Quantum correlation measurements for two qubits in a common squeezed bath

Transcription

1 REVISTA MEXICANA DE FÍSICA S 57 (3) JULIO 011 Quantum correlation measurements for two qubits in a common squeezed bath M. Angeles Gallego and M. Orszag Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago, Chile. Recibido el 11 de enero de 011; aceptado el 18 de marzo de 011 For many years the entangled systems have been associated to the quantum world, while the separable systems with the classical world. Recently quantum discord showed that some separable states posses quantum correlation even when the entanglement is zero. In this work, we compare different features of quantum discord and entanglement of formation for two qubits in a common squeezed reservoir. We relate the quantum correlations with the distance of our initial condition from the decoherence free subspace where the system is not affected by environment. While for some initial conditions, the entanglement presents sudden death and revival, quantum discord does not. Still, is not clear the relation between these two measurements of quantumness. Keywords: Quantm discord; entanglement; squeezed reservoir; decoherence. Durante muchos años los sistemas entrelazados han sido asociados al mundo cuántico, mientras los sistemas separables al mundo clásico. Recientemente la discordia cuántica mostró que algunos sistemas separables poseen correlaciones cuánticas aún cuando el entrelazamiento es nulo. En el presente trabajo comparamos características de la discordia cuántica y el entrelazamiento para dos qubits en un reservorio comprimido común. Se relacionan ambas medidas de correlación con la distancia de la condición incial al subespacio libre de decoherencia, subespacio donde el sistema no es afectado por el medio ambiente. Mientras que para algunas de éstas condiciones inciales el entrelazamiento presenta muerte subita y renacimiento, la discordia cuántica no presenta dichos fenómenos. No obstante, la relación entre estas dos medidas de la cuanticidad no está clara. Descriptores: Discordia cuántica; entrelazamiento; reservorio comprimido; decoherencia. PACS: a; Ta; Ud 1. Introduction The controversial paper of Einstein, Podolski and Rosen [1], on the completeness of quantum mechanics, opened the debate about a new property of quantum mechanics. This feature, called entanglement, is a property of systems that do not interact directly, but interacted in the past. This interaction allows them to maintain strong quantum correlations between them, even when they are spatially separated. Entanglement is one of the most remarkable effects of quantum mechanics and also is a essential tool for various applications, such as quantum teleportation [], quantum cryptography [3] and superdense coding [4]. Entanglement has different quantifiers, for pure and mixed states, some of them are entanglement of formation, entanglement cost, relative entropy of entanglement, etc. All of them are equal to zero for separable states. The density matrix for a separable state of a bipartite system composed of parts A and B, is represented as a sum of product states ρ A ρ B, thus ρ AB = i p i ρ A i ρ B i (1) where the sub index i, represents a statistical (classical) mixture of product density matrices, and ρ A = tr B (ρ AB ), ρ B = tr A (ρ AB ) are the partial matrices of each subsystem. In a product state there is no correlation at all between A and B, because any operator will act in each party separately. Thus, is natural to believe that a sum of product states (a separable state), does not have quantum correlations. Therefore, entanglement is a measure of the inseparability of a density matrix. Entangled versus separable paradigm was explored for a long time. Recently, it was noticed that there are other useful quantum correlations besides entanglement [5]. In particular it was found that there are separable states with quantum correlations, even when entanglement is zero. A better separation between classical and quantum world was proposed by Ollivier and Zurek, they introduced the concept of quantum discord