Non-signaling correlations cannot increase renyi channel capacity
Monotonicity of the Rényi channel capacity under non-signaling assisted channel simulation
Information Theory
Summary
This paper investigates whether special kinds of shared information called non-signaling correlations can help improve how much data a communication channel can reliably send, measured by something called Rényi capacity. The authors show that even with these stronger shared correlations, the capacity does not increase at any measure ranging from zero to infinity. This means that such correlations cannot help make communication better in these specific mathematical terms. Their proof uses a new method involving the Legendre transform to bridge a tricky mathematical gap.
What this means in practice
- •For network schedulers: Avoid relying on non-signaling correlations when optimizing coding schemes since they do not improve communication capacity.
- •For error correction engineers: Focus on traditional channel models without non-signaling assistance when designing codes to achieve limits of reliable communication.
A theory result. No direct application yet.
Authors
Tina Shariat, Matt Hoogsteder-Riera, Marco Tomamichel
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.