Barycentric decomposition of quantum measurements in finite dimensions

We analyze the convex structure of the set of positive operator valued measures POVMs representing quantum measurements on a given finite dimensional quan- tum system, with outcomes in a given locally compact Hausdorff space. The ex- treme points of the convex set are operator valued measures concentrated on a finite set of kd2 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 concen- trated on kd2 points of the outcome space.