Pairwise Quantum Correlations for Superpositions of Dicke States

Transcription

1 Commun. Theor. Phys Vol. 57, No. 5, May 5, Pairwise Quantum Correlations for Superpositions of Dicke States XI Zheng-Jun Ê,,, XIONG Heng-Na, LI Yong-Ming Ó,, and WANG Xiao-Guang ½, College of Computer Science, Shaanxi Normal University, Xi an 76, China Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, Hangzhou 7, China Received October, ; revised manuscript received December, Abstract Pairwise correlation is really an important property for multi-qubit states. For the two-qubit X states extracted from Dicke states and their superposition states, we obtain a compact expression of the quantum discord by numerical check. We then apply the expression to discuss the quantum correlation of the reduced two-qubit states of Dicke states and their superpositions, and the results are compared with those obtained by entanglement of formation, which is a quantum entanglement measure. PACS numbers:.67.-a,.65.ud Key words: quantum correlation, quantum discord, entanglement, Dicke states Introduction It is well known that quantum entanglement is a typical quantum computation resource and plays an important role in quantum information processing. [ ] Much attention has been paid to detect and measure the quantum entanglement see [ 5] and references therein. Traditionally, it is believed that quantum entanglement is synonymous with quantum correlation. However, it is not always the case. Many works showed that there exists a more general quantum correlation [6 9] besides entanglement, as shown in Fig.. That is, some separable states can also possess quantum correlation, which can improve some computational tasks. [] Such the general quantum correlation is quantitatively characterized by quantum discord QD, [ 4] which is defined from the quantum measurement perspective. Using this quantity, it was proved that almost all quantum states actually have quantum correlations. [5] Most recently, operational interpretations of quantum discord were proposed, where quantum discord was shown to be a quantitative measure about the performance in quantum state merging. [6 7] Up to now, QD has been widely studied in many fields. [8 4] Calculating QD involves an optimization process over all possible quantum measurements. So far, a general analytical expression of QD still lacks even for the twoqubit states. Some analytical results were shown only for a special subset of two-qubit X-states. [,,,7 8,] In this article, we consider a kind of two-qubit X-states with exchange and parity symmetries, whose off-diagonal elements are complex, as shown in Eq.. We give an upper bound for the expression of QD. We find that the expression does not depend on the arguments of the complex off-diagonal elements. A typical example of the two-qubit X states shown in Eq. is the reduced two-qubit states of Dicke states and their superposition states. Two motivations inspire us to study the pairwise or two-qubit quantum correlation properties of such kinds of states. One comes from the experimental perspective. Dicke states are fundamental multiqubit states. They can be realized both in atomic systems [4 44] and in photonic systems [45] and references therein. Many multi-qubit states are based on them, such as W states, [46 47] GHZ states [48] and spin coherent states SCSs, [49 5] etc. The importance of the Dicke states in experiment has attracted much theoretical studies. [5 5] Here we would like to examine their pairwise quantum correlation properties in terms of QD. The other motivation comes from the theoretical aspect. The exchange symmetry of a Dicke state ensures that its two-qubit reduced state can be extracted randomly from the global state. That is, the reduced state of the i-th and j-th qubits ρ ij = Tr ij ρ global is invariant for arbitrary positions of i and j. Once the two-qubit reduced state ρ ij is quantum correlated, all of its possible twoqubit states are quantum correlated. Therefore, to some extent, the existence of the two-qubit quantum correla- Supported by the Natural Science Foundation of China under Grant Nos. 557, 95, and 6879, National Foundation Research Program of China under Grant No. CB96; the Fundamental Research Funds for the Central Universities, and as well as the Superior Dissertation Foundation of Shannxi Normal University under Grant No. S9YB, and the Higher School Doctoral Subject Foundation of Ministry of Education of China under Grant No snnuxzj@gmail.com liyongm@snnu.edu.cn xgwang8@zju.edu.cn c Chinese Physical Society and IOP Publishing Ltd

