{n},U^{n}(\hat{a},m_{1}\oplus
m_{2})=\hat{v}^{n}\end{array}\\!\\!\\!}\right)$ (46)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\leq
4\sum_{\underline{m}}\sum_{a_{1},a_{2}}\sum_{\hat{a}\neq\underline{a}^{\oplus}}\sum_{\begin{subarray}{c}(\underline{v}^{n})\in\\\
T_{\delta_{4}}(p_{\underline{V}})\end{subarray}}\sum_{\hat{v}^{n}\in\mathcal{V}^{n}}\\!\\!\\!\\!p_{\underline{M}}(\underline{m})\tr(\pi_{\hat{v}^{n}}\pi_{\rho}\rho_{v_{1}^{n}v_{2}^{n}}^{\otimes
n}\pi_{\rho})\frac{1}{q^{3n}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\leq
4\sum_{\begin{subarray}{c}\underline{m},\\\
a_{1},a_{2}\end{subarray}}\sum_{\hat{a}}\sum_{\hat{v}^{n}\in\mathcal{V}^{n}}\\!\\!\\!\\!p_{\underline{M}}(\underline{m})\tr(\pi_{\hat{v}^{n}}\pi_{\rho})\frac{2^{-n(S(\rho)-H(V_{1},V_{2})-4\delta)}}{q^{3n}}$
(47)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=4\sum_{\begin{subarray}{c}\underline{m},\\\
a_{1},a_{2}\end{subarray}}\sum_{\hat{a}}\sum_{\hat{v}^{n}\in
T_{\delta}(V_{1}\oplus
V_{2})}p_{\underline{M}}(\underline{m})\tr(\pi_{\hat{v}^{n}}\pi_{\rho})\frac{2^{-n(S(\rho)-H(V_{1},V_{2})-4\delta)}}{q^{3n}}$
(48) $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\leq
4\sum_{\begin{subarray}{c}\underline{m},\\\
a_{1},a_{2}\end{subarray}}\sum_{\hat{a}}p_{\underline{M}}(\underline{m})\frac{\exp{-n\left[S(\rho)-H(V_{1},V_{2})-4\delta-\displaystyle\sum_{u}p_{V_{1}\oplus
V_{2}}(u)S(\rho_{u})-H(V_{1}\oplus V_{2})\right]}}{q^{3n}}$ (49)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\leq 4\exp{-n\left\\{\left[\log
q-\left(\displaystyle\sum_{u}p_{V_{1}\oplus V_{2}}(u)S(\rho_{u})+H(V_{1}\oplus
V_{2})-S(\rho)\right)\right]+\left[2\log q-H(V_{1},V_{2})\right]-\frac{3k\log
q}{n}\right\\}}$ (50)

where (i) (36) follows from a summing over possible choices for
$U^{n}(\hat{a},m_{1}\oplus m_{2})$, (ii) (40) follows from evaluating
expectation, enlarging the summation range of $x_{1}^{n},x_{2}^{n}$ and
substituting the distribution of the random code, (iii) (43) follows from the
definitions of $\rho_{v_{1}v_{2}}:\underline{v}\in\underline{\mathcal{V}}$,
(iv) (46) follows as an upper bound since the event in question has been
enlarged, (v) (V) follows from [15, Lemma N.0.21$c$] and the operator
inequality $\sum_{x^{n}\in
T_{\delta}(p_{V})}\pi_{\rho}\rho_{x^{n}}\pi_{\rho}\leq
2^{n(H(p_{V})+\epsilon_{V}(\delta))}\pi_{\rho}\rho^{\otimes n}\pi_{\rho}\leq
2^{n(H(p_{V})+\epsilon_{V}(\delta)-S(\rho))}\pi_{\rho}$ found in [12, Eqn.
20.34, 15.20], (vi) (48) follows from the definition of $\pi_{\hat{v}^{n}}$
which is the $0$ projector if $\hat{v}^{n}$ is not typical wrt $p_{V_{1}\oplus
V_{2}}$, (vii) (49) follows from $\pi_{\rho}\leq I$ and [12, Eqn. 15.77] and,
(viii) (50) collating all the bounds. We now analyze $T_{23}$.

