Wigner function and the successive measurement of position and momentum

TL;DR

This paper explores the relationship between the Wigner function and successive quantum measurements of position and momentum, extending measurement models to both continuous and discrete quantum systems.

Contribution

It introduces a generalized measurement framework linking the Wigner function to Kirkwood distributions through successive probes in both continuous and finite-dimensional quantum systems.

Findings

Wigner function expressed via Kirkwood joint distribution.

Successive measurement model generalizes von Neumann measurement.

Applicable to both continuous and discrete quantum systems.

Abstract

Wigner function is a quasi-distribution that provides a representation of the state of a quantum mechanical system in the phase space of position and momentum. In this paper we find a relation between Wigner function and appropriate measurements involving the system position and momentum which generalize the von Neumann model of measurement. We introduce two probes coupled successively in time to projectors associated with the system position and momentum. We show that one can relate Wigner function to Kirkwood joint quasi-distribution of position and momentum, the latter, in turn, being a particular case of successive measurements. We first consider the case of a quantum mechanical system described in a continuous Hilbert space, and then turn to the case of a discrete, finite-dimensional Hilbert space.