2 77 Communications in Theoretical Physics Vol. 57 tion sufficiently reflects the multi-qubit quantum correlation. Substantial efforts have been made to quantify multi-qubit quantum correlation in terms of its pairwise correlations. [,9 9,5 56] All theses works show that pairwise correlation is really an important property for multi-qubit states. This paper is organized as follows. In Sec., we give the basic concepts of QD and EoF. In Sec., for the two-qubit X states with exchange and parity symmetries, whose off-diagonal elements are complex, an upper bound of QD is analytically derived. In Sec. 4, we make a comparison between the pairwise QD and pairwise EoF for Dicke states and their superposition states. Quantum Correlations. Quantum Discord In classical information, the mutual information measures the correlation between two random variables A and B I c A : B = HA + HB HAB, where HX = x p x log p x are the Shannon entropies for the variable XX = A, B and HA, B = a,b p a,b log p a,b is the joint system AB, p a,b is the joint probability of the variables A and B assuming the values a and b, respectively, and p a = b p a,bp b = a p a,b is the marginal probability of the variable AB assuming the value ab. [57] The classical mutual information can also be expressed in terms of the conditional entropy as J c A : B = HA HA B, where HA B = a,b p a,b log p a b is the conditional entropy of the variable A given that variable B is known, and p a b = p a,b /p b is the conditional probability. In quantum information, to obtain a quantum version of Eq., Ollivier and Zurek employed a complete set of perfect orthogonal projective measurements {Π B k } with k ΠB k = I B on the subsystem B. The reduced state of subsystem B after the measurement is given as ρ A Π B k = p k Tr B [I A Π B k ρab I A Π B k ], where p k = Tr AB I A Π B k ρab is probability for the measurement of the k-th state in the subsystem B. Then, for a known subsystem B, one can define conditional entropy of subsystem A, S {Π B k } ρ A B : = k p ksρ A Π B. Then, k for a bipartite quantum state ρ AB, the quantum analogues for Eqs. and are given as Iρ AB = Sρ A + Sρ B Sρ AB, J ρ AB = Sρ A S {Π B k } ρ A B. 4 where ρ AB =Tr BA ρ AB is the reduced density matrix for A B, and Sρ X = Trρ X log ρ X X = A, B is the von Neumann entropy. [] In general, Eqs. and 4 are not equivalent, [ ] their difference is defined as quantum discord QD, which is a more general characterization of quantum correlation, namely, Dρ AB : = Iρ AB max {Π B k } J ρ AB. 5 Quantum mutual information is usually used to quantify the total amount of the correlation between the two subsystems A and B. [6,] In particular, if one considers positive operator valued measure POVM on subsystem B and maximizes Eq. 4, it may be viewed as classical correlation, which is defined by Henderson and Vedral. [] Hamieh et al. have shown that the projective measurement is POVM that maximizes 4 for two-qubit system. In our article, we will only compute quantum discord for two-qubits states in terms the original definition. [ ] An important property of QD is that, even if ρ AB is a separable state, its QD may be nonzero. That is, QD captures more general quantum correlation than entanglement, as shown in Fig.. It has been experimentally proved that quantum separable states with nonzero QD have played an important role in quantum computation. [9 ] Quantum correlated Quantum entangled Fig. Color online Schematic configuration of the