On the Measurement in Quantum Mechanics: the Consistent Measurement Apparatus

TL;DR

This paper explores the formal conditions for consistent measurement apparatus in quantum mechanics, linking the Copenhagen collapse with a holistic system-apparatus perspective, and extends the analysis to weak measurements.

Contribution

It establishes the criteria for consistent measurement apparatus using Schmidt decomposition and compares two approaches to weak measurement protocols within this framework.

Findings

Consistent measurement apparatus requires non-overlapping supports in the Schmidt basis.

Only one of the two weak measurement formulations satisfies the consistent apparatus condition.

A generalized framework for weak measurements emerges from combining the two approaches.

Abstract

Measurement in quantum mechanics is generally described as an irreversible process that perturbs the wavefunction describing a quantum system. In this work we establish a formal connection between the measurement description within the Copenhagen interpretation (i.e., through the collapse of the wavefunction) compared versus a picture in which the system and the measurement apparatus are considered as a whole. We first consider a projective measurement. In this limiting case, the natural requirements of consistency and equivalence between the two pictures lead to the rigorous definition of consistent measuring apparatus: the orthonormal wavefunctions from the Schmidt decomposition of the system plus apparatus must have non-overlapping supports. This result arises from the comparison of the two pictures (otherwise hidden), and while it seems to be an obvious conclusion in the limit of projective measurements, it has some nontrivial implications as one extends its validity to the domain of weak measurements. In this respect, we argue on the existence of two alternative approaches to mathematically constructing a weak measurement protocol. While the two approaches are equivalent from the system's perspective, they do strongly differ from the apparatus point of view, and hence can be only distinguished one from each other in the picture where system and apparatus are considered as a whole. We show that only one of the two mathematical formulations of the weak measurement fulfills the consistent apparatus condition, while the combination of the two gives rise to a generalized weak measurements framework.