Description of weak measurements and weak values in the phase space representation of quantum mechanics

Abstract

Weak measurements estimate an operator’s value of a quantum system while minimising the perturbation of the state, contrary to the usual strong measurement. The observations from weak measurements with post-selection depend on complex numbers called weak values. When very large or complex, they are called anomalous weak values and indicate a quantum behaviour. To get more intuition about their physical meaning, we describe them in the quantum phase space defined with the Wigner distribution and the Weyl transform. We illustrate the formalism by a weak measurement of two coupled harmonic oscillators. The weak value is interpreted as the average value over an interference between pre-selection and post-selection. The particular case of the momentum operator is studied and an interpretation is proposed, using notions from the literature. Then, we transpose the von Neumann model of measurement in phase space for strong and weak post-selected measurements. Using the Stratonovich-Weyl kernel, we then generalise the phase space formalism to curved configuration spaces, useful to describe constrained spaces. The previous results are extended to this situation.

Date of Award	22 Jun 2022
Original language	English
Awarding Institution	University of Namur
Supervisor	Yves Caudano (Supervisor) & Dominique Lambert (Co-Supervisor)