Closing the problem of which causal structures of up to six total nodes have a classical-quantum gap
Shashaank Khanna, Matthew Pusey, Roger Colbeck
TL;DR
The paper resolves the question of which causal structures with up to six nodes admit a classical-quantum gap by proving that the unique six-node structure G1 supports quantum correlations non-reproducible classically. The authors impose extra correlation constraints that force Bell-locality in the classical case while providing an explicit quantum strategy that violates CHSH, thereby establishing a non-classical gap for G1 and its variant G2. This extends prior results on smaller structures and builds on Fritz’s triangle-gap approach by working directly with probabilities. Together with forthcoming work, the six-node classification is completed, leaving no room for post-quantum correlations in these structures that lack a classical-quantum gap.
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.