Maximum nonlocality in the (3,2,2) scenario

TL;DR

This paper classifies the maximum quantum nonlocality achievable in the (3,2,2) Bell scenario by analyzing entanglement and measurement incompatibility, revealing specific resource combinations and fundamental insights into quantum and almost quantum correlations.

Contribution

It systematically identifies the resource combinations leading to maximum nonlocality in the (3,2,2) scenario and uncovers fundamental differences between quantum and almost quantum correlations.

Findings

Maximum nonlocality depends on specific entanglement and incompatibility combinations.

Adding a third party does not increase nonlocality for certain inequalities.

Almost quantum correlations suggest nonlocality increases with more parties, conflicting with quantum predictions.

Abstract

We identify the simplest combinations of entanglement and incompatibility giving the maximum quantum violation for each of the 46 classes of tight Bell inequalities for the (3,2,2) scenario, i.e., three parties, two measurements per party, and two outcomes per measurement. This allows us to classify the maximum quantum nonlocality according to the simplest resources needed to achieve it. We show that entanglement and incompatibility only produce maximum nonlocality when they are combined in specific ways. For each entanglement class there is, in most cases, just one incompatibility class leading to maximum nonlocality. We also identify two interesting cases. We show that the maximum quantum violation of \'Sliwa inequality 23 only occurs when the third party measures the identity, so nonlocality cannot increase when we add a third party to the bipartite case. Almost quantum correlations predict that adding a new party increases nonlocality. This points out that either almost quantum correlations violate a fundamental principle or that there is a form of tripartite entanglement which quantum theory cannot account for. The other interesting case is the maximum quantum violation of \'Sliwa inequality 26, which, like the Mermin inequality, requires maximum incompatibility for all parties. In contrast, it requires a specific entangled state which has the same tripartite negativity as the W state.