Effectus of Quantum Probability on Relational Structures

Octavio Zapata (UCL)

Most of the work presented in this document can be read as a sequel to previous work of the author and collaborators, which has been published and appears as `The Quantum Monad on Relational Structures’. There, the mathematical description of quantum homomorphisms of graphs and more generally of relational structures, using the language of category theory is given. Specifically, we introduce the concept of ‘quantum’ monad. The main contribution in the present document consists of providing evidence in support of the claim that the quantum monad fits nicely into the categorical framework of effectus theory, developed by Jacobs et al. We show that the Kleisli category of the quantum monad is an effectus. Also we describe certain aspects of effect-theoretic reasoning like states and predicates, validity and conditioning, instantiated in this effectus.