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 \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.

!5 pages, no figures

References in corpus (4)

• Quantization Methods: A Guide for Physicists and Analysts
• Classical randomness in quantum measurements
• How continuous quantum measurements in finite dimension are actually discrete
• Radon-Nikodym derivatives of quantum operations

Cited by in corpus (8)

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• Complete Characterization of Pure Quantum Measurements and Quantum Channels
• Decomposition of any quantum measurement into extremals
• Classical and nonclassical randomness in quantum measurements
• Normal completely positive maps on the space of quantum operations
• Extreme Covariant Quantum Observables in the Case of an Abelian Symmetry Group and a Transitive Value Space
• Entropic partial orderings of quantum measurements
• Barycentric decomposition for quantum instruments