AI 中文总结
本文为每个整数d≥2构造了模d线性系统，具有有限维算子解但无经典解，推广了Mermin-Peres幻方到奇素数情形，并利用广义Pauli群上的多项式相位算子实现。
AI 中文摘要
Mermin-Peres 幻方提供了在 $\mathbb{Z}/2\mathbb{Z}$ 上线性方程组的一个简单例子，该方程组没有经典解，但确实存在有限维算子解。长期以来，人们不知道如何在 $d$ 为奇数时构造 $\mathbb{Z}/d\mathbb{Z}$ 上的类似例子。在本文中，我们为每个整数 $d\ge2$ 构造了 $\mathbb{Z}/d\mathbb{Z}$ 上的一个线性系统，该系统具有有限维算子解但没有经典解。对于奇素数 $p$，我们的算子作用于两个 $p$ 维 qudit 上，并通过将对角多项式相位算子附加到广义 Pauli 群上生成一个有限 $p$-群。经典不一致性源于一个基本的线性论证，该论证比较了阿贝尔子群上的赋值。
英文摘要
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.
Comments15 pages. v2: added Section 6, added citation of Frembs paper, minor improvements to the exposition
