Exponents for Shared Randomness-Assisted Channel Simulation

TL;DR

This paper precisely characterizes the error and strong converse exponents for shared randomness-assisted channel simulation, revealing no critical rates and showing that quantum entanglement does not affect these exponents.

Contribution

It provides a tight, rate-independent characterization of exponents in shared randomness-assisted channel simulation using Rényi mutual information.

Findings

Exact error and strong converse exponents are derived.

No critical rates exist for the exponents.

Quantum entanglement does not alter the exponents.

Abstract

We determine the exact error and strong converse exponents of shared randomness-assisted channel simulation in worst case total-variation distance. Namely, we find that these exponents can be written as simple optimizations over the R\'enyi channel mutual information. Strikingly, and in stark contrast to channel coding, there are no critical rates, allowing a tight characterization for arbitrary rates below and above the simulation capacity. We derive our results by asymptotically expanding the meta-converse for channel simulation [Cao {\it et al.}, IEEE Trans.~Inf.~Theory (2024)], which corresponds to non-signaling assisted codes. We prove this to be asymptotically tight by employing the approximation algorithms from [Berta {\it et al.}, Proc.~IEEE ISIT (2024)], which show how to round any non-signaling assisted strategy to a strategy that only uses shared randomness. Notably, this implies that any additional quantum entanglement-assistance does not change the error or the strong converse exponents.