The zero-error capacity of a channel (or Shannon capacity of a graph) quantifies how much information can be transmitted with no risk of error. In contrast to the Shannon capacity of a channel, the zero-error capacity has not even been shown to be computable: we have no convergent upper bounds. In this work, we present a new quantity, the zero-error {\em unitary} capacity, and show that it can be succinctly represented as the tensor product value of a quantum game. By studying the structure of finite automata, we show that the unitary capacity is within a controllable factor of the zero-error capacity. This allows new upper bounds through the sum-of-squares hierarchy, which converges to the commuting operator value of the game. Under the conjecture that the commuting operator and tensor product value of this game are equal, this would yield an algorithm for computing the zero-error capacity.

翻译：信道的零错误容量（或图的香农容量）量化了在无错误风险下可传输的信息量。与信道的香农容量不同，零错误容量甚至尚未被证明是可计算的：我们缺乏收敛的上界。本文提出一个新的量——零错误幺正容量，并证明其可简洁地表示为量子游戏的张量积值。通过研究有限自动机的结构，我们证明幺正容量与零错误容量之间的差距可控。这使得可通过平方和层级引入新的上界，该层级收敛至游戏的交换算子值。若此游戏的交换算子值与张量积值相等（此为猜想），则将为零错误容量的计算提供算法。