Generalizations of Kochen and Specker’s Theorem and the Effectiveness of Gleason’s Theorem

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 also follow from the same simple consideration. In Pitowsky (1998) some such consequences, in particular the “logical indeterminacy principle”are pointed out, and some are given a direct constructive proof. In this section we shall formulate the general compactness principle underlying these results and mention some new ones. In the second section there are some explicit constructions of finite sets of rays whose existence was inferred indirectly in the first section; in particular, a simpler proof of the logical indeterminacy principle. In the last section we prove that there is an effective (algorithmic) procedure to construct finite sets of rays which force a uniform approximation to quantum states. In particular, we provide a short demonstration that Gleason’s theorem has a constructive proof, a fact previously established by Richman and Bridges (1999). Let H be a Hilbert Space of a finite dimension n ≥ 3 over the complex or real field. A non negative real function p defined on the unit vectors in H is called a state on H if the following conditions hold: 1. p(αx) = p(x) for every scalar α, |α| = 1, and every unit vector x ∈ H. 2. If x1, x2, ..., xn is an orthonormal basis in H then ∑n j=1 p(xj) = 1. Gleason’s theorem characterizes all states: Theorem 1. Given a state p, there is an Hermitian, non negative operator W on H, whose trace is unity, such that p(x) = (x,Wx) for all unit vectors x ∈ H, where (, ) is the inner product. Gleason’s (1957) original proof of the theorem has three parts: The first is to show that every state p on R is continuous. The second part is a proof of the