Linear equations mod $n$ are pseudo-telepathic

Lorenzo Ciardo

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arXiv · 2609.14632

Linear equations mod $n$ are pseudo-telepathic

Lorenzo Ciardo

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Abstract

We prove that the quantum monad in dimension $2n$ admits no natural transformation to the polymorphism clone of linear equations modulo $n$. Consequently, for every $n\geq 2$, there exists an unsatisfiable system of linear equations over $\mathbb{Z}_n$ whose constraint system game admits a perfect finite-dimensional quantum strategy. As a corollary, we completely characterise pseudo-telepathic constraint languages in finite dimension. The proof combines a result of Harding, Jager, and Smith on group-valued measures on subspaces of Hilbert spaces with the polymorphism-minion characterisation of quantum pseudo-telepathy.

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Lorenzo Ciardo. 2026-09-13. Linear equations mod $n$ are pseudo-telepathic. https://arxiv.org/abs/2609.14632

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