{"ID":23577915,"CreatedAt":"2026-09-18T05:07:30.921092366Z","UpdatedAt":"2026-09-18T05:07:30.921092366Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20746","arxiv_id":"2609.20746","title":"Mermin-Peres magic rectangles modulo odd primes","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 $d\\ge2$, 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. Classical inconsistency follows from an elementary linearity argument comparing assignments on abelian subgroups.","short_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...","url_abs":"https://arxiv.org/abs/2609.20746","url_pdf":"https://arxiv.org/pdf/2609.20746v1","authors":"[\"Josse van Dobben de Bruyn\",\"Remy van Dobben de Bruyn\",\"Peter Zeman\"]","published":"2026-09-17T17:34:44Z","proceeding":"quant-ph","tasks":"[\"quant-ph\"]","methods":"[]","has_code":false}