Linear equations mod $n$ are pseudo-telepathic

• URL: http://arxiv.org/abs/2609.14632v1
• Date: Sun, 13 Sep 2026 16:04:54 GMT
• Title: Linear equations mod $n$ are pseudo-telepathic
• Abstract summary: We prove that the quantum monad in dimension $2n$ admits no natural transformation to the polymorphism clone of linear equations modulo $n$.<n>For every $ngeq 2$, there exists an unsatisfiable system of linear equations over $mathbbZ_n$ whose constraint system game admits a perfect finite-dimensional quantum strategy.
• Score: 2.3859169601259347
• License: http://creativecommons.org/licenses/by/4.0/
• 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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