Order Unit Spaces and Probabilistic Models

TL;DR

This paper establishes a functorial relationship between order unit spaces and probabilistic models, demonstrating that the convex-operational approach can be embedded within the test-space framework, and explores unsharp observables via weighted coin tests.

Contribution

It introduces a functor from order unit spaces to probabilistic models that preserves monoidal structures, unifying two approaches to physical theories.

Findings

The functor is monoidal on subcategories with a bilinear composition.

The convex-operational approach is subsumed by the test-space approach.

Weighted coin tests provide insights into unsharp observables.

Abstract

We exhibit a functor from the category OUS of order unit spaces and positive, unit-preserving mappings into the category $\Prob$ of probabilistic models (test spaces with designated state spaces) and morphisms thereof. Restricted to any subcategory of OUS monoidal with respect to a positive, normalized, bilinear composition rule, our functor is also monoidal. This shows that the convex-operational approach to physical theories can be subsumed by the test-space approach, without resort to ``generalized test spaces''. A second construction, equipping a probabilistic model with tests representing ``weighted coins'', also sheds light on the nature of unsharp observables.