$\displaystyle\mathbb{E}_{\mathcal{P}}[T_{23}]=\mathbb{E}_{\mathcal{P}}\left\\{4\sum_{\begin{subarray}{c}\underline{m}\in\mathcal{V}^{2l}\\\
\hat{m}\neq m_{1}\oplus
m_{2}\end{subarray}}\sum_{\begin{subarray}{c}\underline{a}\in\mathcal{V}^{2k}\\\
\hat{a}\in\mathcal{V}^{k}\end{subarray}}\sum_{\begin{subarray}{c}(\underline{v}^{n},\underline{x})\in\\\
T_{\delta_{4}}(p_{\underline{V}\underline{X}})\end{subarray}}p_{\underline{M}}(\underline{m})\tr(\Gamma_{\hat{a},\underline{m}^{\oplus}}\rho_{m_{1}m_{2}}^{\otimes
n})\mathds{1}_{\mathcal{J}}\mathds{1}_{\left\\{A_{j,m_{j}}=a_{j}:j\in[2]\right\\}}\right\\}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=4\\!\\!\\!\\!\sum_{\begin{subarray}{c}\underline{m}\in\mathcal{V}^{2l}\\\
\hat{m}\neq m_{1}\oplus
m_{2}\end{subarray}}\sum_{\begin{subarray}{c}\underline{a}\in\mathcal{V}^{2k}\\\
\hat{a}\in\mathcal{V}^{k}\end{subarray}}\sum_{\begin{subarray}{c}(\underline{v}^{n},\underline{x})\in\\\
T_{\delta_{4}}(p_{\underline{V}\underline{X}})\end{subarray}}\sum_{\hat{v}^{n}\in\mathcal{V}^{n}}p_{\underline{M}}(\underline{m})\tr(\pi_{\hat{v}^{n}}\pi_{\rho}\rho_{x_{1}^{n}x_{2}^{n}}^{\otimes
n}\pi_{\rho})\mathbb{E}_{\mathcal{P}}\left\\{\mathds{1}_{\mathcal{J}}\mathds{1}_{\left\\{\\!\\!\\!\begin{array}[]{c}A_{j,m_{j}}=a_{j}:j\in[2]\\\
U^{n}(\hat{a},\hat{m})=\hat{v}^{n}\end{array}\\!\\!\\!\right\\}}\right\\}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\leq
4\\!\\!\\!\\!\sum_{\begin{subarray}{c}\underline{m}\in\mathcal{V}^{2l}\\\
\hat{m}\neq m_{1}\oplus
m_{2}\end{subarray}}\sum_{\begin{subarray}{c}\underline{a}\in\mathcal{V}^{2k}\\\
\hat{a}\in\mathcal{V}^{k}\end{subarray}}\sum_{\begin{subarray}{c}(\underline{v}^{n})\in\\\
T_{\delta_{4}}(p_{\underline{V}})\end{subarray}}\sum_{\underline{x}^{n}\in\underline{\mathcal{X}}^{n}}\sum_{\hat{v}^{n}\in\mathcal{V}^{n}}p_{\underline{M}}(\underline{m})\left[\prod_{j=1}^{2}p_{X_{j}|V_{j}}(x_{j}^{n}|v_{j}^{n})\right]\tr(\pi_{\hat{v}^{n}}\pi_{\rho}\rho_{x_{1}^{n}x_{2}^{n}}^{\otimes
n}\pi_{\rho})\mathcal{P}\left(\\!\\!\\!{\begin{array}[]{c}V_{j}^{n}(a_{j},m_{j})=v_{j}^{n}\\\
A_{j,m_{j}}=a_{j}:j\in[2]\\\
U^{n}(\hat{a},\hat{m})=\hat{v}^{n}\end{array}\\!\\!\\!}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=4\\!\\!\\!\\!\sum_{\begin{subarray}{c}\underline{m}\in\mathcal{V}^{2l}\\\
\hat{m}\neq m_{1}\oplus
m_{2}\end{subarray}}\sum_{\begin{subarray}{c}\underline{a}\in\mathcal{V}^{2k}\\\
\hat{a}\in\mathcal{V}^{k}\end{subarray}}\sum_{\begin{subarray}{c}(\underline{v}^{n})\in\\\
T_{\delta_{4}}(p_{\underline{V}})\end{subarray}}\sum_{\hat{v}^{n}\in\mathcal{V}^{n}}p_{\underline{M}}(\underline{m})\tr(\pi_{\hat{v}^{n}}\pi_{\rho}\rho_{v_{1}^{n}v_{2}^{n}}^{\otimes
n}\pi_{\rho})\mathcal{P}\left(\\!\\!\\!{\begin{array}[]{c}V_{1}^{n}(a_{1},m_{1})=v_{1}^{n},A_{j,m_{j}}=a_{j}:j\in[2]\\\
V_{2}^{n}(a_{2},m_{2})=v_{2}^{n},U^{n}(\hat{a},\hat{m})=\hat{v}^{n}\end{array}\\!\\!\\!}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\leq
4\\!\\!\\!\\!\sum_{\begin{subarray}{c}\underline{m}\in\mathcal{V}^{2l}\\\
\hat{m}\neq m_{1}\oplus
m_{2}\end{subarray}}\sum_{\begin{subarray}{c}\underline{a}\in\mathcal{V}^{2k}\\\
\hat{a}\in\mathcal{V}^{k}\end{subarray}}\sum_{\begin{subarray}{c}(\underline{v}^{n})\in\\\
T_{\delta_{4}}(p_{\underline{V}})\end{subarray}}\sum_{\hat{v}^{n}\in\mathcal{V}^{n}}p_{\underline{M}}(\underline{m})\tr(\pi_{\hat{v}^{n}}\pi_{\rho}\rho_{v_{1}^{n}v_{2}^{n}}^{\otimes
n}\pi_{\rho})\mathcal{P}\left(\\!\\!\\!{\begin{array}[]{c}V_{1}^{n}(a_{1},m_{1})=v_{1}^{n},\\\
V_{2}^{n}(a_{2},m_{2})=v_{2}^{n},U^{n}(\hat{a},\hat{m})=\hat{v}^{n}\end{array}\\!\\!\\!}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\leq
4\\!\\!\\!\\!\sum_{\begin{subarray}{c}\underline{m}\in\mathcal{V}^{2l}\\\
\hat{m}\neq m_{1}\oplus
m_{2}\end{subarray}}\sum_{\begin{subarray}{c}\underline{a}\in\mathcal{V}^{2k}\\\
\hat{a}\in\mathcal{V}^{k}\end{subarray}}\sum_{\begin{subarray}{c}(\underline{v}^{n})\in\\\
T_{\delta_{4}}(p_{\underline{V}})\end{subarray}}\sum_{\hat{v}^{n}\in\mathcal{V}^{n}}\\!\\!\\!\\!p_{\underline{M}}(\underline{m})\tr(\pi_{\hat{v}^{n}}\pi_{\rho}\rho_{v_{1}^{n}v_{2}^{n}}^{\otimes
n}\pi_{\rho})\frac{1}{q^{3n}}\leq
4\\!\\!\\!\\!\sum_{\begin{subarray}{c}\underline{m}\in\mathcal{V}^{2l}\\\
\hat{m}\neq m_{1}\oplus
m_{2}\end{subarray}}\sum_{\begin{subarray}{c}\underline{a}\in\mathcal{V}^{2k}\\\
\hat{a}\in\mathcal{V}^{k}\end{subarray}}\sum_{\hat{v}^{n}\in\mathcal{V}^{n}}\\!\\!\\!\\!p_{\underline{M}}(\underline{m})\tr(\pi_{\hat{v}^{n}}\pi_{\rho})\frac{2^{-n(S(\rho)-H(V_{1},V_{2})-4\delta)}}{q^{3n}}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=4\sum_{\begin{subarray}{c}\underline{m},\\\
a_{1},a_{2}\end{subarray}}\sum_{\hat{a}}\sum_{\hat{v}^{n}\in
T_{\delta}(V_{1}\oplus
V_{2})}p_{\underline{M}}(\underline{m})\tr(\pi_{\hat{v}^{n}}\pi_{\rho})\frac{2^{-n(S(\rho)-H(V_{1},V_{2})-4\delta)}}{q^{3n}}$
(65) $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\leq
4\sum_{\begin{subarray}{c}\underline{m},\\\
a_{1},a_{2}\end{subarray}}\sum_{\hat{a}}p_{\underline{M}}(\underline{m})\frac{\exp{-n\left[S(\rho)-H(V_{1},V_{2})-4\delta-\displaystyle\sum_{u}p_{V_{1}\oplus
V_{2}}(u)S(\rho_{u})-H(V_{1}\oplus V_{2})\right]}}{q^{3n}}$ (66)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\leq 4\exp{-n\left\\{\left[\log
q-\left(\displaystyle\sum_{u}p_{V_{1}\oplus V_{2}}(u)S(\rho_{u})+H(V_{1}\oplus
V_{2})-S(\rho)\right)\right]+\left[2\log
q-H(V_{1},V_{2})\right]-\frac{(3k+l)\log q}{n}\right\\}}.$ (67)

