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

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.