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

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