{
  "abstract": "The Mermin-Peres magic square provides a simple example of a system of linear equations over $\\mathbb{Z}/2\\mathbb{Z}$ which has no classical solutions but does have a finite-dimensional operator solution. For a long time, it was not known how to construct similar examples over $\\mathbb{Z}/d\\mathbb{Z}$ with $d$ odd. In this paper, we construct, for every integer $d \\geq 2$, a linear system over $\\mathbb{Z}/d\\mathbb{Z}$ that has a finite-dimensional operator solution but no classical solution. For an odd prime $p$, our operators act on two $p$-dimensional qudits and generate a finite $p$-group obtained by adjoining diagonal polynomial phase operators to the generalized Pauli group, thereby forming a natural analogue of the Mermin--Peres magic square. Classical inconsistency follows from an elementary linearity argument comparing assignments on abelian subgroups. Furthermore, we prove a sharp threshold for the required degree of the added polynomial phase operators: our construction uses polynomials up to degree $p - 1$, and we show that the group formed by polynomials up to degree $p - 2$ is noncontextual.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20746",
  "author_total": 3,
  "authors": [
    {
      "name": "Josse van Dobben de Bruyn"
    },
    {
      "name": "Remy van Dobben de Bruyn"
    },
    {
      "name": "Peter Zeman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b71rqhrzfadxz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20746",
  "fulltext_url": null,
  "id": "b71rqhrzfadxz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20746",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "3107|Quantum Mechanics and Applications"
    ]
  ],
  "openalex": "W7213610841",
  "pdf_url": "https://arxiv.org/pdf/2609.20746",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b71rqhrzfadxz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:05Z",
      "native_id": "2609.20746",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20746"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Mermin-Peres magic rectangles modulo odd primes",
  "title_text": "Mermin-Peres magic rectangles modulo odd primes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:05Z",
  "url": "https://research.draconitic.com/paper/b71rqhrzfadxz",
  "venue": "arXiv",
  "versions": []
}