An analysis of Wigner's friend in the framework of quantum mechanics based on the principle of typicality

Abstract

The notion of probability plays a crucial role in quantum mechanics. It appears in quantum mechanics as the Born rule. In modern mathematics which describes quantum mechanics, however, probability theory means nothing other than measure theory, and therefore any operational characterization of the notion of probability is still missing in quantum mechanics. In our former works [K. Tadaki, arXiv:1804.10174], based on the toolkit of algorithmic randomness, we presented a refinement of the Born rule, called the principle of typicality, for specifying the property of results of measurements in an operational way. The Wigner's friend paradox is a Gedankenexperiment regarding when and where the reduction of the state vector occurs in a chain of the measurements by several observers where the state of the consciousness of each observer is measured by the subsequent observer, except for the last observer in the chain. It is one of the central open questions in the measurement problem of quantum mechanics. In this paper, we make an analysis of the Wigner's friend paradox in the framework of quantum mechanics based on the principle of typicality. We then draw common sense conclusions about the Wigner's friend paradox. Deutsch's thought experiment is a variant of the Wigner's friend paradox, which can, in principle, verify the effect of the consciousness of observer on the reduction of the state vector. We make an analysis of Deutsch's thought experiment comprehensively in our framework. We then make a prediction which is testable in principle. In our framework, we can analyze still more complicated situations than the Wigner's friend paradox or Deutsch's thought experiment. In this paper, as such an example, we introduce a combination of these two, called the Wigner-Deutsch collaboration. We make an analysis of it comprehensively in our framework.