2026-10-22 · arXiv Quantum Physics
arXiv:2610.29696v1 Announce Type: new
Abstract: The same classical f-divergence admits inequivalent extensions to quantum states. The question is whether these extensions can be obtained from a common operational variational principle. Here we generalize Matsumoto's reverse-test optimization by introducing a reference prior state. More explicitly, we minimize the statistical distinguishability between a forward and a reverse prepare-and-measure process, related by retrodiction through the chosen reference and subject to prescribed quantum averages. Distinguishability is quantified by the classical f-divergence between their complete records. For faithful states in finite dimension, we prove that the same spectral implementation attains the minimum simultaneously for every operator-convex f. Varying the reference allows us to interpolate between standard and maximal quantum f-divergences of the prescribed states, passing through the geodesic family. More precisely, a flat reference yields the standard quantum f-divergences, thereby giving the Nussbaum-Szko{\l}a distributions a variational characterization as optimal records. Choosing either prescribed state as reference recovers maximal quantum f-divergences and Matsumoto's reverse tests, while power references realize the geodesic interpolation. In the relative-entropy case, this gives the passage from Umegaki to Belavkin-Staszewski relative entropy by changing the prior for retrodiction.
原文链接：https://arxiv.org/abs/2610.29696
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