The capacity of a channel characterizes the maximum rate at which information can be transmitted through the channel asymptotically faithfully. For a channel with multiple senders and a single receiver, computing its sum capacity is possible in theory, but challenging in practice because of the nonconvex optimization involved. To address this challenge, we investigate three topics in our study. In the first part, we study the sum capacity of a family of multiple access channels (MACs) obtained from nonlocal games. For any MAC in this family, we obtain an upper bound on the sum rate that depends only on the properties of the game when allowing assistance from an arbitrary set of correlations between the senders. This approach can be used to prove separations between sum capacities when the senders are allowed to share different sets of correlations, such as classical, quantum or no-signalling correlations. We also construct a specific nonlocal game to show that the approach of bounding the sum capacity by relaxing the nonconvex optimization can give arbitrarily loose bounds. Owing to this result, in the second part, we study algorithms for non-convex optimization of a class of functions we call Lipschitz-like functions. This class includes entropic quantities, and hence these results may be of independent interest in information theory. Subsequently, in the third part, we show that one can use these techniques to compute the sum capacity of an arbitrary two-sender MACs to a fixed additive precision in quasi-polynomial time. We showcase our method by efficiently computing the sum capacity of a family of two-sender MACs for which one of the input alphabets has size two. Furthermore, we demonstrate with an example that our algorithm may compute the sum capacity to a higher precision than using the convex relaxation.

翻译：信道容量刻画了通过信道渐近保真地传输信息的最大速率。对于具有多个发送端和一个接收端的信道，理论上可计算其和容量，但由于涉及非凸优化，实际中极具挑战性。为应对这一难题，本文研究三个主题。第一部分，研究从非局部博弈获得的一类多址接入信道的和容量。对于该类中的任意信道，当允许发送端共享任意关联集时，我们得到仅依赖于博弈性质的和速率上界。该方法可用于证明当发送端允许共享不同关联集（如经典、量子或无信号关联）时的和容量分离。我们亦构造特定非局部博弈，表明通过松弛非凸优化来界定和容量的方法可能产生任意松散的上界。基于此结果，第二部分研究一类称为利普希茨型函数的非凸优化算法。该类函数包含熵量，因此这些结果可能对信息论具有独立意义。随后第三部分表明，可利用这些技术在拟多项式时间内计算任意双发送端多址接入信道的和容量至固定加性精度。我们通过高效计算一族输入字母集大小为二的双发送端多址接入信道和容量来展示该方法，并举例说明该算法可能比凸松弛方法达到更高的和容量计算精度。