The capacity of private classical communication over point-to-point quantum channels has been characterized in [2, 3]. While only a regularized expression of this capacity is known, a single-letter expression has been obtained in the case of degradable quantum channels , and coincides with the coherent information of the channel. In this paper, we define private classical communication over quantum multiple-access channels, and determine a regularized expression of the capacity region for an arbitrary number of transmitters. As formally described in the next sections, we consider message indistinguishability as privacy metric. Our proposed setting can be seen as a quantum counterpart to the classical multiple-access wiretap channel, first introduced in  and further studied in [6, 7, 8, 9, 10]. Note that for the special case of classical communication over multiple-access quantum channels without privacy constraint, the capacity region has already been characterized in .
Often, for simplicity and to facilitate the design of good codes, coding for multiple-access channels is reduced to point-point coding techniques, for instance, with successive decoding or rate-splitting [12, 13]. However, in the presence of a privacy constraint these techniques are challenging to apply. In a successive decoding approach, the transmitters’ messages are decoded one after another at the receiver. This approach works well in the absence of privacy constraints  because the capacity region is a polymatroid. Unfortunately, in the presence of privacy constraints, this task is challenging, even in the classical case and for only two transmitters , because the capacity region is not known to be a polymatroid in general. With a rate-splitting approach, again, the presence of privacy constraints renders the technique challenging to apply, even in the classical case and for only two transmitters, because the rate-splitting procedure may result in negative “rates” for some virtual users .
Instead of relying on successive decoding or rate-splitting, we investigate another method (because of the challenges described above) but will still only rely on point-to-point coding techniques. Specifically, our approach in this paper relies on ideas from random binning techniques, first developed in , which have demonstrated that three primitives are sufficient to build good codes for classical point-to-point wiretap channels. Namely, source coding with side information at the decoder , privacy amplification 
(which may or may not be implemented with universal hashing), and distribution approximation, i.e., the problem of creating from a random variable that is uniformly distributed, another random variable whose distribution is close (for instance with respect to relative entropy or variational distance) to a fixed target distribution, e.g.. Random binning ideas has been successfully applied to construct optimal coding schemes for point-to-point private classical communication over quantum channels  from universal hash functions (used to implement privacy amplification and distribution approximation) and schemes for source coding with quantum side information [21, 22]. Random binning ideas have also been put forward in  as a means to prove the existence of good codes for classical wiretap channels, and have been applied in the context of polar coding to provide efficient and optimal codes for several classical point-to-point wiretap channel models [24, 25, 26]. Note that a capacity-achieving approach that separately handles the reliability constraint and the privacy constraint in the classical point-to-point wiretap channel and the classical-quantum wiretap channel has also been developped in  and , respectively.  and  handle the reliability constraint via channel coding and the privacy constraint via universal hashing. We remark that the approaches in  and  differ from a random binning approach in that  and  rely on channel coding to handle the reliability constraint, whereas the random binning approach relies on source coding. Despite this difference, we believe that both approaches are interesting: The approach based on channel coding seems more natural as the wiretap channel model is a generalization of a channel coding problem, whereas the approach based on source coding uses a simpler building block, since source coding with quantum side information can be used to obtain classical-quantum channel coding, e.g., .
In this paper, following random binning ideas, we establish the sufficiency of the three same primitives (source coding with quantum side information, privacy amplification, and distribution approximation) to achieve the capacity region of private classical communication over quantum multiple-access channels. Additionally, universal hashing will be sufficient to handle privacy amplification and distribution approximation. More specifically, in our coding scheme, the reliability and privacy constraints are decoupled and handled via source coding with quantum side information at the receiver, and two-universal hash functions , respectively. The challenge for the transmitters is to encode their private messages without the knowledge of the other users messages, and still guarantee privacy for all the messages jointly. We establish a distributed version of the leftover hash lemma against quantum side information as a tool for this task. While simultaneously smoothing the min-entropies that appears in the distributed leftover hash lemma is challenging , we are still able to approximate these min-entropies by Von Neumann entropies in the case of product states. Next, to ensure reliability of the messages at the receivers we design and appropriately combine with universal hashing a multiple-access channel code designed from distributed source coding with quantum side information at the decoder. The crux of our analysis is to precisely control the joint state of the encoders output by ensuring a close trace distance between this joint state and a fixed target state in the different steps of the coding scheme, as it not only affects the rates at which the users can transmit but also the privacy guarantees. Finally, a non-trivial Fourier-Motzkin elimination that leverages submodularity properties associated with our achievable rates is performed to obtain the final expression of our achievability region.
