This content has been downloaded from IOPscience. Please scroll down to see the full text. Download details: IP Address: 186.217.234.114 This content was downloaded on 05/02/2014 at 12:09 Please note that terms and conditions apply. Multipartite quantum correlations in open quantum systems View the table of contents for this issue, or go to the journal homepage for more 2013 New J. Phys. 15 043023 (http://iopscience.iop.org/1367-2630/15/4/043023) Home Search Collections Journals About Contact us My IOPscience iopscience.iop.org/page/terms http://iopscience.iop.org/1367-2630/15/4 http://iopscience.iop.org/1367-2630 http://iopscience.iop.org/ http://iopscience.iop.org/search http://iopscience.iop.org/collections http://iopscience.iop.org/journals http://iopscience.iop.org/page/aboutioppublishing http://iopscience.iop.org/contact http://iopscience.iop.org/myiopscience Multipartite quantum correlations in open quantum systems ZhiHao Ma1,2, ZhiHua Chen3 and Felipe Fernandes Fanchini 4,5 1 Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, People’s Republic of China 2 Department of Physics and Astronomy, University College London, Gower Street, WC1E 6BT London, UK 3 Department of Science, Zhijiang College, Zhejiang University of Technology, Hangzhou 310024, People’s Republic of China 4 Departamento de Fı́sica, Faculdade de Ciências, Universidade Estadual Paulista, Bauru SP, CEP 17033-360, Brazil E-mail: fanchini@fc.unesp.br New Journal of Physics 15 (2013) 043023 (11pp) Received 21 November 2012 Published 16 April 2013 Online at http://www.njp.org/ doi:10.1088/1367-2630/15/4/043023 Abstract. In this paper, we present a measure of quantum correlation for a multipartite system, defined as the sum of the correlations for all possible partitions. Our measure can be defined for quantum discord (QD), geometric quantum discord or even for entanglement of formation (EOF). For tripartite pure states, we show that the multipartite measures for the QD and the EOF are equivalent, which allows direct comparison of the distribution and the robustness of these correlations in open quantum systems. We study dissipative dynamics for two distinct families of entanglement: a W state and a GHZ state. We show that, for the W state, the QD is more robust than the entanglement, while for the GHZ state, this is not true. It turns out that the initial genuine multipartite entanglement present in the GHZ state makes the EOF more robust than the QD. 5 Author to whom any correspondence should be addressed. Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. New Journal of Physics 15 (2013) 043023 1367-2630/13/043023+11$33.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft mailto:fanchini@fc.unesp.br http://www.njp.org/ http://creativecommons.org/licenses/by/3.0 2 Contents 1. Introduction 2 2. Quantum correlations 3 2.1. Entanglement of formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2. Usual quantum discord . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3. Geometric measure of quantum discord . . . . . . . . . . . . . . . . . . . . . 4 3. Multipartite measure of quantum correlations (MMQC) 4 4. MMQC analytical solution for three-qubit pure states 5 5. MMQC for three-qubit mixed states 5 6. MMQC in open quantum systems 7 7. Conclusion 10 Acknowledgments 10 References 10 1. Introduction Entanglement in composite quantum systems leads to many puzzling paradoxes in quantum theory [1–4]. The importance of entanglement is universally recognized, but it is well known that even separable quantum states possess correlations that cannot be simulated by classical systems—for instance, the nonclassical correlations captured by quantum discord (QD) [5]. For bipartite states, many attempts have been made to detect and measure QD [6, 7] and to find the connection between it and entanglement [8–10]. Furthermore, attention is also being paid to the measurement of quantum correlation in multipartite systems [11, 12]. Such measures help us to