Measuring processes and repeatability hypothesis

TL;DR

This paper explores the extension of the Srinivas collapse postulate in quantum mechanics, constructing measurement processes for continuous spectrum observables that depend on different invariant means and involve non-normal initial states.

Contribution

It introduces a method to realize various invariant means in quantum measurements by extending von Neumann processes with non-normal apparatus states.

Findings

Constructed measurement processes for any invariant mean.

Linked invariant means to momentum distributions of non-normal states.

Extended von Neumann measurement model to include non-normal initial states.

Abstract

Srinivas [Commun. Math. Phys. 71 (1980), 131-158] proposed a postulate in quantum mechanics that extends the von Neumann-Lueders collapse postulate to observables with continuous spectrum. His collapse postulate does not determine a unique state change, but depends on a particular choice of an invariant mean. To clear the physical significance of employing different invariant means, we construct different measuring processes of the same observable satisfying the Srinivas collapse postulate corresponding to any given invariant means. Our construction extends the von Neumann type measuring process with the meter being the position observable to the one with the apparatus prepared in a non-normal state. It is shown that the given invariant mean corresponds to the momentum distribution of the apparatus in the initial state, which is determined as a non-normal state, called a Dirac state, such that the momentum distribution is the given invariant mean and that the position distribution is the Dirac measure.