theorem and the effectiveness of Gleason’s

Abstract. Kochen and Specker’s theorem can be seen as a consequence of Gleason’s theorem and logical compactness. Similar compactness arguments lead to stronger results about finite sets of rays in Hilbert space, which we also prove by a direct construction. Finally, we demonstrate that Gleason’s theorem itself has a constructive proof, based on a generic, finite, effectively generated set of rays, on which every quantum state can be approximated. 1. Gleason’s Theorem and Logical Compactness Kochen and Specker’s (1967) theorem (KS) puts a severe constraint on possible hidden-variable interpretations of quantum mechanics. Often it is considered an improvement on a similar argument derived from Gleason (1957) theorem (see, for example, Held. 2000). This is true in the sense that KS provide an explicit construction of a finite set of rays on which no two-valued homomorphism exists. However, the fact that there is such a finite set follows from Gleason’s theorem using a simple logical compactness argument (Pitowsky 1998, a similar point is made in Bell 1996). The existence of finite sets of rays with other interesting features

Meyer originally raised the question of whether non-contextual hidden variable models can, despite the Kochen-Specker theorem, simulate the predictions of quantum mechanics to within any fixed finite experimental

Abstract not found

A particular incomplete Kochen-Specker colouring, suggested by Appleby in dimension three, is generalized to arbitrary dimension. We investigate its effectivity as a function of dimension, using two different measures. A limit is derived for the fraction of the sphere that can be coloured using the generalized Appleby construction as the number of dimensions approaches infinity. The second, and physically more relevant measure of effectivity, is to look at the fraction of properly coloured ON-bases. Using this measure, we derive a ’lower bound for the upper bound ’ in three and four real dimensions. 1 1