understand the distribution of quantum correlations and provide a way of studying dissipative dynamics in many-part quantum systems. Although it is claimed that the QD is more robust than the entanglement in open quantum systems, little concrete evidence has been published to corroborate this claim as far as multipartite systems are concerned. Actually, for two qubits, there is a good deal of evidence that this is true [13], but can we extend this rule to multipartite quantum systems? For three qubits, for example, there exist two distinct families of entanglement [14]; how robust, then, is the QD for members of each family? Could entanglement be more robust than discord, against noise arising from the environment, in multipartite systems? Little has been written on these questions in the literature [15] and that is the focus of this paper. To investigate the robustness of the quantum correlations in open quantum systems for tripartite states, we define a multipartite measure of quantum correlation slightly different from that employed by Rulli and Sarandy [11]. Those authors define a measure of global multipartite QD as the maximum of the quantum correlations that exist among all possible bipartitions. Here, to attain an average measure, we define global QD as the sum of correlations for all possible bipartitions. Despite being only subtly different, our measure now accounts for how the quantum correlation is distributed in the tripartite system and certainly gives a better insight into the robustness of these correlations in open quantum systems. This paper is organized as follows. In section 2, we present the formal definition of the entanglement of formation (EOF), the usual QD and the geometric measure of quantum discord New Journal of Physics 15 (2013) 043023 (http://www.njp.org/) http://www.njp.org/ 3 (GQD). In section 3, we define our multipartite measure of quantum correlations (MMQC) based on the sum of correlations for all possible bipartitions. In section 4, we present an analytical solution to the MMQC for a three-qubit pure state and we show that for a general tripartite pure state the MMQC based on the usual QD and on the EOF are equivalent. In section 5, we extend our analysis to a tripartite mixed state, taking the dissipative dynamics of three qubits into account, and in section 6, we summarize our results. 2. Quantum correlations In this paper, we consider three well-known measures of quantum correlations: EOF, usual QD and GQD. Here, we present the formal definition of each of these measurements. 2.1. Entanglement of formation EOF is a measure of entanglement defined more than 15 years ago by Bennett et al [16]. Although very different from QD, EOF is connected with the latter by a monogamic relation [8, 17] and has a nice operational interpretation. It is defined as follows. Given a bipartite system A and B, consider all possible pure-state decompositions of the density matrix ρAB , that is, all ensembles of states |9i〉with probability pi such that ρAB = ∑ i pi |9i〉〈9i |. For each pure state, the entanglement is defined as the von Neumann entropy of either of the two subsystems A and B, such that E(9)= SA = SB , where SA := S(ρA)=−Tr(ρA log ρA), ρA being the partial trace over B, and there is an analogous expression for SB . The EOF for a mixed state is the average entanglement of the pure states minimized over all possible decompositions, i.e. E(ρAB)=min ∑ i pi E(9i). (1) Although this is very hard to calculate for a general bipartite system, for two qubits there is an analytical solution given in the seminal Wootters [18] paper. 