Comparison of Channels: Criteria for Domination by a Symmetric Channel

TL;DR

This paper provides a criterion to determine when a given channel can be considered more noisy than a symmetric channel, using divergence measures and matrix properties, with implications for information inequalities.

Contribution

It introduces a new criterion based on the minimum entry of the channel matrix for domination by symmetric channels, extending to additive noise channels over finite Abelian groups.

Findings

A characterization of channel domination using $oldsymbol{ extit{ ext{chi}}^2}$-divergence.

A simple matrix-based criterion for domination by symmetric channels.

Domination implies a logarithmic Sobolev inequality for the channel.

Abstract

This paper studies the basic question of whether a given channel $V$ can be dominated (in the precise sense of being more noisy) by a $q$-ary symmetric channel. The concept of "less noisy" relation between channels originated in network information theory (broadcast channels) and is defined in terms of mutual information or Kullback-Leibler divergence. We provide an equivalent characterization in terms of $\chi^2$-divergence. Furthermore, we develop a simple criterion for domination by a $q$-ary symmetric channel in terms of the minimum entry of the stochastic matrix defining the channel $V$. The criterion is strengthened for the special case of additive noise channels over finite Abelian groups. Finally, it is shown that domination by a symmetric channel implies (via comparison of Dirichlet forms) a logarithmic Sobolev inequality for the original channel.