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Classical Explanations in (and of) General Probabilistic Theories

John Harding, Alex Wilce

Abstract

We introduce a notion of the ``explanation" of one (generalized) probabilistic model by another as particular kind of span in the category $\Prob$ of probabilistic models and morphisms. We show that explanations compose under a standard pullback construction (notwithstanding that $\Prob$ does not support arbitrary pullbacks). We then show that every locally-finite probabilistic model has a canonical, sharp classical explanation. The construction is functorial, so every locally-finite probabilistic theory has a canonical, sharp classical (though of course, usually non-local) representation.

Classical Explanations in (and of) General Probabilistic Theories

Abstract

We introduce a notion of the ``explanation" of one (generalized) probabilistic model by another as particular kind of span in the category of probabilistic models and morphisms. We show that explanations compose under a standard pullback construction (notwithstanding that does not support arbitrary pullbacks). We then show that every locally-finite probabilistic model has a canonical, sharp classical explanation. The construction is functorial, so every locally-finite probabilistic theory has a canonical, sharp classical (though of course, usually non-local) representation.
Paper Structure (13 sections, 6 theorems, 26 equations)

This paper contains 13 sections, 6 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Probabilistic models and theories
  3. Classical and Quantum Models
  4. Categories of Probabilistic Models
  5. Embeddings, Quotients, and Explanations
  6. Classical Explanations
  7. Weak-$\ast$ integrals and barycenters
  8. Borelification
  9. Ontological Representations
  10. Locality
  11. Composites and entanglement
  12. Bell-locality
  13. Conclusion

Key Result

Proposition 2.11

Sub-quotients have pullbacks, which are explanations.

Theorems & Definitions (26)

  • Definition 2.1
  • Example 2.2: Borel models
  • Example 2.3: Projective quantum models
  • Definition 2.4: Unsharp Borel manuals
  • Definition 2.5
  • Example 2.6: Morphisms of Borel manuals
  • Example 2.7: Morphisms of projection manuals
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10: Explanations
  • ...and 16 more