Effectus of Quantum Probability on Relational Structures

Abstract: The notion of effectus from categorical logic is relevant in the emerging field of categorical probability theory. In some cases, stochastic maps are represented by maps in the Kleisli category of some probability monad. Quantum homomorphisms from combinatorics and quantum information theory are the Kleisli maps of certain sort of quantum monad. We show that the Kleisli category of this quantum monad is an effectus. This gives rise to notions of quantum probabilistic reasoning as predicates, validity, conditioning, and channels.