On the complementarity of the quadrature observables

TL;DR

This paper explores the coupling and complementarity of quadrature observables, demonstrating their relation to phase space distributions, and provides methods for joint measurement schemes of these observables.

Contribution

It establishes the coupling properties of quadrature observables, relates them to phase space and tomography, and introduces a measurement scheme for complementary pairs.

Findings

Quadrature observables share coupling properties with position-momentum pairs.

Quadrature distributions are Radon transforms of the Wigner function.

A method for joint measurement of complementary quadrature pairs is proposed.

Abstract

In this paper we investigate the coupling properties of pairs of quadrature observables, showing that, apart from the Weyl relation, they share the same coupling properties as the position-momentum pair. In particular, they are complementary. We determine the marginal observables of a covariant phase space observable with respect to an arbitrary rotated reference frame, and observe that these marginal observables are unsharp quadrature observables. The related distributions constitute the Radon tranform of a phase space distribution of the covariant phase space observable. Since the quadrature distributions are the Radon transform of the Wigner function of a state, we also exhibit the relation between the quadrature observables and the tomography observable, and show how to construct the phase space observable from the quadrature observables. Finally, we give a method to measure together with a single measurement scheme any complementary pair of quadrature observables.