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

TL;DR

This paper explores the limitations of quantum probabilities by examining third-order interference and specific states like Wright's pentagon state, proposing an alternative derivation based on physical principles beyond Hilbert space formalism.

Contribution

It introduces a new derivation showing that Wright's pentagon state and Popescu-Rohrlich box require third-order interference, using principles like conditional probability calculus and the absence of third-order interference.

Findings

Wright's pentagon state and Popescu-Rohrlich box require third-order interference.

Third-order interference is incompatible with quantum mechanics.

Alternative derivation based on physical principles without Hilbert space formalism.

Abstract

Quantum probabilities differ from classical ones in many ways, e.g., by violating the well-known Bell and CHSH inequalities or another simple inequality due to R. Wright. The latter one 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 any more 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 its 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 phyisical 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.