Rényi Divergences as Weighted Non-commutative Vector-Valued $$L_p$$ L p -Spaces

Rényi Divergences as Weighted Non-commutative Vector-Valued $$L_p$$ L p -Spaces We show that Araki and Masuda’s weighted non-commutative vector-valued $$L_p$$ 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 $$\alpha = \frac{p}{2}$$ α = 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 $$\alpha $$ α . We thereby also give new proofs for the corresponding finite-dimensional properties. We discuss the limiting cases $$\alpha \rightarrow \{\frac{1}{2},1,\infty \}$$ α → { 1 2 , 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 $$L_p$$ L p -spaces and an Araki–Lieb–Thirring inequality for states on von Neumann algebras. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales Henri Poincaré Springer Journals

Rényi Divergences as Weighted Non-commutative Vector-Valued $$L_p$$ L p -Spaces

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Physics; Theoretical, Mathematical and Computational Physics; Dynamical Systems and Ergodic Theory; Quantum Physics; Mathematical Methods in Physics; Classical and Quantum Gravitation, Relativity Theory; Elementary Particles, Quantum Field Theory
ISSN
1424-0637
eISSN
1424-0661
D.O.I.
10.1007/s00023-018-0670-x
Publisher site
See Article on Publisher Site

Abstract

We show that Araki and Masuda’s weighted non-commutative vector-valued $$L_p$$ 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 $$\alpha = \frac{p}{2}$$ α = 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 $$\alpha $$ α . We thereby also give new proofs for the corresponding finite-dimensional properties. We discuss the limiting cases $$\alpha \rightarrow \{\frac{1}{2},1,\infty \}$$ α → { 1 2 , 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 $$L_p$$ L p -spaces and an Araki–Lieb–Thirring inequality for states on von Neumann algebras.

Journal

Annales Henri PoincaréSpringer Journals

Published: Mar 17, 2018

References

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