Uniqueness of all fundamental noncontextuality inequalities

Contextuality is one way of capturing the non-classicality of quantum theory. The contextual nature of a theory is often witnessed via the violation of non-contextuality 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 [PRA 88, 032104 (2013); Annals of mathematics, 51-299 (2006)] 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 anti-cyclically exclusive. Thus, the non-contextuality inequalities whose underlying exclusivity structure is as stated, either cyclic or anti-cyclic, are fundamental to quantum theory. We show that there is a unique non-contextuality inequality for each non-trivial cycle and anti-cycle. 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.Ministry of Education (MOE)National Research Foundation (NRF)Published versionA.S.A. and J.R. acknowledge financial support from the Belgian Fonds de la Recherche Scientifique (FNRS) under Grants No. F.4515.16 (QUICTIME) and No. R.50.05.18.F (QuantAlgo). A.S.A. further acknowledges the FNRS for support through Grant No. F3/5/5–MCF/XH/FC– 16749 FRIA. K.B. acknowledges the Centre for Quantum Technologies (CQT) Graduate Scholarship. K.B. and L.C.K. are grateful to the National Research Foundation and the Ministry of Education, Singapore for financial support