×

Compatibility of any pair of 2-outcome measurements characterizes the Choquet simplex. (English) Zbl 1479.46009

Let \(K\) be a compact convex subset of a locally convex Hausdorff space and \(A(K)\) the set of continuous real affine functionals on \(K\). A measurement on \(A(K)\) is a finite sequence of positive elements in \(A(K)\) with sum equal to 1. It is proved that \(K\) is a Choquet simplex if and only if any pair of 2-outcome measurements are compatible, i.e., the measurements are given as the marginals of a single measurement. This generalizes the finite-dimensional result of M. Plávala [“All measurements in a probabilistic theory are compatible if and only if the state space is a simplex”, Phys. Rev. A 94, No. 4, Article ID 042108, 7 p. (2016; doi:10.1103/PhysRevA.94.042108)] obtained in the context of the foundations of quantum theory. The proof of Plávala depends on the notion of maximal face and is not straightforwardly applicable to infinite-dimensional convex compact sets.

MSC:

46A55 Convex sets in topological linear spaces; Choquet theory
46B40 Ordered normed spaces
81P16 Quantum state spaces, operational and probabilistic concepts
81P15 Quantum measurement theory, state operations, state preparations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alfsen, EM, Compact Convex Sets and Boundary Integrals (1971), Berlin: Springer, Berlin
[2] Ellis, AJ, The duality of partially ordered normed linear spaces, J. London Math. Soc., 39, 1, 730-744 (1964) · Zbl 0131.11302
[3] Heinosaari, T.; Miyadera, T.; Ziman, M., An invitation to quantum incompatibility, J. Phys. A Math. Theor., 49, 12, 123001 (2016) · Zbl 1347.81024
[4] Olubummo, Y.; Cook, TA, The predual of an order-unit Banach space, Int. J. Theor. Phys., 38, 12, 3301-3303 (1999) · Zbl 0959.46007
[5] Plávala, M., All measurements in a probabilistic theory are compatible if and only if the state space is a simplex, Phys. Rev. A, 94, 042108 (2016)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.