A measure of non-Gaussianity for quantum states

TL;DR

This paper introduces a new measure of non-Gaussianity for quantum states based on the shape of the quasi-probability $Q$ function, invariant under certain transformations, and compares it with existing measures.

Contribution

It proposes a shape-invariant non-Gaussianity measure for quantum states and demonstrates its properties and differences from previous measures.

Findings

The measure is computed for various quantum states.

It remains invariant under displacements, passive linear transformations, and scaling.

The non-Gaussianity of photon-added thermal states is shown to be temperature-independent.

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 $Q(\alpha), \alpha\in{\cal C}^n$. 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: $Q(\alpha)\to \lambda^{2n}Q(\lambda \alpha)$. 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.