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arXiv:0805.1645 · Physical Review A

Quantifying the non-Gaussian character of a quantum state by quantum relative entropy

Marco G. Genoni, Matteo G. A. Paris, Konrad Banaszek

Published Dec 18, 2008• 209 citations
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Abstract

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.

DOI
10.1103/physreva.78.060303

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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.
We address the quantification of non-Gaussianity (nG) of states and operations in continuous-variable systems and its use in quantum information. We start by illustrating in detail the properties and the relationships of two recently proposed measures of nG based on the Hilbert-Schmidt distance and the quantum relative entropy (QRE) between the state under examination and a reference Gaussian state. We then evaluate the non-Gaussianities of several families of non-Gaussian quantum states and show that the two measures have the same basic properties and also share the same qualitative behavior in most of the examples taken into account. However, we also show that they introduce a different relation of order; that is, they are not strictly monotone to each other. We exploit the nG measures for states in order to introduce a measure of nG for quantum operations, to assess Gaussification and de-Gaussification protocols, and to investigate in detail the role played by nG in entanglement-distillation protocols. Besides, we exploit the QRE-based nG measure to provide different insight on the extremality of Gaussian states for some entropic quantities such as conditional entropy, mutual information, and the Holevo bound. We also deal with parameter estimation and present a theorem connecting the QRE nG to the quantum Fisher information. Finally, since evaluation of the QRE nG measure requires the knowledge of the full density matrix, we derive some experimentally friendly lower bounds to nG for some classes of states and by considering the possibility of performing on the states only certain efficient or inefficient measurements.
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The quantum relative entropy between a quantum state and its Gaussian equivalent is a quantifying function of the system's non-Gaussianity, a useful resource in several applications, such as quantum communication and computation. One of its most fundamental properties is to be monotonically decreasing under Gaussian evolutions. In this paper, we develop the conditions for a non-Gaussian quantum channel to preserve the monotonically decreasing property. We propose a necessary condition to classify between Gaussian and non-Gaussian channels and use it to define a class of quantum channels that decrease the system's non-Gaussianity. We also discuss how this property, combined with a restriction on the states at the channel's input, can be applied to the security analysis of continuous-variable quantum key distribution protocols.
We consider how to quantify non-Gaussianity for the correlation of a bipartite quantum state by using various measures such as relative entropy and geometric distances. We first show that an intuitive approach, i.e., subtracting the correlation of a reference Gaussian state from that of a target non-Gaussian state, fails to yield a non-negative measure with monotonicity under local Gaussian channels. Our finding clearly manifests that quantum-state correlations generally have no Gaussian extremality. We therefore propose a different approach by introducing relevantly averaged states to address correlation. This enables us to define a non-Gaussianity measure based on, e.g., the trace-distance and the fidelity, fulfilling all requirements as a measure of non-Gaussian correlation. For the case of the fidelity-based measure, we also present readily computable lower bounds of non-Gaussian correlation.
We introduce a measure of quantum non-Gaussianity (QNG) for those quantum states not accessible by a mixture of Gaussian states in terms of quantum relative entropy. Specifically, we employ a convex-roof extension using all possible mixed-state decompositions beyond the usual pure-state decompositions. We prove that this approach brings a QNG measure fulfilling the properties desired as a proper monotone under Gaussian channels and conditional Gaussian operations. As an illustration, we explicitly calculate QNG for the noisy single-photon states and demonstrate that QNG coincides with non-Gaussianity of the state itself when the single-photon fraction is sufficiently large.
We consider a quantity that is the differential relative entropy between a generic Wigner function and a Gaussian one. We prove that said quantity is minimized with respect to its Gaussian argument, if both Wigner functions in the argument of the Wigner differential entropy have the same first and second moments, i.e., if the Gaussian argument is the Gaussian associate of the other, generic Wigner function. Therefore, we introduce the differential relative entropy between any Wigner function and its Gaussian associate and we examine its potential as a non-Gaussianity measure. The proposed, phase-space based non-Gaussianity measure is complex-valued, with its imaginary part possessing the physical meaning of the Wigner function’s negative volume. At the same time, the real part of this measure provides an extra layer of information, rendering the complex-valued quantity a measure of non-Gaussianity, instead of a quantity pertaining only to the negativity of the Wigner function. We prove that the measure (both the real and imaginary parts) is faithful, invariant under Gaussian unitary operations, and find a sufficient condition on its monotonic behavior under Gaussian channels. We provide numerical results supporting the aforesaid condition. In addition, we examine the measure’s usefulness to non-Gaussian quantum state engineering with partial measurements.