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Renormalization of quantum discord and Bell nonlocality in the XXZ model with Dzyaloshinskii–Moriya interaction
Annals of Physics 349 (2014) 220–231
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Renormalization of quantum discord and Bell nonlocality in the XXZ model with Dzyaloshinskii–Moriya interaction Xue-ke Song a , Tao Wu b , Shuai Xu a , Juan He a , Liu Ye a,∗ a
School of Physics & Material Science, Anhui University, Hefei, 230039, China
b
School of Physics & Electronics Science, Fuyang Normal College, Fuyang, 236037, China
article
info
Article history: Received 23 January 2013 Accepted 10 June 2014 Available online 14 June 2014 Keywords: Quantum phase transition Geometric quantum discord Bell–CHSH inequality
abstract In this paper, we have investigated the dynamical behaviors of the two important quantum correlation witnesses, i.e. geometric quantum discord (GQD) and Bell–CHSH inequality in the XXZ model with DM interaction by employing the quantum renormalization group (QRG) method. The results have shown that the anisotropy suppresses the quantum correlations while the DM interaction can enhance them. Meanwhile, using the QRG method we have studied the quantum phase transition of GQD and obtained two saturated values, which are associated with two different phases: spin-fluid phase and the Néel phase. It is worth mentioning that the block–block correlation is not strong enough to violate the Bell–CHSH inequality in the whole iteration steps. Moreover, the nonanalytic phenomenon and scaling behavior of Bell inequality are discussed in detail. As a byproduct, the conjecture that the exact lower and upper bounds of Bell inequality versus GQD can always be established for this spin system although the given density matrix is a general X state. © 2014 Elsevier Inc. All rights reserved.
1. Introduction Quantum entanglement, as one of the most intriguing traits of quantum information theory, plays a crucial role in quantum information processing [1,2]. However, it has been demonstrated that
∗
Corresponding author. Tel.: +86 13956016055. E-mail address: [email protected] (L. Ye).
http://dx.doi.org/10.1016/j.aop.2014.06.006 0003-4916/© 2014 Elsevier Inc. All rights reserved.
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quantum entanglement fails to be considered as the unique measure of quantum correlations since there exist other types of nonclassical correlations even in some mixed separable (not entangled) states [3–8]. As a result, the general entanglement–separability framework seems to be not suitable in the sense of quantifying and characterizing nonclassical correlations. Therefore, it is natural for us to search for new methods to quantify them. Quantum discord, introduced as a novel method to measure the nonclassical correlations by Ollivier and Zurek [6], is nonzero in almost all quantum states. Nevertheless, owing to the drastic difficulties of computation, quantum discord of arbitrary two-qubit states cannot be obtained [9]; Dakić et al. [10] propose an alternative measure of quantum correlation, called ‘‘geometric quantum discord’’, which can be used to get an analytical formula of universal two-qubit states in contrast to the original one. Subsequently, the dynamics of quantum discord attracts a great deal of attention [11–18]. On the other hand, quantum nonlocality which is generally captured by the violation of Bell-type inequality also stands for correlation measures of many systems [19,20]. Intuitively, Bell nonlocality can display some connection with other quantum correlation measures since they are used to characterize different aspects of nonclassical correlations. Great efforts have been intensively made to investigate their relations in recent years [21–27]. In particular, quantum entanglement can be regarded as an effective indicator of quantum phase transition (QPT) which is induced by the change of an external parameter or coupling constant [28] in several critical systems. This change occurs at absolute zero temperature where the quantum fluctuations play a dominant role while all the thermal fluctuations become frozen [29–33]. On the other hand, it has been proven that QPT is closely related to the entanglement of the ground state in the critical system, and the property of the ground state will be changed qualitatively at the critical point. Generally, performance of pairwise quantum entanglement in QPT is studied by using the quantum renormalization group method. Invoked by the method, it has been shown that implementation of the QRG method is helpful for detecting