Monotonicity of the R\'enyi channel capacity under non-signaling assisted channel simulation

2026-09-28

AI Summary

This study addresses whether non-signaling correlations can enhance the Rényi capacity of classical channels, a question complicated by the mismatch between existing linear constraints and nonlinear capacity formulations. To resolve this, the work proposes a variational approach based on the Legendre transform, which recasts the nonlinear capacity optimization as an extremal problem over affine functionals, thereby effectively handling the linear constraints. The authors rigorously prove that non-signaling correlations cannot increase the Rényi capacity of any order, nor can they improve the random coding exponent or reduce the strong converse exponent. These results establish monotonicity bounds on channel capacity and provide new analytical tools for channel coding theory.

Abstract

We ask whether non-signaling correlations shared between the sender and receiver can increase the R\'enyi capacity of a classical channel. Non-signaling boxes form a strictly larger class than that of encoder and decoder pairs assisted by shared randomness, so the usual argument based on independent preprocessing and postprocessing is insufficient, and the constraints defining such boxes are linear in the box, while the R\'enyi capacity depends on the channel nonlinearly. We resolve this mismatch with a variational approach based on the Legendre transform, which recasts 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 prove that no non-signaling box can raise the R\'enyi capacity at any order, from order zero through the Shannon capacity to the limiting order at infinity. Consequently, non-signaling simulation cannot increase the random-coding or sphere-packing exponent below capacity, or decrease the strong-converse exponent above capacity.

Problem

• Rényi channel capacity
• non-signaling correlations
• channel simulation
• classical channel

Innovation

• Rényi channel capacity
• non-signaling correlations
• Legendre transform
• variational approach
• channel simulation