von Neumann's Formula, Measurements and the Lindblad Equation

TL;DR

This paper extends von Neumann's measurement formula for continuous spectra, demonstrates its complete positivity, and confirms that measurement dynamics can be modeled by Lindblad equations as an open quantum system.

Contribution

It generalizes von Neumann's measurement formula to continuous spectra, shows its complete positivity, and links measurement evolution to Lindblad equations.

Findings

Extended von Neumann's formula involves detector resolution.

The post-measurement state map is completely positive.

Measurement dynamics can be described by Lindblad equations.

Abstract

We previously remarked that when an observable A has a continuous spectrum, then von Neumann's formula for the post-measurement state needs to be extended and the correct formula ineluctably involves the resolution of the detector used in the measurement. We generalize previous results to compute the uncertainties in successive measurements of more general pairs of observables. We also show that this extended von Neumann's formula for the post-measurement state is a completely positive map and, moreover, that there is a completely-positive interpolation between the pre- and post-measurement states. Weinberg has advocated that the time-evolution during the measurement process should be modeled as an open quantum system and governed by a Lindblad equation. We verify that this is indeed the case for an arbitrary observable, A, and a fairly general class of interpolations.