Quantum Sufficiency in the Operator Algebra Framework

Andrzej Łuczak

doi:10.1007/s10773-013-1747-4

Andrzej Łuczak

Open Access

https://doi.org/10.1007/s10773-013-1747-4

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Abstract

The paper is devoted to the investigation of the notion of sufficiency in quantum statistics. Three kinds of this notion are considered: plain sufficiency (called simply: sufficiency), Petz’s sufficiency, and Umegaki’s sufficiency. The problem of the existence and structure of the minimal sufficient subalgebra is analyzed in some detail, conditions yielding equivalence of the three modes of sufficiency are considered, and quantum Basu’s theorem is obtained. Moreover, it is shown that an interesting “factorization theorem” of Jencova and Petz needs some corrections to hold true.

Highlights

Let M be a von Neumann algebra, let N be its von Neumann subalgebra, and let {ρθ : θ ∈ Θ} be a family of normal states on M
The most general notion of sufficiency of the subalgebra N for the family {ρθ : θ ∈ Θ} was introduced by Petz in [5, 6] as a generalization of sufficiency in Umegaki’s sense considered earlier in [8, 9]. It was further investigated in [1, 2]. In this setup the sufficiency of N means the existence of a two-positive map α : M → N such that ρθ ◦ α = ρθ, θ ∈ Θ. (Note that if the map α is a conditional expectation we get sufficiency in Umegaki’s sense.) it seems interesting to investigate a natural generalization of this notion which would consist in giving up the, rather technical, requirement of two-positivity
We examine various questions concerning the notion of minimality, show that under the additional assumption of completeness all the three notions of sufficiency: the one considered in the paper, Petz’s sufficiency and Umegaki’s sufficiency coincide, and obtain a quantum version of Basu’s theorem

Summary

Introduction

Let M be a von Neumann algebra, let N be its von Neumann subalgebra, and let {ρθ : θ ∈ Θ} be a family of normal states on M. The most general notion of sufficiency of the subalgebra N for the family {ρθ : θ ∈ Θ} was introduced by Petz in [5, 6] as a generalization of sufficiency in Umegaki’s sense considered earlier in [8, 9]. It was further investigated in [1, 2]. The same argument applies if one considers the minimax risk instead of the Bayes one The investigation of this general form of sufficiency is the purpose of the paper. It is worth noting that the analysis of minimality in the first part of the paper can be adapted to Petz’s definition of sufficiency yielding a new description of the minimal sufficient subalgebra

Preliminaries and Notation

Minimal Sufficient Subalgebras