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

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 $χ<sup>2$-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.

• The paper establishes universal upper bounds on contraction rates for quantum f-divergences in primitive channels using SDPI constants.
• It leverages a local reverse Pinsker inequality to demonstrate quadratic behavior near the stationary state and achieve tightness under detailed balance.
• The results impact quantum mixing times, error correction, and open quantum systems by elucidating the spectral and algebraic properties of quantum channels.

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

Introduction and Context

This paper investigates the contraction rates of quantum channels acting on finite-dimensional density matrices, as measured via quantum $f$-divergences and their associated strong data-processing inequality (SDPI) constants. Quantum $f$-divergences generalize the classical Csiszár $f$-divergence framework to the non-commutative setting, enabling the quantitative analysis of distinguishability loss under CPTP maps. The authors study the asymptotic contraction rates of primitive quantum channels to their unique stationary state, establishing universal upper bounds in terms of the SDPI constants for suitably chosen quantum $\chi^2$-divergences. They further provide conditions for tightness, leveraging quantum detailed balance and a fine-grained analysis of the local behavior of quantum $f$-divergences.

Framework: Quantum $f$-Divergences, SDPI Constants, and $\chi^2$-Divergences

Quantum $f$-divergences, defined for convex functions $f$, generalize the operational and mathematical properties of classical $f$0-divergences. For a quantum channel $f$1 and stationary state $f$2, the SDPI constant $f$3 quantifies the strongest contraction of the $f$4-divergence induced by $f$5 relative to $f$6. This notion underpins the analysis of mixing times and convergence rates of quantum Markov processes analogous to the classical setting.

A broad family of quantum $f$7-divergences, parameterized by standard monotone functions $f$8, arises from non-commutative inversions. They capture the second-order local behavior of quantum $f$9-divergences and serve pivotal roles in the quantitative characterization of SDPI constants. Notably, the maximal quantum $f$0-divergence, $f$1, upper-bounds all other monotone quantum $f$2-divergences, which is key for establishing general upper bounds.

Main Results: Asymptotic Contraction Bounds and Tightness Conditions

The central theorem proved is that for a primitive quantum channel $f$3 with full-rank fixed point $f$4 and for any quantum $f$5-divergence (with $f$6 sufficiently smooth and Pinsker-type lower bound), the asymptotic contraction rate is upper-bounded by the SDPI constant for any associated quantum $f$7-divergence: $f$8 This result is non-trivial in the quantum setting due to non-commutativity and the lack of a universal, unique quantum $f$9-divergence. The proof leverages the establishment of a local reverse Pinsker inequality for quantum $f$0-divergences, showing quadratic local behavior around the stationary point.

Tightness is achieved when the quantum $f$1-divergence is locally equivalent to a specific quantum $f$2-divergence and the channel $f$3 is reversible under the corresponding generalized quantum detailed balance condition. Under these circumstances,

$f$4

This extends classical tightness instances for irreducible, aperiodic Markov chains to the quantum domain, encompassing genuinely non-commutative regimes.

Applications to Common Quantum $f$5-Divergences

The theory is specialized to several key constructions:

• HT (Hirche-Tomamichel) $f$6-divergences ($f$7): The asymptotic contraction rate matches the SDPI constant under $f$8-detailed balance.
• Matsumoto $f$9-divergences (maximal case): The contraction rate aligns with that for $\chi^2$0 in the presence of corresponding detailed balance.
• Petz $\chi^2$1-divergences (operator convex $\chi^2$2): The upper bound improves on existing results by replacing the generic $\chi^2$3 bound with one tailored to $\chi^2$4, and equality holds under the associated quantum detailed balance.

The arguments clarify which quantum $\chi^2$5-divergence locally controls the contraction for each quantum $\chi^2$6-divergence, solidifying the interpretation of $\chi^2$7-divergence contraction in terms of spectrally and algebraically meaningful quantum structures.

Implications and Future Directions

These results deepen the understanding of distinguishability decay in quantum information processing, particularly in non-commutative Markov processes and quantum mixing scenarios. The universal bounds and tightness criteria delineate a sharp distinction between commutative and genuinely quantum regimes, highlighting the nuanced role of quantum detailed balance and the spectral structure of quantum channels.

Practically, these contraction rates have immediate relevance for quantum algorithmic mixing time estimates, quantum error correction thresholds, and operational tasks in quantum statistical mechanics. Theoretically, the established connection between local third-order smoothness of quantum $\chi^2$8-divergences, Pinsker-type inequalities, and the algebraic structure of quantum detailed balance points toward further classifications of mixing phenomena in open quantum systems. This work suggests new research in the exploration of non-Markovian extensions, infinite-dimensional settings, and the interplay between reversibility and functional inequalities in quantum dynamics.

Conclusion

The paper provides a rigorous, general theory of contraction rates for primitive quantum channels with respect to quantum $\chi^2$9-divergences, establishing universally valid upper bounds, explicit conditions for tightness, and comprehensive application to prominent divergence families. These advances clarify the structural determinants of quantum mixing rates and contribute significant technical tools for both quantum probability and quantum information theory (2605.06452).