Duan and Winter studied the one-shot zero-error classical capacity of a quantum channel assisted by quantum non-signalling correlations, and formulated this problem as a semidefinite program depending only on the Kraus operator space of the channel. For the class of classical-quantum channels, they showed that the asymptotic zero-error classical capacity assisted by quantum non-signalling correlations, minimized over all classicalquantum channels with a confusability graph G, is exactly log ϑ(G), where ϑ(G) is the celebrated Lov´asz theta function. In this paper, we show that the one-shot capacity for a classical-quantum channel, induced from a circulant graph G defined by equal-sized cyclotomic cosets, is logbϑ(G)c, which further implies that its asymptotic capacity is log ϑ(G). This type of graphs include the cycle graphs of odd length, the Paley graphs of prime vertices, and the cubit residue graphs of prime vertices. Examples of other graphs are also discussed. This gives Lov´asz ϑ function another operational meaning in zero-error classical-quantum communication.
The Private Classical Capacity and Quantum Capacity of a Quantum Channel