Sufficient positive maps between von Neumann algebras: R\'enyi divergences

Abstract:We prove recovery theorems for $\alpha$-$z$ Rényi divergences under normal unital positive maps between von Neumann algebras. Previous results in this setting required 2-positivity, while recent finite dimensional work showed that this assumption can be relaxed to mere positivity. Our proof uses sufficiency for JW*-subalgebras and $L^p$-spaces over them to characterize equality in the data processing inequality. As a further application of our methods, we solve a problem discussed by Haagerup and Stormer: Every conditional expectation of a von Neumann algebra onto a JW*-subalgebra factors through a conditional expectation onto the generated von Neumann subalgebra.

Quantum Physics (quant-ph); Operator Algebras (math.OA)

arXiv:2608.28510 [quant-ph]

(or arXiv:2608.28510v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.28510 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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