Mermin-Peres magic rectangles modulo odd primes
Abstract
The Mermin-Peres magic square provides a simple example of a system of linear equations over ℤ/2ℤ 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 ℤ/dℤ with d odd. In this paper, we construct, for every integer d ≥ 2, a linear system over ℤ/dℤ 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.
Related
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BibTeX
@misc{bruyn2026merminperes,
title = {{Mermin-Peres magic rectangles modulo odd primes}},
author = {Josse van Dobben de Bruyn and Remy van Dobben de Bruyn and Peter Zeman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.20746},
eprint = {2609.20746},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20746},
}
RIS
TY - UNPB
TI - Mermin-Peres magic rectangles modulo odd primes
AU - Josse van Dobben de Bruyn
AU - Remy van Dobben de Bruyn
AU - Peter Zeman
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.20746
UR - https://arxiv.org/abs/2609.20746
AB - The Mermin-Peres magic square provides a simple example of a system of linear equations over ℤ/2ℤ 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 ℤ/dℤ with d odd. In this paper, we construct, for every integer d ≥ 2, a linear system over ℤ/dℤ 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.
ER -