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 $Q(α), α\in{\cal C}^n$. 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: $Q(α)\to λ^{2n}Q(λα)$. 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.

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