Density cubes and higher-order interference theories

TL;DR

This paper introduces density cubes as a framework for higher-order interference theories, extending quantum theory with new states and dynamics, and demonstrates their non-quantum features and violations of classical bounds.

Contribution

It develops a theory of density cubes for third-order interference, embedding quantum theory within it and exploring non-quantum states and dynamics.

Findings

Density cubes naturally embed quantum states.

Density cubes exhibit genuine third-order interference.

They violate Leggett-Garg inequality beyond Tsirelson bound.

Abstract

Can quantum theory be seen as a special case of a more general probabilistic theory, similarly as classical theory is a special case of the quantum one? We study here the class of generalized probabilistic theories defined by the order of interference they exhibit as proposed by Sorkin. A simple operational argument shows that the theories require higher-order tensors as a representation of physical states. For the third-order interference we derive an explicit theory of "density cubes" and show that quantum theory, i.e. theory of density matrices, is naturally embedded in it. We derive the genuine non-quantum class of states and non-trivial dynamics for the case of three-level system and show how one can construct the states of higher dimensions. Additionally to genuine third-order interference, the density cubes are shown to violate the Leggett-Garg inequality beyond the quantum Tsirelson bound for temporal correlations.