Dichotomy between Deterministic and Probabilistic Models in Countably Additive Effectus Theory

Kenta Cho
(National Institute of Informatics, Japan)
Bas Westerbaan
(University College London)
John van de Wetering
(Radboud Universiteit Nijmegen)

Effectus theory is a relatively new approach to categorical logic that can be seen as an abstract form of generalized probabilistic theories (GPTs). While the scalars of a GPT are always the real unit interval [0,1], in an effectus they can form any effect monoid. Hence, there are quite exotic effectuses resulting from more pathological effect monoids.

In this paper we introduce sigma-effectuses, where certain countable sums of morphisms are defined. We study in particular sigma-effectuses where unnormalized states can be normalized. We show that a non-trivial sigma-effectus with normalization has as scalars either the two-element effect monoid 0,1 or the real unit interval [0,1]. When states and/or predicates separate the morphisms we find that in the 0,1 case the category must embed into the category of sets and partial functions (and hence the category of Boolean algebras), showing that it implements a deterministic model, while in the [0,1] case we find it embeds into the category of Banach order-unit spaces and of Banach pre-base-norm spaces (satisfying additional properties), recovering the structure present in GPTs.

Hence, from abstract categorical and operational considerations we find a dichotomy between deterministic and convex probabilistic models of physical theories.

In Benoît Valiron, Shane Mansfield, Pablo Arrighi and Prakash Panangaden: Proceedings 17th International Conference on Quantum Physics and Logic (QPL 2020), Paris, France, June 2 - 6, 2020, Electronic Proceedings in Theoretical Computer Science 340, pp. 91–113.
Published: 6th September 2021.

ArXived at: http://dx.doi.org/10.4204/EPTCS.340.5 bibtex PDF
References in reconstructed bibtex, XML and HTML format (approximated).
Comments and questions to: eptcs@eptcs.org
For website issues: webmaster@eptcs.org