Some remarks on the theorems of Gleason and Kochen-Specker

A Gleason-type theorem is proved for two restricted classes of informationally complete POVMs in the qubit case. 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 of this. In particular, we will derive a limit for the fraction of the sphere that can be satisfactorily 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 possible ON-bases properly coloured. Using this measure, we will derive a 'lower bound for the upper bound' in three and four real dimensions.