Covariant operator bases for continuous variables

A. Z. Goldberg1,2, A. B. Klimov3, G. Leuchs4, and L. L. Sanchez-Soto4,5

1National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario K1N 5A2, Canada
2Department of Physics, University of Ottawa, 25 Templeton Street, Ottawa, Ontario K1N 6N5, Canada
3Departamento de Física, Universidad de Guadalajara, 44420 Guadalajara, Jalisco, Mexico
4Max-Planck-Institut für die Physik des Lichts, 91058 Erlangen, Germany
5Departamento de Óptica, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain

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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.

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