α-z-Renyi Divergences in von Neumann Algebras: Data Processing Inequality, Reversibility, and Monotonicity Properties in α, z

We study the alpha-z-Renyi divergences D-alpha,D-z (psi & Vert;phi) where alpha, z > 0 (alpha not equal 1) for normal positive functionals 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-z-Renyi divergences. We establish the sufficiency theorem for D-alpha,D-z(psi & Vert;phi), saying that for (alpha, z) inside the DPI bounds, the equality D-alpha,D-z(psi degrees gamma & Vert;phi degrees gamma) = D-alpha,D-z(psi & Vert;phi) < infinity 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,D-z(psi & Vert;phi) in the parameters alpha, z and their limits to the normalized relative entropy as alpha NE arrow 1 and alpha SE arrow 1.