The Platonic solids and fundamental tests of quantum mechanics

TL;DR

This paper explores the connection between Platonic solids and quantum mechanics by constructing Bell inequalities based on their symmetries, revealing new robust inequalities and discussing the balance between mathematical elegance and experimental practicality.

Contribution

It introduces novel Bell inequalities derived from Platonic solids and extends the concept to other polyhedra, highlighting their potential in quantum tests.

Findings

Bell inequalities with maximal violations aligned with Platonic vertices

A Bell inequality more noise-robust than CHSH

Insights into the trade-off between mathematical beauty and experimental feasibility

Abstract

The Platonic solids is the name traditionally given to the five regular convex polyhedra, namely the tetradron, the octahedron, the cube, the icosahedron and the dodecahedron. Perhaps strongly boosted by the towering historical influence of their namesake, these beautiful solids have, in well over two millenia, transcended traditional boundaries and entered the stage in a range of disciplines. Examples include natural philosophy and mathematics from classical antiquity, scientific modeling during the days of the european scientific revolution and visual arts ranging from the renaissance to modernity. Motivated by mathematical beauty and a rich history, we consider the Platonic solids in the context of modern quantum mechanics. Specifically, we construct Bell inequalities whose maximal violations are achieved with measurements pointing to the vertices of the Platonic solids. These Platonic Bell inequalities are constructed only by inspecting the visible symmetries of the Platonic solids. We also construct Bell inequalities for more general polyhedra and find a Bell inequality that is more robust to noise than the celebrated Clauser-Horne-Shimony-Holt Bell inequality. Finally, we elaborate on the tension between mathematical beauty, which was our initial motivation, and experimental friendliness, which is necessary in all empirical sciences.