Classical and quantum-mechanical state reconstruction

Abstract

The aim of this paper is to present the subject of state reconstruction in classical and in quantum physics, a subject that deals with the experimentally acquired information that allows the determination of the physical state of a system. Our first purpose is to explain a method for retrieving a classical state in phase space, similar to that used in medical imaging known as computer-aided tomography. It is remarkable that this method can be taken over to quantum mechanics, where it leads to a description of the quantum state in terms of the Wigner function which, although it may take on negative values, plays the role of the probability density in phase space in classical physics. We then present another approach to quantum state reconstruction based on the notion of mutually unbiased bases—a notion of current research interest, for which we give explanatory remarks—and indicate the relation between these two approaches. Since the subject of state reconstruction is rarely considered at the level of textbooks, the presentation in this paper is aimed at graduate-level readers.