Divergence Inequalities with Applications in Ergodic Theory

The data processing inequality is central to information theory and motivates the study of monotonic divergences. However, it is not clear operationally we need to consider all such divergences. We establish a simple method for Pinsker inequalities as well as general bounds in terms of $\chi^{2}$-divergences for twice-differentiable $f$-divergences. These tools imply new relations for input-dependent contraction coefficients. We use these relations to show for many $f$-divergences the rate of contraction of a time homogeneous Markov chain is characterized by the input-dependent contraction coefficient of the $\chi^{2}$-divergence. This is efficient to compute and the fastest it could converge for a class of divergences. We show similar ideas hold for mixing times. Moreover, we extend these results to the Petz $f$-divergences in quantum information theory, albeit without any guarantee of efficient computation. These tools may have applications in other settings where iterative data processing is relevant.