$\alpha$-$z$-Rényi divergences in von Neumann algebras: data-processing inequality, reversibility, and monotonicity properties in $\alpha ,z$

Abstract

We study the $\alpha$-$z$-Rényi divergences $D_{\alpha ,z}(\psi \Vert \phi )$ where $\alpha ,z>0$ ($\alpha \ne 1$) for normal positive functionals $\psi ,\phi$ on general von Neumann algebras, introduced in [S.~Kato and Y.~Ueda, arXiv:2307.01790] and [S.~Kato, arXiv:2311.01748]. We prove the variational expressions and the data processing inequality (DPI) for the $\alpha$-$z$-Rényi divergences. We establish the sufficiency theorem for $D_{\alpha ,z}(\psi \Vert \phi )$, saying that for $(\alpha ,z)$ inside the DPI bounds, the equality $D_{\alpha ,z}(\psi \circ \gamma \Vert \phi \circ \gamma )=D_{\alpha ,z}(\psi \Vert \phi )<\infty$ in the DPI under a quantum channel (or a normal $2$-positive unital map) $\gamma$ implies the reversibility of $\gamma$ with respect to $\psi ,\phi$. Moreover, we show the monotonicity properties of $D_{\alpha ,z}(\psi \Vert \phi )$ in the parameters $\alpha ,z$ and their limits to the normalized relative entropy as $\alpha \nearrow 1$ and $\alpha \searrow 1$.

43 pages, the final version to appear in Comm. Math. Phys