Basic exclusivity graphs in quantum correlations

TL;DR

This paper characterizes which noncontextuality inequalities are violated by quantum theory using exclusivity graphs, identifying specific graph structures that determine quantum violations.

Contribution

It proves that quantum violations occur only for inequalities with exclusivity graphs containing odd cycles or their complements, linking graph structure to quantum contextuality.

Findings

Quantum theory violates inequalities with odd cycle or complement graphs.

Maximum quantum violations are characterized for these inequalities.

Evidence suggests the exclusivity principle determines the quantum violation limit.

Abstract

A fundamental problem is to understand why quantum theory only violates some noncontextuality (NC) inequalities and identify the physical principles that prevent higher-than-quantum violations. We prove that quantum theory only violates those NC inequalities whose exclusivity graphs contain, as induced subgraphs, odd cycles of length five or more, and/or their complements. In addition, we show that odd cycles are the exclusivity graphs of a well-known family of NC inequalities and that there is also a family of NC inequalities whose exclusivity graphs are the complements of odd cycles. We characterize the maximum noncontextual and quantum values of these inequalities, and provide evidence supporting the conjecture that the maximum quantum violation of these inequalities is exactly singled out by the exclusivity principle.