Relative entropy is an exact measure of non-Gaussianity

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.Comment: 6 pages, no figures. Comments are welcom

Einstein-Podolsky-Rosen-like separability indicators for two-mode Gaussian states

We investigate the separability of the two-mode Gaussian states by using the variances of a pair of Einstein-Podolsky-Rosen (EPR)-like observables. Our starting point is inspired by the general necessary condition of separability introduced by Duan {\em et al.} [Phys. Rev. Lett. {\bf 84}, 2722 (2000)]. We evaluate the minima of the normalized forms of both the product and sum of such variances, as well as that of a regularized sum. Making use of Simon's separability criterion, which is based on the condition of positivity of the partial transpose (PPT) of the density matrix [Phys. Rev. Lett. {\bf 84}, 2726 (2000)], we prove that these minima are separability indicators in their own right. They appear to quantify the greatest amount of EPR-like correlations that can be created in a two-mode Gaussian state by means of local operations. Furthermore, we reconsider the EPR-like approach to the separability of two-mode Gaussian states which was developed by Duan {\em et al.} with no reference to the PPT condition. By optimizing the regularized form of their EPR-like uncertainty sum, we derive a separability indicator for any two-mode Gaussian state. We prove that the corresponding EPR-like condition of separability is manifestly equivalent to Simon's PPT one. The consistency of these two distinct approaches (EPR-like and PPT) affords a better understanding of the examined separability problem, whose explicit solution found long ago by Simon covers all situations of interest.Comment: Very close to the published versio

Bures distance as a measure of entanglement for symmetric two-mode Gaussian states

We evaluate a Gaussian entanglement measure for a symmetric two-mode Gaussian state of the quantum electromagnetic field in terms of its Bures distance to the set of all separable Gaussian states. The required minimization procedure was considerably simplified by using the remarkable properties of the Uhlmann fidelity as well as the standard form II of the covariance matrix of a symmetric state. Our result for the Gaussian degree of entanglement measured by the Bures distance depends only on the smallest symplectic eigenvalue of the covariance matrix of the partially transposed density operator. It is thus consistent to the exact expression of the entanglement of formation for symmetric two-mode Gaussian states. This non-trivial agreement is specific to the Bures metric.Comment: published versio

Continuous-variable teleportation: a new look

In contrast to discrete-variable teleportation, a quantum state is imperfectly transferred from a sender to a remote receiver in a continuous-variable setting. We recall the ingenious scheme proposed by Braunstein and Kimble for teleporting a one-mode state of the quantum radiation field. By analyzing this protocol, we have previously proven the factorization of the characteristic function of the output state. This indicates that teleportation is a noisy process that alters, to some extent, the input state. Teleportation with a two-mode Gaussian EPR state can be described in terms of the superposition of a distorting field with the input one. Here we analyze the one-mode Gaussian distorting-field state. Some of its most important properties are determined by the statistics of a positive EPR operator in the two-mode Gaussian resource state. We finally examine the fidelity of teleportation of a coherent state when using an arbitrary resource state.Comment: Contribution to the special issue of Romanian Journal of Physics dedicated to the centenary of Serban Titeica (1908-1985), the founder of the school of theoretical physics in Romani

Gaussification through decoherence

We investigate the loss of nonclassicality and non-Gaussianity of a single-mode state of the radiation field in contact with a thermal reservoir. The damped density matrix for a Fock-diagonal input is written using the Weyl expansion of the density operator. Analysis of the evolution of the quasiprobability densities reveals the existence of two successive characteristic times of the reservoir which are sufficient to assure the positivity of the Wigner function and, respectively, of the $P$ representation. We examine the time evolution of non-Gaussianity using three recently introduced distance-type measures. They are based on the Hilbert-Schmidt metric, the relative entropy, and the Bures metric. Specifically, for an $M$-photon-added thermal state, we obtain a compact analytic formula of the time-dependent density matrix that is used to evaluate and compare the three non-Gaussianity measures. We find a good consistency of these measures on the sets of damped states. The explicit damped quasiprobability densities are shown to support our general findings regarding the loss of negativities of Wigner and $P$ functions during decoherence. Finally, we point out that Gaussification of the attenuated field mode is accompanied by a nonmonotonic evolution of the von Neumann entropy of its state conditioned by the initial value of the mean photon number.Comment: Published version. Comments are welcom

Optimal purifications and fidelity for displaced thermal states

We evaluate the Uhlmann fidelity between two one-mode displaced thermal states as the maximal probability transition between appropriate purifications of the given states. The optimal purifications defining the fidelity are proved to be two-mode displaced Gaussian states.Comment: published versio