to quantify the degree of quantumness of a system [6]. Their formula is based on Mutual Information, which is an entropy based measurement for quantify the total amount of correlations between two random variables [8]. Using this concept, they define QD as the difference between two different definitions of quantum mutual information, which are only equal in the classical theory of information by using the Bayes rule. Also Henderson and Vedral based in the idea that total mutual information of a system composed by two parties, can be split in two parts: classical correlations and quantum discord, arrived to the same formula as Ollivier and Zurek for QD. The essential idea of Vedral is that in the classical world, the state is no affected at all by a measurement [10], but in the quantum world the measure process produces a change in the state of the system. Therefore, the classical correlation represents the amount of information, which can be obtained by measuring one of the system parts, i.e. that can be achieved in a classical manner. And, in certain way, the QD denotes the magnitude of change produced by the measurement. While the entanglement and the quantum discord are the same for a pure state, their relation (if there is any) remains unclear. There are a few examples of QD as [11, 1], and recently last year appeared a concrete way to compute

2 QUANTUM CORRELATION MEASUREMENTS FOR TWO QUBITS IN A COMMON SQUEEZED BATH 49 it [13-15]. In this paper we present a new example of quantum discord comparing it with the entanglement. We confirm, as proposed theoretically, that for some cases the quantum discord is not null, even when the entanglement disappears. This particular example, consists in two two-level atoms that interact with a common squeezed reservoir. We use the master equation to study the temporal evolution of the system, and how it is affected by decoherence. The phenomenon of decoherence is caused when, in nature, a initially pure state interacts intentionally or unexpectedly, with the environment (other quantum degrees of freedom), resulting in a nonunitary evolution [16, 17]. In other words, the system evolves to a mixed state. In this work, our main task is investigate the effect of the decoherence in the entanglement and quantum discord. To make this possible we use as a initial condition, a state belonging to the basis of the Decoherence-free subspaces. DFS is the term used by Lidar et al. [18], to refer to robust states against perturbations, in the context of Markovian Master equations. One uses the symmetry of the systemenvironment coupling to find a quiet corner in the Hilbert Space not experiencing this interaction [19]. The study of the effect of decoherence in the entanglement for this specific model was made by Hernandez and Orszag in Ref. 0. They show that entanglement has surprising features such as sudden death and revival. In other words, the entanglement can spontaneously disappear and then reappear. This work includes discussion of the decoherence effect on quantum discord. Contrary to the entanglement, in our model the quantum discord does not present the phenomena of sudden death and rebirth. Apparently, the quantum discord would be more resistant to the environment than the entanglement. This paper is organized as follows, first, we explain the specific model used for two atoms in a common squeezed reservoir. Subsequently, we show explicitly the formulas for entanglement and QD. In general the quantum discord involves a complicated maximization. In our particular case, the density matrix evolves as a X-form matrix, which facilitates the calculation. To study the effect of decoherence we manipulate two parameters, the field-qubit coupling, and ɛ this last parameter relates initial state to decoherence-free space. Finally, we compare the behavior of both measures of quantumness varying one parameter at a time. We show analytical and numerical results.. Correlations.1. Entanglement of formation. Concurrence One of the most popular measurements of mixedness of the density operator is the von Neumann