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 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. This result completes one by Ramanathan et al. [Phys. Rev. Lett. 109, 050404 (2012)], points out the fundamental importance of the results of Ara\'ujo et al. [Phys. Rev. A 88, 022118 (2013)], and allows us to identify all experimental scenarios with quantum contextuality, unveiling new interesting cases.