Minimal sufficient statistical experiments on von Neumann algebras

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.

28 pages. Example 1 is added. Typos are corrected

References in corpus (2)

• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Sharp and fuzzy observables on effect algebras

Cited by in corpus (6)

• Quantum incompatibility of channels with general outcome operator algebras
• Matrix majorization in large samples
• Accessible information without disturbing partially known quantum states on a von Neumann algebra
• Directed-completeness of quantum statistical experiments in the randomization order
• Sufficiency of Rényi divergences
• Entanglement-breaking channels with general outcome operator algebras