A remark on the Kochen-Specker Theorem

Abstract

A famous theorem of Andrew Gleason states that every finite measure on the lattice of closed subspaces of a Hilbert space H of dimension ≥ 3 is of the form μ(K) = Tr(WP_K), where W is a non-negative operator of trace class, uniquely determined by the measure, and P_K is the projection operator whose range is the subspace K of H. By a finite measure one means in this context a function μ: $\mu : L(H) \to {{\mathbb{R}}_{ + }} ({{\mathbb{R}}_{ + }}$ the non-negative real numbers, L (H) the lattice of closed subspaces of H) with the property that $\sum {\mu ({{K}_{n}}) = \mu ( \oplus {{K}_{n}})}$ for every finite or countably infinite system {K_n} of mutually orthogonal closed subspaces of H. If dim(K) = 1, and $f \in K$ is a unit vector, then the theorem yields μ(K) = (f, Wf). This shows that if μu is restricted to one-dimensional subspaces only, then its range is a connected set of real numbers. Indeed, if f and g are two independent unit vectors, and α = (f ,Wf) , β= (g,Wg) > α say, then for any γ in the interval (α,β) one can find a suitable unit vector h in the span of f and g such that γ = (h,Wh). The only way the range of μ (restricted to one-dimensional subspaces) can be a finite set is when dim(H) < ∞ and W is a multiple of the identity operator. In particular, no measure has the range {0,1}.