Classical and nonclassical randomness in quantum measurements

TL;DR

This paper explores the structure of quantum measurements represented by positive operator-valued measures, introducing a transform to analyze classical and non-classical aspects and characterizing extremal points in this space.

Contribution

It introduces a transform linking POVMs to completely positive maps and provides an integral representation for certain linear maps, advancing the understanding of quantum measurement convexity.

Findings

Characterization of extremal points of POVMs.

Development of a transform connecting POVMs and linear maps.

Integral representation for unital completely positive maps.

Abstract

The space of positive operator-valued measures on the Borel sets of a compact (or even locally compact) Hausdorff space with values in the algebra of linear operators acting on a d-dimensional Hilbert space is studied from the perspectives of classical and non-classical convexity through a transform $\Gamma$ that associates any positive operator-valued measure with a certain completely positive linear map of the homogeneous C*-algebra $C(X)\otimes B(H)$ into $B(H)$. This association is achieved by using an operator-valued integral in which non-classical 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 $\Omega$ for $\Gamma$ yields an integral representation, along the lines of the classical Riesz Representation Theorem for certain linear functionals on $C(X)$, of certain (but not all) unital completely positive linear maps $\phi:C(X)\otimes B(H) \rightarrow B(H)$. The extremal and C*-extremal points of the space of POVMS are determined.