Decay of Gaussian correlations in local thermal reservoirs

In this paper we examine the decay of quantum correlations for the radiation field in a two-mode squeezed thermal state in contact with local thermal reservoirs. Two measures of the evolving quantum correlations are compared: the entanglement of formation and the quantum discord. We derive analytic expressions of the entanglement-death time in two special cases: when the reservoirs for each mode are identical, as well as when a single reservoir acts on the first mode only. In the latter configuration, we show that all the pure Gaussian states lose their entanglement at the same time determined solely by the field-reservoir coupling. Also investigated is the evolution of the Gaussian quantum discord for the same choices of thermal baths. We notice that the discord can increase in time above its initial value in a special situation, namely, when it is defined by local measurements on the attenuated mode and the input state is mixed. This enhancement of discord is stronger for zero-temperature reservoirs and increases with the input degree of mixing.

Hellinger distance as a measure of Gaussian discord

Especially investigated in recent years, the Gaussian discord can be quantified by a distance between a given two-mode Gaussian state and the set of all the zero-discord two-mode Gaussian states. However, as this set consists only of product states, such a distance captures all the correlations (quantum and classical) between modes. Therefore it is merely un upper bound for the geometric discord, no matter which is the employed distance. In this work we choose for this purpose the Hellinger metric that is known to have many beneficial properties recommending it as a good measure of quantum behaviour. In general, this metric is determined by affinity, a relative of the Uhlmann fidelity with which it shares many important features. As a first step of our work, the affinity of a pair of $n$-mode Gaussian states is written. Then, in the two-mode case, we succeeded in determining exactly the closest Gaussian product state and computed the Gaussian discord accordingly. The obtained general formula is remarkably simple and becomes still friendlier in the significant case of symmetric two-mode Gaussian states. We then analyze in detail two special classes of two-mode Gaussian states of theoretical and experimental interest as well: the squeezed thermal states and the mode-mixed thermal ones. The former are separable under a well-known threshold of squeezing, while the latter are always separable. It is worth stressing that for symmetric states belonging to either of these classes, we find consistency between their geometric Hellinger discord and the originally defined discord in the Gaussian approach. At the same time, the Gaussian Hellinger discord of such a state turns out to be a reliable measure of the total amount of its cross correlations.

The Chern character of the Verlinde bundle over the moduli space of stable curves

We prove an explicit formula for the total Chern character of the Verlinde bundle over the moduli space of pointed stable curves in terms of tautological classes. The Chern characters of the Verlinde bundles define a semisimple CohFT (the ranks, given by the Verlinde formula, determine a semisimple fusion algebra). According to Telemans classification of semisimple CohFTs, there exists an element of Giventals group transforming the fusion algebra into the CohFT. We determine the element using the first Chern class of the Verlinde bundle on the moduli space of nonsingular curves and the projective flatness of the Hitchin connection.

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.

On a power series involving classical orthogonal polynomials

We investigate a class of power series occurring in some problems in quantum optics. Their coefficients are either Gegenbauer or Laguerre polynomials multiplied by binomial coefficients. Although their sums have been known for a long time, we employ here a different method to recover them as higher-order derivatives of the generating function of the given orthogonal polynomials. The key point in our proof consists in exploiting a specific functional equation satisfied by the generating function in conjunction with Cauchys integral formula for the derivatives of a holomorphic function. Special or limiting cases of Gegenbauer polynomials include the Legendre and Chebyshev polynomials. The series of Hermite polynomials is treated in a straightforward way, as well as an asymptotic case of either the Gegenbauer or the Laguerre series. Further, we have succeeded in evaluating the sum of a similar power series which is a higher-order derivative of Mehlers generating function. As a prerequisite, we have used a convenient factorization of the latter that enabled us to employ a particular Laguerre expansion. Mehlers summation formula is then applied in quantum mechanics in order to retrieve the propagator of a linear harmonic oscillator.

Entanglement of formation for an arbitrary two-mode Gaussian state

We write the optimal pure-state decomposition of any two-mode Gaussian state and show that its entanglement of formation coincides with the Gaussian one. This enables us to develop an insightful approach of evaluating the exact entanglement of formation. Its additivity is finally proven.

Non-Gaussian continuous-variable teleportation

We have recently shown that the output field in the Braunstein-Kimble protocol of teleportation is a superposition of two fields: the input one and a field created by Alices measurement and by displacement of the state at Bobs station by using the classical information provided by Alice. We study here the noise added by teleportation and compare its influence in the Gaussian and non-Gaussian settings.

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.

Gaussian entanglement of symmetric two-mode Gaussian states

A Gaussian degree of entanglement for a symmetric two-mode Gaussian state can be defined as its distance to the set of all separable two-mode Gaussian states. The principal property that enables us to evaluate both Bures distance and relative entropy between symmetric two-mode Gaussian states is the diagonalization of their covariance matrices under the same beam-splitter transformation. The multiplicativity property of the Uhlmann fidelity and the additivity of the relative entropy allow one to finally deal with a single-mode optimization problem in both cases. We find that only the Bures-distance Gaussian entanglement is consistent with the exact entanglement of formation.

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.