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 \& Masuda, Publ. Res. Inst. Math. Sci., 18:339 (1982)] correspond to an algebraic generalization of the sandwiched Rényi divergences with parameter $\hat{I}\pm =\frac{p}{2}$. 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 $\hat{I}\pm$. We thereby also give new proofs for the corresponding finite-dimensional properties. We discuss the limiting cases $\hat{I}\pm \to \{\frac{1}{2},1,\infty \}$ 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.

v2: 20 pages, published version