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Pré-Publication, Document De Travail Année : 2023

Unified framework for continuity of sandwiched Rényi divergences

Angela Capel
  • Fonction : Auteur
Paul Gondolf
  • Fonction : Auteur
Tim Möbus
  • Fonction : Auteur

Résumé

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 axiomatic 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 ALAAF method, developed in a previous article by some of the authors, to study the stability of approximate quantum Markov chains.

Dates et versions

hal-04201945 , version 1 (11-09-2023)

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Andreas Bluhm, Angela Capel, Paul Gondolf, Tim Möbus. Unified framework for continuity of sandwiched Rényi divergences. 2023. ⟨hal-04201945⟩
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