# Minimal sufficient statistical experiments on von Neumann algebras

## TL;DR

This paper introduces the concept of minimal sufficiency for statistical experiments on von Neumann algebras, establishing their uniqueness and relationships with subalgebras, and applies these ideas to quantum channels and POVMs.

## Contribution

It defines minimal sufficiency for operator algebraic statistical experiments and proves their uniqueness, connecting these concepts with quantum channels and POVMs.

## Key findings

- Any statistical experiment is equivalent to a unique minimal sufficient one.
- The minimal sufficiency condition for experiments aligns with subalgebra conditions.
- Characterization of POVM discreteness via quantum-classical channels.

## Abstract

A statistical experiment on a von Neumann algebra is a parametrized family of normal states on the algebra. This paper introduces the concept of minimal sufficiency for statistical experiments in such operator algebraic situations. We define equivalence relations of statistical experiments indexed by a common parameter set by completely positive or Schwarz coarse-graining and show that any statistical experiment is equivalent to a minimal sufficient statistical experiment unique up to normal isomorphism of outcome algebras. We also establish the relationship between the minimal sufficiency condition for statistical experiment in this paper and those for subalgebra. These concepts and results are applied to the concatenation relation for completely positive channels with general input and outcome von Neumann algebras. In the case of the quantum-classical channel corresponding to the positive-operator valued measure (POVM), we prove the equivalence of the minimal sufficient condition previously proposed by the author and that in this paper. We also give a characterization of the discreteness of a POVM up to postprocessing equivalence in terms of the corresponding quantum-classical channel.