Compact convex structure of measurements and its applications to simulability, incompatibility, and convex resource theory of continuous-outcome measurements

Understand

We introduce the post-processing preorder and equivalence relations for general measurements on a possibly infinite-dimensional general probabilistic theory described by an order unit Banach space $E$ with a Banach predual.

• We define the measurement space $\mathfrak{M}(E)$ as the set of post-processing equivalence classes of continuous measurements on $E .$ We define the weak topology on $\mathfrak{M} (E)$ as the weakest topology in which the state discrimination probabilities for any finite-label ensembles are continuous and show that $\mathfrak{M}(E)$ equipped with the convex operation corresponding to the probabilistic mixture of measurements can be regarded as a compact convex set regularly embedded in a locally convex Hausdorff space.
• We also prove that the measurement space $\mathfrak{M}(E) $ is infinite-dimensional except when the system is $1$-dimensional and give a characterization of the post-processing monotone affine functional.
• We apply these general results to the problems of simulability and incompatibility of measurements.