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 state-independent noncontextuality inequality from a proof of the KS theorem.
Highlights
Quantum contextuality [1] as a natural generalization of Bell nonlocality [2] is one of the fundamental notions in quantum mechanics, and has drawn a lot of interest recently
KS theorem was found from the inconsistency of the noncontextual hidden variable (NCHV) model with quantum mechanics
To examine the inconsistency of the NCHV model with quantum mechanics, Kochen and Specker assumed that the algebraic structure of compatible observables in quantum mechanics is preserved in the NCHV model, which leads to the sum rule and product rule [12, 13]
Summary
Quantum contextuality [1] as a natural generalization of Bell nonlocality [2] is one of the fundamental notions in quantum mechanics, and has drawn a lot of interest recently. It was found that there always exists a finite set of observables for any n-dimensional Hilbert space with n ≥ 3, such that all elements in the set cannot simultaneously have values satisfying the sum rule and product rule This finding is usually called the KS theorem, which shows that the NCHV model is not compatible with quantum mechanics. Our results show that every proof of the KS theorem, based on rays or other observables, can always be converted to a state-independent noncontextuality inequality. To summarize our approach briefly, one may derive a state-independent noncontextuality inequality from a proof of the KS theorem expressed by a set of observables S = {A1, A2, . We will take two well-known kinds of proofs of the KS theorem, proofs based on rays and proofs based on parity arguments, as examples to illustrate the approach
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