The above sequence of steps is analogous to those used to derive an upper
bound on $T_{22}$ and follow from the same set of arguments as provided for
the bounds (36) - (50). This completes the proof of the claimed statement. ∎

We conclude this section with our main result in regards to decoding sum of
sources. The proof of the following theorem follows from the discussion
provided just prior to Definition 4. We therefore omit a detailed proof but
just state the encoding and decoding techniques for completeness.

######  Theorem 2.

The sum of a pair of sources distributed with PMF $\mathbb{W}_{S_{1}S_{2}}$
can be reconstructed on a CQ-MAC
$\rho_{\underline{\mathcal{X}}}=(\rho_{x_{1}x_{2}}\in\mathcal{D}(\mathcal{H}_{\underline{Y}}):(x_{1},x_{2})\in\mathcal{X}_{1}\crossproduct\mathcal{X}_{2})$
if $H(S_{1}\oplus_{q}S_{2})<\mathscr{R}(\rho_{\underline{\mathcal{X}}},q)$.

###### Proof.

We begin with an outline of our coding scheme. As stated in Sec. III, we
propose a ‘separation based approach’ with two modules - source and channel.
The source coding module employs a (distributed) Körner Marton (KM) source
code. Specifically, [4] guarantees the existence of a parity check matrix
$h\in\mathcal{F}_{q}^{l\times n}=\mathcal{S}^{l\times n}$ and a decoder map
$d:\mathcal{F}_{q}^{l}\rightarrow\mathcal{S}^{n}$ such that
$\sum_{\underline{s}^{n}\in\underline{\mathcal{S}}^{n}}\mathbb{W}_{\underline{S}}^{n}(\underline{s}^{n})\mathds{1}_{\\{d(hs_{1}^{n}\oplus_{q}hs_{2}^{n})\neq
s_{1}^{n}\oplus_{q}s_{2}^{n}\\}}\leq\epsilon$, for any $\epsilon>0$, and
sufficiently large $n$, so long as
$\frac{l\log_{2}q}{n}>H(S_{1}\oplus_{q}S_{2})$.

