Graph-theoretic strengths of contextuality

Cabello-Severini-Winter and Abramsky-Hardy (building on the framework of Abramsky-Brandenburger) both provide classes of Bell and contextuality inequalities for very general experimental scenarios using vastly different mathematical techniques. We review both approaches, carefully detail the links between them, and give simple, graph-theoretic methods for finding inequality-free proofs of nonlocality and contextuality and for finding states exhibiting strong nonlocality and/or contextuality. Finally, we apply these methods to concrete examples in stabilizer quantum mechanics relevant to understanding contextuality as a resource in quantum computation.

Figure 1

The exclusivity graph for the standard Bell scenario (left); labels of measurement settings and outcomes (right). Straight lines connect exclusive events. The standard CHSH inequality [33] is derived by giving the grey vertices the weight 1 and the others 0; the independence number is 3. Circular vertices are the possible events of a Hardy model. Grey vertices are the possible events of a PR box [34].