We derive inequalities for -partite states under the assumption that the hidden-variable theoretical joint probability distribution for any pair of commuting observables is equal to the quantum mechanical one. Fine showed that this assumption is connected to the no-hidden-variables theorem of Kochen and Specker (KS theorem). These inequalities give a way to experimentally test the KS theorem. The fidelity to the Bell states which is larger than is sufficient for the experimental confirmation of the KS theorem. Hence, the Werner state is enough to test experimentally the KS theorem. Furthermore, it is possible to test the KS theorem experimentally using uncorrelated states. An -partite uncorrelated state violates the -partite inequality derived here by an amount that grows exponentially with .
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We know that every proposition is true if the presupposition is false [see Eq. (3.6)]. Therefore, one might think that theorem (4.2) and theorem (5.16) are trivial. However, this is not the matter of our argument. We have used a quantum mechanical rule in the proof of the theorem (3.6). But, we have not used the quantum mechanical rule in the proof of the theorem (4.2). Likewise, a quantum mechanical rule , is needless to prove the theorem (5.16), while we have used the quantum mechanical rule in the proof of the theorem (3.6). Obviously, , is needless to prove the theorem (5.16). Of course, Gleason’s theorem is needless. Therefore, we can derive these inequalities (4.2) and (5.16) from more precise and weaker presupposition which should not be necessarily false.
