Minimal ring extensions of the integers exhibiting Kochen-Specker contextuality

TL;DR

This work develops an algebraic framework for quantum contextuality by studying Kochen-Specker colorings in partial rings of symmetric matrices over rational subrings of the integers. It combines explicit KS-uncolorable integer vector sets with a computational ILP approach to determine when no algebraic hidden state exists, yielding sharp results: the minimal base ring for $d=3$ is $\\mathbb{Z}[1/6]$ and for $d\\ge6$ it is $\\mathbb{Z}$. The results translate KS uncolorability into the nonexistence of prime partial ideals and morphisms to nonzero commutative rings, while also detailing colorability phenomena over various number-theoretic rings via projections. Together, these findings tightly couple quantum contextuality to arithmetic properties of base rings, clarifying when algebraic hidden states can or cannot exist across dimensions and rings.

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.

Theorems & Definitions (20)

• Definition
• Theorem 1
• Theorem 3
• proof
• Lemma 4
• proof
• Theorem 5
• proof
• Proposition 6
• proof : Proof of Proposition \ref{['prop:dim 4 and 5']}