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arXiv:2410.07984 · IEEE Transactions on Information Theory

Large Deviation Analysis for the Reverse Shannon Theorem

Shi-Bing Li, Ke Li, Lei Yu

Published May 25, 2026• 1 citation
arXivHTMLPDF

Abstract

We consider the problem of simulating a noisy channel using noiseless channels with unlimited shared randomness. This can be interpreted as the reverse problem of Shannon’s noisy channel coding theorem. In contrast to previous works, we employ Rényi divergence (with the parameter α ∈ [0,∞]) to measure the level of approximation, and we obtain the reverse Shannon theorem under this measure, which characterizes the Rényi simulation rate, the minimum communication rate required for the Rényi divergence vanishing asymptotically. Our derivation is done by a precise large-deviation analysis. When the communication rate is above the Rényi simulation rate, we provide a complete characterization of the convergence exponent for the Rényi divergence, called the reliability function. When the communication rate is below the Rényi simulation rate, we determine the linear increasing rate for the Rényi divergence, which implies the strong converse exponent for the order-α fidelity.

DOI
10.1109/tit.2026.3696715

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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 Rényi 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.
We give an alternative variational proof that non-signaling correlations shared between the sender and receiver cannot increase the Rényi capacity of a classical channel. This monotonicity follows from the data-processing inequality for channel divergences established by Oufkir, Yao and Berta. Our approach works directly with the information-radius representation of the capacity. The constraints defining non-signaling boxes are linear in the box, while the Rényi capacity depends on the channel nonlinearly. We use a Legendre transform to express the capacity as an extremum over a family of affine functionals in which the channel appears linearly. Combining this method with separate arguments for the Shannon capacity and endpoint orders, we recover monotonicity at every order from zero to infinity. We then derive consequences for standard coding exponent functions: non-signaling simulation cannot increase the random-coding or sphere-packing exponent below capacity, or decrease the strong-converse exponent above capacity.
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