relations between quantum correlation and quantum entanglement. The pink circle represents the whole set of quantum states. They can be split into quantum correlated ones the blue circle and classical correlated ones. They can also be divided into quantum entangled ones the red circle and quantum separable ones. In general, the states with quantum entanglement must be quantum correlated. However, a quantum correlated state may not be quantum entangled. That is, there exists a class of quantum separable states which are also quantum correlated, as shown by the shadow area.. Entanglement of Formation For an arbitrary two-qubit state ρ AB, concurrence is one of the most widely used measurements of entanglement, which is defined as, [5] Cρ AB = max{, λ λ λ λ 4 }, 6 where λ, λ, λ, λ 4 are the decreasing ordered eigenvalues of the matrix R = ρ AB σ y σ y ρ AB σ y σ y with σ y the pauli matrix, and ρ AB is the complex conjugation of ρ AB. To compare quantum entanglement with QD, in the following, we would like to use the entropy function of concurrence EoF as the entanglement measure. The EoF is defined as [5] E F ρ AB = min {p i, φ i } [ i ] p i STr A φ i φ i, 7

3 No. 5 Communications in Theoretical Physics 77 where the minimum is taken over all possible decompositions {p i, φ i } with ρ AB = i p i φ i φ i. For a general two-qubit state, the EoF can be expressed by concurrence C [5] + C E F ρ AB = H, 8 where the binary entropy is Hx = hx + h x, 9 with the function hx = xlog x x. There is a closed relation between QD and EoF for a bipartite pure state. Since the conditional density operator ρ A Π B is also a pure state, then the quantum conditional k entropy S {Π B k } ρ A B =. Therefore, QD is reduced to EoF, which is equal to the von Neumann entropy of the subsystem A, i.e., Dρ AB = E F ρ AB = Sρ A. However, for a general mixed state, the two concepts show totally different behaviors due to the difference between their definitions. The QD describes the quantum correlation in terms of quantum measurements. It is based on the idea that a classical correlated state remains unchanged under a quantum measurement on one of the subsystems. However, the quantum entanglement such as EoF is defined mathematically opposite to quantum separability. shown in Eq.. The density matrix of a two-qubit state with exchange and parity symmetries takes the form [56] v + u ρ AB y y = y y, u v in the basis {,,, }, where the elements v +, v, and y are real numbers, u is a complex number, and u is the complex conjugate of u. First, the joint entropy is easily given by Sρ AB = hλ + hλ + + hλ, with λ is the eigenvalues of ρab λ = y, λ ± = v + + v ± v + v + 4 u. Meanwhile, after tracing over the degree of the qubit A, we obtain the reduced density matrix for the qubit B ρ B = Tr A ρ AB v+ + y =, 4 v + y with the von Neumann entropy Sρ B = Hv + + y. 5 The quantum conditional entropy S {Π B k } ρ A B involves all possible one qubit projective measurements {Π B k }. Here we choose the measurements Quantum Discord for Two-Qubit X States with Exchange and Parity Symmetries Here, we would like to consider the case that the offdiagonal elements of the density matrix are complex as Π B ± = I ± n σ, 6 where n = sin θ cosφ, sin θ sin φ, cosθ with θ [, π] and φ [, π. Then the conditional density operators ρ A ± ρ A Π B ± are ρ A ± θ, φ = v+ + y ± v + ycosθ ±y e iφ + u e iφ sin θ p ± θ ±y e iφ + u e iφ, 7 sin θ v + y v ycosθ with the probabilities p ± θ = [ ± v + v cosθ]. 8 Obviously, ρ A θ, φ = ρa + π θ, φ, and p θ = p + π θ. Therefore, the conditional entropy is given by S A B θ, φ S {Π B k } ρ A B θ, φ + κθ, φ = p + θh + κπ θ, φ + p θh, 9 where κθ, φ is one of the eigenvalues of conditional state ρ A +θ, φ, and κ θ, φ = [ p +θ 4 v + v + 4ycosθ ] + u + y + yreue iφ sin θ. The following work is to minimize the conditional entropy S A B θ, φ over the parameters of θ and φ. First, we note that the probabilities p ± θ are independent of φ, taking the derivative S A B θ, φ over φ. From the equation S A B θ, φ/ φ =, we get the value of φ making S A B θ, φ minimum at φ m = argu or argu + π, where argu [, π is the argument of the complex number u. Thus, + κθ min S A B θ, φ = p + θh φ + κπ θ +p θh, with κ θ = [ p + θ 4 v + v + 4ycosθ ] + u + y + y u sin θ, which is independent of the argument of u. So far, the minimization of S A B θ, φ over φ is done completely. However, its optimization over θ is so difficult that we only give an upper bound min θ,φ S A Bθ, φ min{s, S },