entropy S(ρ)= tr(ρ log ρ). For a pure state, this entropy vanishes, and for a maximally mixed state, gives log d, d being the dimension of the Hilbert space. The entropy is a convex function, which implies that it always increases by further mixing. This motivates the next definition. Given a state ψ, we define the entropy of entanglement E(ψ) as the Von-Neumann entropy of the reduced density operator. So using the above discussion d E(ψ) = S(ρ A ) = S(ρ B ) = λ k log (λ k ), () k=1 thus, once more we see that the more mixed the reduced density operator is, the more entangled the original state is. This definition is only valid for pure states. For mixed states, the quantification of entanglement becomes more complex. The Entanglement of Formation was originally proposed by Bennett et al. in 1996 [1], and it is a direct generalization of entropy of entanglement applied to mixed states. A mixed state can be realized by a large number of pure state ensembles, with different entanglement of formation. Thus, for a given ensemble of pure states {p i, ψ i }, is the average entropy of entanglement over a set of states that minimizes this average over all possible decompositions of ρ. E(ρ) = min i p i E(ψ i ) (3) where the entanglement E(ψ) is defined as the entropy of either of two subsystems A or B, i.e. E(ψ) = T r(ρ A log ρ A ) = T r(ρ B log ρ B ). (4) Here ρ A, ρ B are the reduced density matrices. But is very difficult to know which ensemble {p i, Ψ i } is the one that minimizes the entropy, so a concept closely related to the entanglement of formation is the concurrence [, 3]. For a general mixed state ρ AB of two qubits, we define ρ to be the spin-flipped state ρ AB = (σ y σ y )ρ AB(σ y σ y ), (5) where ρ is the complex conjugate of ρ, and σ y is the Pauli matrix. The concurrence is defined as C (ρ) = max{0, λ 1 λ λ 3 λ 4 }, (6) where {λ i } are the square roots, in decreasing order of the eigenvalues of the non-hermitian matrix ρ ρ. We use C to differentiate concurrence of classical correlations named C. For separable qubits C = 0 and for maximally entangled ones C = 1. E and C both range from 0 to 1, and E is monotonically increasing function of C, so that C itself is a kind of measurement of entanglement. Finally, the entanglement of formation is related to concurrence, via: E(ρ AB ) = E(C (ρ AB )), (7) Rev. Mex. Fís. S 57 (3) (011) 48 55

3 50 M. ANGELES GALLEGO AND M. ORSZAG with [ 1 E(C ) = H + 1 ] 1 C, H(x) = x log x (1 x) log (1 x). (8).. Quantum discord The total correlation of a quantum system is quantified by the quantum mutual information I(ρ). I(ρ) = S(ρ A ) + S(ρ B ) S(ρ) (9) where S = tr(ρ log ρ) is the von Neumann quantum equivalent for Shannon classical entropy. The total correlation can be separated into classical and quantum correlations. I(ρ) = C(ρ) + Q(ρ) (10) It is clear that in the case of a pure state S(ρ) = 0. In the case of a product state I(ρ) = 0, due to the system parts do not share information. In search of a formula for classical correlation, Vedral proposes a list of conditions that a classical correlation must satisfy [9]. The obtained expression that fulfills all the conditions is: C(ρ AB ) = max {B k } [S(ρA ) S(ρ {B k })] (11) The maximization is done over all possible measurements of B, i.e. we are looking for the measurement that disturbs the least the overall quantum state. For a classical state, the system is not perturbed by the measurement at all, in which case we obtain the maximum classical correlation value. The difference with classical theory of information is the conditional entropy S(ρ {B k }). In quantum physics the state of the system changes every time we measure. After a set of von Neumann measurements {B k }, made in subsystem B, the state of the total system is: ρ k = 1 p k (I B k )ρ(i B k ) (1) where p k = tr(i B k )ρ(i B k ) is the probability for obtaining the outcome k after the measurement. Thus the quantum conditional entropy is defined as: S(ρ {B k }) = k p k S(ρ k ) (13) where {ρ k, p k } is the ensemble of possible results for the outcome. With this definition for classical