Super quantum probabilities and three-slit experiments - Wright's pentagon state and the Popescu-Rohrlich box require third-order interference

Quantum probabilities differ from classical ones in many ways, e.g. by violating the well-known Bell and Clauser-Horne-Shimony-Holt inequalities or another simple inequality due to R Wright. The latter has recently regained attention because of its equivalence to a novel noncontextual inequality by Klyachko et al. On the other hand, quantum probabilities still obey many limitations which need not hold in more general probabilistic theories (super quantum probabilities). Wright, Popescu and Rohrlich identified states which are included in such theories, but impossible in quantum mechanics, and they showed this using the Hilbert space formalism. Recently, Fritz et al and Cabello detected that the impossibility of these states can be derived from very general principles (local orthogonality and global exclusive disjunction, respectively) without using Hilbert space techniques. In the paper, an alternative derivation from rather different physical principles will be presented. These are a reasonable calculus of conditional probability (i.e. a model for the quantum measurement process) and the absence of third-order interference. The concept of third-order interference was introduced by Sorkin, who also recognized its impossibility in quantum mechanics