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.

翻译：数据处理不等式是信息论的核心，并推动了对单调散度的研究。然而，从操作层面看，我们是否需要考虑所有此类散度尚不明确。我们为二次可微的$f$-散度建立了一种简单的Pinsker不等式方法，以及基于$\chi^{2}$-散度的一般界。这些工具蕴含了输入相关收缩系数的新关系。利用这些关系，我们证明对于许多$f$-散度，时间齐次马尔可夫链的收缩速率可由$\chi^{2}$-散度的输入相关收缩系数刻画。该系数易于计算，且对于一类散度而言，其收敛速度可达最快。我们证明了类似思想也适用于混合时间。此外，我们将这些结果推广到量子信息论中的Petz $f$-散度，尽管无法保证其计算效率。这些工具可能在涉及迭代数据处理的其他场景中具有应用价值。