Both encoders of this KM source coding module employ one such parity check
matrix $h\in\mathcal{F}_{q}^{l\times n}$. The decoder of the KM source code
employs the corresponding decoder map $d$. KM Source encoder $j$ outputs
$M_{j}^{l}=h(S_{j}^{n})$. If the KM source decoder is provided
$M_{1}^{l}\oplus_{q}M_{2}^{l}$, then it can reconstruct
$S_{1}^{n}\oplus_{q}S_{2}^{n}$ with reliability at least $1-\epsilon$. The
task of the CQ-MAC channel coding module is to make
$M_{1}^{l}\oplus_{q}M_{2}^{l}$ available to the KM source decoder. We are thus
confronted with the problem of designing a CQ-MAC channel coding module that
can reliably communicate the sum of the messages indices that are input at the
encoders.

Specifically, this channel coding module must communicate
$M_{1}^{l}\oplus_{q}M_{2}^{l}\in\mathcal{F}_{q}^{l}$ within $n$ channel uses.
If we can prove that there exists a MAC channel coding module for sufficiently
large $n$ so long as

$\displaystyle\frac{l\log_{2}q}{n}<\min\\{\\!H(V_{1}),\\!H(V_{2})\\!\\}\\!-\\!H(U)\\!+\\!\chi(\\{p_{U};\rho_{u}\\})$

for any choice of auxiliary

The source module employs the KM code. The corresponding KM decoder at the
receiver only requires the sum of the message indices output by the KM code.
The CQ-MAC channel coding module needs to communicate only the sum of the two
message indices input by the two encoders. Given $\epsilon_{c}$, we seek to
identify a CQ-MAC code $c=(n,e_{1},e_{2},\mathcal{M})$ such that
$\overline{\xi}(c)\leq\epsilon_{c}$.

Encoding: The process of mapping source sequences to the CQ-MAC channel inputs
is divided into two stages. In the first stage, a distributed source code
proposed in [4] is employed which maps the $n-$length source sequences to
message indices taking values over $\mathcal{F}_{q}^{l}$. For the second stage
we develop functions mapping these message indices to channel input codewords.
We begin by defining the first stage of encoding which relies on Lemma 1 of
[9]. This lemma guarantees the existence of a parity check matrix
$h\in\mathcal{F}_{q}^{l(n)\crossproduct n}$ and a map
$d:\mathcal{F}_{q}^{l}(n)\rightarrow\mathcal{F}_{q}^{n}$, for a sufficiently
large $n$, such that (i) $\frac{l(n)}{n}\leq
H(S_{1}\oplus_{q}S_{2})+\frac{\epsilon_{c}}{2}$ and (ii)
$\mathbb{P}(d(hS_{1}^{n}\oplus hS_{2}^{n})\neq S_{1}^{n}\oplus
S_{2}^{n})\leq\frac{\epsilon_{c}}{2}$. We use one such parity matrix which
satisfies the above conditions and define
$M_{j}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}hS_{j},$
for $j=1,2$. This forms our first stage encoder.

Moving on to the second stage encoder, let us denote the maps of the two
encoders as
$\kappa_{j}:\mathcal{F}_{q}^{l}\rightarrow\mathcal{X}_{j}^{n}:j=1,2.$. For
this stage, we use the NCC encoding developed in Section IV for a CQ-PTP.
Consider two NCCs with parameters
$(n,k,l,g_{I},g_{O/I},b_{j}^{n}):j\in\\{1,2\\}$ with range spaces as
$v_{j}^{n}(a,m_{j})\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}ag_{I}\oplus
m_{j}g_{O/I}\oplus b_{j}^{n}:j\in\\{1,2\\},$ respectively. Note that the two
NCCs share the common $g_{I}$ and $g_{O/I},$ but not necessarily the bias
vector $b_{j}^{n}$. Encoder $j$ then constructs its NCC CQ-PTP code
$(n,\mathcal{I}=\mathcal{F}_{q}^{l},e_{j},\lambda^{j}_{\mathcal{I}})$ using
the corresponding NCC $(n,k,l,g_{I},g_{O/I},b_{j}^{n})$ as described in
Definition 1. This defines the second stage encoding. Integrating the two
stages, we obtain the following. To transmit the source sequence pair
$(s_{1}^{n},s_{2}^{n})$ the sequence pair
$(e_{1}(hs_{1}^{n}),e_{2}(hs_{2}^{n}))$ is send over the CQ-MAC channel which
produces the quantum state $\rho_{e_{1}(hs_{1}^{n}),e_{2}(hs_{2}^{n})}$ as the
output.

