Rényi divergences as weighted non-commutative vector valued L_p-spaces

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 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 α. We thereby also give new proofs for the corresponding finite-dimensional properties. We discuss the limiting cases 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-spaces and an Araki–Lieb–Thirring inequality for states on von Neumann algebras.