Comparison of Quantum Binary Experiments

A quantum binary experiment consists of a pair of density operators on a finite-dimensional Hilbert space. An experiment ℰ is called ∈-deficient with respect to another experiment ℱ if, up to ∈, 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 oneself 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 another. 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.