The discussion on noncontextual hidden variable models as an underlying description for the quantum-mechanical predictions started in ernest with 1967 paper by Kochen and Specker. There, it was shown that no noncontextual hidden-variable model can give these predictions. The proof used in that paper is complicated, but recently, a paper by Yu and Oh [PRL, 2012] proposes a simpler statistical proof that can also be the basis of an experimental test. Here we report on a sharper version of that statistical proof, and also explain why the algebraic upper bound to the expressions used are not reachable, even with a reasonable contextual hidden variable model. Specifically, we show that the quantum mechanical predictions reach the maximal possible value for a contextual model that keeps the expectation value of the measurement outcomes constant.
Maximal violation of state-independent contextuality inequalities
Jan-Åke Larsson, Matthias Kleinmann, Constantino Budroni, Otfried Gühne, Adán Cabello; Maximal violation of state-independent contextuality inequalities. AIP Conf. Proc. 18 December 2012; 1508 (1): 265–274. https://doi.org/10.1063/1.4773138
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