On the One-Shot Zero-Error Classical Capacity of Classical-Quantum Channels Assisted by Quantum Non-signalling Correlations

Abstract

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 classical-quantum channels with a confusability graph $G$, is exactly $\log \vartheta (G)$, where $\vartheta (G)$ is the celebrated Lovász 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 $\log \lfloor \vartheta (G)\rfloor$, which further implies that its asymptotic capacity is $\log \vartheta (G)$. 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 endows the Lovász $\theta$ function with a more straightforward operational meaning.

13 pages, 6 figures

References in corpus (4)

• Graph-Theoretic Approach to Quantum Correlations
• No-Signalling Assisted Zero-Error Capacity of Quantum Channels and an Information Theoretic Interpretation of the Lovasz Number
• Super-Activation of Zero-Error Capacity of Noisy Quantum Channels
• Entanglement between Two Uses of a Noisy Multipartite Quantum Channel Enables Perfect Transmission of Classical Information

Cited by in corpus (1)

• No-Signalling Assisted Zero-Error Capacity of Quantum Channels and an Information Theoretic Interpretation of the Lovasz Number