Optimality of Meta-Converse for Channel Simulation

We study the effect of shared non-signaling correlations for the problem of simulating a channel using noiseless communication in the one-shot setting. For classical channels, we show how to round any non-signaling-assisted simulation strategy - which exactly corresponds to the meta-converse for channel simulation - to a strategy that only uses shared randomness. For quantum channels, we round any non-signaling-assisted simulation strategy to a strategy that only uses shared entanglement. As our main result, we prove a guarantee on the ratio of success probabilities of at least $(1-\frac{-1}{\mathbf{e}})$, for both the classical and the quantum setting. We further - show this ratio to be optimal. It can be improved to $(1-\frac{1}{t})$ using $o$ (ln $(t)$) additional bits (qubits) of communication.