Zero-Error Classical Channel Capacity and Simulation Cost Assisted by Quantum Non-Signalling Correlations

We study the one-shot zero-error classical capacity of quantum channels assisted by quantum non-signalling correlations, and the reverse problem of simulation. Both lead to simple semi-definite programmings whose solutions can be given in terms of the conditional min-entropies. We show that the asymptotic simulation cost is precisely the conditional min-entropy of the Choi-Jamiołkowski matrix of the given channel. For classical-quantum channels, the asymptotic capacity is reduced to a quantum fractional packing number suggested by Harrow, which leads to an operational interpretation of the celebrated Lovász θ function as the zero-error classical capacity of a graph assisted by quantum nonsignalling correlations.