Quantum state divergences unified by a new statistical principle
Quantum f-divergences from retrodiction: a unifying classical variational principle
Information Theory
Summary
Measuring differences between quantum states is tricky because classical methods don't directly apply. The authors introduce a new way to extend classical measures to quantum systems by comparing forward and backward measurement processes using a chosen reference state. This approach unifies various known quantum divergences and offers a smoother way to understand transitions between them. It helps explain and connect complex quantum distinctions with a common underlying idea.
What this means in practice
- •For quantum computing engineers: Design more consistent quantum algorithms by using a unified principle to measure similarity between quantum states under different operational assumptions.
- •For quantum communication developers: Improve protocols by selecting optimal state distinguishability metrics that interpolate between standard and maximal quantum divergences.
A theory result. No direct application yet.
Authors
Francesco Buscemi
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-Szkoła 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.