A measure of non-Gaussianity for quantum states

Abstract

We propose a measure of non-Gaussianity for quantum states of a system of n oscillator modes. Our measure is based on the quasi-probability Q(α),α ∈ Cⁿ . Since any measure of non-Gaussianity is necessarily an attempt at making a quantitative statement on the departure of the shape of the Q function from Gaussian, any good measure of non-Gaussianity should be invariant under transformations which do not alter the shape of the Q functions, namely displacements, passage through passive linear systems, and uniform scaling of all the phase space variables: Q(α)→λ ²ⁿ Q(λα). Our measure which meets this 'shape criterion' is computed for a few families of states, and the results are contrasted with existing measures of non-Gaussianity. The shape criterion implies, in particular, that the non-Gaussianity of the photon-added thermal states should be independent of temperature.