4 774 Communications in Theoretical Physics Vol. 57 where S = S A B θ =, φ m, S = S A B θ = π, φ m. 4 They are obtained at θ =, π/ in terms of the fact that S A B θ, φ m = S A B π θ, φ m. That is, S A B θ, φ m is symmetric around θ = π/. Thus S and S are two not the only two extremes of the conditional entropy S A B θ, φ over the parameters θ and φ. From Eqs. and, the explicit expressions for S and S are easily derived as + κ+ S = v + + yh + κ S = H with + v + yh + κ,, 5 κ ± = v ± y v ± + y, κ = v + v + 4y + u. 6 Many works have shown that the upper bound is tight, i.e., min θ,φ S A B θ, φ = min{s, S }. [,7,] However, for some two-qubit X-state, this upper bound is not necessary reachable, a counterexample is given in [58] see the equation 8 therein. Fortunately, for the two-qubit density matricies, which have been extracted from Dicke states and their superpositions, the matrix elements are shown in Eq. 9, our numerical results show that min θ,φ S A B θ, φ = S. That is, the minimum of S A B θ, φ over θ is just obtained at the symmetric point θ = π/ see exemplifications in Figs. and 5. Therefore, we get the compact expression of the for the two-qubit reduced density matrix as Dρ AB = Sρ B Sρ AB + S, 7 where Sρ AB, Sρ B, and S are given in Eqs., 5, and 5 respectively. In the following, we will apply Eq. 7 to study the pairwise quantum correlation of Dicke states and their superpositions. Meanwhile, the for the reduced density matrix is given by Eq. 8 with the concurrence expressed in [59], { u y, u y; C = y v + v, y 8 v + v. 4 Symmetric Multi-Qubit States The state can be obtained from the two-qubit reduced density matrix of Dicke states or their superposition states. In this case, the elements of the density matrix can be expressed in terms of the expectation values of the collective spin operators v ± = N N + 4 Jz ± 4 J z N, 4NN y = N 4 Jz 4NN, u = J+ NN. 9 where the collective spin operators are defined as J γ = N i= σ iγ/ γ = x, y, z with N the total spin number and σ iγ the pauli operator on i-th site of spin. In the following, in terms of the shown in Eq. 7, we would like to study the pairwise correlation of the X states with the special elements shown in Eq. 9. In addition, the results will be compared with those obtained from, which is obtained by inserting Eq. 8 into Eq Dicke State An N-qubit Dicke state is defined as n N = N, N N + n, n =,...,N, and N = N/, N/ indicates that all spins are pointing down. N is the total spin number, and n is the excitation number of spins. [6] The expressions for the relevant spin expectation values can be easily obtained as, [56] J z = n N, J z = n N, J + =. From Eq. 9, it is easy to see that the matrix elements of the reduced density matrix ρ AB are given by nn v + = NN, v N nn n =, NN nn n y =, u =. NN In Fig., we give a numerical check that, for any N and n, the minimum of the conditional entropy min φ S A B θ, φ [as shown in Eq. ] are always obtained at θ = π/. The expression of for Dicke states can be obtained by inserting Eq. into Eq. 7. The corresponding expression of is obtained by inserting into Eq. 8 and Eq. 8. The two expressions are a little lengthy, so we do not explicitly write them down. First, we fixed the spin number N to see the behavior of and for different excitation number n. For n = or N, the Dicke state becomes a product state, which has zero and zero, i.e., Dρ AB n N = E F ρ AB n N =. In the following, we are interested in the cases that n [, N ]. As shown in Fig., when N is even, the reaches to its maximum at n = N/, and the Dicke state reads n N = N/ N, which has equal numbers of spins pointing up and down. When N is odd, the arrives its maximum at the point n = N ± /, where the Dicke state n N = N/ ± / N has the minimum different numbers of spins pointing up and down. However, no matter N is even or odd, the maximum value of is obtained at the points n = or N, where the Dicke states have the maximum different numbers of spins pointing up and down, which are identical with the W state as shown in Eq. 4. The different behaviors of and come from the different definitions of them, as illustrated in Sec.. Therefore, a state with

