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.