Tight relations and equivalences between smooth relative entropies

Bartosz Regula; Ludovico Lami; Nilanjana Datta

arxiv: 2501.12447 · v5 · submitted 2025-01-21 · 🪐 quant-ph · cs.IT· math-ph· math.IT· math.MP

Tight relations and equivalences between smooth relative entropies

Bartosz Regula , Ludovico Lami , Nilanjana Datta This is my paper

Pith reviewed 2026-05-23 04:43 UTC · model grok-4.3

classification 🪐 quant-ph cs.ITmath-phmath.ITmath.MP

keywords smooth relative entropieshypothesis testing relative entropysmooth max-relative entropyinformation spectrum divergenceone-shot quantum informationduality relations

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The pith

Hypothesis testing relative entropy equals the measured smooth max-relative entropy via information spectrum divergence.

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

The paper establishes an equivalence showing that the hypothesis testing relative entropy matches a variant of the smooth max-relative entropy defined through the information spectrum divergence. This same quantity can be viewed as the measured smooth max-relative entropy. The authors strengthen this link by refining the fundamental lemma that relates different versions of the smooth max-relative entropy, using matrix geometric means together with a tightened gentle measurement lemma. These connections produce strictly tighter one-shot bounds and duality relations than previously known, and they sharpen inequalities linking the max-relative entropy to Rényi divergences.

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.

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.

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.

Where Pith is reading between the lines

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

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.

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.

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.

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.

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

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 main news is that the hypothesis testing relative entropy is now equivalent to the information-spectrum (or measured) version of the smooth max-relative entropy, and they have a cleaner proof of the lemma that links the different smooth-max variants. The new argument uses matrix geometric means plus a sharpened gentle-measurement step, which removes some of the previous slack and produces strictly better one-shot bounds and duality relations. They also sharpen a few Renyi-divergence inequalities as a byproduct. That is concrete progress inside the existing one-shot framework rather than a wholesale rewrite of the quantities themselves. The bounds being provably tight is the part that matters for applications; earlier versions left a gap that this closes. The equivalence is only to the measured/information-spectrum variant, not to the usual smooth max-relative entropy, so it refines the picture without making the standard quantity obsolete. No obvious circularity or hidden fitting shows up in the claims. The work sits squarely in the line of Datta-Renner and subsequent one-shot papers, so the citation pattern looks normal. This is useful for anyone who already works with these entropies for concrete tasks; the tighter constants will propagate into follow-up bounds. It is not foundational enough to change how the field defines the quantities, but the technical improvement is real and worth checking. I would send it to a serious referee in quantum information theory.

Referee Report

0 major / 3 minor

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

0 responses · 0 unresolved

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

0 steps flagged

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.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claims rest on standard definitions of smooth entropies and the validity of the cited gentle measurement lemma.

pith-pipeline@v0.9.0 · 5726 in / 1090 out tokens · 29004 ms · 2026-05-23T04:43:51.679193+00:00 · methodology

Tight relations and equivalences between smooth relative entropies

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)