\(\alpha \)-z-Rényi Divergences in von Neumann Algebras: Data Processing Inequality, Reversibility, and Monotonicity Properties in \(\alpha ,z\)

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