A proof of the Kochen–Specker theorem can always be converted to a state-independent noncontextuality inequality

Abstract

Quantum contextuality is one of the fundamental notions in quantum mechanics. Proofs of the Kochen-Specker theorem and noncontextuality inequalities are two means for revealing the contextuality phenomenon in quantum mechanics. It has been found that some proofs of the Kochen-Specker theorem, such as those based on rays, can be converted to a state-independent noncontextuality inequality, but it remains open whether this is true in general, i.e., whether any proof of the Kochen-Specker theorem can always be converted to a noncontextuality inequality. In this paper, we address this issue. We prove that all kinds of proofs of the Kochen-Specker theorem, based on rays or any other observables, can always be converted to state-independent noncontextuality inequalities. Besides, our constructive proof also provides a general approach for deriving a stateindependent noncontextuality inequality from a proof of the KS theorem.

Quantum Contextuality

“…Connections between contextuality and universal quantum computation [32] and steering [33] that have recently been established ask for a quantification of properties of contextual sets, e.g., robustness to noise [34], size of maximal independent sets of stabilizer states [32], or suitability for implementation in general. It has been shown that inequalities are an efficient tool for the pur-pose [1,30,[35][36][37][38][39][40]. Yu, Guo, and Tong prove [40] that operator formulations of Kochen-Specker contextual sets can always be converted to state-independent noncontextuality inequalities.…”

“…It has been shown that inequalities are an efficient tool for the pur-pose [1,30,[35][36][37][38][39][40]. Yu, Guo, and Tong prove [40] that operator formulations of Kochen-Specker contextual sets can always be converted to state-independent noncontextuality inequalities. The problem with the inequalities in these references is that either no definite inequality is given or that they were given for chosen particular contextual sets previously specified via sets of vectors/rays, or that they have not been formulated for probabilities applicable to genuine YES-NO quantum experiments.…”

Algebraic structures identified with bivalent and non-bivalent semantics of experimental quantum propositions

“…The KS theorem was proved on the set L(H) of the closed linear subspaces of H such that each pair of the subspaces in L(H) has a meet (greatest lower bound) corresponding to their set-theoretic intersection. Since the original demonstration for |S| = 117 and N = 3, more and more proofs of the KS theorem have been found for the same or higher N but lesser |S| (see [5,6,7,8], to cite but a few examples), and the task has become to reduce the technical difficultness required to prove the KS theorem in order to make the issues involved clearer.…”

Generalized Householder transformations

“…For all other multi-context configurations allowing two-valued states-even with a nonseparable or unital set of two-valued states-the translation from {0, 1}-states into two-valued {−1, 1}-observables there is no state-independent quantum contextuality. For other operator-valued assignments see, for instance, references [4,12].…”