Optimal demonstration of generalized quantum contextuality

Finding a set of empirical criteria fulfilled by any theory that satisfies the generalized notion of noncontextuality is a challenging task of both operational and foundational importance. The conventional approach of deriving facet inequalities from the relevant noncontextual polytope is computationally demanding. Specifically, the noncontextual polytope is a product of two polytopes, one for preparations and the other for measurements, and the dimension of the former typically increases polynomially with the number of measurements. This work presents an alternative methodology for constructing a polytope that encompasses the actual noncontextual polytope while ensuring that the dimension of the polytope associated with the preparations remains constant regardless of the number of measurements and their outcome size. In particular, the facet inequalities of this polytope serve as necessary conditions for noncontextuality. To demonstrate the efficacy of our methodology, we apply it to nine distinct contextuality scenarios involving four to nine preparations and two to three measurements to obtain the respective sets of facet inequalities. Additionally, we retrieve the maximum quantum violations of these inequalities. Our investigation uncovers many novel non-trivial noncontextuality inequalities and reveals intriguing aspects and applications of quantum contextual correlations.