Fixed-ray escort representations of sandwiched and $\alpha$--$z$ R\'enyi divergences on von Neumann algebras

Abstract:We represent sandwiched and $\alpha$-$z$ Rényi divergences as averages of ordinary relative entropy. The $\alpha$-$z$ Rényi divergence is shown to be an integral over the relative entropy of a canonical family of fixed-ray escort states along the ray $z=c\alpha$. We prove this representation for normal states on an arbitrary von Neumann algebra, using Haagerup non-commutative $L^p$ spaces and interpolation. The formula holds for every $z>0$: for $0<\alpha<1$ it holds when the support of the first state is contained in that of the reference state, and for $\alpha>1$ it holds whenever the divergence is finite. When the lower-order support condition fails, we identify the exact fixed-ray support-boundary term. The representation yields a monotone escort profile and a convex order potential. We use these to reformulate one-shot testing converses, exact sandwiched strong-converse exponents, and work-extraction reliability as signed-area or level-crossing statements, and discuss a restricted two-parameter pair-conversion rate.

Quantum Physics (quant-ph); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Operator Algebras (math.OA)

arXiv:2608.21214 [quant-ph]

(or arXiv:2608.21214v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2608.21214 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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