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The Contextual Modal Logic of a Wigner's Friend Generalization

Felipe Dilho Alves, João Carlos Alves Barata

Abstract

Quantum mechanics has been subject to logical scrutiny since its inception. The behavior of quantum systems, which are fundamentally dissimilar from classical systems, often appears to point to a logical inconsistency in quantum mechanics, allegedly leading to contradictions in the prediction of experimental measurements--though such contradictions have never materialized. A recent example of this type of inquiry into the logical well-posedness of quantum mechanics is the Frauchiger-Renner Gedankenexperiment, which purports to demonstrate that quantum mechanics is logically inconsistent. In this article, we show that by considering the property of contextuality in quantum systems--as predicted by the Kochen-Specker theorem--the supposed contradiction proposed by Frauchiger and Renner becomes logically inaccessible.

The Contextual Modal Logic of a Wigner's Friend Generalization

Abstract

Quantum mechanics has been subject to logical scrutiny since its inception. The behavior of quantum systems, which are fundamentally dissimilar from classical systems, often appears to point to a logical inconsistency in quantum mechanics, allegedly leading to contradictions in the prediction of experimental measurements--though such contradictions have never materialized. A recent example of this type of inquiry into the logical well-posedness of quantum mechanics is the Frauchiger-Renner Gedankenexperiment, which purports to demonstrate that quantum mechanics is logically inconsistent. In this article, we show that by considering the property of contextuality in quantum systems--as predicted by the Kochen-Specker theorem--the supposed contradiction proposed by Frauchiger and Renner becomes logically inaccessible.

Paper Structure

This paper contains 7 sections, 1 theorem, 46 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Generalized Wigner's Friend Gedankenexperiment
  3. Formal Deductions
  4. Frauchinger-Renner No-Go Result
  5. Complications of generalizing to QFTs
  6. Conclusion
  7. ACKNOWLEDGMENTS

Key Result

Theorem 4.1

Any theory that satisfies the assumptions given by Born's rule (Q), the ability to generalize an assertion (C) about the information available to an agent, with the possibility of simplifying information without any obstructions related to the receptacle of this information; and the univaluation (S)

Figures (2)

  • Figure 1: Representation of Wigner's friend Gedankenexperiment, where the quantum system $S$ is measured by $F$ in the laboratory $L$ and $W$ measures the entirety of $L$.
  • Figure 2: Representation of the Frauchinger-Renner Gedankenexperiment as a variation in the Wigner's friend Gedankenexperiment.

Theorems & Definitions (9)

  • Definition 3.1: Kripke structure
  • Definition 3.2: Distribution axiom
  • Definition 3.3: Knowledge generalization axiom
  • Definition 3.4: Positive introspection axiom
  • Definition 3.5: Negative introspection axiom
  • Theorem 4.1: Frauchinger-Renner
  • proof
  • Definition 4.1: Contextual information axiom
  • Definition 4.2: Contextual distribution axiom