The Platonic solids and fundamental tests of quantum mechanics

Armin Tavakoli and Nicolas Gisin

Département de Physique Appliquée, Université de Genève, CH-1211 Genève, Switzerland

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The Platonic solids is the name traditionally given to the five regular convex polyhedra, namely the tetrahedron, 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 millennia, 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.

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Cited by

[1] Armin Tavakoli, "Semi-Device-Independent Certification of Independent Quantum State and Measurement Devices", arXiv:2003.03859, Physical Review Letters 125 15, 150503 (2020).

[2] Armin Tavakoli, Ingemar Bengtsson, Nicolas Gisin, and Joseph M. Renes, "Compounds of symmetric informationally complete measurements and their application in quantum key distribution", arXiv:2007.01007.

[3] H. Chau Nguyen, Sébastien Designolle, Mohamed Barakat, and Otfried Gühne, "Symmetries between measurements in quantum mechanics", arXiv:2003.12553.

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