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

• URL: http://arxiv.org/abs/2404.07617v2
• Date: Mon, 30 Sep 2024 06:33:47 GMT
• Title: $α$-$z$-Rényi divergences in von Neumann algebras: data-processing inequality, reversibility, and monotonicity properties in $α,z$
• Authors: Fumio Hiai, Anna Jenčová,
• Abstract summary: We prove the variational expressions and the data processing inequality (DPI) for the $alpha$-$z$-R'enyi divergences. We show the monotonicity properties of $D_alpha,z(psi|varphi)$ in the parameters $alpha,z$ and their limits to the normalized relative entropy.
• Score: 2.3020018305241337
• License:
• Abstract: We study the $\alpha$-$z$-R\'enyi divergences $D_{\alpha,z}(\psi\|\varphi)$ where $\alpha,z>0$ ($\alpha\ne1$) for normal positive functionals $\psi,\varphi$ 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 $\alpha$-$z$-R\'enyi divergences. We establish the sufficiency theorem for $D_{\alpha,z}(\psi\|\varphi)$, saying that for $(\alpha,z)$ inside the DPI bounds, the equality $D_{\alpha,z}(\psi\circ\gamma\|\varphi\circ\gamma)=D_{\alpha,z}(\psi\|\varphi)<\infty$ 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 $D_{\alpha,z}(\psi\|\varphi)$ in the parameters $\alpha,z$ and their limits to the normalized relative entropy as $\alpha\nearrow1$ and $\alpha\searrow1$.

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