Rényi Divergences as Weighted Non-commutative Vector-Valued L<inf>p</inf> -Spaces

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Journal Article
Annales Henri Poincare, 2018, 19 (6), pp. 1843 - 1867
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© 2018, Springer International Publishing AG, part of Springer Nature. We show that Araki and Masuda’s weighted non-commutative vector-valued Lp-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 α=p2. 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 α. We thereby also give new proofs for the corresponding finite-dimensional properties. We discuss the limiting cases α→{12,1,∞} 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 Lp-spaces and an Araki–Lieb–Thirring inequality for states on von Neumann algebras.
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