2.2. Usual quantum discord QD is a well-known measure of quantum correlation defined by Ollivier and Zurek [5] about 10 years ago. It is defined as δ←AB = IAB − J←AB, (2) where IAB = SA + SB − SAB is the mutual information and J←AB is the classical correlation [19]. Explicitly, J←AB =max {5k} [ SA− ∑ k pk S(ρA|k) ] , (3) where ρA|k = TrB(5kρAB5k)/TrAB(5kρAB5k) is the local post-measurement state after obtaining the outcome k in B with probability pk . QD measures the amount of mutual information that is not accessible locally [20, 21] and generally is not symmetric, i.e. δ←AB 6= δ←B A. New Journal of Physics 15 (2013) 043023 (http://www.njp.org/) http://www.njp.org/ 4 2.3. Geometric measure of quantum discord Intuitively, QD can be viewed as a measure of the minimum loss of correlation due to measurements in the sense of quantum mutual information. A state with zero discord, i.e. δ←AB = 0, is a state whose information is not disturbed by local measurements; it is known as a classical-quantum (CQ) state. A CQ state is of the form [5] χ = m∑ k=1 pk|k〉〈k| ⊗ ρB k , (4) where {pk} is a probability distribution, |k〉 an arbitrary orthonormal basis in subsystem A and ρB k a set of arbitrary density matrices in subsystem B. Denoting the set of all CQ states on HA⊗ HB as �0, it is natural to think that the farther a state ρ is from �0, the higher is its QD. Indeed, we can use the distance from ρ to the nearest state in �0 as a measure of discord for state ρ, and this is the idea behind GQD [22]. Thus, GQD was introduced as D(ρ)= min χ∈�0 ||ρ−χ ||2, (5) where �0 denotes the set of zero-discord states and ||X − Y ||2 = tr(X − Y )2 stands for the squared Hilbert–Schmidt norm. Note that the maximum value reached by the GQD is 1/2 for two-qubit states, so it is appropriate to consider 2D as a measure of GQD hereafter, in order to compare it with other measures of correlation [23]. Interestingly, an explicit expression for GQD in a general two-qubit state can be written [22]. In the Bloch representation, any two-qubit state ρ can be represented as follows: ρ = 1 4 I⊗ I+ 3∑ i=1 xiσ i ⊗ I+ 3∑ i=1 yiI⊗ σ i + 3∑ i, j=1 ti jσ i ⊗ σ j  , (6) where I is the identity matrix, σ i (i = 1, 2, 3) are the three Pauli matrices, xi = tr(σ i ⊗ I)ρ and yi = tr(I⊗ σ i)ρ are the components of the local Bloch vectors Ex and Ey, respectively, and ti j = tr(σ i ⊗ σ j)ρ are components of the correlation matrix T . Then the GQD of ρ is given by D(ρ)= 1 4 ( ||Ex ||2 + ||T ||2− λmax ) , (7) λmax being the largest eigenvalue of the matrix K = Ex Ex t + T T t and ||T ||2 = tr(T T t). The superscript t denotes the transpose of vectors or matrices. Furthermore, it is important to mention that an analytical solution for a bipartite system of dimension 2× N has been given in [24]. 3. Multipartite measure of quantum correlations (MMQC) Definition 1. For an arbitrary N-partite state ρ̂1,...,N , the multipartite measure of quantum correlation Q(ρ̂1,...,N ) is defined as follows. Let ρ be an N-partite state, and µ and ν be any subsets among all possible partitions. The MMQC is defined as the sum of the quantum correlations for all possible bipartitions, Q ( ρ̂1,...,N ) = N∑ µ 6=ν=1 Mµ(ν), (8) New Journal of Physics 15 (2013) 043023 (http://www.njp.org/) http://www.njp.org/ 5 where Mµ(ν) is a measure of quantum correlation that can be given by the GQD, the usual QD or the EOF. Here, the subset between (·) is the measured one in the case of GQD or QD and can be ignored for the EOF. It can be seen that the measure defined in equation (8) is symmetrical and, more importantly, it is zero if the state has just classical–classical correlations. To elucidate the MMQC defined above, let us consider the case of a tripartite state. Explicitly, Q ( ρ̂ABC ) = MA(B) + MB(A) + MA(C) + MC(A) + MB(C) + MC(B) + MA(BC) + MBC(A) + MB(AC) + MAC(B) + MC(AB) + MAB(C). (9) 4. MMQC analytical solution for three-qubit pure states Our starting point is an analytical solution to the MMQC for three-qubit pure states. For the case of GQD, an analytical solution can be found with the help of the result presented in [24]. Actually, from that result, an analytical solution can be obtained for GQD even for three-qubit mixed states. The analytical solution for three qubits can also be obtained for the usual EOF. With the help of concurrence, each entanglement measurement