Less Noisy Domination By Symmetric Channels

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 qnSCs 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 x(2)-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 17 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.