correlation, we get the Quantum Discord as: Q(ρ) = I(ρ) C(ρ) (14) By replacing I(ρ) and (Cρ) we obtain the exact formula: Q(ρ AB ) = S(ρ B ) S(ρ AB ) + min {B k } S(ρ {B k}) (15) For pure states, this formula coincides with entanglement of formation. We would like to point out the problem that arises with the classical correlation (11), when we measure the system A instead of system B, we obtain different correlation values. However this problem disappears for systems where S(ρ A ) = S(ρ B ). In a future work we would like to include this discussion, and use a definition for classical correlation that does not depend which system we are measuring. The exact formula for Entanglement of Formation and QD are developed in the Results section..3. The model We consider, two two-level atoms that interact with a common squeezed reservoir, and we will focus on the evolution of the entanglement and quantum discord, using as a basis, the Decoherence Free Subspace states, as defined in Ref. 18, 19, and 4. We write now, a general master equation for the density matrix in the interaction picture, assuming that the correlation time between the atoms and the reservoirs is much shorter than the characteristic time of the dynamical evolution of the atoms, so that the Markov approximation is valid, ˆρ t = Γ [(N + 1)(σ i ˆρσ j σ i σ j ˆρ ˆρσ i σ j) i,j=1 + N(σ i ˆρσ j σ i σ j ˆρ ˆρσ iσ j ) M(σ i ˆρσ j σ i σ j ˆρ ˆρσ i σ j ) M (σ i ˆρσ j σ i σ j ˆρ ˆρσ i σ j )], (16) where Γ is the decay constant of the qubits, and σ + i = 1 i 0 and σ i = 0 i 1 are the raising (+) and lowering ( ) operators of the ith atom. It should be pointed out that in Eq. (16), the i = j terms describe the atoms interacting with independent local reservoirs, while the i j terms denote the couplings between the modes induced by the common bath. It is simple to show that this master equation can also be written in the Lindblad form with a single Lindblad operator S with ρ t = 1 Γ(SρS S Sρ ρs S), (17) S = N + 1(σ 1 + σ ) Ne iψ (σ 1 + σ ) = cosh(r)(σ 1 + σ ) sinh(r)e iψ (σ 1 + σ ), (18) where the squeeze parameters are Ψ, and N = sinh r. Here we consider M= N(N + 1). The Decoherence Free Subspace consists of the eigenstates of S with zero eigenvalue. The states defined in this way, form a two-dimensional plane Rev. Mex. Fís. S 57 (3) (011) 48 55

4 QUANTUM CORRELATION MEASUREMENTS FOR TWO QUBITS IN A COMMON SQUEEZED BATH 51 in Hilbert Space and are not affected by decoherence when the system interacts with environment. Two orthogonal vectors in this plane are: φ 1 = 1 N + M (N Me iψ ), (19) φ = 1 ( + + ). (0) We can also define the states φ 3 and φ 4 orthogonal to the { φ 1, φ } plane: φ 3 = 1 ( ), (1) φ 4 = 1 N + M (M + + Ne iψ ). () We solve analytically the master equation by using the { φ 1, φ, φ 3, φ 4 } basis, however, we use the standard basis to calculate the concurrence and discord. For simplicity we will consider Γ = Analytical results In this section, we develop the exact formula of entanglement and QD, for a particular system composed for two two-level atoms, with a X form density matrix. It has been extensively studied how decoherence and entanglement are closely related phenomena, mainly because decoherence is responsible for the fragility of the entanglement in systems interacting with reservoirs [5]. We want to add here the discussion about the relation between decoherence and quantum discord. In order to study the relation between decoherence with entanglement and quantum discord, as in Ref. 0 we consider as initial states superpositions of the form: Ψ 1 = ɛ φ ɛ φ 4 (3) Ψ = ɛ φ + 1 ɛ φ 3 (4) where ɛ is a variable amplitude of one of the states belonging to the DFS, which allows us to vary the initial state. We evolve the density matrix using the master equation, obtaining for all times a X-form matrix. The density matrix written in the base 1 = 11, = 10, 3 = 01, 4 = 00, is: ρ ρ 14 0 ρ ρ ρ 3 ρ 33 0 ρ ρ 44 (5) For this kind of density matrix the concurrence can be easily found [7] : C (ρ) = max {0, C 1(ρ), C (ρ)} (6) where: C 1(ρ) = ( ρ 3 ρ 3 ρ 11 ρ 44 ) (7) C (ρ) = ( ρ 14 ρ 