Simultaneous measurement of two non-commuting quantum variables: Solution of a dynamical model

TL;DR

This paper models the simultaneous measurement of two non-commuting quantum variables using a dynamical Curie-Weiss model, revealing non-ideal measurement outcomes but maintaining full informational content through repeated experiments.

Contribution

It provides a detailed dynamical analysis of simultaneous quantum measurements for non-commuting variables, extending the Curie-Weiss model to this context.

Findings

Measurement is non-ideal with joint statistics differing from separate measurements.

The spin density matrix does not collapse in either measured observable.

Expected spin values can be accurately estimated over many runs.

Abstract

The possibility of performing simultaneous measurements in quantum mechanics is investigated in the context of the Curie-Weiss model for a projective measurement. Concretely, we consider a spin-$\frac{1}{2}$ system simultaneously interacting with two magnets, which act as measuring apparatuses of two different spin components. We work out the dynamics of this process and determine the final state of the measuring apparatuses, from which we can find the probabilities of the four possible outcomes of the measurements. The measurement is found to be non-ideal, as (i) the joint statistics do not coincide with the one obtained by separately measuring each spin component, and (ii) the density matrix of the spin does not collapse in either of the measured observables. However, we give an operational interpretation of the process as a generalised quantum measurement, and show that it is fully informative: The expected value of the measured spin components can be found with arbitrary precision for sufficiently many runs of the experiment.