The Born rule as a natural transformation of functors

Abstract: In this work, we show that the quantum mechanical notions of density operator, positive operator-valued measure (POVM), and the Born rule, are all simultaneously encoded in the categorical notion of a natural transformation of functors. In particular, we show that given a fixed quantum system, there exists an explicit bijection from the set of density operators on the associated Hilbert space to the set of natural transformations between the canonical measurement and probability functors associated with the system, which formalize the way in which quantum effects (i.e., POVM elements) and their associated probabilities are additive with respect to a coarse-graining of measurements.