After performing the measurement and decoding the message $\hat{m}$, the
decoder then employs the KM decoder $d(.)$ to obtain the sum of sources
$d(\hat{m})$. An analysis of this coding scheme is provided in the Proof of
Lemma 1.

∎

##  VI Decoding arbitrary functions over CQ-MAC

Leveraging the technique developed in Theorem 2, we provide the following
proposition to reconstruct an arbitrary function of the sources

######  Proposition 2.

The function $f:\underline{\mathcal{S}}\rightarrow\mathcal{S}$ of sources
$\mathbb{W}_{S_{1}S_{2}}$ can be reconstructed on a CQ-MAC
$(\rho_{x_{1}x_{2}}\in\mathcal{D}(\mathcal{H}_{\underline{Y}}):(x_{1},x_{2})\in\mathcal{X}_{1}\crossproduct\mathcal{X}_{2})$
if there exists functions $h_{j}:\mathcal{S}_{j}\rightarrow\mathcal{F}_{q}$
for $j:1,2$, a function $g:\mathcal{F}_{q}\rightarrow\mathcal{S},$ and a PMF
$p_{V_{1}V_{2}X_{1}X_{2}}=p_{V_{1}X_{1}}p_{V_{2}X_{2}}$ on
$\mathcal{V}_{1}\crossproduct\mathcal{X}_{1}\crossproduct\mathcal{V}_{2}\crossproduct\mathcal{X}_{2}$,
where $\mathcal{V}_{1}=\mathcal{V}_{1}=\mathcal{F}_{q}$,such that
$f(s_{1},s_{2})=g(h_{1}(s_{1})\oplus h_{2}(s_{2}))$ and
$H(h_{1}(S_{1})\oplus_{q}h_{2}(S_{2}))\leq\min\\{H(V_{1}),H(V_{2})\\}\\!-H(U)+\chi(\\{p_{U};\rho_{u}\\}),$
where $p_{U}$ and $\rho_{u}$ are as defined in Theorem 2.

###### Proof.

The proof follows from the proof of Theorem 2. ∎

######  Example 1.

Let
$\mathcal{X}_{1}=\mathcal{X}_{2}=\mathcal{S}_{1}=\mathcal{S}_{2}=\mathcal{X}=\\{0,1\\}$,
$\mathcal{H}=\mathbb{C}^{2}$, and $\rho_{x_{1},x_{2}}=(1-q)\sigma_{(x_{1}\lor
x_{2})}+q\sigma_{(\bar{x}_{1}\land\bar{x}_{2})}$, where
$\sigma_{0},\sigma_{1}\in\mathcal{D}(\mathcal{H})$ be arbitrary. Let
$\rho(q)\triangleq(1-q)\sigma_{0}+q\sigma_{1}$. Consider correlated symmetric
individually uniform sources with
$\mathbb{W}_{S_{1}|S_{2}}(1|0)=\mathbb{W}_{S_{1}|S_{2}}(0|1)=p$ for
$p\in(0,1)$. Let $f(S_{1},S_{2})=S_{1}\lor S_{2}$. Consider the sufficient
conditions given by the unstructured coding scheme:
$H(S_{1},S_{2})<\chi(\\{P_{X_{1},X_{2}},\rho_{x_{1},x_{2}}\\})$, with $X_{1}$
and $X_{2}$ being independent, which can be simplified as
$1+h_{b}(p)<S(\rho(0.5))-S(\rho(q))$. This implies that the $f$ is not
reconstructible using the unstructured codes. We embed $f$ in the ternary
field. In other words, the encoders and decoder work toward reconstructing
$S_{1}\oplus_{3}S_{2}$. The sufficient condition given by the algebraic coding
scheme turns out to be

$H(S_{1}\oplus_{3}S_{2})\\!<\\!H(X_{1}\\!)-H(X_{1}\oplus_{3}X_{2}\\!)+\chi(\\!\\{p_{X_{1}\oplus_{3}X_{2}},\rho_{x_{1}\oplus_{3}x_{2}}\\}\\!),$

for some $p_{X_{1},X_{2}}$, which can be simplified as

$\displaystyle
h_{b}(2p-p^{2})+(2p-p^{2})h_{b}(p/(2-p))\\!<\\!\max_{\theta}[h_{b}(\theta)\\!-\\!h_{b}(2\theta\\!-\\!\theta^{2})-(2\theta\\!-\\!\theta^{2})h_{b}(\theta/(2-\theta))+S(\rho((2\theta-\theta^{2})*q))-S(\rho(q))].$

One can show that there exists choices for $p$, $q$, $\sigma_{0}$ and
$\sigma_{1}$ such that this condition is satisfied.

##  Appendix A Characterization of Certain High Probable Subspaces

In this appendix, we characterize certain high probability subspaces of tensor
product quantum states. The statements we prove here are colloquially referred
to as ‘pinching’ [7] in the literature. We prove statements in a form that can
be used for use in the proof of both Theorems 1, Lemma 2. We begin with
definitions of typical and conditional typical projectors. We adopt strong
(frequency) typicality. All statements hold for most of the variants of notion
of typicality. For concreteness, the reader may refer to [6, App. A].