involving two sets of one subsystem (e.g. EB(C)) can be obtained trivially by calculating the Wootters formula [18]. Furthermore, for tripartite pure states, we note that the entanglement measure involving a set of one subsystem and a set of two (e.g. E A(BC)) is given by the von Neumann entropy of one of the partitions. For example, E A(BC) = SA = SBC . Finally, to calculate the MMQC for the usual QD, we note the result given in [21], where the authors show that the sum of the QD for all possible bipartitions involving sets of one subsystem is equal to the sum of the EOF for all possible bipartitions, E A(B) + EB(A) + E A(C) + EC(A) + EB(C) + EC(B) = δ←A(B) + δ←B(A) + δ←A(C) + δ←C(A) + δ←B(C) + δ←C(B). (10) Moreover, since we are considering pure states, the QD measure involving a set of one and a set of two subsystems is given by the entropy of one of the partitions, exactly as the EOF. For example, E A(BC) = δ←A(BC) = δ←BC(A) = SA = SBC . Thus, for general tripartite pure states, the MMQC defined in equation (9) is identical for the QD and the EOF, resulting in an analytical solution for the MMQC for three-qubit pure states, for the QD as well. To confirm this, we note the result given in equation (10), which is valid irrespective of the system dimension of the subsystems. On the other hand, for general tripartite states (not three-qubit), an analytical solution does not exist, either for the EOF or for the QD. 5. MMQC for three-qubit mixed states To calculate the MMQC for three-qubit mixed states, we limit ourselves to studying a rank-2 density matrix. In this case, as we show below, a simple strategy can be used to calculate all terms of equation (9). Since we are considering three qubits, to calculate the MMQC for sets of one subsystem (E A(B), E A(C), δ←A(B), δ←A(C), etc) is trivial. Since in this case the terms are composed of two qubits, the EOF can be calculated analytically by means of concurrence [18], New Journal of Physics 15 (2013) 043023 (http://www.njp.org/) http://www.njp.org/ 6 and the QD can be calculated numerically by using positive-operator valued measurements (POVMs)6. So, the question is: how can we calculate, for a three-qubit mixed state, the MMQC for the terms that involve a set of one subsystem and a set of two, i.e. E A(BC), EB(AC), δ←A(BC), δ←B(AC), etc? In this case, in contrast to that of pure states, the von Neumann entropy cannot be used to calculate these terms. As we will show below, the answer to the above question is given by the monogamic relation between EOF and QD [8, 17]. In the dissipative dynamics that we will study below, at all times ρABC is a rank-2 density matrix. As a consequence, the extra subsystem that purifies ρABC is a two-level subsystem that we define here as E . Then, to calculate the MMQC, for the terms that involve a set of one subsystem and a set of two, the strategy is to calculate the quadripartite pure state ρABC E that involves four qubits and, in sequence, use the monogamic relation. The monogamic relation implies that the EOF between two partitions is connected with the QD between one of the partitions and the third one that purifies the pair, E A(BC) = δ←A(E) + SA|E . (11) To purify ABC , the first step is to write ρABC in its diagonal form, ρABC = λ1|81〉〈81|+ λ2|82〉〈82|, (12) where λi and |8i〉 are, respectively, the density matrix eigenvalues and eigenvectors, for i = 1, 2. The pure state is then written as |9ABC E〉 = √ λ1|81〉|0〉+ √ λ2|82〉|1〉, (13) where the states |0〉 and |1〉 pertain to the two-level system Hilbert space of E . It is easy to verify that TrE{|9ABC E〉〈9ABC E |} = ρABC . Given the purification procedure, below we present in detail the strategy used to calculate the QD and the EOF for the rank-2 tripartite mixed state ρABC . Here, we show how to calculate just the terms E A(BC), δ←A(BC) and δ←BC(A), but the strategy is analogous for all the other terms in equation (9) involving a set of one subsystem and a set of two. To calculate the