Data-processing inequalities capture the phenomenon that two probability distributions can only become less distinguishable under any common post-processing. For more fine-grained inequalities, one turns to strong data-processing inequality (SDPI) constants, which give the strongest inequalities for a given channel and reference state for a fixed measure of distinguishability. These quantities have been used to quantify the rate at which time-homogeneous Markov chains contract towards a fixed point both in the classical and quantum setting. In this work, we establish that quantum $f$-divergences satisfy a local reverse Pinsker inequality, which implies the asymptotic contraction rate of a primitive channel to its stationary state is upper bounded by the SDPI constant of any non-commutative $χ^2$-divergence. Using quantum-detailed balance, we establish a sufficient condition for these bounds to be tight. Finally, we apply these results to Petz, Matsumoto, and Hirche-Tomamichel $f$-divergences, establishing new and strengthening previously known results.

翻译：数据处理不等式描述了在任意共同后处理下，两个概率分布只能变得难以区分的现象。为了获得更精细的不等式，研究者转向强数据处理不等式常数，该常数给出了固定区分度度量下给定通道和参考态的最强不等式。这些量已被用于量化时间和齐次马尔可夫链在经典和量子设置中向不动点收缩的速率。在本工作中，我们证明了量子$f$散度满足局部逆平斯克尔不等式，这意味着原始通道向其稳态的渐近收缩率受到任何非交换$\chi^2$散度的强数据处理不等式常数的上界限制。利用量子细致平衡条件，我们建立了这些界为紧致的充分条件。最后，我们将这些结果应用于Petz、Matsumoto和Hirche-Tomamichel $f$散度，建立了新的结论并强化了先前已知的结果。