arxiv: 2605.06452 · v1 · submitted 2026-05-07 · 🪐 quant-ph · cs.IT· math.IT

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Tight Contraction Rates for Primitive Channels under Quantum f-Divergences

Matthew Simon Tan , Marco Tomamichel , Ian George

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Pith reviewed 2026-05-08 10:59 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath.IT

keywords quantum f-divergencesstrong data processing inequalitycontraction ratesprimitive quantum channelsreverse Pinsker inequalityquantum detailed balancechi-squared divergenceasymptotic mixing

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The pith

Quantum f-divergences satisfy a local reverse Pinsker inequality that upper bounds the contraction rate of primitive channels by the SDPI constant of non-commutative chi-squared divergences.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that quantum f-divergences obey a local reverse Pinsker inequality near a stationary state. This inequality shows that the rate at which a primitive quantum channel brings states closer to its unique stationary distribution is limited by the strong data-processing inequality constant for a non-commutative version of the chi-squared divergence. The authors also find that when the channel satisfies quantum detailed balance, these upper bounds become tight for several specific f-divergences. These findings apply to Petz, Matsumoto, and Hirche-Tomamichel divergences and extend previous contraction rate results in quantum information.

Core claim

Quantum f-divergences satisfy a local reverse Pinsker inequality, which implies that the asymptotic contraction rate of a primitive channel to its stationary state is upper bounded by the SDPI constant of any non-commutative χ²-divergence. Using quantum-detailed balance, a sufficient condition for these bounds to be tight is established, and the results are applied to Petz, Matsumoto, and Hirche-Tomamichel f-divergences.

What carries the argument

The local reverse Pinsker inequality for quantum f-divergences, which connects the f-divergence to the chi-squared divergence in a small neighborhood around the stationary state, allowing control of asymptotic contraction rates through SDPI constants.

If this is right

The asymptotic contraction rate is upper bounded by the SDPI constant of the χ²-divergence.
The bounds are tight under the condition of quantum detailed balance.
Strengthened contraction rate results hold for Petz, Matsumoto, and Hirche-Tomamichel f-divergences.
Fine-grained distinguishability measures can be used to quantify mixing rates in quantum Markov chains.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This approach might simplify calculations of mixing times for quantum channels by reducing them to computing SDPI constants.
The results could extend to bounding errors in quantum algorithms or processes that rely on channel mixing.
Connections may exist to classical results on contraction rates, allowing unified treatments of quantum and classical cases.

Load-bearing premise

The quantum channel must be primitive, having a unique stationary state to which it contracts asymptotically, and the local reverse Pinsker inequality must apply near that state.

What would settle it

A counterexample would be a primitive quantum channel and an f-divergence where the observed asymptotic contraction rate exceeds the SDPI constant of the associated non-commutative chi-squared divergence.

read the original 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 $\chi^2$-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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a local reverse Pinsker inequality for quantum f-divergences that upper-bounds the asymptotic contraction rate of primitive channels by the SDPI constant of a non-commutative chi-squared divergence, with a sufficient tightness condition from quantum detailed balance.

read the letter

The main new piece is the local reverse Pinsker inequality for quantum f-divergences. From that they get an upper bound on the long-term contraction rate of any primitive channel to its stationary state, controlled by the SDPI constant of a chi-squared type divergence. They also supply a quantum detailed balance condition that makes the bound tight in some cases, then check the three families of Petz, Matsumoto, and Hirche-Tomamichel divergences and recover or improve earlier contraction-rate results. That chain is clean and stays within standard monotonicity properties of f-divergences, so the logic holds together on paper. The applications are concrete enough to be useful for anyone tracking mixing times or equilibration in quantum Markov processes. The primitivity assumption is standard and necessary for a unique fixed point and asymptotic contraction, but it does limit the scope to channels that mix. The tightness condition is only sufficient, not necessary, which is honest but leaves open whether other channels achieve equality without detailed balance. Without the full derivations in front of me I cannot rule out small gaps in the local-to-asymptotic step, though nothing in the abstract suggests circularity or hidden fitting. This is for readers already comfortable with quantum SDPI constants and f-divergences; they will see the most value. It is worth sending to a serious referee because the claims are specific, the method is reproducible in principle, and the results tighten existing bounds in a field where such refinements matter.

Referee Report

0 major / 2 minor

Summary. The paper establishes that quantum f-divergences satisfy a local reverse Pinsker inequality near the stationary state. This inequality is used to prove that the asymptotic contraction rate of a primitive quantum channel to its fixed point is upper-bounded by the SDPI constant of any associated non-commutative χ²-divergence. Quantum detailed balance supplies a sufficient condition for these bounds to be tight. The framework is then applied to the Petz, Matsumoto, and Hirche-Tomamichel families of f-divergences, yielding new bounds and strengthening some previously known contraction-rate results.

Significance. If the local inequality and its passage to the asymptotic rate hold, the work supplies a unified approach to obtaining tight asymptotic mixing bounds for primitive quantum Markov chains under a wide class of f-divergences. The explicit tightness criterion via quantum detailed balance is a concrete strength that identifies cases where the SDPI constant is achieved. The applications to concrete divergences make the abstract bounds immediately usable for convergence analysis in quantum information.

minor comments (2)

The transition from the local reverse Pinsker inequality to the global asymptotic rate (presumably in the main theorem) would benefit from an explicit statement of the neighborhood size in which the local inequality is assumed to hold.
Notation for the non-commutative χ²-divergence and its SDPI constant should be introduced with a self-contained definition in the preliminaries rather than only by reference.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as for the recommendation of minor revision. The report correctly identifies the key contributions: the local reverse Pinsker inequality for quantum f-divergences, the resulting upper bound on asymptotic contraction rates via non-commutative χ²-SDPI constants, the tightness criterion under quantum detailed balance, and the applications to the Petz, Matsumoto, and Hirche-Tomamichel families. Since the report contains no specific major comments, we have no individual points to address. We will incorporate any minor editorial or clarification suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The central results—a local reverse Pinsker inequality for quantum f-divergences and the implied bound on asymptotic contraction rates via the SDPI constant of a non-commutative χ²-divergence—are derived from standard monotonicity properties of f-divergences and an explicit inequality derivation within the paper. The sufficient condition for tightness is obtained using quantum detailed balance, which is applied as an independent assumption rather than a self-referential definition. No step reduces by construction to the target contraction rate or relies on a load-bearing self-citation chain; the argument remains self-contained against external benchmarks such as data-processing inequalities and primitivity of the channel.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on established properties of quantum channels and divergences without introducing new free parameters or postulated entities.

axioms (2)

domain assumption Quantum f-divergences are monotone under completely positive trace-preserving maps
Standard property invoked to obtain data-processing inequalities for the channel.
domain assumption Primitive quantum channels possess a unique stationary state and contract to it asymptotically
Required to define the asymptotic contraction rate to a fixed point.

pith-pipeline@v0.9.0 · 5477 in / 1479 out tokens · 46454 ms · 2026-05-08T10:59:10.132132+00:00 · methodology

discussion (0)

Reference graph

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