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Order Unit Spaces and Probabilistic Models

John Harding, Alex Wilce

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.

Order Unit Spaces and Probabilistic Models

Abstract

We exhibit a functor from the category OUS of order unit spaces and positive, unit-preserving mappings into the category 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.
Paper Structure (9 sections, 13 theorems, 38 equations)

This paper contains 9 sections, 13 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. ${\boldsymbol A}$-valued weights on Test Spaces
  3. Test spaces associated with an OUS
  4. Processes and Channels
  5. Composites
  6. Sub-order unit spaces
  7. Markov Kernels
  8. Monoidality
  9. Rolling Dice

Key Result

Lemma 3.2

Suppose that for all $i, j \in \bigcup \boldsymbol{\mathscr J}$, there exists some $r \in \bigcup \boldsymbol{\mathscr J}$ such that $\{i,r\}, \{j,r\} \in {\mathcal{E}}(\boldsymbol{\mathscr J})$. Then every probability weight on ${\mathcal{M}}^{\boldsymbol{\mathscr J}}({\boldsymbol A})$ has the form

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • Lemma 3.5
  • Remark 3.6
  • Definition 4.1
  • Proposition 4.2
  • Corollary 4.3
  • ...and 14 more