Tight Contraction Rates for Primitive Channels under Quantum $f$-Divergences

• URL: http://arxiv.org/abs/2605.06452v1
• Date: Thu, 07 May 2026 15:48:23 GMT
• Title: Tight Contraction Rates for Primitive Channels under Quantum $f$-Divergences
• Abstract summary: We show that quantum $f$-divergences satisfy a local reverse Pinsker inequality.<n>We apply these results to Petz, Matsumoto, and Hirche-Tomamichel $f$-divergences.
• Score: 10.960289997471085
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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