Teaching ideal quantum measurement, from dynamics to interpretation

TL;DR

This paper presents a comprehensive graduate course on ideal quantum measurements, analyzing them as dynamical processes involving system-apparatus interactions within quantum statistical mechanics, and addresses foundational issues like the measurement problem and Born's rule.

Contribution

It introduces a detailed dynamical model for ideal quantum measurements, clarifies the measurement problem, and derives Born's rule from conservation laws within a quantum statistical mechanics framework.

Findings

Dynamical equations for ideal measurements can be solved in simple models.

Conservation laws lead to independent relaxation mechanisms: truncation and registration.

Born's rule emerges from the conservation law for the measured observable.

Abstract

We present a graduate course on ideal measurements, analyzed as dynamical processes of interaction between the tested system S and an apparatus A, described by quantum statistical mechanics. The apparatus A=M+B involves a macroscopic measuring device M and a bath B. The requirements for ideality of the measurement allow us to specify the Hamiltonian of the isolated compound system S+M+B. The resulting dynamical equations may be solved for simple models. Conservation laws are shown to entail two independent relaxation mechanisms: truncation and registration. Approximations, justified by the large size of M and of B, are needed. The final density matrix $\hat{\cal D}(t_f)$ of S+A has an equilibrium form. It describes globally the outcome of a large set of runs of the measurement. The measurement problem, i.e., extracting physical properties of individual runs from $\hat{\cal D}(t_f)$, then arises due to the ambiguity of its splitting into parts associated with subsets of runs. To deal with this ambiguity, we postulate that each run ends up with a distinct pointer value $A_i$ of the macroscopic M. This is compatible with the principles of quantum mechanics. Born's rule then arises from the conservation law for the tested observable; it expresses the frequency of occurrence of the final indications $A_i$ of M in terms of the initial state of S. Von Neumann's reduction amounts to updating of information due to selection of $A_i$. We advocate the terms $q$-probabilities and $q$-correlations when analyzing measurements of non-commuting observables. These ideas may be adapted to different types of courses.