| (in English) |
In this talk, we discuss the infinite-dimensional generalization of the result in [M. Pl'avala, Phys. Rev. A textbf{94}, 042108 (2016)]
which states that a finite-dimensional compact convex set $Omega,$ which corresponds to the state space of a generalized probabilistic theory (GPT), is a simplex if and only if any pair of binary measurements on $Omega$ is compatible (i.e. jointly measurable).
In the context of the GPT, this implies that the existence of a pair of incompatible measurements characterizes the non-classical theories.
In the infinite-dimensional setting, there are two possibilities: in the first possibility,
$Omega$ is a compact convex set and we consider each continuous affine functional as an observable, while in the second we only assume the norm-closedness of $Omega$ and consider each bounded affine functional as an observable on $Omega .$
In the former and latter settings, we should replace the mere simplex condition in the finite-dimensions to the Choquet or Bauer simplex condition, respectively. |