5 No. 5 Communications in Theoretical Physics 775 maximum quantum correlation may not have maximum quantum entanglement. Conditional entropy..5 5 n Fig. Color online Conditional entropy min φ S A B θ, φ [as shown in Eq. ] of Dicke state n N as a function of n and θ for fixed N =. The minimum of min φ S A B θ, φ for fixed n and N is obtained at θ = π/. θ reduces more slowly than. That is, for an arbitrary Dicke state, the general pairwise quantum correlation characterized by the is more robust against the increasing of the total particle number of the Dicke ststes. This result is consistent with those obtained for the reduced two-qubit states under a decoherence environment. [8 ],.6.4., EoF.6.4.,.4..., N 5 5 N Fig. 4 Color online and of Dicke state n N for different particle numbers N with fixed excitation n = left and right. 6 9 n 4 8 n Fig. Color online and of Dicke state n N for different excitation number n [, N ] with the fixed particle number N = 9 left and right. Second, we discuss the behaviors of and for different spin number N with fixed excitation number n. For n =, the Dicke state becomes a generic W state N = + + N +, 4 and the reduces to Dρ AB N = H H N N N + N H. 5 N On the other hand, the corresponding is given by E F ρ AB N = max E Fρ AB n n [,N ] N N + N 4 = H. 6 N It is easy to check that Dρ AB N E F ρ AB N. 7 For n, the similar relation between the and the are numerically shown in Fig. 4. Both of them decrease as the particle number N increases, but the 4. Superposition of Dicke States Then we consider a simple superposition of Dicke states as ψ D = cosα n N + e iδ sin α n + N, 8 where n =,...,N, the angle α [, π and the relative phase δ [, π. The expressions of the relevant spin expectations are J z = n N J z = n N cos α + n + N sin α, cos α + n + N sin α, J + = eiδ sin α µ n, 9 with µ n = n + n + N nn n. For the state ψ D, the minimization of the conditional entropy S A B θ, φ over the measurement phase φ is obtained at φ = /δ or /δ + π according to Eq.. Thus, the final expression of the is independent of the superposition phase δ from Eq.. In Fig. 4, we also numerically check that the conditional entropy min φ S A B θ, φ [as shown in Eq. ] of the state ψ D arrives its minimum at θ = π/ for fixed N, n, and α. Therefore, the analytical expression 7 for the superpo-

6 776 Communications in Theoretical Physics Vol. 57 sitions of Dicke states is still valid. N = and n =, the elements of the reduced density matrix are v + = y = sin α e, v = cos iδ sin α α, u = Conditional entropy α θ And for N = and n =, they are v = y = sin α, v + = cos α, u = Both of them satisfy the relation that e iδ sin α v + = y, v + v = u. 48 This ensures the joint entropy Sρ AB ψ D equals the reduced entropy Sρ A ψ D, i.e., Fig. 5 Color online Conditional entropy min φ S A B θ, φ [as shown in Eq. ] of the superposition of Dicke states ψ D as a function of the angles α and θ for fixed N = 5 and n =. The minimum of min φ S A B θ, φ is obtained at θ = π/ for fixed n, N and α. There is a symmetry property of the or for ψ D. From Eq. 9, we observe that if we let n N n and α π/ + α, then J z J z and J+ J +, and furthermore v + v, v v +, and u. Finally, two meaningful relations are obtained as DN, n, α, δ = D N, N n +, π + α, φ, 4 E F N, n, α, δ = E F N, N n +, π. + α, φ 4 That is, both the and are symmetrical about n = N/ and α = π/4. These are useful properties for the following analysis. Next we would like to discuss the relation between the and for the state ψ D. First, when N = and n =, the superposition of Dicke states becomes a two-qubit GHZ-like state, ψ D = cosα + e iδ sin α. 4 This is a bipartite pure state so that the equals, i.e., D = E F = Hcos α. 4 This is also the result for the case that N = and n = according to the symmetry properties 4 and 4. In fact, for a general multi-qubit GHZ-like state with N GHZ = cosα N N + e iδ sin α N, 44 there is no quantum correlation for the two-qubit reduced state, i.e., D = E F =. 