Contextuality: the exclusivity-graph approach

Abstract

In this chapter we present the exclusivity-graph approach to contextuality, where the main concerns are the exclusivity relations of a finite set of measurement events. We will see that in this approach the convex sets obtained when the probabilities are computed with classical, quantum and general probability theories satisfying the so called Exclusivity Principle are related to important convex sets of graph theory. Noncontextuality inequalities also arise from the geometric description of the classical set and the classical and quantum bounds are related to graph invariants of the exclusivity graph. This is also the case for the bound obtained with probability theories satisfying the exclusivity principle. We then discuss two refinements of the exclusivity-graph approach. First, we present the coloured-graph approach to non-locality, where each party exclusivities in a Bell scenario are encoded in a different colour for the edges of the exclusivity graph. Second, we present the exclusivity-hypergraph approach, where the idea is to take into account not only the exclusivity relations among the measurement events but also to include the information of which measurement gave rise to each specific exclusivity.