Contraction Coefficients of Product Symmetric Channels

Contraction coefficients for Kullback-Leibler (KL) divergence are channel-dependent constants that are used to sharpen data processing inequalities for KL divergence into strong data processing inequalities. In this work, we present a method to analytically compute the KL contraction coefficients of product q-ary symmetric channels. Although such contraction coefficients have many applications in information theory as well as probability and statistics more broadly, deriving exact values is considered to be challenging for complex channels such as product channels. While recent discoveries have reduced the search space of the optimization problems defining contraction coefficients significantly, the problem is still far from trivial. We find that the symmetry inherent in channels such as the q-ary symmetric channel allows for further simplification, making an exact closed-form expression for contraction coefficients for KL divergence possible to derive. We begin by obtaining the KL contraction coefficients of product binary symmetric channels, and then generalize our results to larger alphabets. Our methods are not specific to product q-ary symmetric channels and can be used for wider classes of channels that satisfy certain symmetry properties.