Inadequacy of Modal Logic in Quantum Settings

Abstract

We test the principles of classical modal logic in fully quantum settings. Modal logic models our reasoning in multi-agent problems, and allows us to solve puzzles like the muddy children paradox. The Frauchiger-Renner thought experiment highlighted fundamental problems in applying classical reasoning when quantum agents are involved; we take it as a guiding example to test the axioms of classical modal logic. In doing so, we find a problem in the original formulation of the Frauchiger-Renner theorem: a missing assumption about unitarity of evolution is necessary to derive a contradiction and prove the theorem. Adding this assumption clarifies how different interpretations of quantum theory fit in, i.e., which properties they violate. Finally, we show how most of the axioms of classical modal logic break down in quantum settings, and attempt to generalize them. Namely, we introduce constructions of trust and context, which highlight the importance of an exact structure of trust relations between agents. We propose a challenge to the community: to find conditions for the validity of trust relations, strong enough to exorcise the paradox and weak enough to still recover classical logic.Draco said out loud, "I notice that I am confused." Your strength as a rationalist is your ability to be more confused by fiction than by reality... Draco was confused. Therefore, something he believed was fiction. Eliezer Yudkowsky, Harry Potter and the Methods of Rationality

Multi-agent paradoxes beyond quantum theory

“…This was first pointed out by Frauchiger and Renner, in a thought experiment where agents who can measure each others memories (modelled as quantum systems) and reason about shared and individual knowledge may reach contradictory conclusions [1]. We will not review the original experiment here, apart from a very brief description in figure 1 4 ; a pedagogical exposition can be found in our paper [2], but is not necessary to follow this article.…”

“…Condition (1) was explained in previous paragraphs. Condition (2) is required because the trivial map which entails doing nothing to the system and memory (i.e. the identity) satisfies Condition (1) even though such an operation should certainly not be regarded as a memory update.…”

“…Further discussion on axioms of modal logic and their application in quantum mechanics can be found in our paper [2].…”

Quantum Information Versus Epistemic Logic: An Analysis of the Frauchiger–Renner Theorem

“…Finally, it is an additional assumption [cf. 23,272] that observables defined over joint spaces can be evolved unitarily:2 U Suppose agent x has established: (i) System s x is in state |v s at time t 0 . Suppose furthermore that x knows: (ii) A value for O is obtained by a measurement of the projectors {π o } o∈σ(O) on s at t 1 > t 0 .…”

Unitary-Only Quantum Theory Cannot Consistently Describe the Use of Itself: On the Frauchiger–Renner Paradox

“…I hasten to add that the unitarity assumption underlying the FR paradox is pointed out (at least) by Sudbery (2019), Nurgalieva and del Rio (2019), and Araújo (2018); I make no claim that the crucial role of unitarity in deriving the contradiction has not been noticed before. Indeed, it is only 'by linearity' that Frauchiger and Renner arrive at their 'absurd superposition' (to be discussed in more detail below).…”