EOF, we use the monogamic relation given by equation (11) where, as pointed out above, A(E) involves a two-qubit system since ABC is of rank 2. In this case, to calculate E A(BC), we compute δ←A(E) numerically and SA|E analytically. To calculate the QD, on the other hand, the result can be reached analytically by the expression δ←A(BC) = E A(E) + SA|E , (14) where E A(E) can be computed by means of Wootter’s concurrence. So, the remaining question is: how can we calculate δ←BC(A)? Once more, we use the monogamic relation. First we note that δ←BC(A) = EBC(E) + SBC |E , (15) which relates the QD with the entanglement between BC and E . To calculate EBC(E), we recall that the EOF is symmetric, i.e. EBC(E) = EE(BC), and use the monogamic relation EE(BC) = δ←E(A) + SE |A. (16) 6 It is important to observe that to calculate QD for the two-qubit state, in the maximization of equation (3), it is necessary to use general POVMs rather than projection measurements to evaluate the maximum of the classical correlation. However, as well noted in [25], in this case, the difference is tiny. New Journal of Physics 15 (2013) 043023 (http://www.njp.org/) http://www.njp.org/ 7 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 Q ua nt um C or re la tio ns time Figure 1. The red curve (solid) shows the EOF dynamics, while the blue curve (dotted) shows QD and the cyan curve (traced) the GQD: initial W state subjected to independent amplitude-damping channels. Thus, substituting the equation above in equation (15) and noting that SBC E = SA, we obtain δ←BC(A) = δ←E(A) + SA|E , (17) which connects δ←BC(A) with δ←E(A). Note that the latter term involves two qubits and can be calculated numerically. Thus, following the recipe above, it is straightforward to calculate the quantum correlations for all terms of equation (9). 6. MMQC in open quantum systems To study the MMQC in open quantum systems, we analyze two special situations: firstly, a three- qubit W state subjected to independent amplitude-damping channels and secondly, the GHZ state with independent phase damping. The reason for this specific choice is that, throughout the whole dissipative process, we have a rank-2 density matrix [26]. In this case, we can use the strategy explained in section 5 to calculate the MMQC for the usual QD and the EOF. For these specific channels, the dissipative dynamics can be calculated straightforwardly by means of Kraus operators [27]. Since we assume independent environments for each qubit, given an initial state for three qubits ρ(0), its evolution can be written as ρ(t)= ∑ α,β,γ Eα,β,γ ρ(0)E† α,β,γ , (18) where the so-called Kraus operators Eα,β,γ ≡ Eα⊗ Eβ ⊗ Eγ satisfy ∑ α,β,γ E† α,β,γ Eα,β,γ = I for all t . The operators E{α} describe the one-qubit quantum channel effects. We first consider a W state subjected to independent amplitude damping. This damping describes the exchange of energy between the system and the environment and is described by the Kraus operators E0 = √ p(σx + iσy)/2 and E1 = diag(1, √ 1− p), where p = 1− e−0t , 0 denoting the decay rate, and σx and σy are Pauli matrices. In figure 1, we show the dissipative dynamics of the MMQC for an initial state given by |W 〉 = (|100〉+ |010〉+ |001〉)/ √ 3. We see that in this New Journal of Physics 15 (2013) 043023 (http://www.njp.org/) http://www.njp.org/ 8 0 1 2 3 4 5 6 0 1 2 3 4 Q ua nt um C or re la tio ns time Figure 2. The red curve (solid) shows the EOF dynamics, whereas the blue curve (dotted) shows QD and the cyan curve (traced) represents the GQD: initial GHZ state subjected to independent phase-damping channels. situation the QD is actually more robust than EOF and GQD. For a very short time, the EOF resists, but it decays fast while QD maintains greater robustness. This result corroborates the idea that the QD is more robust than the EOF in open quantum systems. This peculiar situation occurs because of the conservative relation between the EOF and the QD [8], E A(BC) + E