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

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 Plavala (Phys Rev A 94:042108, 2016) obtained in the context of the foundations of quantum theory.