Sufficiency in quantum statistical inference

TL;DR

This paper develops a theory of sufficiency for quantum statistical inference within non-commutative algebras, extending classical ideas to quantum settings and providing characterizations and applications.

Contribution

It introduces a framework for sufficiency in quantum statistics, including characterizations, a non-commutative factorization theorem, and applications to entropy and exponential families.

Findings

Characterization of sufficient coarse-grainings

Non-commutative factorization theorem established

Applications to von Neumann entropy and quantum exponential families

Abstract

This paper attempts to develop a theory of sufficiency in the setting of non-commutative algebras parallel to the ideas in classical mathematical statistics. Sufficiency of a coarse-graining means that all information is extracted about the mutual relation of a given family of states. In the paper sufficient coarse-grainings are characterized in several equivalent ways and the non-commutative analogue of the factorization theorem is obtained. Among the applications the equality case for the strong subadditivity of the von Neumann entropy, the Imoto-Koashi theorem and exponential families are treated. The setting of the paper allows the underlying Hilbert space to be infinite dimensional.