Minimal ring extensions of the integers exhibiting Kochen-Specker contextuality

Abstract

This paper is a contribution to the algebraic study of contextuality in quantum theory. As an algebraic analogue of Kochen and Specker's no-hidden-variables result, we investigate rational subrings over which the partial ring of $d \times d$ symmetric matrices ($d \geq 3$) admits no morphism to a commutative ring, which we view as an "algebraic hidden state." For $d = 3$, the minimal such ring is shown to be $\mathbb{Z}[1/6]$, while for $d \geq 6$ the minimal subring is $\mathbb{Z}$ itself. The proofs rely on the construction of new sets of integer vectors in dimensions 3 and 6 that have no Kochen-Specker coloring.