Closing the problem of which causal structures of up to six total nodes have a classical-quantum gap

TL;DR

This paper identifies all causal structures with up to six nodes that support quantum correlations exceeding classical limits, completing the classification of such structures in quantum foundations.

Contribution

It proves the existence of non-classical quantum correlations in the only unresolved causal structure with six or fewer nodes, using a novel method involving additional correlation restrictions.

Findings

Quantum correlations exist in the specific six-node causal structure.

The classification of causal structures supporting quantum non-locality is complete up to six nodes.

The method can be applied to other causal structures for similar analysis.

Abstract

The discovery of Bell that there exist quantum correlations that cannot be reproduced classically is one of the most important in the foundations of quantum mechanics, as well as having practical implications. Bell's result was originally proven in a simple bipartite causal structure, but analogous results have also been shown in further causal structures. Here we study the only causal structure with six or fewer nodes in which the question of whether or not there exist quantum correlations that cannot be achieved classically was open. In this causal structure we show that such quantum correlations exist using a method that involves imposing additional restrictions on the correlations. This hence completes the picture of which causal structures of up to six nodes support non-classical quantum correlations. We also provide further illustrations of our method using other causal structures.