Comparison of quantum binary experiments

TL;DR

This paper investigates the conditions under which quantum binary experiments can be compared in terms of deficiency, revealing that classical properties hold only when the experiments commute, and exploring the existence of quantum mappings for zero deficiency.

Contribution

It establishes that classical deficiency properties hold only for commuting quantum experiments and shows that zero deficiency implies a completely positive mapping exists, but not necessarily trace-preserving.

Findings

Classical deficiency properties hold only for commuting densities.

Zero deficiency implies existence of a completely positive mapping.

Such mappings are not necessarily trace-preserving.

Abstract

A quantum binary experiment consists of a pair of density operators on a finite dimensional Hilbert space. An experiment E is called \epsilon-deficient with respect to another experiment F if, up to \epsilon, its risk functions are not worse than the risk functions of F, with respect to all statistical decision problems. It is known in the theory of classical statistical experiments that 1. for pairs of probability distributions, one can restrict to testing problems in the definition of deficiency and 2. that 0-deficiency is a necessary and sufficient condition for existence of a stochastic mapping that maps one pair onto the other. We show that in the quantum case, the property 1. holds precisely if E consist of commuting densities. As for property 2., we show that if E is 0-deficient with respect to F, then there exists a completely positive mapping that maps E onto F, but it is not necessarily trace preserving.