Uniqueness of all fundamental noncontextuality inequalities

Abstract

Contextuality is one way of capturing the nonclassicality of quantum theory. The contextual nature of a theory is often witnessed via the violation of noncontextuality inequalities—certain linear inequalities involving probabilities of measurement events. Using the exclusivity graph approach (one of the two main graph theoretic approaches for studying contextuality), it was shown [Cabello et al. Phys. Rev. A 88, 032104 (2013)10.1103/PhysRevA.88.032104; Chudnovsky et al. Ann. Math. 164, 51 (2006)10.4007/annals.2006.164.51] that a necessary and sufficient condition for witnessing contextuality is the presence of an odd number of events (greater than three) which are either cyclically or anticyclically exclusive. Thus, the noncontextuality inequalities the underlying exclusivity structure of which is as stated, either cyclic or anticyclic, are fundamental to quantum theory. We show that there is a unique noncontextuality inequality for each nontrivial cycle and anticycle. In addition to the foundational interest, we expect this to aid the understanding of contextuality as a resource to quantum computing and its applications to local self-testing.