A Kochen–Specker inequality from a SIC

Yu and Oh (eprint) have given a state-independent proof of the Kochen–Specker theorem in three dimensions using only 13 rays. The proof consists of showing that a non-contextual hidden variable theory necessarily leads to an inequality that is violated by quantum mechanics. We give a similar proof making use of 21 rays that constitute a SIC (symmetric informationally-complete positive operator-valued measure) and a complete set of MUB (mutually unbiased bases). A theory-independent inequality is also presented using the same 21 rays, as required for experimental tests of contextuality. -- Highlights: ► We find a state-independent Kochen–Specker inequality in dimension 3 with 21 rays. ► The rays constitute a SIC (9 rays) and a complete set of MUB (12 rays). ► Orthogonalities among the rays produce the Hesse configuration. ► The rays also give a state-independent non-contextual hidden variable inequality. ► We show that both inequalities are violated by quantum mechanics.