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arXiv · 2510.12502

The logic of quantum mechanics

Eric Buffenoir
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Abstract

The quantum logic program originated in a 1936 article by G. Birkhoff and J. von Neumann. This program is generally disregarded due to no-go theorems restricting the existence of the tensor product of elementary quantum logics and, above all, the impossibility of considering entangled states and Bell non-local states within the framework of these composite quantum logics. We revisit this study from the beginning and reverse the perspective. Here, the existence of a tensor product and a star involution are the only prerequisites for the definition of the state spaces. Surprisingly, the quantum logics constructed in this way turn out to have a close connection with irreducible Hilbert geometries, even though we did not impose this sort of structure ab initio. Endly, the existence of some basic quantum-like properties is explicitly proven in our framework : contextuality, no-broadcasting theorem, and Bell non-locality. These elements demonstrate that our quantum logic program is capable of achieving G. Birkhoff and J. von Neumann's initial ambition of founding quantum theory.

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Eric Buffenoir. 2025-10-14. The logic of quantum mechanics. https://arxiv.org/abs/2510.12502

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