Categorical Operational Physics

Abstract

Many insights into the quantum world can be found by studying it from amongst more general operational theories of physics. In this thesis, we develop an approach to the study of such theories purely in terms of the behaviour of their processes, as described mathematically through the language of category theory. This extends a framework for quantum processes known as categorical quantum mechanics (CQM) due to Abramsky and Coecke. We first consider categorical frameworks for operational theories. We introduce a notion of such theory, based on those of Chiribella, D'Ariano and Perinotti (CDP), but more general than the probabilistic ones typically considered. We establish a correspondence between these and what we call "operational categories", using features introduced by Jacobs et al. in effectus theory, an area of categorical logic to which we provide an operational interpretation. We then see how to pass to a broader category of "super-causal" processes, allowing for the powerful diagrammatic features of CQM. Next we study operational theories themselves. We survey numerous principles that a theory may satisfy, treating them in a basic diagrammatic setting, and relating notions from probabilistic theories, CQM and effectus theory. We provide a new description of superpositions in the category of pure quantum processes, using this to give an abstract construction of the category of Hilbert spaces and linear maps. Finally, we reconstruct finite-dimensional quantum theory itself. More broadly, we give a recipe for recovering a class of generalised quantum theories, before instantiating it with operational principles inspired by an earlier reconstruction due to CDP. This reconstruction is fully categorical, not requiring the usual technical assumptions of probabilistic theories. Specialising to such theories recovers both standard quantum theory and that over real Hilbert spaces.