41 ρ ρ 33 ) (8) and again entanglement is given by Eq. (8). In order to evaluate the quantum discord, we follow the procedure of reference [14]. The maximization of the classical correlation is taken over the measurement for k = 0, 1 where, Π k = k k, B k = V Π k V V = ti + i γ σ and V is a general transformation with t, γ 1, γ, γ 3 R, and t + γ 1 + γ + γ 3 = 1. This means that we have three independent parameters, each belonging to [ 1, 1]. Now, for the calculation of QD given in Eq. (15), with the elements S(ρ AB ), S(ρ A ), S(ρ B ) as in Ref. 14, we must obtain the expression min {Bk } S(ρ {B k }) taking the minimization over the parameters t and γ. Taking k = 0, 1 we have: where: where S(ρ 0 ) = 1 θ S(ρ 1 ) = 1 θ θ = θ = S(ρ {B k }) = p 0 S(ρ 0 ) + p 1 S(ρ 1 ) (9) p 0 = (ρ 11 + ρ 33 )k + (ρ + ρ 44 )l (30) p 1 = (ρ 11 + ρ 33 )l + (ρ + ρ 44 )k (31) log 1 θ log 1 θ 1 + θ 1 + θ log 1 + θ log 1 + θ (3) (33) [(ρ 11 ρ 33 )k + (ρ ρ 44 )l] + Θ [(ρ 11 + ρ 33 )k + (ρ + ρ 44 )l] (34) [(ρ 11 ρ 33 )l + (ρ ρ 44 )k] + Θ [(ρ 11 + ρ 33 )l + (ρ + ρ 44 )k] (35) Θ = 4kl[ ρ 14 + ρ 3 + Re(ρ 14 ρ 3 )] 16mRe(ρ 14 ρ 3 ) + 16nIm(ρ 14 ρ 3 ). The parameters m,n, k and l are defined as: m = (tγ 1 + γ γ 3 ), n = (tγ γ 1 γ 3 )(tγ 1 + γ γ 3 ) k = t + γ 3, l = γ 1 + γ (36) Rev. Mex. Fís. S 57 (3) (011) 48 55

5 5 M. ANGELES GALLEGO AND M. ORSZAG With k + l = 1, and notice that k [0, 1], m [0, 1/4] and n [ 1/8, 1/8]. To find the minimum of Eq. (9), respect the parameters k, l, m, n we have to look for two kinds of critical points: those where the partial derivatives are equal to zero, and those where the derivatives are not well defined (this happens at the end points of each interval). Observing that the expression is symmetric under the interchange of k and l = 1 k, it is therefore, an even function of (k l), thus the partial derivative vanishes at k = l = 1/. On the other hand, we see that is not well defined when k = 0 or k = 1 (this two values are equivalent by the symmetry). Further, with Θ involving m and n in a linear way, extreme values are only attained to the end of each interval: m = 0, 1/4, n = ±1/8. Looking at the definition of k, and m we see that k = 0, 1 implies t = γ 3 = 0 or γ 1 = γ = 0 and therefore m = n = 0. The parameter n is not required in this case, since the density matrix will always be real. Using the above set of parameters, we have three possible minimums for quantum discord: Thus, Q 1 (ρ) = Q(k = 0, 1, m = 0) Q (ρ) = Q(k = 1/, m = 0) Q 3 (ρ) = Q(k = 1/, m = 1/4) Q(ρ) = min{q 1 (ρ), Q (ρ), Q 3 (ρ)} (37) It is noteworthy that for the initial state φ 1, we obtain the match ρ = ρ 33, therefore S(ρ A ) = S(ρ B ). This equality implies that classical correlations are independent of the subsystem we choose to measure. On the other hand, for an initial state φ, we obtain an inequality ρ ρ 33, therefore, we do not get the same result when measuring in A or B. Here, we chose to measure at subsystem B. Now we present some analytical results obtained, for our specific model, using the Eq. (6) for entanglement, and Eq. (37) for quantum discord. We took as initial conditions, states of the form (3) and (4). Next, we discuss quantum discord in connection with the ESD and ESB phenomena. 1. The first observation arises when starting from an initial state in the DFS plane. The local and non-local coherences are not affected by the environment, thus it experiences no decoherence and the entanglement stays constant in time. The density matrix of any bipartite pure state: ρ= φ φ can be written in the Schmidt decomposition, where the state is φ = j α j j j,thus the distribution of quantum information corresponds to: I(ρ) = S, C(ρ) = S, Q(ρ) = S (38) where S= j α j log α j is the reduced von Neumann entropy of each subsystem S(ρ A )=S(ρ B )=S, and S(ρ AB ) = 0. For the initial state φ 1, the concurrence does increase with the squeeze parameter N, getting a maximally entangled Bell state φ 1 1 ( ) for N. In the case of the quantum discord, we get that S increases with N, getting the maximum value of S = 1 for N This means that this reservoir is not acting as