######  Lemma 2.

Suppose (i) $\mathcal{A},\mathcal{B}$ are finite sets, (ii) $p_{AB}$ is a PMF
on $\mathcal{A}\times\mathcal{B}$, (iii)
$(\rho_{b}\in\mathcal{D}(\mathcal{H}):b\in\mathcal{B})$ is a collection of
density operators,
$\rho_{a}\mathrel{\ensurestackMath{\stackon[1pt]{=}{\scriptstyle\Delta}}}\sum_{b\in\mathcal{B}}p_{B|A}(b|a)\rho_{b}$
for $a\in\mathcal{A}$ and
$\rho=\displaystyle\sum_{a\in\mathcal{A}}p_{A}(a)\rho_{a}=\displaystyle\sum_{b\in\mathcal{B}}p_{B}(b)\rho_{b}$.
There exists a strictly positive $\mu>0$, whose value depends only on
$p_{AB}$, such that for every $\delta>0$, there exists a
$N(\delta)\in\mathbb{N}$ such that for all $n\geq N(\delta)$, we have

$\displaystyle\tr(\Pi_{\rho}^{\delta}\Pi_{a^{n}}^{\delta}\Pi_{\rho}^{\delta}\rho_{b^{n}})\geq
1-\exp\\{-n\lambda\delta^{2}\\}$

whenever $(a^{n},b^{n})\in T_{\frac{\delta}{4}}^{n}(p_{AB})$ where
$\Pi_{a^{n}}^{\delta}$ is the conditional typical projector of
$\rho_{a^{n}}=\otimes_{t=1}^{n}\rho_{a_{t}}$ [7, Defn. 15.2.4] and
$\Pi_{\rho}^{\delta}$ is the unconditional typical projector [7, Defn. 15.1.3]
of $\rho^{\otimes n}$ .

###### Proof.

We rename $\mathcal{A}=\mathcal{V}$, $\mathcal{B}=\mathcal{X}$,
$p_{AB}=p_{VX}$, $a$ as $v$ and $b$ as $x$. We have

$\displaystyle\tr(\Pi_{\rho}^{\delta}\Pi_{v^{n}}^{\delta}\Pi_{\rho}^{\delta}\rho_{x^{n}})$
$\displaystyle=\tr(\Pi_{\rho}^{\delta}\Pi_{v^{n}}^{\delta}\rho_{x^{n}}\Pi_{\rho}^{\delta})$
$\displaystyle\geq\tr(\Pi_{v^{n}}^{\delta}\rho_{x^{n}})-\frac{1}{2}\left\|\rho_{x^{n}}-\Pi_{\rho}^{\delta}\rho_{x^{n}}\Pi_{\rho}^{\delta}\right\|.$
(68)

In the following we derive a lower bound on
$\tr(\Pi_{v^{n}}^{\delta}\rho_{x^{n}})$ and derive an upper bound on
$\left\|\rho_{x^{n}}-\Pi_{\rho}^{\delta}\rho_{x^{n}}\Pi_{\rho}^{\delta}\right\|$.
Toward the deriving the former, we recall that we have $(v^{n},x^{n})\in
T_{\delta/2}{{}^{n}}(p_{VX})$. Let us define:

$p_{Y|XV}(y|x,v):=\innerproduct{e_{y|v}|\rho_{x}|e_{y|v}}{e_{y|v}|\rho_{x}|e_{y|v}},$

for all $(x,v,y)\in\mathcal{X}\times\mathcal{V}\times\mathcal{Y}$.

Clearly, we have $p_{Y|XV}(y|x,v)\geq 0$, and
$\sum_{y\in\mathcal{Y}}p_{Y|XV}(y|x,v)=\sum_{y\in\mathcal{Y}}\innerproduct{e_{y|v}|\rho_{x}|e_{y|v}}{e_{y|v}|\rho_{x}|e_{y|v}}=\tr(\rho_{x})=1$.
Hence we see that $p_{Y|XV}$ is a stochastic matrix.

Next we note that

$\displaystyle\sum_{x\in\mathcal{X}}p_{Y|XV}(y|x,v)p_{XV}(x,v)$
$\displaystyle=\sum_{x\in\mathcal{X}}p_{XV}(x,v)\innerproduct{e_{y|v}|\rho_{x}|e_{y|v}}{e_{y|v}|\rho_{x}|e_{y|v}}$
$\displaystyle=p_{V}(v)\innerproduct{e_{y|v}|\sum_{x\in\mathcal{X}}p_{X|V}(x|v)\rho_{x}|e_{y|v}}{e_{y|v}|\sum_{x\in\mathcal{X}}p_{X|V}(x|v)\rho_{x}|e_{y|v}}$
$\displaystyle=p_{V}(v)\innerproduct{e_{y|v}|\rho_{v}|e_{y|v}}{e_{y|v}|\rho_{v}|e_{y|v}}=p_{V}(v)q_{Y|V}(y|v),$
(69)

where we have used the spectral decomposition of $\rho_{v}$.

