发表机构
Paul G. Allen School of Computer Science and Engineering, University of Washington（华盛顿大学保罗·G·艾伦计算机科学与工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了 $I_{3322}$ 贝尔不等式在有限维度下无法达到最大违反，而无限维度可以，从而证实了长期猜想，并提供了完整证明。
AI 中文摘要
$I_{3322}$ 不等式是最简单的二分贝尔不等式之一，每方有三个可能的问题和两个可能的答案。然而，尽管其简单性，Pál 和 Vértesi（《物理评论A》，2010年）长期以来一直猜想它具有以下奇特性质：没有任何有限维量子策略能够达到其最大违反，但无限维策略可以做到。此后，Coladangelo 和 Stark（《自然·通讯》，2020年）发现了一种贝尔关联，每方有五个问题和三个答案，可证明具有相同性质。然而，证明 $I_{3322}$ 本身（它存在于可能发生此类现象的最简单贝尔场景中）也具有相同性质，一直难以实现。在此，我们提供了这一猜想的证明。一个包含所有主要思想的近似证明，由 GPT 5.5 Pro 经过多轮交互后生成。该证明的呈现方式由作者为完整性、正确性和清晰性进行了大幅修订。
英文摘要
The $I_{3322}$ inequality is one of the simplest bipartite Bell inequalities, with three possible questions and two possible answers per party. Yet, despite its simplicity, it has long been conjectured by Pál and Vértesi (Physical Review A, 2010) to possess the following quirk: no finite-dimensional quantum strategy can attain its maximal violation, but an infinite-dimensional strategy can. A Bell correlation, with five questions and three answers per party, that provably possesses the same property has since been discovered by Coladangelo and Stark (Nature Communications, 2020). However, proving that the same property holds for $I_{3322}$, which lives in the simplest Bell scenario in which such a phenomenon could occur, has remained elusive. Here, we provide a proof of this conjecture. An approximate proof, containing all the main ideas, was produced by GPT 5.5 Pro after various rounds of interaction. The presentation of this proof was revised substantially by the author for completeness, correctness, and clarity.
Comments79 pages