a thermal one, in the sense that introduces randomness. On the contrary, a common squeezed bath tends to enhance the quantum correlations, as we increase the parameter N. On the other hand, if we start with the initial state φ, this state is independent of N and it is also maximally entangled, so C = 1 and again S = 1 for all times and all N s.. Now, we consider other situations with initial states outside the DFS. We consider as initial states the superpositions given in (3) and (4), where we vary ε between 0 and 1 for a fixed value of the parameter N = 0.1. We study the asymptotic limit with initial Ψ 1 The stationary state, is of the form: Ψ 1 limt =α 11 +β 00, does not depend on ɛ, and is given by: N α = 1 + N, β = N + 4N + 1N 3 (1 + N)( 1 + 1N + 1N ) Since the above state is pure, the entropy is given by: S = β log β α log α (39) and the entanglement of formation is equal to quantum discord. When we have Ψ as initial state, our steady state is mixed, without decoherence. The entanglement is constant and has different values for every ɛ. Also, the classical and quantum correlations are constant. we observe that asymptotically the classical correlations (Fig. 6) are bigger than the quantum discord (Fig. 4). 3. Now we take ɛ=ɛ c. For Ψ 1 as initial state, when ε is equal to the critical value ε c = N(N+1)/(N+1), we get Ψ 1 = 11 >. The entanglement appears after a time where the concurrence is null and eventually goes to its steady-state value. For the initial state Rev. Mex. Fís. S 57 (3) (011) 48 55

6 QUANTUM CORRELATION MEASUREMENTS FOR TWO QUBITS IN A COMMON SQUEEZED BATH 53 Ψ, the critical value of ε is ε c = (1/ ), and, unlike the Ψ 1 case, it is independent of N, in this case Ψ = 01 > In both critical initial states, and taking ɛ = ɛ c we are in presence of a product state. In this case the distribution of information is: I(ρ) = 0, C(ρ) = 0, Q(ρ) = 0, C (ρ) = 0. (40) The result above is expected, since a product state has no correlations. 4. Numerical results and discussion Recently Yu and Eberly [6] investigated the dynamics of disentanglement of a bipartite qubit system due to spontaneous emission, where the two two level atoms (qubits) were coupled individually to two cavities (environments). They found that the quantum entanglement may vanish in a finite time, while local decoherence takes a infinite time. They called this phenomena Entanglement Sudden Death (ESD). FIGURE 3. Time evolution of quantum discord for Ψ 1 as initial condition and N = 0.1: ɛ = 0 (thick line), ɛ = 0.9 (dotted line), ɛ c = 0.5 (dashed line), ɛ = 0.9 (long-dashed line), ɛ 1 (dotted-dashed line). FIGURE 1. Time evolution of entanglement for Ψ 1 as initial condition and N = 0.1: ɛ = 0 (thick line),ɛ = 0.9 (dotted line), ɛ c = 0.5 (dashed line), ɛ = 0.9 (long-dashed line), ɛ = 1 (dotteddashed line) FIGURE. Time evolution of entanglement for Ψ as initial condition and N = 0.1: ɛ = 0.1 (thick line), ɛ = 0.4 (dotted line), ɛ c = 0.54 (dashed line), ɛ = 0.6 (long-dashed line), ɛ = (dashed-dotted line), ɛ = 0.9 (spaced-dotted line). FIGURE 4. Time evolution of quantum discord for Ψ as initial condition and N = 0.1: ɛ = 0.1 (thick line), ɛ = 0.4 (dotted line), ɛ c = 0.54 (dashed line), ɛ = 0.6 (long-dashed line), ɛ = (dashed-dotted line), ɛ = 0.9 (spaced-dotted line). ESD is not unique to systems of independent atoms. It can also occur for atoms coupled to a common reservoir, in which case we also observe the effect of the revival of the entanglement that has already been destroyed [7]. The effect of global noise on entanglement decay may depend on whether the initial two-party state belongs to a decoherence free subspace (DFS) or not. As opposed to the ESD and against our intuition, it has been shown that under certain conditions, the process of spontaneous emission can entangle qubits that were initially unentangled [8], and in some cases the creation of entanglement can occur some time after the system-reservoir interaction has been turned on. The authors in Ref. 9 call this phenomenon delayed sudden birth of entanglement. Comparing entanglement (Fig. 1 and Fig. ) with quantum discord (Fig. 3 and Fig. 4) we see that for 0 ε < ε c Entanglement presents sudden death and sudden birth while Quantum Discord does not. Looking at the formulas for concurrence (8), we see that whenever the coherences are lower Rev. Mex. Fís. S 57 (3) (011) 48 55