A(E) = δ←A(BC) + δ←A(E), (19) where ABC E is a pure state with the three-qubit system represented by ABC and the environment by E . For the W state subjected to the amplitude-damping channel, figure 1 shows that QD is sustained in the system since the QD MMQC becomes greater than the entanglement MMQC. It means that, to maintain the conservative relation, the subsystems become entangled with the environment but create less discord. In equation (19), for example, if we have E A(BC) < δ←A(BC), i.e. the QD is greater than the EOF in the system ABC , necessarily E A(E) > δ←A(E), i.e. the entanglement between parts of the system and the environment is greater. Indeed, this property is valid for any bipartition present in the definition of equation (9), since it is impossible to create or destroy some amount of EOF (QD) in the system without destroying or creating the same amount of QD (EOF) with the environment. In the second, and more important case, we consider an initial GHZ state subjected to phase damping. The dephasing channel induces a loss of quantum coherence without any energy exchange. In this case the Kraus operators are given by E0 = diag(1/ √ 2, √ 1− p) and E1 = diag(1/ √ 2, √ p). In figure 2 we show the dissipative dynamics of the MMQC for an initial state given by |GHZ〉 = (|000〉+ |111〉)/ √ 2. Here, we see a very interesting result. Contrary to what is claimed in the literature, the EOF is in fact more robust than the QD for this kind of initial condition and quantum channel. EOF is sustained for a longer time than QD. This occurs because the phase-damping channel does not create any entanglement or discord between two parts of the subsystem. In other words, for any given time, E A(B) = E A(C) = EB(C) = 0, δ←A(B) = δ←A(C) = δ←B(C) = 0 and δ←B(A) = δ←C(A) = δ←C(B) = 0. With this peculiar property, in this case New Journal of Physics 15 (2013) 043023 (http://www.njp.org/) http://www.njp.org/ 9 the MMQC for the EOF or the QD is given by Q ( ρ̂ABC ) = MA(BC) + MBC(A) + MB(AC) + MAC(B) + MC(AB) + MAB(C). (20) To calculate equation (20) for the QD, we use, as explained above, the monogamic relation between the EOF and the QD. In this case, each of the six terms is given by δ←A(BC) = δ←BC(A) = E A(E) + SA|E , δ←B(AC) = δ←AC(B) = EB(E) + SB|E , (21) δ←C(AB) = δ←AB(C) = EC(E) + SC |E , where the QD is symmetric because in this situation SA = SBC , SB = SAC and SC = SAB . Given the results in the equations above, what can we say about the entanglement of each part of the system (A, B or C) with the environment E , when the quantum state is initially in a GHZ state and is subjected to independent phase damping? In other words, are E A(E), EB(E) and EC(E) different from zero during the dissipative dynamics process? To answer this question, we calculate explicitly the density matrices ρAE , ρB E and ρC E . When an initial three- qubit GHZ state is subjected to independent dephasing environments the global state (system plus environment) is pure and is given by equation (13). In this situation, the eigenvalues are λ1(t)= 1− e−30t and λ2(t)= 1 + e−30t and the eigenvectors are |81〉 = (|111〉− |000〉)/ √ 2 and |82〉 = (|111〉+ |000〉)/ √ 2. Thus, |9ABC E(t)〉 = √ λ1(t) ( |111〉− |000〉 √ 2 ) |0〉+ √ λ2(t) ( |111〉+ |000〉 √ 2 ) |1〉. (22) Defining the density matrix ρABC E(t)= |9ABC E(t)〉〈9ABC E(t)| and tracing out the subsystems B and C , the density matrix ρAE(t)= TrBC{ρABC E(t)} is given by ρAE(t)= 1 2 |φ1(t)〉〈φ1(t)|+ 1 2 |φ2(t)〉〈φ2(t)| (23) with |φ1(t)〉 = |1〉( √ λ1(t)|0〉+ √ λ2(t)|1〉) and |φ2(t)〉 = −|0〉( √ λ1(t)|0〉− √ λ2(t)|1〉). Thus, since |φ1(t)〉 and |φ2(t)〉 are separable states for any time t , ρAE(t) is also separable. Furthermore, tracing out the subsystems A and C or A and B from ρABC E(t) we note that ρAE(t)= ρB E(t)= ρC E(t) which proves that E A(E) = EB(E) = EC(E) = 0. So, the proof above shows that none of the subsystems of the GHZ state becomes entangled with the environment, if each one is subjected to an independent phase-damping