Barycentric decomposition of quantum measurements in finite dimensions

TL;DR

This paper studies the structure of quantum measurement sets (POVMs) in finite dimensions, showing they can be represented as convex combinations of simpler, finite-outcome measurements, with implications for understanding quantum measurement processes.

Contribution

It provides a barycentric decomposition framework for POVMs in finite-dimensional quantum systems, extending classical convex analysis to quantum measurement sets.

Findings

Extreme POVMs are concentrated on at most d^2 points

Any POVM admits a representation as a barycenter of extreme POVMs

Krein-Milman theorem applies to general POVMs in this context

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 \le d^2 points of the outcome space, d< \infty 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 \le d^2 points of the outcome space.