Contextuality: the compatibility-hypergraph approach

Abstract

In this chapter we present the compatibility-graph approach, where the main concerns are the compatibility relations of a finite set of measurements. We will see in detail the different convex sets that arise when we compute the probabilities of the joint measurement of each context using general probability theories that satisfy the nondisturbing condition, which include classical and quantum probability theories as special cases. With this characterization, noncontextuality inequalities arise naturally from the geometric description of the classical set. We analyse the relation between noncontextuality inequalities and the exclusivity graph of the contextuality scenario and how the classical and quantum bounds to these inequalities are related to graph invariants of this graph. We then look for the special case in which every context consists of at most two binary measurements, in which both nondisturbing and noncontextual sets can be equivalently described in different ways that lead to familiar polytopes from graph theory. We finish this chapter with a brief introduction to the sheaf theoretical aspects of contextuality, which provide a direct and unified characterization of both contextuality and non-locality, along with different new tools, insights and results.