Joint source-channel coding for a quantum multiple access channel

Suppose that two senders each obtain one share of the output of a classical, bivariate, correlated information source. They would like to transmit the correlated source to a receiver using a quantum multiple access channel. In prior work, Cover, El Gamal, and Salehi provided a combined sourcechannel coding strategy for a classical multiple access channel which outperforms the simpler “separation” strategy where separate codebooks are used for the source coding and the channel coding tasks. In the present paper, we prove that a coding strategy similar to the Cover-El Gamal-Salehi strategy and a corresponding quantum simultaneous decoder allow for the reliable transmission of a source over a quantummultiple access channel, as long as a set of information inequalities involving the Holevo quantity hold. Suppose that a correlated information source, embodied in many realizations of two random variables U and V distributed independently and identically as p (u, v), is in the possession of two spatially separated senders, such that one sender has U and the other V . Suppose further that a multiple access channel, modeled as a conditional probability distribution p (y|x1, x2), connects these two senders to a single receiver. When is it possible for the senders to transmit the output of the correlated source reliably over the multiple access channel? A separation strategy would have the senders first use a Slepian-Wolf compression code [26] to compress the source, followed by encoding the compressed bits using a multiple access channel code [14, 1]. The receiver would first decode the multiple access channel code and then decode the Slepian-Wolf compression code in order to recover the output of the information source. This strategy will work provided that the following information inequalities hold: H (U |V ) ≤ I (X1;Y |X2) , H (V |U) ≤ I (X2;Y |X1) , H (UV ) ≤ I (X1X2;Y ) , where the information quantities are with respect to a distribution of the form p (u, v) p (x1) p (x2) p (y|x1, x2). The codewords for the multiple access channel code in this scheme are generated independently according to the product distribution p (x1) p (x2). Cover, El Gamal, and Salehi (CES) demonstrated the failure of a separation strategy in the above scenario [3], in spite of a separation strategy being optimal for the case of a single sender and single receiver [25]. They found a simple example of a source with p (0, 0) = p (0, 1) = p (1, 1) = 1/3 and p (1, 0) = 0 and a channel Y = X1 + X2 such that the above information inequalities do not hold (H (UV ) = log2 3 = 1.58 while maxp(x1)p(x2) I (X1X2;Y ) = 1.5), whereas the simple “joint” strategy of sending the source directly over the channel succeeds perfectly (set X1 = U and X2 = V so that the receiver can determine U and V from Y ). More generally, the main result of their paper is a joint source-channel coding strategy to allow for the senders to transmit a source reliably over the channel, provided that the following information