45 In summary, for a multi-qubit GHZ-like state, the is always equal to the. Their values are always zero except for N =. Second, when N =, it is interesting that, for any n, the state ψ D has equal and. In fact, for Therefore, the is with Sρ AB ψ D = Sρ A ψ D. 49 D = min θ,φ S A Bθ, φ = S = H + κ, 5 κ = v + v + 4y + u. 5 Meanwhile, from Eq. 48, the corresponding concurrence is given by Obviously, Thus, is equal to, i.e., C = 4y u. 5 κ = C. 5 D = E F. 54 Combining the symmetry properties 4 and 4, this equivalence can be extended to any n for N =. This indicates that for certain kinds of mixed state, quantum entanglement may also represent the whole quantum correlation. Both of the above two cases with N = and N = show that s are equal to s for pure and some special mixed two-qubit states. However, when N 4, the equivalence does not always hold, as shown in Fig. 6. It seems that the is always greater than or equal to. This is similar to the case of Dicke states. In addition, the results are a little more complex than that of Dicke state. Every possible cases exists. The state with maximum minimum may have or not have the maximum minimum as shown in Figs. 6a and 6b for small particle number N. While for large N, as shown in Figs. 6c and 6d, the state with maximum minimum approaches to the one with the maxi-

7 No. 5 Communications in Theoretical Physics 777 mum minimum.,,.5 an/4 n/ a,,.5 b N/6 n/ a c N/ n/ d N/5 n/ a a Fig. 6 Color online and for superposition of Dicke state ψ D as the function of α for different N 4 and n. 4. Spin Coherent States Finally, we consider two more complex superpositions of Dicke states, i.e., the superpositions of the form n C n n N with the excitation numbers n even and odd respectively. They can be given by SCSs [49,6 6] η ± = η ± η, 55 ± γn with γ = η / + η. The ± corresponds to the so called even SCSs ESCSs and odd SCSs OSCSs, respectively. A SCS is obtained by a rotation of the Dicke state N, η = + η N/ expj + η N N / N = + η N/ η n n N, 56 n n= where the parameter η [, ]. The expectations for the ESCSs and OSCSs are J z ± = N γ ± γ N ± γ N, J z ± = N 4 ± NN η υ η ± γ N, J + ± = ± NN η υ ± η ± γ N, 57 with υ ± η = γ N η ± + η. Similar to the above two cases shown in Subsecs. 4. and 4., we also numerically check that the analytical expression 7 of can be reliably accepted. Here the expression of is so lengthy that we only numerically show its behaviors for different parameters, as displayed in Fig. 7. We see that, when η is small, the QD of OSCSs is always greater than that of ESCSs for fixed particle number N. However, when η becomes large and approaches, they become gradually equal to each other. Especially, when N, they coincide with each other even for small η. All these results can be explained analytically from the three special cases in the following. When η, the of the OSCS is different from that of the ESCS. The OSCS reduces to the Dicke state η = = N the W state shown in Eq. 4, whose is nonzero as shown in Eq. 5. Meanwhile, the ESCS reduces to product state η = + = N, then D =, as shown in Fig η.6.4. a N=.6 b N=5 Even.4 Even.5. η..5. η.6 c N= d N=5 Even.4 Even..5. η Fig. 7 Color online for odd and even SCSs along with the parameter η [, ] for different spin numbers N. However, when η, the s of OSCSs and ESCSs coincide with each other. In this case, both of the OSCSs and ESCSs reduce to GHZ states in the x direction, [6] η = ± = N N x ± N x. 58 If N, the reduced density matrix of the OSCSs and ESCSs are diagonalized ρ AB η= = +, 59 and we have D =. Finally, when N, for both the OSCSs and ESCSs, the reduced density matrices are almost independent of N, ρ AB N = + η and can be given by η η η η η η η 4, 6 Dρ AB N η = H + η H + η + η + + 4η + H 4 + η 8 + η. 6 Then will also become independent of N. As a result, s of OSCSs and ESCSs become close to each other even for small η, as shown in Fig. 7d. In particular, we obtain E F ρ AB N =. This implies that QD is a more general measure of quantum correlation than

