Comment on "Quantum phase for an arbitrary system with finite-dimensional Hilbert space"

TL;DR

This paper critically analyzes a recent construction of covariant quantum phase observables for finite-dimensional systems, highlighting its limitations for quasiperiodic systems and contrasting it with previously defined covariant time observables.

Contribution

It demonstrates that the recent phase construction is a rescaling of the canonical 'time' observable and discusses its shortcomings for quasiperiodic Hamiltonians.

Findings

The phase construction reduces to a rescaled 'time' observable for periodic systems.

The construction has undesirable features for quasiperiodic systems, such as trivial probability density.

A previously defined covariant time observable avoids these issues.

Abstract

A construction of covariant quantum phase observables, for Hamiltonians with a finite number of energy eigenvalues, has been recently given by D. Arsenovic et al. [Phys. Rev. A 85, 044103 (2012)]. For Hamiltonians generating periodic evolution, we show that this construction is just a simple rescaling of the known canonical 'time' or 'age' observable, with the period T rescaled to 2\pi. Further, for Hamiltonians generating quasiperiodic evolution, we note that the construction leads to a phase observable having several undesirable features, including (i) having a trivially uniform probability density for any state of the system, (ii) not reducing to the periodic case in an appropriate limit, and (iii) not having any clear generalisation to an infinite energy spectrum. In contrast, we note that a covariant time observable has been previously defined for such Hamiltonians, which avoids these features. We also show how this 'quasiperiodic' time observable can be represented as the well-defined limit of a sequence of periodic time observables.