Less noisy domination by symmetric channels

Abstract

Consider the family of all q-ary symmetric channels (q-SCs) with capacities decreasing from log(q) to 0. This paper addresses the following question: what is the member of this family with the smallest capacity that dominates a given channel V in the 'less noisy' preorder sense. When the q-SCs are replaced by q-ary erasure channels, this question is known as the 'strong data processing inequality.' We provide several equivalent characterizations of the less noisy preorder in terms of x2-divergence, Lowner (PSD) partial order, and spectral radius. We then illustrate a simple criterion for domination by a q-SC based on degradation, and mention special improvements for the case where V is an additive noise channel over an Abelian group of order q. Finally, as an application, we discuss how logarithmic Sobolev inequalities for q-SCs, which are well-studied, can be transported to an arbitrary channel V.