Minimal ring extensions of the integers exhibiting Kochen-Specker contextuality
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914164550664192 |
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| author | Cortez, Ida Morales, Camilo Reyes, Manuel |
| author_facet | Cortez, Ida Morales, Camilo Reyes, Manuel |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_13216 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Minimal ring extensions of the integers exhibiting Kochen-Specker contextuality Cortez, Ida Morales, Camilo Reyes, Manuel Number Theory Mathematical Physics Quantum Physics 05C15, 81P13 (Primary), 08A55, 11C20 (Secondary) 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. |
| title | Minimal ring extensions of the integers exhibiting Kochen-Specker contextuality |
| topic | Number Theory Mathematical Physics Quantum Physics 05C15, 81P13 (Primary), 08A55, 11C20 (Secondary) |
| url | https://arxiv.org/abs/2211.13216 |