8 778 Communications in Theoretical Physics Vol. 57 quantum entanglement. For a set of quantum separable states, they do have quantum correlations. In Fig. 8, we display the relation of maximum s between the OSCSs and ESCSs over the parameter scale η [, ]. It shows that the maximum of OSCSs is always greater than or equal to that of ESCSs for fixed particle number N. When N, their maximum values attain a constant, which can also be seen from Fig Even N Fig. 8 Color online Maximum of the ESCSs and OSCSs over the parameter scale η [, ] for different N. 5 Conclusion In terms of, we have investigated the pairwise quantum correlations in Dicke states and their superpositions in terms of its two-qubit density matrix, whose elements may be complex. A general expression for is derived according to the numerical proof. For the Dicke states, our analytical and numerical results show that the is always greater than or equal to the. This further proves that QD is a more general measure of quantum correlation than quantum entanglement. For the superpositions of Dicke states, it is interesting that the QD is always equal to EoF when N = GHZ-like states and N = W states, which indicates that for some kinds of mixed states, quantum correlation can also be fully described by quantum entanglement. For OSCSs and ESCSs, the former always shows more quantum correlations than the latter. In addition, the is generally more robust against the enlarging of the particle number of the multi-qubit states, which implies that has an advantage over pairwise entanglement in characterizing the quantum correlation in the multi-qubit states. To some extent, the pairwise quantum correlations of the reduced states may reflect the multi-qubit quantum correlations of the global states. Acknowledgments We are grateful to Xiao-Ming Lu, Xiaoqian Wang, and Jian Ma for fruitful discussions. References [] M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum information, Cambridge University Press, Cambridge. [] V. Vedral, Rev. Mod. Phys [] M.B. Plenio and S. Virmani, Quant. Inf. Comp [4] R. Horodecki, et al., Rev. Mod. Phys [5] W.K. Wootters, Phys. Rev. Lett ; W.K. Wootters, Quant. Inf. Comp. 7. [6] B. Groisman, et al., Phys. Rev. A [7] M. Horodecki, et al., Phys. Rev. A [8] C.H. Bennett, et al., Phys. Rev. A [9] B.P. Lanyon, et al., Phys. Rev. Lett [] A. Datta, et al., Phys. Rev. A ; A. Datta and G. Vidal, Phys. Rev. A ; A. Datta, et al., Phys. Rev. Lett [] H. Ollivier and W.H. Zurek, Phys. Rev. Lett [] W.H. Zurek, Ann. Phys ; W.H. Zurek, Phys. Rev. A 67. [] L. Henderson and V. Vedral, J. Phys. A [4] V. Vedral, Phys. Rev. Lett [5] A. Ferraro, et al., Phys. Rev. A [6] D. Cavalcanti, et al., Phys. Rev. A 8 4. [7] V. Madhok and A. Datta, Phys. Rev. A 8. [8] T. Werlang, et al., Phys. Rev. A [9] J. Maziero, et al., Phys. Rev. A 8 6. [] F.F. Fanchini, et al., Phys. Rev. A [] M. Piani, et al., Phys. Rev. Lett. 55 [] R. Dillenschneider, Phys. Rev. B [] M.S. Sarandy, Phys. Rev. A [4] Y. Chen and S. Li, Phys. Rev. A 8. [5] T. Werlang and G. Rigolin, Phys. Rev. A [6] A. Shabani and D.A. Lidar, Phys. Rev. Lett [7] S. Luo, Phys. Rev. A ; S. Luo, Phys. Rev. A [8] J. Maziero, et al., Phys. Rev. A 8 6. [9] A. Datta and S. Gharibian, Phys. Rev. A ; A. Datta, Phys. Rev. A [] S. Wu, et al., Phys. Rev. A [] Csar A Rodríguez-Rosario, et al., J. Phys. A: Math. Theor [] M. Ali, et al., Phys. Rev. A [] A. Brodutch and D.R. Terno, Phys. Rev. A 8 6. [4] A. SaiToh, et al., Phys. Rev. A

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