Comparison of quantum statistical models: equivalent conditions for sufficiency

TL;DR

This paper extends the classical Blackwell-Sherman-Stein theorem to quantum statistical models, establishing conditions for sufficiency using quantum channels and introducing statistical morphisms for simplified analysis.

Contribution

It generalizes the BSS theorem to quantum models, unifies classical and quantum sufficiency results, and introduces statistical morphisms as a weaker, more easily characterized sufficiency criterion.

Findings

Extended BSS theorem to quantum models using completely positive maps

Unified classical and quantum sufficiency conditions

Introduced statistical morphisms as a weaker sufficiency criterion

Abstract

A family of probability distributions (i.e. a statistical model) is said to be sufficient for another, if there exists a transition matrix transforming the probability distributions in the former to the probability distributions in the latter. The Blackwell-Sherman-Stein (BSS) theorem provides necessary and sufficient conditions for one statistical model to be sufficient for another, by comparing their information values in statistical decision problems. In this paper we extend the BSS theorem to quantum statistical decision theory, where statistical models are replaced by families of density matrices defined on finite-dimensional Hilbert spaces, and transition matrices are replaced by completely positive, trace-preserving maps (i.e. coarse-grainings). The framework we propose is suitable for unifying results that previously were independent, like the BSS theorem for classical statistical models and its analogue for pairs of bipartite quantum states, recently proved by Shmaya. An important role in this paper is played by statistical morphisms, namely, affine maps whose definition generalizes that of coarse-grainings given by Petz and induces a corresponding criterion for statistical sufficiency that is weaker, and hence easier to be characterized, than Petz's.