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A Formal Theory for Finite-Dimensional Possibilistic Quantum Mechanics

Olivier Brunet

Abstract

In this work, we present a logical formalism for reasoning about quantum systems in finite dimension. Contrary to the usual approach in quantum logic, our formalism is based classical first-order logic, which allows us to use the tools of model theory in our study. In particular, we show that our formal theory is complete, meaning that it entirely determines the behaviour of quantum systems. Moreover, we provide a characterization of the models of our formal theory, thus providing new insights in the study of hidden variable models of quantum theory.

A Formal Theory for Finite-Dimensional Possibilistic Quantum Mechanics

Abstract

In this work, we present a logical formalism for reasoning about quantum systems in finite dimension. Contrary to the usual approach in quantum logic, our formalism is based classical first-order logic, which allows us to use the tools of model theory in our study. In particular, we show that our formal theory is complete, meaning that it entirely determines the behaviour of quantum systems. Moreover, we provide a characterization of the models of our formal theory, thus providing new insights in the study of hidden variable models of quantum theory.
Paper Structure (13 sections, 16 theorems, 55 equations, 5 figures)

This paper contains 13 sections, 16 theorems, 55 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Describing Quantum Systems and Their Evolution
  3. A Formal Theory for Reasoning About Quantum Systems
  4. Completeness of the formal theory
  5. From subspaces to vectors
  6. From any model to subspaces
  7. From Vect to any model
  8. Wrapping up
  9. Conclusion and perspectives
  10. Proofs
  11. Proof of proposition \ref{['prop:valid_axioms']}
  12. Proof of proposition \ref{['prop:no_two_atoms']}
  13. Proof of proposition \ref{['prop:incompatible']}

Key Result

Proposition 1

All the axioms listed in figure fig:rules following from the previous rules.

Figures (5)

  • Figure 1: Creation of a Bell state
  • Figure 2: Creation of a Bell state
  • Figure 3: Rules for circuits
  • Figure 4: The axioms of $\mathrm{PQM}$
  • Figure 5: The axioms of $\mathrm{PQM}_d$, revised for $d \geq 3$

Theorems & Definitions (32)

  • Definition 1
  • Proposition 1
  • proof
  • Definition 2
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Corollary 4
  • Theorem 5
  • ...and 22 more