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Unified framework for continuity of sandwiched Rényi divergences

Andreas Bluhm, Ángela Capel, Paul Gondolf, Tim Möbus

TL;DR

This work proves uniform continuity bounds for entropic quantities related to the sandwiched R\'enyi divergences such as the sandwiching R‐enyi conditional entropy and uses the ALAAF method, developed in a previous article by some of the authors, to study the stability of approximate quantum Markov chains.

Abstract

In this work, we prove uniform continuity bounds for entropic quantities related to the sandwiched Rényi divergences such as the sandwiched Rényi conditional entropy. We follow three different approaches: The first one is the "almost additive approach", which exploits the sub-/ superadditivity and joint concavity/ convexity of the exponential of the divergence. In our second approach, termed the "operator space approach", we express the entropic measures as norms and utilize their properties for establishing the bounds. These norms draw inspiration from interpolation space norms. We not only demonstrate the norm properties solely relying on matrix analysis tools but also extend their applicability to a context that holds relevance in resource theories. By this, we extend the strategies of Marwah and Dupuis as well as Beigi and Goodarzi employed in the sandwiched Rényi conditional entropy context. Finally, we merge the approaches into a mixed approach that has some advantageous properties and then discuss in which regimes each bound performs best. Our results improve over the previous best continuity bounds or sometimes even give the first continuity bounds available. In a separate contribution, we use the ALAFF method, developed in a previous article by some of the authors, to study the stability of approximate quantum Markov chains.

Unified framework for continuity of sandwiched Rényi divergences

TL;DR

This work proves uniform continuity bounds for entropic quantities related to the sandwiched R\'enyi divergences such as the sandwiching R‐enyi conditional entropy and uses the ALAAF method, developed in a previous article by some of the authors, to study the stability of approximate quantum Markov chains.

Abstract

In this work, we prove uniform continuity bounds for entropic quantities related to the sandwiched Rényi divergences such as the sandwiched Rényi conditional entropy. We follow three different approaches: The first one is the "almost additive approach", which exploits the sub-/ superadditivity and joint concavity/ convexity of the exponential of the divergence. In our second approach, termed the "operator space approach", we express the entropic measures as norms and utilize their properties for establishing the bounds. These norms draw inspiration from interpolation space norms. We not only demonstrate the norm properties solely relying on matrix analysis tools but also extend their applicability to a context that holds relevance in resource theories. By this, we extend the strategies of Marwah and Dupuis as well as Beigi and Goodarzi employed in the sandwiched Rényi conditional entropy context. Finally, we merge the approaches into a mixed approach that has some advantageous properties and then discuss in which regimes each bound performs best. Our results improve over the previous best continuity bounds or sometimes even give the first continuity bounds available. In a separate contribution, we use the ALAFF method, developed in a previous article by some of the authors, to study the stability of approximate quantum Markov chains.
Paper Structure (23 sections, 39 theorems, 218 equations, 2 figures)

This paper contains 23 sections, 39 theorems, 218 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Notation and basic concepts
  5. Divergences and entropic quantities
  6. Technical tools and main theorems
  7. Almost additive approach
  8. Operator spaces approach
  9. Mixed approach
  10. Continuity bounds for sandwiched Rényi divergences
  11. Continuity bounds for the sandwiched Rényi conditional entropy
  12. Continuity bounds for the sandwiched Rényi mutual information
  13. Continuity bounds for the sandwiched Rényi conditional mutual information
  14. Continuity bounds in the first argument for sandwiched Rényi divergences
  15. Divergence bounds for sandwiched Rényi divergences
  16. ...and 8 more sections

Key Result

Theorem 1

Let $\rho, \sigma, \tau \in \mathop{\mathrm{\mathcal{S}}}\nolimits(\mathop{\mathrm{\mathcal{H}}}\nolimits_A \otimes \mathop{\mathrm{\mathcal{H}}}\nolimits_B)$Note that this includes the case $S(\mathop{\mathrm{\mathcal{H}}}\nolimits)$ by choosing $\mathop{\mathrm{\mathcal{H}}}\nolimits_A=\mathop{\ma

Figures (2)

  • Figure 1: Continuity bounds for $\tilde{H}^{\uparrow}_\alpha(A|B)_\rho$ proven by the almost additive, operator space, and mixed approach, where the visible colour indicates the tightest bound.
  • Figure 2: A comparison of the continuity bounds for $\tilde{H}_\alpha(A|B)_\rho$ proven by the almost-additive, operator space, and mixed approach. The value of $d_A$ is according to the title of each plot. The visible colour indicates where the respective bound outperforms (is tighter than) the others.

Theorems & Definitions (93)

  • Theorem
  • Corollary
  • Definition 3.1: Sandwiched Rényi divergence
  • Remark 3.2
  • Proposition 3.3
  • Definition 3.4: Sandwiched Rényi conditional entropy
  • Definition 3.5: Sandwiched Rényi mutual information
  • Definition 3.6: Sandwiched Rényi conditional mutual information
  • Definition 3.7: Umegaki relative entropy
  • Definition 3.8: Quantum conditional entropy
  • ...and 83 more