Dichotomy between deterministic and probabilistic models in countably additive effectus theory
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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 σ-effectuses, where certain countable sums of morphisms are defined. We study in particular σ-effectuses where unnormalized states can be normalized. We show that a non-trivial σ-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.
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