Monotonicity of the Rényi channel capacity under non-signaling assisted channel simulation

Abstract

We ask whether non-signaling correlations shared between the sender and receiver can increase the Rényi 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 Rényi 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 Rényi 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.