Barycentric decomposition of quantum measurements in finite dimensions

Details Barycentric decomposition of quantum measurements in finite dimensions Barycentric decomposition of quantum measurements in finite dimensions

2008-07-30

arxiv.org/abs/0807.4803v1

J. Math. Phys. 51, 022111 (2010)

Physics

Mathematical Physics

!5 pages, no figures

Scientific paper

10.1063/1.3298681

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.

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