Classical and nonclassical randomness in quantum measurements

Abstract

The space POVM H(X) of positive operator-valued probability measures on the Borel sets of a compact (or even locally compact) Hausdorff space X with values in B(H), the algebra of linear operators acting on a d-dimensional Hilbert space H, is studied from the perspectives of classical and nonclassical convexity through a transform Γ that associates any positive operator-valued measure ν with a certain completely positive linear map Γ(ν) of the homogeneous C*-algebra C(X)⊗B(H) into B(H). This association is achieved by using an operator-valued integral in which nonclassical random variables (that is, operator-valued functions) are integrated with respect to positive operator-valued measures and which has the feature that the integral of a random quantum effect is itself a quantum effect. A left inverse Ω for Γ yields an integral representation, along the lines of the classical Riesz representation theorem for linear functionals on C(X), of certain (but not all) unital completely positive linear maps φ:C(X)⊗B(H)→B(H). The extremal and C*-extremal points of POVM H(X) are determined.