Comparison of channels: criteria for domination by a symmetric channel

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 give equivalent characterizations in terms of χ-divergence, Löwner (PSD) partial order, and spectral radius. Furthermore, we develop a simple criterion for domination by a 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. Finally, it is shown that domination by a symmetric channel implies (via comparison of Dirichlet forms) a logarithmic Sobolev inequality for the original channel.