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The logic of quantum mechanics

Eric Buffenoir

TL;DR

The paper develops an operational, Chu-space based formulation of quantum mechanics that foregrounds tensor-product composition and a star involution as core prerequisites for state spaces. By introducing real structures that separate real from hidden states and detailing ontic completions, it provides a mechanism to model both deterministic and indeterministic quantum-like behavior while preserving a close link to Hilbert-geometric structures. It then constructs minimal and ontic tensor products to describe bipartite systems, proving that these constructions can reproduce key quantum features such as contextuality and non-locality within a purely operational, information-theoretic framework. The work further clarifies how contextuality arises from the compatibility structure of measurements and shows that the proposed tensor-product formalisms can describe both local realism and quantum-like correlations without assuming Hilbert spaces from the outset. Overall, the approach realizes the original ambition of Birkhoff and von Neumann to ground quantum theory in logical and operational principles, while illuminating the roles of ontic completions and generalized probabilistic constructs in quantum foundations.

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.

The logic of quantum mechanics

TL;DR

The paper develops an operational, Chu-space based formulation of quantum mechanics that foregrounds tensor-product composition and a star involution as core prerequisites for state spaces. By introducing real structures that separate real from hidden states and detailing ontic completions, it provides a mechanism to model both deterministic and indeterministic quantum-like behavior while preserving a close link to Hilbert-geometric structures. It then constructs minimal and ontic tensor products to describe bipartite systems, proving that these constructions can reproduce key quantum features such as contextuality and non-locality within a purely operational, information-theoretic framework. The work further clarifies how contextuality arises from the compatibility structure of measurements and shows that the proposed tensor-product formalisms can describe both local realism and quantum-like correlations without assuming Hilbert spaces from the outset. Overall, the approach realizes the original ambition of Birkhoff and von Neumann to ground quantum theory in logical and operational principles, while illuminating the roles of ontic completions and generalized probabilistic constructs in quantum foundations.

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.
Paper Structure (33 sections, 136 theorems, 400 equations)

This paper contains 33 sections, 136 theorems, 400 equations.

Table of Contents

  1. Introduction
  2. The operational formalism
  3. States/Effects Chu spaces
  4. Morphisms
  5. Real structures
  6. First results on real structures
  7. Real morphisms
  8. Determinism vs. Indeterminism
  9. Characterization of hidden states
  10. Preliminary remarks
  11. Reconstruction of hidden states
  12. Ontic completions
  13. Morphisms and ontic completions
  14. Bipartite experiments
  15. Building principles for the description of bipartite experiments
  16. ...and 18 more sections

Key Result

Lemma 2.7

The evaluation map $\epsilon^{ {S}}$ satisfies the following properties : and

Theorems & Definitions (333)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Lemma 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 323 more