Quantum f-divergences from retrodictive tests: a unifying classical variational principle

Abstract

The same classical f-divergence admits inequivalent extensions to quantum states. The question is whether these extensions can be obtained from a common classical variational principle with an operational interpretation. Here we generalize Matsumoto's reverse-test optimization by introducing retrodictive tests relative to a reference state. A retrodictive test compares a forward and a reverse stochastic process, realized by quantum preparation and measurement, related by Petz transposition relative to the chosen reference, and subject to prescribed quantum averages. Equivalently, it compares two reverse tests sharing a reference output, whose labels are coupled by retrodiction. For faithful states in finite dimension, we prove that the same spectral implementation attains the minimum classical f-divergence over all admissible implementations simultaneously for every operator-convex f. A flat reference yields the standard quantum f-divergences, thereby giving the Nussbaum-Szkoła distributions a variational characterization. Choosing either prescribed state as reference recovers maximal quantum f-divergences and Matsumoto's reverse-test optimization, while power references interpolate between the standard and maximal divergences along the geodesic family. In the relative-entropy case, this gives the passage from Umegaki to Belavkin-Staszewski relative entropy by changing the reference state. The same optimization extends to arbitrary quantum channels and, for channels with faithful Choi operator, gives a variational characterization of quantum entropy production.