# 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.

## Key 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.