channel. In this case, the MMQC for the QD can be calculated analytically by means of the conditional entropy. Indeed, QQ D ( ρ̂ABC ) = 2(SA|E + SB|E + SC |E). (24) For the EOF, on the other hand, the situation is a little different. In this case, the six terms are given by E A(BC) = EBC(A) = δ←A(E) + SA|E , EB(AC) = E AC(B) = δ←B(E) + SB|E , (25) EC(AB) = E AB(C) = δ←C(E) + SC |E . The crucial difference between equations (21) and (25) is that, while each part of the system does not become entangled with the environment, it does create QD with it. In other words, New Journal of Physics 15 (2013) 043023 (http://www.njp.org/) http://www.njp.org/ 10 the phase-damping channel, acting independently over each qubit, creates QD between the subsystems and the environment. Furthermore, given the initial symmetry of the GHZ state, we find that δ←A(E) = δ←B(E) = δ←C(E) 6= 0 and, consequently, QE O F ( ρ̂ABC ) =QQ D ( ρ̂ABC ) + 2 ( δ←A(E) + δ←B(E) + δ←C(E) ) . This is an important result and a direct consequence of the conservative relation between the EOF and the QD where we are concerned with the quadripartite system ABC E (three qubits plus environment). For the GHZ state subjected to phase damping, there is no entanglement between each subsystem (A, B or C) and the environment (E) but there is QD. Thus, the QD that is created with the environment needs to be compensated by the entanglement retained in the system, making the EOF more robust than the QD in this particular situation. It must be emphasized that this is a direct consequence of the GHZ being a genuine multipartite entangled state, which means that the EOF between any set of two parts (E A(B), E A(E), EB(C), etc) is always zero during dissipative dynamics. 7. Conclusion In this paper, we have presented an alternative measure of multipartite quantum correlations. Our measure gives a novel and intuitive means of comparing the robustness of entanglement and discord in multipartite systems, against the detrimental interaction with the environment. We analyze two distinct initial conditions, which involve different kinds of multipartite entanglement. We show that the robustness of the EOF depends on the family of entanglement present in the initial state, raising the question of whether it is greater than the robustness of QD in open quantum systems. Actually, for a three-qubit W state, QD proves to be more robust, but the same cannot be said about the GHZ state. We show that this behavior is related to the way that the multipartite quantum state is quantum correlated with the environment. For the GHZ state subjected to independent phase-damping channels, the individual qubits do not become entangled with the environment, but to create QD with it. Thus entanglement is preserved for a longer time than the QD. We believe that the discussion presented here may contribute further to the understanding of the distribution of entanglement and discord in open quantum systems. Acknowledgments ZM was supported by the NSF of China (10901103) and by the Foundation of China Scholarship Council (2010831012). ZC was supported by the NSF of China (11201427). FFF was supported by Brazilian agencies FAPESP and CNPq, through the Brazilian National Institute for Science and Technology of Quantum Information (INCT-IQ). Note added. While finishing this paper we became aware of related work [28] where the author reached some complementary conclusions to those presented here. References [1] Peres A 1995 Quantum Theory: Concepts and Methods 1st edn (Berlin: Springer) [2] Horodecki R, Horodecki P, Horodecki M and Horodecki K 2009 Rev. Mod. Phys. 81 865 [3] Gühne O and Toth G 2009 Phys. Rep. 474 1 New Journal of Physics 15 (2013) 043023 (http://www.njp.org/) http://dx.doi.org/10.1103/RevModPhys.81.865 http://dx.doi.org/10.1016/j.physrep.2009.02.004 http://www.njp.org/ 11 [4] Fuchs C A 2011 Coming of Age with Quantum Information (Cambridge: Cambridge University Press) [5] Ollivier H and Zurek W H 2001 