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

TL;DR

This paper investigates the zero-error classical capacity of quantum channels with no-signalling assistance, providing SDP characterizations, and offers an information-theoretic interpretation of the Lovasz number as a capacity measure.

Contribution

It introduces SDP formulations for zero-error capacities with no-signalling assistance and links the Lovasz number to quantum channel capacities.

Findings

Zero-error classical simulation cost equals the conditional min-entropy of the Choi matrix.

Asymptotic zero-error capacity for classical-quantum channels matches a simplified SDP.

Lovasz number interpreted as the zero-error capacity assisted by quantum no-signalling correlations.

Abstract

We study the one-shot zero-error classical capacity of a quantum channel assisted by quantum no-signalling correlations, and the reverse problem of exact simulation of a prescribed channel by a noiseless classical one. Quantum no-signalling correlations are viewed as two-input and two-output completely positive and trace preserving maps with linear constraints enforcing that the device cannot signal. Both problems lead to simple semidefinite programmes (SDPs) that depend only on the Kraus operator space of the channel. In particular, we show that the zero-error classical simulation cost is precisely the conditional min-entropy of the Choi-Jamiolkowski matrix of the given channel. The zero-error classical capacity is given by a similar-looking but different SDP; the asymptotic zero-error classical capacity is the regularization of this SDP, and in general we do not know of any simple form. Interestingly however, for the class of classical-quantum channels, we show that the asymptotic capacity is given by a much simpler SDP, which coincides with a semidefinite generalization of the fractional packing number suggested earlier by Aram Harrow. This finally results in an operational interpretation of the celebrated Lovasz $\vartheta$ function of a graph as the zero-error classical capacity of the graph assisted by quantum no-signalling correlations, the first information theoretic interpretation of the Lovasz number.