Moments of non-Gaussian Wigner distributions and a generalized uncertainty principle: I. The single-mode case

Abstract

The non-negativity of the density operator of a state is faithfully coded in its Wigner distribution, and this coding places on the moments of the Wigner distribution constraints arising from the non-negativity of the density operator. Working in a monomial basis for the algebra $\hat{\mathcal A}$ of operators on the Hilbert space of a bosonic mode, we formulate these constraints in a canonically covariant form which is both concise and explicit. Since the conventional uncertainty relation is such a constraint on the first and second moments, our result constitutes a generalization of the same to all orders. The structure constants of $\hat{\mathcal A}$, in the monomial basis, are shown to be essentially the SU(2) Clebsch–Gordan coefficients. Our results have applications in quantum state reconstruction using optical homodyne tomography and, when generalized to the n-mode case, which will be done in the second part of this work, will have applications also for continuous variable quantum information systems involving non-Gaussian states.