Journal of Physics A Mathematical and General

Approximations to the quantum phase operator

Mark A. Hennings, T B Smith, Daniel A. Dubin

Published Dec 7, 1995• 10 citations

Abstract

In a series of papers, Barnett, Pegg (1986, 1988, 1989, 1992) and various co-authors have proposed a description of quantum phase by means of a collection of s-dimensional states and operators, s>or=1. We analyse the limiting procedure they employ for large s, which is known not to be compatible with quantum mechanics in the usual sense. Further, we supply a rigorous demonstration of the asymptotic limits of the `mean` and `variance` of their system of operators in coherent states. These values had previously been given but not justified mathematically. Our analysis, based on the asymptotic analysis of certain random variables, shows that the physical deductions that can be drawn from these limits are limited. We also prove that the (s+1)-dimensional `pure phase` LHW-states they consider form a sequence of approximate eigenvectors for the Weyl-quantized angle operator Delta ( phi ) and the Toeplitz phase operator X proposed by Garrison and Wong (1970), Popov and Yarunin (1992), and others. These states X s ( theta ), to our knowledge first introduced by Lerner, Huang and Waiters (1970), can be used to construct a system of measurement as in the usual quantum theory, sensitive to certain qualities of phase, but not all. Indeed, a feature of the Barnett-Pegg method, when it gives finite answers, is the construction of associated measurement systems for different observables. We give examples of sequences of (s+1)-dimensional devices which represent measurements significantly closer to ideal for X. This serves as a model for corresponding devices for Delta ( phi ), or indeed, any observable with a continuous spectrum, contingent on its spectral decomposition being obtained explicitly.

DOI: 10.1088/0305-4470/28/23/026

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