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

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 Choi-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-Jamiołkowski 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 Lovász ϑ 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 Lovász number.