Barycentric decomposition of quantum measurements in finite dimensions

Abstract

We analyze the convex structure of the set of positive operator valued measures (POVMs) representing quantum measurements on a given finite dimensional quantum system, with outcomes in a given locally compact Hausdorff space. The extreme points of the convex set are operator valued measures concentrated on a finite set of k≤d2 points of the outcome space, d<∞ being the dimension of the Hilbert space. We prove that for second-countable outcome spaces any POVM admits a Choquet representation as the barycenter of the set of extreme points with respect to a suitable probability measure. In the general case, Krein–Milman theorem is invoked to represent POVMs as barycenters of a certain set of POVMs concentrated on k≤d2 points of the outcome space.