In this paper we study the Platonic Bell inequalities for all possible dimensions. There are five Platonic solids in three dimensions, but there are also solids with Platonic properties (also known as regular polyhedra) in four and higher dimensions. The concept of Platonic Bell inequalities in the three-dimensional Euclidean space was introduced by Tavakoli and Gisin [Quantum 4, 293 (2020)]. For any three-dimensional Platonic solid, an arrangement of projective measurements is associated where the measurement directions point toward the vertices of the solids. For the higher dimensional regular polyhedra, we use the correspondence of the vertices to the measurements in the abstract Tsirelson space. We give a remarkably simple formula for the quantum violation of all the Platonic Bell inequalities, which we prove to attain the maximum possible quantum violation of the Bell inequalities, i.e. the Tsirelson bound. To construct Bell inequalities with a large number of settings, it is crucial to compute the local bound efficiently. In general, the computation time required to compute the local bound grows exponentially with the number of measurement settings. We find a method to compute the local bound exactly for any bipartite two-outcome Bell inequality, where the dependence becomes polynomial whose degree is the rank of the Bell matrix. To show that this algorithm can be used in practice, we compute the local bound of a 300-setting Platonic Bell inequality based on the halved dodecaplex. In addition, we use a diagonal modification of the original Platonic Bell matrix to increase the ratio of quantum to local bound. In this way, we obtain a four-dimensional 60-setting Platonic Bell inequality based on the halved tetraplex for which the quantum violation exceeds the $\sqrt 2$ ratio.

中文翻译：

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所有维度的柏拉图贝尔不等式

在本文中，我们研究了所有可能维度的柏拉图贝尔不等式。三维有五个柏拉图立体，但也有四维或更高维度的具有柏拉图性质的立体（也称为正多面体）。Tavakoli 和 Gisin [Quantum 4, 293 (2020)] 介绍了三维欧几里得空间中的柏拉图贝尔不等式的概念。对于任何三维柏拉图立体，投影测量的排列是相关的，其中测量方向指向立体的顶点。对于更高维的正多面体，我们使用顶点与抽象 Tsirelson 空间中的测量值的对应关系。对于所有柏拉图贝尔不等式的量子违反，我们给出了一个非常简单的公式，我们证明了最大可能的量子违反贝尔不等式，即 Tsirelson 界。要构建具有大量设置的贝尔不等式，有效计算局部边界至关重要。通常，计算局部边界所需的计算时间随着测量设置的数量呈指数增长。我们找到了一种方法来精确计算任何二分二结果贝尔不等式的局部界限，其中依赖性变为多项式，其度数是贝尔矩阵的秩。为了证明该算法可以在实践中使用，我们基于减半的十二重链计算了 300 个设置的柏拉图贝尔不等式的局部界限。此外，我们使用原始柏拉图贝尔矩阵的对角线修改来增加量子与局部界限的比率。这样，