7 54 M. ANGELES GALLEGO AND M. ORSZAG FIGURE 5. Time evolution of classical correlations for Ψ 1 as initial condition and N = 0.1: ɛ = 0 (thick line), ɛ = 0.9 (dotted line), ɛ c = 0.5 (dashed line), ɛ = 0.9 (long-dashed line), ɛ = 1 (dotted-dashed line). than the diagonal elements, ρ 14 ρ ρ 33, ρ 3 ρ 11 ρ 44, the concurrence vanishes, and therefore the density matrix is separable. But, a zero concurrence does not imply that ρ 3 and ρ 14 ar null. On the other hand, QD is zero if all coherences disappear (for other cases of zero QD see [30]). In this sense QD is more accurate, because, in this example, is only zero when the matrix is diagonal (in the computational basis). When we get near to the DFS (ε c < ε 1), the system shows no disentanglement and these phenomenona of sudden death and revival disappear. As we mentioned before for ε = ε c at t = 0 we are in the presence of a product state and the subsystems do not share information at all, but the interaction with the common bath forces them to interact; for an initial Ψ the concurrence and quantum discord appears immediately after t = 0, however for Ψ 1 the quantum discord appears immediately but the entanglement takes some time before appearing. This happens because when N is small the predominant interaction between atoms and the reservoir is the doubly excited state via two photon spontaneous emission. A similar effect was studied in Ref. 1. We see that quantum discord does not show the phenomenon of sudden death and revival. This FIGURE 6. Time evolution of classical correlations for Ψ as initial condition and N = 0.1: ɛ = 0.1(thick line), ɛ = 0.4 (dotted line), ɛ c = 0.54 (dashed line), ɛ = 0.6 (long-dashed line), ɛ = 0.707(dashed-dotted line), ɛ = 0.9 (spaced-dotted line). means that even when the concurrence is null, there still are non vanishing quantum correlations. Indeed, quantum discord is never null except when we have a product state, in which case the classical correlations also vanish (Fig. 5 and Fig. 6). In this paper we provided another example that coincides with the theoretical expected results. Quantum discord is stronger than Entanglement when the system is affected by decoherence. Even when in some periods the entanglement is null, QD, is not. This research is still unfinished. The quantum discord is a measurement that has an annoying feature: it is asymmetric under exchange of subsystems A and B, which is not a desirable feature for a quantifier of quantum and classical correlations. Thus, in a future work we will explore other correlation quantifiers that are actually independent of the particular measured party. Acknowledgments One of us (M. Orszag) would like to thank Fondecyt for partial support (Project # ). 1. A. Einstein, B. Podolosky, and N. Rosen, Phys. Rev. 47 (1935) C.H. Bennett et al.,phys. Rev. Lett. 70 (1993) A.K. Ekert, Phys. Rev. Lett. 67 (1991) 661; D. Deutsch et al., Phys. Rev. Lett. 77 (1996) C.H. Bennett and S.J. Wiesner, Phys. Rev. Lett. 69 (199) C. Bennett et al., Phys. Rev. A 59 (1999) 1070; M. Horodecki et al., Phys. Rev. A 71 (005) 06307; J. Niset and N. Cerf, Phys. Rev. A 74 (006) 05103; S.L. Braunstein et al., Phys. Rev. Lett. 83 (1999) 1054; D.A. Meyer, Phys. Rev. A 85 (000) 014; A. Datta and G. Vidal, ibid. 75 (007) H. Ollivier and W. Zurek, Phys. Rev. Lett. 88 (001) M. Ikram, F. Li, and M. Zubairy, Phys. Rev. A 75 (007) V. Vedral, Introduction to Quantum Information Science (Oxford University Press Inc., New York 006); S.M. Barnett, Quantum Information (Oxford University Press Inc., New York, 009). 9. L. Henderson and V. Vedral, J. Phys. A 34 (001) V. Vedral, Phys. Rev. Lett. 90 (003) T. Werlang, S. Souza, F. Fanchini, and C. Villas Boas, Phys. Rev. A 80 (009) Rev. Mex. Fís. S 57 (3) (011) 48 55

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