Observe that if $(x^{n},v^{n})\in T_{\delta/4}^{n}(p_{XV})$, and $y^{n}\in
T_{\delta}^{n}(p_{XV}p_{Y|XV}|x^{n},v^{n})$, then we have
$(x^{n},v^{n},y^{n})\in T_{\delta}^{n}(p_{XV}p_{Y|XV})$. This implies that we
have $(v^{n},y^{n})\in T_{\delta}^{n}(p_{VY})$, where $p_{VY}$ is the marginal
of $p_{XV}p_{Y|XV}$. Using this and (69), we see that $(v^{n},y^{n})\in
T_{\delta}^{n}(p_{V}q_{Y|V})$. In summary, we see that if $(x^{n},v^{n})\in
T_{\delta/4}(p_{XV})$, then we have

$T_{\delta}^{n}(p_{XV}p_{Y|XV}|x^{n},v^{n})\subseteq\left\\{y^{n}:(v^{n},y^{n})\in
T_{\delta}^{n}(p_{V}q_{Y|V})\right\\}.$

We are now set to provide the promised lower bound. Consider

$\displaystyle\tr(\Pi_{v^{n}}\rho_{x^{n}})$
$\displaystyle=\tr\left(\left[\sum_{y^{n}:(v^{n},y^{n})\in
T_{\delta}^{n}(p_{V}q_{Y|V})}\bigotimes_{t=1}^{n}\ket{e_{y_{t}|v_{t}}}\bra{e_{y_{t}|v_{t}}}\right]\left[\bigotimes_{j=1}^{n}\rho_{x_{j}}\right]\right)$
(70) $\displaystyle=\tr\left(\left[\sum_{y^{n}:(v^{n},y^{n})\in
T_{\delta}^{n}(p_{V}q_{Y|V})}\bigotimes_{t=1}^{n}\ket{e_{y_{t}|v_{t}}}\bra{e_{y_{t}|v_{t}}}\rho_{x_{t}}\right]\right)$
(71) $\displaystyle=\sum_{y^{n}:(v^{n},y^{n})\in
T_{\delta}^{n}(p_{V}q_{Y|V})}\prod_{t=1}^{n}\innerproduct{e_{y_{t}|v_{t}}|\rho_{x_{t}}|e_{y_{t}|v_{t}}}{e_{y_{t}|v_{t}}|\rho_{x_{t}}|e_{y_{t}|v_{t}}}$
(72) $\displaystyle\geq\sum_{y^{n}\in
T_{\delta}^{n}(p_{XV}p_{Y|XV}|x^{n},v^{n}))}\prod_{t=1}^{n}p_{Y|XV}(y_{t}|x_{t},v_{t})$
(73) $\displaystyle\geq
1-2|\mathcal{X}||\mathcal{Y}||\mathcal{V}|\exp\left\\{-\frac{2n\delta^{2}p_{XVY}(x^{*},v^{*},y^{*})}{4(\log(|\mathcal{X}||\mathcal{Y}||\mathcal{V}|))^{2}}\right\\},$
(74)

where we used the definition (A) in the last equality.

We next provide the upper bound. Note from the Gentle measurements lemma [7,
Lemma 9.4.2], we have
$\|\rho_{x^{n}}-\Pi_{\rho}^{\delta}\rho_{x^{n}}\Pi_{\rho}^{\delta}||\leq
3\sqrt{\epsilon}$ if $\tr(\Pi_{\rho}^{\delta}\rho_{x^{n}})\geq 1-\epsilon$. In
the following we provide a lower bound on
$\tr(\Pi_{\rho}^{\delta}\rho_{x^{n}})$. Recall that
$\Pi_{\rho}^{\delta}=\sum_{y^{n}\in
T_{\delta}^{n}(s_{Y})}\bigotimes_{t=1}^{n}\ket{g_{y_{t}}}\bra{g_{y_{t}}}$,
where

$\rho=\sum_{y\in\mathcal{Y}}s_{Y}(y)\ket{g_{y}}\bra{g_{y}},$

is the spectral decomposition of $\rho$, and
$\rho=\sum_{x\in\mathcal{X}}p_{X}(x)\rho_{x}$. Let
$\hat{p}_{Y|X}(y|x):=\innerproduct{g_{y}|\rho_{x}|g_{y}}{g_{y}|\rho_{x}|g_{y}}$,
for all $(x,y)\in\mathcal{X}\times\mathcal{Y}$. Note that $\hat{p}_{Y|X}$ is
not related to $p_{Y|X}$ defined previously. We note that
$\hat{p}_{Y|X}(y|x)\geq 0$, and
$\sum_{y\in\mathcal{Y}}\hat{p}_{Y|X}(y|x)=\sum_{y\in\mathcal{Y}}\innerproduct{g_{y}|\rho_{x}|g_{y}}{g_{y}|\rho_{x}|g_{y}}=\tr(\rho_{x})=1$
for all $x\in\mathcal{X}$. Thus we see that $\hat{p}_{Y|X}$ is a stochastic
matrix. It can also be noted that