We summarize our main contributions as follows. (i) We first derive a regularized expression for the private classical capacity region of quantum multiple-access channels for an arbitrary number of transmitters. (ii) Then, we derive a single-letter expression of the best achievable sum-rate for degradable channels by leveraging properties of the polymatroidal structure of the regularized capacity region. (iii) We establish that the latter quantity is also equal to the best achievable sum-rate for quantum communication over degradable quantum multiple-access channels. (iv) As a byproduct of independent interest, we derive a distributed version of the leftover hash lemma against quantum side information, that is used in our analysis of distributed hashing to ensure privacy. (v) Finally, our achievability scheme, which decouples reliability and privacy via distributed source coding and distributed hashing, establishes that the multi-user coding problem under consideration can be handled solely via point-to-point coding techniques. Namely, source coding with quantum side information between two parties and universal hashing. Even in the classical case, i.e., the classical multiple-access wiretap channel, the reduction of this multi-user coding problem to point-to-point coding techniques was only established for two transmitters but not an arbitrary number of transmitters.
Finally, we refer to the recent work  for the study of a one-shot achievability scheme for the problem considered in this paper in the case of two transmitters.
The remainder of the paper is organized as follows. We formally define the problem in Section III and present our main results in Section IV. Before we prove our inner bound for the capacity region in Section VI, we present in Section V preliminary results that will be used in our achievability scheme. Specifically, in Section V, we discuss distributed universal hashing against quantum side information, distributed source coding with quantum side information, and classical data transmission over classical-quantum multiple-access channels from distributed source coding. We prove an outer bound for the capacity region in Section VII. We prove our results regarding the best achievable sum-rate in Section VIII. Finally, we provide concluding remarks in Section IX.
For , define and . For , a finite-dimensional Hilbert space, let be the set of positive semi-definite operators on . Then, let and be the set of normalized and subnormalized, respectively, quantum states. Let also denote the space of bounded linear operators on . For any and , the min-entropy of relative to  is defined as where denotes the identity operator on , and the max-entropy of  is defined as For any , define the quantum entropy , the conditional quantum entropy , the quantum mutual information , the quantum conditional mutual information , and the coherent information
. For two probability distributionsand defined over the same finite alphabet , define the variational distance between and as . Finally, the power set of a set is denoted by .
Iii Problem Statement
Let and define . Consider a quantum multiple-access channel with transmitters, where . Let be an isometric extension of the channel such that the complementary channel to the environment satisfies for .
An private classical multiple-access code for the channel consists of
message sets , ;
encoding maps , ;
A decoding positive operator-valued measure (POVM) , where ;
and operates as follows: Transmitter selects a message and prepares the state , which is sent over . The channel output is where and . The decoding POVM is then used at the receiver to detect the messages sent. The complementary channel output is denoted by .
A rate-tuple is achievable if there exists a sequence of private classical multiple-access codes such that for some sequence of constant states , we have
The private classical capacity region of a quantum multiple-access channel is defined as the closure of the set of achievable rate-tuples .
Iv Main results
We first propose a regularized expression for the private classical capacity region.
The private classical capacity region of a quantum multiple-access channel is
where cl denotes the closure operator and is the set of rate-tuples that satisfy
for some classical-quantum state of the form
and with an isometric extension of , and the notation for any .
In the next result, for the case of degradable channels, we propose a single-letter expression for the best achievable sum-rate in the private classical capacity region.
Consider a degradable quantum multiple-access channel , i.e., there exists a channel such that . Define as the supremum of all achievable sum-rates in . Then, we have
where the maximization is over classical-quantum states that have the same form as in Theorem 1.
See Section VIII. ∎
We now propose another single-letter characterization of for degradable channels. We first define the quantity .
Consider a quantum multiple-access channel . Define
where the maximization is over states of the form with , , a pure state, and
Note that by , is a regularized expression for the largest achievable sum-rate for quantum communication over quantum multiple-access channels.
Consider a degradable quantum multiple-access channel . Then, we have
See Section VIII. ∎
V Preliminary results
V-a Distributed leftover hash lemma against quantum side information
Define . Consider the random variables , defined over the Cartesian product
with probability distribution, and a quantum system whose state depends on , described by the following classical-quantum state:
where and with the state of the system conditioned on the realization . Next, consider a hash function chosen uniformly at random in a family , , of two-universal hash functions , i.e.,
For any , define , , , , and for , , . The hash functions outputs , the state of the quantum system, and the choice of the functions are described by the following operator
where , , and .
Lemma 1 (Distributed leftover hash lemma).
