Compatibility of any pair of 2-outcome measurements characterizes the Choquet simplex

Abstract

For a compact convex subset $K $ of a locally convex Hausdorff space, a measurement on $A(K)$ is a finite family of positive elements in $A(K)$ normalized to the unit constant $1_K$, where $A(K)$ denotes the set of continuous real affine functionals on $K$. It is proved that a compact convex set $K$ 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 [Pl\'avala M 2016 Phys.\ Rev.\ A \textbf{94}, 042108] obtained in the context of the foundations of quantum theory.