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Annales Henri Poincaré

, Volume 19, Issue 6, pp 1843–1867 | Cite as

Rényi Divergences as Weighted Non-commutative Vector-Valued \(L_p\)-Spaces

  • Mario Berta
  • Volkher B. Scholz
  • Marco Tomamichel
Article

Abstract

We show that Araki and Masuda’s weighted non-commutative vector-valued \(L_p\)-spaces (Araki and Masuda in Publ Res Inst Math Sci Kyoto Univ 18:339–411, 1982) correspond to an algebraic generalization of the sandwiched Rényi divergences with parameter \(\alpha = \frac{p}{2}\). Using complex interpolation theory, we prove various fundamental properties of these divergences in the setup of von Neumann algebras, including a data-processing inequality and monotonicity in \(\alpha \). We thereby also give new proofs for the corresponding finite-dimensional properties. We discuss the limiting cases \(\alpha \rightarrow \{\frac{1}{2},1,\infty \}\) leading to minus the logarithm of Uhlmann’s fidelity, Umegaki’s relative entropy, and the max-relative entropy, respectively. As a contribution that might be of independent interest, we derive a Riesz–Thorin theorem for Araki–Masuda \(L_p\)-spaces and an Araki–Lieb–Thirring inequality for states on von Neumann algebras.

Mathematics Subject Classification

Primary 81P45, 81R15, 94A17 Secondary 46L52 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Mario Berta
    • 1
  • Volkher B. Scholz
    • 2
  • Marco Tomamichel
    • 3
  1. 1.Department of ComputingImperial College LondonLondonUK
  2. 2.Department of PhysicsGhent UniversityGhentBelgium
  3. 3.Centre for Quantum Software and InformationUniversity of Technology SydneySydneyAustralia

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