Covariant operator bases for continuous variables

TL;DR

This paper introduces a covariant operator basis for continuous-variable quantum systems, enabling a canonical, explicit, and symplectically well-behaved representation of quantum states that facilitates analysis of properties like quantumness and Gaussianity.

Contribution

It develops a new basis of monomials on basic observables that behaves well under symplectic transformations, providing a covariant and concise representation of quantum states.

Findings

The basis allows for a canonical, explicit state representation.

It connects tomographic measurements with quasiprobability distributions.

The approach aids in assessing quantum properties like quantumness and Gaussianity.

Abstract

Coherent-state representations are a standard tool to deal with continuous-variable systems, as they allow one to efficiently visualize quantum states in phase space. Here, we work out an alternative basis consisting of monomials on the basic observables, with the crucial property of behaving well under symplectic transformations. This basis is the analogue of the irreducible tensors widely used in the context of SU(2) symmetry. Given the density matrix of a state, the expansion coefficients in that basis constitute the multipoles, which describe the state in a canonically covariant form that is both concise and explicit. We use these quantities to assess properties such as quantumness or Gaussianity and to furnish direct connections between tomographic measurements and quasiprobability distribution reconstructions.