Construction of State-independent Proofs for Quantum Contextuality

Since the enlightening proofs of quantum contextuality first established by Kochen and Specker, and also by Bell, various simplified proofs have been constructed to exclude the non-contextual hidden variable theory of our nature at the microscopic scale. The conflict between the non-contextual hidden variable theory and quantum mechanics is commonly revealed by Kochen-Specker (KS) sets of yes-no tests, represented by projectors (or rays), via either logical contradictions or noncontextuality inequalities in a state-(in)dependent manner. Here we first propose a systematic and programmable construction of a state-independent proof from a given set of nonspecific rays in $\mC^3$ according to their Gram matrix. This approach brings us a greater convenience in the experimental arrangements. Besides, our proofs in $\mC^3$ can also be generalized to any higher dimensional systems by a recursive method.