Discrimination between States of a Quantum System by Observations

Let M be a von Neumann algebra with normal states φ and ω, and let αi:Ai → M be a net of positive unital mappings of finite dimensional algebras. The paper studies the existence of a net βi:M → Ai such that ω ∘ αi ∘ βi → ω and φ ∘ αi ∘ βi → φ. A necessary and sufficient condition is given for the existence of βi which may fail to hold when Ai are commutative and ω does not commute with φ. Reformulation in terms of relative entropy is given and the study of the entropic version is extended to the case when M, φ, and ω are infinite tensor products. All these mathematical problems are motivated by quantum statistical considerations.