α-z-R\'enyi divergences in von Neumann algebras: data-processing inequality, reversibility, and monotonicity properties in α,z

We study the α-z-R\'enyi divergences Dα,z(ψ\|φ) where α,z>0 (α≠1) for normal positive functionals ψ,φ 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 α-z-R\'enyi divergences. We establish the sufficiency theorem for Dα,z(ψ\|φ), saying that for (α,z) inside the DPI bounds, the equality Dα,z(ψ∘γ\|φ∘γ)=Dα,z(ψ\|φ)<∞ in the DPI under a quantum channel (or a normal $2$-positive unital map) γ implies the reversibility of γ with respect to ψ,φ. Moreover, we show the monotonicity properties of Dα,z(ψ\|φ) in the parameters α,z and their limits to the normalized relative entropy as α1 and α1.