Phys. Rev. Lett. 88 017901 [6] Girolami D and Adesso G 2012 Phys. Rev. Lett. 108 150403 [7] Yu S, Zhang C, Chen Q and Oh C H 2011 arXiv:1102.4710 Zhang C, Yu S, Chen Q and Oh C H 2011 Phys. Rev. A 84 052112 [8] Fanchini F F, Cornelio M F, de Oliveira M C and Caldeira A O 2011 Phys. Rev. A 84 012313 [9] Piani M and Adesso G 2012 Phys. Rev. A 85 040301 [10] Bellomo B, Giorgi G L, Galve F, Lo Franco R, Compagno G and Zambrini R 2012 Phys. Rev. A 85 032104 [11] Rulli C C and Sarandy M S 2011 Phys. Rev. A 84 042109 [12] Okrasa M and Walczak Z 2011 Europhys. Lett. 96 60003 Ma Z-H and Chen Z-H 2011 arXiv:1108.4323 Saguia A, Rulli C C, de Oliveira T R and Sarandy M S 2011 Phys. Rev. A 84 042123 Giorgi G L, Bellomo B, Galve F and Zambrini R 2011 Phys. Rev. Lett. 107 190501 Xu J 2013 Phys. Lett. A 377 238–42 Bai Y-K, Zhang N, Ye M-Y and Wang Z D 2012 arXiv:1206.2096 Heydari H 2006 Quantum Inform. Comput. 6 166 [13] Werlang T, Souza S, Fanchini F F and Boas C J V 2009 Phys. Rev. A 80 024103 Maziero J, Celeri L C, Serra R M and Vedral V 2009 Phys. Rev. A 80 044102 Maziero J, Werlang T, Fanchini F F, Celeri L C and Serra R M 2010 Phys. Rev. A 81 022116 Hu M-L and Fan H 2012 Ann. Phys. 327 851 [14] Dur W, Vidal G and Cirac J I 2000 Phys. Rev. A 62 062314 [15] Ramzan M 2012 arXiv:1205.3133 [16] Bennett C H, DiVincenzo D P, Smolin J and Wootters W K 1996 Phys. Rev. A 54 3824 [17] Koashi M and Winter A 2004 Phys. Rev. A 69 022309 [18] Wootters W K 1998 Phys. Rev. Lett. 80 2245 [19] Henderson L and Vedral V 2001 J. Phys. A: Math. Gen. 34 6899 [20] Zurek W H 2003 Phys. Rev. A 67 012320 [21] Fanchini F F, Castelano L K, Cornelio M F and de Oliveira M C 2012 New J. Phys. 14 013027 [22] Dakic B, Vedral V and Brukner C 2010 Phys. Rev. Lett. 105 190502 [23] Girolami D and Adesso G 2011 Phys. Rev. A 83 052108 [24] Rana S and Parashar P 2012 Phys. Rev. A 85 024102 [25] Galve F, Giorgi G and Zambrini R 2011 Eur. Phys. Lett. 96 40005 [26] de Oliveira T R 2009 Phys. Rev. A 80 022331 [27] Nielsen M A and Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press) [28] Campbell S 2013 Quantum Inform. Process. doi:10.1007/s11128-013-0548-2 New Journal of Physics 15 (2013) 043023 (http://www.njp.org/) http://dx.doi.org/10.1103/PhysRevLett.88.017901 http://dx.doi.org/10.1103/PhysRevLett.108.150403 http://arxiv.org/abs/1102.4710 http://dx.doi.org/10.1103/PhysRevA.84.052112 http://dx.doi.org/10.1103/PhysRevA.84.012313 http://dx.doi.org/10.1103/PhysRevA.85.040301 http://dx.doi.org/10.1103/PhysRevA.85.032104 http://dx.doi.org/10.1103/PhysRevA.84.042109 http://dx.doi.org/10.1209/0295-5075/96/60003 http://arxiv.org/abs/1108.4323 http://dx.doi.org/10.1103/PhysRevA.84.042123 http://dx.doi.org/10.1103/PhysRevLett.107.190501 http://dx.doi.org/10.1016/j.physleta.2012.11.054 http://arxiv.org/abs/1206.2096 http://dx.doi.org/10.1103/PhysRevA.80.024103 http://dx.doi.org/10.1103/PhysRevA.80.044102 http://dx.doi.org/10.1103/PhysRevA.81.022116 http://dx.doi.org/10.1016/j.aop.2011.11.001 http://dx.doi.org/10.1103/PhysRevA.62.062314 http://arxiv.org/abs/1205.3133 http://dx.doi.org/10.1103/PhysRevA.54.3824 http://dx.doi.org/10.1103/PhysRevA.69.022309 http://dx.doi.org/10.1103/PhysRevLett.80.2245 http://dx.doi.org/10.1088/0305-4470/34/35/315 http://dx.doi.org/10.1103/PhysRevA.67.012320 http://dx.doi.org/10.1088/1367-2630/14/1/013027 http://dx.doi.org/10.1103/PhysRevLett.105.190502 http://dx.doi.org/10.1103/PhysRevA.83.052108 http://dx.doi.org/10.1103/PhysRevA.85.024102 http://dx.doi.org/10.1209/0295-5075/96/40005 http://dx.doi.org/10.1103/PhysRevA.80.022331 http://dx.doi.org/10.1007/s11128-013-0548-2 http://www.njp.org/ 1. Introduction 2. Quantum correlations 2.1. Entanglement of formation 2.2. Usual quantum discord 2.3. Geometric measure of quantum discord 3. Multipartite measure of quantum correlations (MMQC) 4. MMQC analytical solution for three-qubit pure states 5. MMQC for three-qubit mixed states 6. MMQC in open quantum systems 7. Conclusion Acknowledgments References