The Mermin-Peres magic square is a celebrated example of a system of Boolean linear equations that is not (classically) satisfiable but is satisfiable via linear operators on a Hilbert space of dimension four. A natural question is then, for what kind of problems such a phenomenon occurs? Atserias, Kolaitis, and Severini answered this question for all Boolean Constraint Satisfaction Problems (CSPs): For 0-Valid-SAT, 1-Valid-SAT, 2-SAT, Horn-SAT, and Dual Horn-SAT, classical satisfiability and operator satisfiability is the same and thus there is no gap; for all other Boolean CSPs, these notions differ as there are gaps, i.e., there are unsatisfiable instances that are satisfiable via operators on Hilbert spaces. We generalize their result to CSPs on arbitrary finite domains and give an almost complete classification: First, we show that NP-hard CSPs admit a separation between classical satisfiability and satisfiability via operators on finite- and infinite-dimensional Hilbert spaces. Second, we show that tractable CSPs of bounded width have no satisfiability gaps of any kind. Finally, we show that tractable CSPs of unbounded width can simulate, in a satisfiability-gap-preserving fashion, linear equations over an Abelian group of prime order $p$; for such CSPs, we obtain a separation of classical satisfiability and satisfiability via operators on infinite-dimensional Hilbert spaces. Furthermore, if $p=2$, such CSPs also have gaps separating classical satisfiability and satisfiability via operators on finite- and infinite-dimensional Hilbert spaces.

翻译：Mermin-Peres 幻方是一个著名的例子，它展示了一个布尔线性方程组在经典意义上不可满足，但通过在四维希尔伯特空间上的线性算子却可满足。一个自然的问题是：这种现象会在何种问题中出现？Atserias、Kolaitis 和 Severini 针对所有布尔约束满足问题（CSP）回答了这个问题：对于 0-Valid-SAT、1-Valid-SAT、2-SAT、Horn-SAT 和 Dual Horn-SAT，经典可满足性与算子可满足性相同，因此不存在间隙；对于所有其他布尔 CSP，这些概念存在差异，即存在不可满足的实例可以通过希尔伯特空间上的算子满足。我们将他们的结果推广到任意有限域上的 CSP，并给出了一个近乎完整的分类：首先，我们证明 NP 难 CSP 在经典可满足性与有限维及无限维希尔伯特空间上的算子可满足性之间存在分离。其次，我们证明具有有界宽度的易处理 CSP 不存在任何类型的可满足性间隙。最后，我们证明无界宽度的易处理 CSP 能够以保持可满足性间隙的方式模拟素数阶 $p$ 阿贝尔群上的线性方程组；对于此类 CSP，我们得到了经典可满足性与无限维希尔伯特空间上的算子可满足性之间的分离。此外，若 $p=2$，此类 CSP 还存在分离经典可满足性与有限维及无限维希尔伯特空间上的算子可满足性的间隙。