Relative entropy is an exact measure of non-Gaussianity

TL;DR

This paper proves that the relative entropy provides an exact measure of non-Gaussianity for quantum states, establishing a precise way to quantify how much a state deviates from Gaussianity.

Contribution

It demonstrates that the relative entropy to the closest Gaussian state, determined by covariance and displacement, is an exact measure of non-Gaussianity, with explicit evaluation for Fock-diagonal states.

Findings

Relative entropy is an exact non-Gaussianity measure.

Closest Gaussian state is characterized by covariance and displacement.

Explicit evaluation for Fock-diagonal states confirms the measure.

Abstract

We prove that the closest Gaussian state to an arbitrary $N$-mode field state through the relative entropy is built with the covariance matrix and the average displacement of the given state. Consequently, the relative entropy of an $N$-mode state to its associate Gaussian one is an exact distance-type measure of non-Gaussianity. In order to illustrate this finding, we discuss the general properties of the $N$-mode Fock-diagonal states and evaluate their exact entropic amount of non-Gaussianity.