$\sum_{x\in\mathcal{X}}\hat{p}_{Y|X}(y|x)p_{X}(x)=\innerproduct{g_{y}|\sum_{x\in\mathcal{X}}p_{X}(x)\rho_{x}|g_{y}}{g_{y}|\sum_{x\in\mathcal{X}}p_{X}(x)\rho_{x}|g_{y}}=\innerproduct{g_{y}|\rho|g_{y}}{g_{y}|\rho|g_{y}}=s_{Y}(y),$

for all $y\in\mathcal{Y}$. This implies that the condition $y^{n}\in
T_{\delta}^{n}(s_{Y})$ is equivalent to the condition $y^{n}\in
T_{\delta}^{n}(\hat{p_{Y}})$, where
$\hat{p_{Y}}(y)=\sum_{x\in\mathcal{X}}\hat{p}_{Y|X}(y|x)p_{X}(x)$. Moreover,
if $x^{n}\in T_{\delta/2}^{n}(p_{X})$, and $y^{n}\in
T_{\delta}^{n}(p_{X}\hat{p}_{Y|X}|x^{n})$, then we have $(x^{n},y^{n})\in
T_{\delta}^{n}(p_{X}\hat{p}_{Y|X})$. Consequently, we have $y^{n}\in
T_{\delta}^{n}(\hat{p_{Y}})$, which in turn implies that $y^{n}\in
T_{\delta}^{n}(s_{Y})$. In essence, we have that if $x^{n}\in
T_{\delta/2}^{n}(p_{X})$ then
$T_{\delta}^{n}(p_{X}\hat{p}_{Y|X}|x^{n})\subseteq T_{\delta}^{n}(s_{Y})$. Now
we are set to provide the lower bound on
$\tr(\Pi_{\rho}^{\delta}\rho_{x^{n}})$ as follows:

$\displaystyle\tr(\Pi_{\rho}^{\delta}\rho_{x^{n}})$
$\displaystyle=\tr\left(\sum_{y^{n}\in
T_{\delta}(s_{Y})}\bigotimes_{t=1}^{n}\ket{g_{y_{t}}}\bra{g_{y_{t}}}\rho_{x_{t}}\right)=\sum_{y^{n}\in
T_{\delta}(s_{Y})}\prod_{t=1}^{n}\innerproduct{g_{y_{t}}|\rho_{x_{t}}|g_{y_{t}}}{g_{y_{t}}|\rho_{x_{t}}|g_{y_{t}}}$
$\displaystyle=\sum_{y^{n}\in
T_{\delta}(s_{Y})}\prod_{t=1}^{n}\hat{p}_{Y|X}(y_{t}|x_{t})\geq\sum_{y^{n}\in
T_{\delta}(\hat{p}_{Y|X}p_{X}|x^{n})}\prod_{t=1}^{n}\hat{p}_{Y|X}(y_{t}|x_{t})$
$\displaystyle\geq
1-2|\mathcal{X}||\mathcal{Y}|\exp\left\\{-\frac{2n\delta^{2}p_{X}^{2}(x^{*})\hat{p}_{Y|X}^{2}(y|x)}{4(\log(|\mathcal{X}||\mathcal{Y}|))^{2}}\right\\}.$
(75)

We therefore have

$\|\rho_{x^{n}}-\Pi_{\rho}^{\delta}\rho_{x^{n}}\Pi_{\rho}^{\delta}\|\leq
6|\mathcal{X}||\mathcal{Y}|\exp\left\\{-\frac{2n\delta^{2}p_{X}^{2}(x^{*})\hat{p}_{Y|X}^{2}(y|x)}{4(\log(|\mathcal{X}||\mathcal{Y}|))^{2}}\right\\},$

and

$\tr(\Pi_{v^{n}}\rho_{x^{n}})\geq
1-2|\mathcal{X}||\mathcal{Y}|||\mathcal{V}|\frac{2n\delta^{2}p_{X}^{2}(x^{*})\hat{p}_{Y|X}^{2}(y|x)}{4(\log(|\mathcal{X}||\mathcal{Y}|))^{2}},$

thereby permitting us to conclude that

$\tr(\Pi_{\rho}^{\delta}\Pi_{v^{n}}^{\delta}\Pi_{\rho}^{\delta}\rho_{x^{n}})\geq\tr(\Pi_{v^{n}}^{\delta}\rho_{x^{n}})-\frac{1}{2}\|\rho_{x^{n}}-\Pi_{\rho}^{\delta}\rho_{x^{n}}\Pi_{\rho}^{\delta}\|\geq
1-\frac{2n\delta^{2}p_{X}^{2}(x^{*})\hat{p}_{Y|X}^{2}(y|x)}{4(\log(|\mathcal{X}||\mathcal{Y}|))^{2}},$

if $(x^{n},v^{n})\in T_{\delta/2}^{n}(p_{XV})$. ∎

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