REVIEW 3 minor 2 cited by
Tight relations and equivalences between smooth relative entropies
T0 review · 0 major / 3 minor · reviewed 2026-05-23 · grok-4.3
Pith's one-line read Hypothesis testing relative entropy equals the measured smooth max-relative entropy via information spectrum divergence.
desk verdict They turn the hypothesis-testing to smooth-max connection into an equivalence for the information-spectrum variant and tighten the Datta-Renner lemma with matrix geometric means, yielding provably tight bounds. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The equivalence between the hypothesis testing relative entropy and the information-spectrum smooth max-relative entropy (equivalently, the measured smooth max-relative entropy), proved via matrix geometric means and a tightened gentle measurement lemma.
What would settle it
An explicit pair of quantum states and a measurement where the numerical value of the hypothesis testing relative entropy differs from the information-spectrum smooth max-relative entropy by more than the equivalence margin allows.
Extended reading notes
Core claim
The hypothesis testing relative entropy is equivalent to the variant of the smooth max-relative entropy based on the information spectrum divergence, which is also the measured smooth max-relative entropy. An improved lemma connects the variants of the smooth max-relative entropy without gaps, yielding provably tight bounds and duality relations between the hypothesis testing relative entropy and the smooth max-relative entropy, plus refined inequalities with Rényi divergences.
Load-bearing premise
The modified proof using matrix geometric means and the tightened gentle measurement lemma correctly equates the different smooth max-relative entropy variants without new gaps.
Editorial extensions
If this is right
- One-shot bounds between the smooth max-relative entropy and hypothesis testing relative entropy become provably tight.
- Duality relations between these quantities hold with equality in the improved form.
- Bounds connecting the max-relative entropy to Rényi divergences are sharpened.
- Operational one-shot characterizations in quantum tasks gain tighter expressions.
Reading between the lines
- The measured formulation may simplify explicit calculations for finite-block tasks such as quantum hypothesis testing.
- The matrix-geometric-mean technique could apply to other smooth entropy relations outside the current setting.
- Tighter bounds may improve finite-resource estimates in quantum channel discrimination.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript establishes an equivalence between the hypothesis testing relative entropy and a variant of the smooth max-relative entropy defined via the information spectrum divergence (equivalently, a measured smooth max-relative entropy). It strengthens the Datta-Renner connection lemma between variants of the smooth max-relative entropy by means of a matrix-geometric-mean argument and a tightened gentle-measurement lemma. These relations are then applied to obtain strictly tighter one-shot bounds and duality relations between the smooth max-relative entropy and the hypothesis testing relative entropy, as well as sharpened inequalities linking the max-relative entropy to Rényi divergences.
Significance. If the central equivalences and tightened bounds hold, the work supplies sharper, provably tight characterizations for one-shot quantum tasks governed by smooth entropies, including hypothesis testing and related operational settings. The improved lemma and matrix-geometric-mean technique constitute reusable technical tools. The paper ships explicit, parameter-free derivations and falsifiable tightening statements rather than fitted constants.
minor comments (3)
- §3.2, after Eq. (18): the statement that the new gentle-measurement bound is 'strictly tighter' would benefit from an explicit numerical comparison (or a short remark) showing the improvement factor relative to the original Datta-Renner constant for a simple qubit example.
- Notation: the symbol D_∞^ε is used both for the information-spectrum variant and the measured variant; a single clarifying sentence or footnote would prevent reader confusion.
- Figure 1 caption: the plotted curves are not labeled with the precise ε values used; adding them would make the visual comparison self-contained.
Simulated Author's Rebuttal
We thank the referee for their positive evaluation of the manuscript, including the summary of our contributions and the recommendation for minor revision. No specific major comments were raised in the report.
Circularity Check
No significant circularity
full rationale
The paper establishes equivalences between hypothesis testing relative entropy and information-spectrum/measured variants of smooth max-relative entropy via explicit constructions and an improved proof of the Datta-Renner lemma using matrix geometric means and a tightened gentle measurement lemma. These steps rely on standard one-shot quantum information techniques and independent mathematical arguments rather than self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the central claims to prior inputs. The derivation chain remains self-contained against external benchmarks in quantum information theory.
Assumptions & free parameters
Cite this review
Pith. "Pith review of Tight relations and equivalences between smooth relative entropies." pith.science (2026). https://pith.science/paper/2501.12447
@misc{pith2026250112447,
author = {Pith},
title = {Pith review of: Tight relations and equivalences between smooth relative entropies},
year = {2026},
howpublished = {\url{https://pith.science/paper/2501.12447}},
note = {Machine review of arXiv:2501.12447}
}
read the original abstract
The precise one-shot characterisation of operational tasks in classical and quantum information theory relies on different forms of smooth entropic quantities. A particularly important connection is between the hypothesis testing relative entropy and the smooth max-relative entropy, which together govern many operational settings. We first strengthen this connection into a type of equivalence: we show that the hypothesis testing relative entropy is equivalent to a variant of the smooth max-relative entropy based on the information spectrum divergence, which can be alternatively understood as a measured smooth max-relative entropy. Furthermore, we improve a fundamental lemma due to Datta and Renner that connects the different variants of the smooth max-relative entropy, introducing a modified proof technique based on matrix geometric means and a tightened gentle measurement lemma. We use the unveiled connections and tools to strictly improve on previously known one-shot bounds and duality relations between the smooth max-relative entropy and the hypothesis testing relative entropy, establishing provably tight bounds between them. The results then allow us to refine other divergence inequalities, in particular sharpening bounds that connect the max-relative entropy with R\'enyi divergences.
Forward citations
Cited by 2 Pith papers
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Erasure cost of a quantum process: A thermodynamic meaning of the dynamical min-entropy
The adversarial erasure cost of a quantum channel equals, in the zero-error limit, the negative of the channel's min-entropy times k_B T ln 2.
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Convex Split Lemma without Inequalities
An exact collision-mutual-information version of the convex split lemma and a universal, dimension-independent bound on smoothed max mutual information are proven, with applications to quantum state splitting and chan...
Reviewed May 23, 2026 · model on record in the stance chip above.
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