Measurement of a quantum system with a classical apparatus using ensembles on configuration space

TL;DR

This paper presents a detailed model of classical-quantum measurement interactions using ensembles on configuration space, demonstrating how classical apparatus can measure and update the state of a quantum particle, with implications for metrology.

Contribution

It introduces a novel formalism for classical-quantum measurement using ensembles on configuration space, addressing fundamental issues and incorporating uncertainties.

Findings

Classical apparatus can measure quantum particle positions accurately.

Measurement updates the quantum wave function based on pointer observations.

The formalism is suitable for quantum metrology applications.

Abstract

Finding a physically consistent approach to modelling interactions between classical and quantum systems is a highly nontrivial task. While many proposals based on various mathematical formalisms have been made, most of these efforts run into difficulties of one sort or another. One of the first detailed descriptions was given by Sudarshan and his collaborators who, motivated by the measurement problem in quantum mechanics, proposed a Hilbert space formulation of classical-quantum interactions which made use of the Koopman-von Neumann description of classical systems. Here we use the approach of ensembles on configurations space to give a detailed account of a classical apparatus measuring the position of a quantum particle that is prepared in a superposition of two localized states. We show that the probability of the pointer of the classical apparatus is left in a state that corresponds to the probability of the quantum particle. A subsequent observation of the pointer leads to an update of its probability density. From this we can obtain information about the position of the quantum particle, leading to an update of its wave function. Since this formalism incorporates uncertainties and finite measurement precision, it is well suited for metrological applications. Furthermore, it resolves fundamental issues that appear in the case of a quantum description of the apparatus.