Sufficiency in quantum statistical inference. A survey with examples

TL;DR

This survey explores the concept of sufficiency in quantum statistical inference, drawing parallels with classical theory, and introduces a new characterization of sufficiency using quantum Fisher information.

Contribution

It provides a comprehensive overview of sufficiency in quantum statistics, including classical analogues, key examples, and a novel characterization via quantum Fisher information.

Findings

Classical examples have non-commutative analogues

Main examples include Weyl algebra and exponential family of states

New characterization of sufficiency using quantum Fisher information

Abstract

This paper attempts to give an overview about sufficiency in the setting of quantum statistics. The basic concepts are treated paralelly to the the measure theoretic case. It turns out that several classical examples and results have a non-commutative analogue. Some of the results are presented without proof (but with exact references) and the presentation is intended to be self-contained. The main examples discussed in the paper are related the Weyl algebra and to the exponential family of states. The characterization of sufficiency in terms of quantum Fisher information is a new result.