Necessary and sufficient condition for contextuality from incompatibility

Measurement incompatibility is the most basic resource that distinguishes quantum from classical physics. Contextuality is the critical resource behind the power of some models of quantum computation and is also a necessary ingredient for many applications in quantum information. A fundamental problem is thus identifying when incompatibility produces contextuality. Here, we show that, given a structure of incompatibility characterized by a graph in which nonadjacent vertices represent incompatible ideal measurements, the necessary and sufficient condition for the existence of a quantum realization producing contextuality is that this graph contains induced cycles of size larger than three.

Figure 1

All the compatibility graphs corresponding to scenarios that can produce quantum contextuality with up to six ideal measurements. They are of three types: (a1)–(a5) can be realized in Bell scenarios (dots of the same color denote measurements performed by the same party); (b1)–(b8) can be realized in multipartite scenarios, but at the cost that at least one party has at least two compatible measurements, and (c1)–(c11) cannot be realized in multipartite scenarios.