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This paper extends the Mermin-Peres magic square concept from binary systems to linear equations over odd primes, demonstrating that classical inconsistencies can be constructed using generalized Pauli operators on qudits. The work advances foundational quantum information theory by providing a unified framework for non-classical correlations across arbitrary dimensions without relying on qubits alone.
Mermin-Peres magic rectangles modulo odd primes
arXiv:2609.20746v1 Announce Type: new 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.
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.
arXiv:2609.20746 [quant-ph]
(or arXiv:2609.20746v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2609.20746 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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