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Tight relations and equivalences between smooth relative entropies

Bartosz Regula, Ludovico Lami, Nilanjana Datta

TL;DR

The paper strengthens the bridge between one-shot smooth divergences by introducing the modified max-relative entropy $ ilde{D}^ ext{ε}_{ ext{max}}$ and proving precise equivalences with the hypothesis testing relative entropy $D^{1-ε}_H$, enabling exact reconstruction of one from the other. It upgrades a foundational Datta–Renner lemma using the operator geometric mean, derives tight one-shot dualities between $D^ ext{ε}_{ ext{max}}$ and $D^{1-ε}_H$, and yields sharp trace/purified-distance bounds. The authors further connect these results to Rényi divergences and the information spectrum, develop simultaneous smoothing techniques, and provide strengthened versions of the quantum substate theorem and Frenkel-type integral representations. Collectively, these contributions sharpen one-shot bounds, refine dualities, and illuminate the structure of quantum divergences beyond the asymptotic regime, with potential impact on privacy, coding, and hypothesis testing tasks.

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 smoothed 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 smoothed 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. We use these results to refine other divergence inequalities, in particular sharpening bounds that connect the max-relative entropy with Rényi divergences.

Tight relations and equivalences between smooth relative entropies

TL;DR

The paper strengthens the bridge between one-shot smooth divergences by introducing the modified max-relative entropy and proving precise equivalences with the hypothesis testing relative entropy , enabling exact reconstruction of one from the other. It upgrades a foundational Datta–Renner lemma using the operator geometric mean, derives tight one-shot dualities between and , and yields sharp trace/purified-distance bounds. The authors further connect these results to Rényi divergences and the information spectrum, develop simultaneous smoothing techniques, and provide strengthened versions of the quantum substate theorem and Frenkel-type integral representations. Collectively, these contributions sharpen one-shot bounds, refine dualities, and illuminate the structure of quantum divergences beyond the asymptotic regime, with potential impact on privacy, coding, and hypothesis testing tasks.

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 smoothed 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 smoothed 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. We use these results to refine other divergence inequalities, in particular sharpening bounds that connect the max-relative entropy with Rényi divergences.
Paper Structure (26 sections, 27 theorems, 79 equations)

This paper contains 26 sections, 27 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Summary of main results
  3. Smooth divergences
  4. Hypothesis testing relative entropy and max-relative entropy
  5. Hypothesis testing relative entropy and max-relative entropy
  6. Modified max-relative entropy $\boldsymbol{\widetilde{D}^\varepsilon_{\max}}$
  7. On normalised and subnormalised smoothing
  8. Equivalences
  9. Classical $\boldsymbol{\widetilde{D}^\varepsilon_{\max}}$ and $\boldsymbol{{D}^\varepsilon_{\max}}$
  10. $\boldsymbol{\widetilde{D}^\varepsilon_{\max}}$ as measured $\boldsymbol{D^\varepsilon_{\max}}$
  11. $\boldsymbol{D^\varepsilon_H}$ as optimised $\boldsymbol{\widetilde{D}^\varepsilon_{\max}}$
  12. Improving a useful lemma
  13. Tightened bounds and relations
  14. Relating $\boldsymbol{\widetilde{D}^\varepsilon_{\max}}$ and $\boldsymbol{D^\varepsilon_H}$
  15. Connecting $\boldsymbol{D^\varepsilon_{\max}}$ and $\boldsymbol{\widetilde{D}^\varepsilon_{\max}}$
  16. ...and 11 more sections

Key Result

Lemma 1

For any two quantum states $\rho$ and $\sigma$ and any $\varepsilon \in [0,1]$, it holds that As a consequence, $D_{\max}^{ \IfEqCase{t}{ {t}{ \IfEqCase{n}{ {norm}{ \varepsilon,\,T,= } {n}{ \varepsilon,\,T,= } {sub}{ \varepsilon,\,T,\leq} {s}{ \varepsilon,\,T,\leq} } } {p}{ \IfEqCase{n}{ {norm}{ \varepsilon,\,P,= } {n}{ \varepsilon,\,P,= } {sub}{ \varepsilon,\,P

Theorems & Definitions (27)

  • Lemma 1
  • Lemma 2: dwork_2014abdelhadi_2020
  • Proposition 3
  • Theorem 4
  • Theorem 5: (Tightened Datta--Renner lemma)
  • Lemma 6: (A gentler measurement lemma)
  • Lemma 7
  • Corollary 8
  • Lemma 9
  • Lemma 10
  • ...and 17 more