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arXiv:1308.2939 · Physical Review A

Relative entropy is an exact measure of non-Gaussianity

Paulina Marian, Tudor A. Marian

Published Jul 23, 2013• 70 citations
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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.

DOI
10.1103/physreva.88.012322

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We prove that the closest Gaussian state to an arbitrary NNN-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 NNN-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 NNN-mode Fock-diagonal states and evaluate their exact entropic amount of non-Gaussianity.
We introduce and investigate a distance-type measure of non-Gaussianity based on the quantum fidelity. This new measure can readily be evaluated for all pure states and the mixed ones that are diagonal in the Fock basis. In particular, for an M -photon-added thermal state, an analysis of the Bures degree of non-Gaussianity is performed in comparison with two previous measures built with the Hilbert–Schmidt metric and the relative entropy. We obtain a compact analytic formula for the Hilbert–Schmidt measure of non-Gaussianity and find good consistency between the three examined measures.
We consider the class of the one-mode Gaussian states of the quantum radiation field. The relative entropy of such a state with respect to a similar one is derived. We analyze the entropic amount of nonclassicality of a Gaussian state defined as the minimal relative entropy of any classical Gaussian state with respect to it. A similar quantity built with the Hilbert-Schmidt distance is also calculated. Both nonclassicality measures are then compared with the Bures-metric degree of nonclassicality evaluated previously. The properties of the closest classical Gaussian state, as well as the decrease of the nonclassicality under thermal noise mappings are carefully examined in each case. For mixed states we find that only the Bures-distance amount of nonclassicality is equivalent to the nonclassical depth.
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We introduce a measure to quantify the non-Gaussian character of a quantum state: the quantum relative entropy between the state under examination and a reference Gaussian state. We analyze in detail the properties of our measure and illustrate its relationships with relevant quantities in quantum information such as the Holevo bound and the conditional entropy; in particular, a necessary condition for the Gaussian character of a quantum channel is also derived. The evolution of non-Gaussianity is analyzed for quantum states undergoing conditional Gaussification toward twin beams and de-Gaussification driven by Kerr interaction. Our analysis allows us to assess non-Gaussianity as a resource for quantum information and, in turn, to evaluate the performance of Gaussification and de-Gaussification protocols.