Simple unified form for the major no-hidden-variables theorems.

Two examples are given that substantially simplify the no-hidden-variables theorem of Kochen and Specker, greatly reducing the number of observables considered and requiring no intricate geometric argument. While one of the examples also obeys a more powerful version of Bell's theorem, the other does not. The examples provide a new perspective on both of these fundamental theorems and on the relation between them. The Kochen-Specker (KS) theorem ' demonstrates that it is, in general, impossible to ascribe to an individual quantum system a definite value for each of a set of observables not all of which necessarily commute. Of course elementary quantum metaphysics insists that we cannot assign definite values to noncommuting observ-ables; the point of the KS theorem is to extract this directly from the quantum-mechanical formalism, rather than merely appealing to precepts enunciated by the founders. If such an assignment of values turned out to be possible in spite of those precepts, then uncertainty relations for the results of measuring noncommuting ob-servables could be viewed as a manifestation of the statistical scatter of these definite values in many different individual realizations of the identical quantum state. The state vector alone would not provide complete information about a system, and the additional values in particular realizations of the same quantum state could be regarded as "hidden variables. " A similar conclusion against hidden variables is reached by Bell's Theorem, but in a rather different way. The violation of quantum dogma contemplated by Bell is weaker than that tested by Kochen and Specker, noncommuting observables only being provided with simultaneous values when required to have them by the simple locality condition of Einstein, Podolsky, and Rosen (EPR). On the other hand, Bell's refutation has a strongly statistical character that the argument of Ko-chen and Specker does not. I describe below a simple system for which one can prove both a KS and a Bell-EPR theorem. The KS theorem is substantially simpler than the original argument of Kochen and Specker; the Bell-EPR theorem eliminates the statistical aspect of Bell s original argument ; and the applicability of both theorems to a single system clarifies their relationship, clearly revealing the Bell-EPR result to be the stronger of the two. The discussion that follows is inspired by a new version of Bell's Theorem due to Greenberger, Horne, and Zeilinger (GHZ), by the observation of Stairs' that GHZ can also be made the basis …