Minimal ring extensions of the integers exhibiting Kochen-Specker contextuality

• URL: http://arxiv.org/abs/2211.13216v3
• Date: Sat, 12 Apr 2025 00:38:28 GMT
• Title: Minimal ring extensions of the integers exhibiting Kochen-Specker contextuality
• Authors: Ida Cortez, Camilo Morales, Manuel L. Reyes,
• Abstract summary: We investigate rational subrings over which the partial ring of $d times d$ symmetric matrices admits no morphism to a commutative ring.<n>For $d = 3$, the minimal such ring is shown to be $mathbbZ[1/6]$, while for $d geq 6$ the minimal subring is $mathbbZ$ itself.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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