Let be the fully mixed state on . Define for any , . For any , we have
See Appendix A. ∎
Note that a similar lemma was known in the classical case, e.g., , and had found applications to oblivious transfer [35, 34, 36], secret generation [37, 38, 39], and multiple-access channel resolvability . We are now interested in deriving a distributed leftover hash lemma for product states. We will use the following result on product probability distributions, which is a kind of asymptotic equipartition property (AEP) that holds simultaneously for a set of min-entropies.
Consider the random variables , defined over with probability distribution . In this lemma, let denote the Shannon entropy for random variables following or its marginals. For any , there exists a subnormalized non-negative function defined over such that and
where , , .
See Appendix B. ∎
Lemma 3 (Distributed leftover hash lemma for product states).
See Appendix C. ∎
V-B Distributed classical source coding with quantum side information
Consider , defined over with probability distribution , and a quantum system whose state depends on the random variable , described by the following classical-quantum state
where with the state of the system conditioned on the realization , and we have used the same notation as in Section V-A.
A distributed source code for a classical-quantum product state consists of
sets , ;
encoders , ;
One decoder , where .
A rate-tuple is said to be achievable when the average error probability satisfies , where for all , . Let be the set of all achievable rate-tuples.
Lemma 4 ().
Note that the set associated with the set function defines a contrapolymatroid. Using the fact that its dominant face, i.e., is the convex hull of its extreme points , one can easily verify that the region is achievable using source coding with quantum side information for two parties  and time-sharing. This is exactly the coding technique employed in  to prove Lemma 4.
V-C Multiple-access channel coding from distributed source coding
Consider finite sets , , such that for some and define . Consider a classical-quantum multiple-access channel, i.e., a map , which maps to the state . Let describe the input and output of when the input is uniformly distributed over , and where we have used the notation .
Lemma 5 (Multiple-access channel coding from distributed source coding).
Consider uniformly distributed messages , where for some , . If there exists a distributed source code (as defined in Definition 4) for the classical-quantum product state , then there exist encoders , , and one decoder such that one can choose as , , and , where .
See Appendix D. ∎
Note that this lemma recovers [20, Lemma 2], which treats the case of point-to-point channels.
Vi Achievability of Theorem 1
Consider a classical-quantum multiple-access wiretap channel, i.e., a map , which maps to . The achievability part of Theorem 1 reduces to another achievability result (with a slight adaptation of Definitions 1, 2) for this classical-quantum multiple-access wiretap channel. Specifically, we show in this section that, for any probability distribution , the following region is achievable
where . Note that, compared to the setting of Section III, the signal states sent by the transmitters are now part of the channel definition. Hence, achievability of and regularization lead to the achievability part of Theorem 1.
Vi-a Coding scheme
The main idea of the coding scheme is to combine distributed source coding and distributed randomness extraction to emulate a random binning-like proof. We proceed in three steps.
: We create a stochastic channel that simulates the inversion of multiple hash functions while approximating the joint distribution of the inputs and outputs of the hash functions. Approximating this joint distribution is crucial for the message indistinguishability analysis. In the special case of a single hash function, this operation is referred to as shaping in and distribution approximation in .
Consider distributed according to some arbitrary product distribution , and two-universal hash functions uniformly distributed over , where we use the same notation as in Section V-A. The output lengths of the hash functions, denoted by , will be defined later. Let
be the channel described by the conditional probability distributionand be the channel described by the conditional probability distribution , . For , let be uniformly distributed over , and define
where is the uniform distribution over with the same notation as in Section V-C. Hence, denotes the joint probability distribution of the input and output of the channel . To simplify notation in the following, we write instead of by redefining and including in its definition.
Step 2: Using Lemma 5, we construct a multiple-access channel code for jointly uniform input distributions (in the absence of any privacy constraint) for the channel .
and where belongs to . Then, by Lemma 5, there exist encoders and one decoder where we have defined for , such that as , and
Step 3: We combine Step 1 and Step 2 to define our encoders and decoder for the classical-quantum multiple-access wiretap channel. Specifically, the encoders are defined as
and the decoder is defined as
In Step 2, Lemma 4 cannot be directly applied to as it is not a product state.
Vi-B Coding scheme analysis
Vi-B1 Average reliability
Vi-B2 Average message indistinguishability
Note that by a random choice of the encoder in the proof of Lemma 5, is uniformly distributed, hence, follows a product distribution and is a product state, which one can write , where
Next, define the following classical-quantum state
Then, for the fully mixed state on and the fully mixed state on , we have