Quantum information vs.\ epistemic logic: An analysis of the Frauchiger-Renner theorem

TL;DR

This paper refines the Frauchiger-Renner theorem using multi-agent modal logic, showing that the contradiction arises under various logical systems and discussing implications for quantum interpretations.

Contribution

It provides an extended modal logic reconstruction of the theorem, demonstrating the contradiction under different logical frameworks and analyzing its implications.

Findings

Contradiction holds in reflexive and serial frames.

Stronger theorem applies in doxastic logic systems.

Implications for quantum probability interpretations.

Abstract

A recent no-go theorem (Frauchiger and Renner, 2018) establishes a contradiction from a specific application of quantum theory to a multi-agent setting. The proof of this theorem relies heavily on notions such as 'knows' or `is certain that'. This has stimulated an analysis of the theorem by Nurgalieva and del Rio (2018), in which they claim that it shows the "[i]nadequacy of modal logic in quantum settings" (ibid.). In this paper, we will offer a significantly extended and refined reconstruction of the theorem in multi-agent modal logic. We will then show that a thorough reconstruction of the proof as given by Frauchiger and Renner requires the reflexivity of access relations (system $\mathsf{\mathbf{T}}$). However, a stronger theorem is possible that already follows in serial frames, and hence also holds in systems of \emph{doxastic} logic (such as $\mathsf{\mathbf{KD45}}$). After proving this, we will discuss the general implications for different interpretations of quantum probabilities as well as several options for dealing with the result.