Colouring the rational quantum sphere and the Kochen-Specker theorem

Abstract

We review and extend recent findings of Godsil and Zaks, who published a constructive colouring of the rational unit sphere with the property that for any orthogonal tripod formed by rays extending from the origin of the points of the sphere, exactly one ray is red, one white and one black. They also showed that any consistent colouring of the real sphere requires an additional colour. We discuss some of the consequences for the Kochen--Specker theorem. PACS numbers: 0365B, 0210B, 0210G 1. Colourings In what follows we shall consider `rational rays'. A `rational ray' is the linear span of a non-zero vector of Q n # R n . Let p be a prime number. A colouring of the rational rays of R n , n # 1, using p n-1 + p n-2 + + 1 colours can be constructed in a straightforward manner. We refer to [3--5] for the theoretical background of the following construction. Each rational ray is the linear span of a vector (x 1 ,x 2 ,...,x n ) # Z n , where x 1 ,x 2 ,...,x n are coprime. Such a vector is unique up to a factor 1.