the nonanalytic phenomenon and the scaling behavior near the critical points in some physical systems. Furthermore, since quantum discord is introduced as a new crucial resource for quantum computation and Bell nonlocality is found to display certain connection with other quantum correlation witnesses, it is indispensable for us to clarify the roles played by them in quantum phase transition. To the best of our knowledge, many surveys have already illustrated that quantum discord and Bell nonlocality as well as quantum entanglement can observe the quantum critical points associated with QPT after enough iterations of the renormalization of coupling constants in one-dimensional Ising and Heisenberg models [34–40]. However, the dynamical properties of quantum correlations in the XXZ model with Dzyaloshinskii–Moriya (DM) interaction seem to have been seldom studied before. The DM interaction [41–43], both the Ising model in the transverse field and anisotropic XXZ models [44] can be supplemented with a magnetic term, arising from the spin–orbit coupling. It can restore the spoiled quantum entanglement via creation of the quantum fluctuation and influence the quantum phase transitions of many systems. Motivated by this, the main aim of this paper is to investigate quantum correlation properties of the ground state in the XXZ model with DM interaction by employing the quantum renormalization group method. In this paper, the effect of the DM interaction and anisotropy parameter, respectively, on renormalized geometric quantum discord and Bell nonlocality in quantum phase transition are discussed in detail. To serve as a further comparison, the relationship between the violation of Bell–CHSH inequality and geometric measure of quantum discord in this spin system is also taken into consideration. The paper is organized as follows. In Section 2 we will first briefly review the quantum renormalization group method in the XXZ model with DM interaction. In Section 3 the dynamical behaviors of the renormalized GQD and Bell nonlocality in QPT are investigated in detail. In Section 4, we are devoted to studying the nonanalytic and the scaling behavior of the system. Finally conclusion is given in Section 5. 2. Quantum renormalization of the XXZ model with Dzyaloshinskii–Moriya interaction In this section, we are committed to recalling the quantum renormalization group method and its application in the XXZ model with Dzyaloshinskii–Moriya interaction. Actually, the main idea of the QRG method is the mode elimination or the thinning of the degrees of freedom followed by exploring
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a recursive procedure, and purpose of the procedure is to reduce a certain amount of variables step by step until reaching a fixed point. In Kadanoff’s approach, the lattice is split into blocks in which the Hamiltonian is exactly diagonalized. Each block is treated independently to obtain the lower energy renormalized Hilbert subspace. Consequently, an effective Hamiltonian H eff is achieved by the full Hamiltonian being mapped into the renormalized space. The Hamiltonian of the XXZ model with DM interaction in the z direction on a periodic chain of N site is N J
H ( J , ∆) =
σix σix+1 + σiy σiy+1 + 1σiz σiz+1 + D σix σiy+1 − σiy σix+1
4
(1)
i
where J , ∆, D > 0, J is the exchange coupling constant, ∆ is the anisotropy parameter, and D is the χ strength of z component of DM interaction. σi (χ = x, y, z ) are the standard Pauli operators at site i. In the case of D = 0, the model becomes the known one-dimensional anisotropic XXZ model. To get a self-similar Hamiltonian after each QRG step, it is essential to divide the spin chain into three-site blocks. The corresponding block Hamiltonian has two degenerate ground states as follows [45]: 1 2 |φ0 ⟩ = {2(D + 1)|↓↓↑⟩ − (1 − iD)(q + ∆)| ↓↑↓⟩ 2q (q + ∆) 1 + D2
− 2[2iD + (D2 + 1)]|↑↓↓⟩} |φ0′ ⟩ =
1 2q (q + ∆) 1 +
D2
(2)
2 {2(D + 1)|↓↑↑⟩ − (1 − iD)(q + ∆)|↑↓↑⟩
− 2[2iD + (D2 + 1)]|↑↑↓⟩} where |↑⟩, |↓⟩ are the eigenstates of σ z Pauli matrix and q = ∆2 + 8 D2 + 1 .
(3)
(4)
Then the effective Hamiltonian of the renormalized XXZ chain with DM interaction can be expressed as H eff =
N J′
4
σix σix+1 + σiy σiy+1 + ∆′ σiz σiz+1 + D′ σix σiy+1 − σiy σix+1
(5)
i
where the renormalized couplings are ′
2
J =J
2 q
2
1+D
∆ ∆ = 1 + D2
,
′
∆+q
2
4
,
D′ = D.
′ By considering ∆c = √ ∆ = ∆, we can obtain a phase boundary √ phase ∆ < 1 + D2 from the Néel phase ∆ > 1 + D2 .
(6)
√
1 + D2 that separates the spin-fluid
3. Dynamical properties of renormalized geometric quantum discord and Bell nonlocality In order to study the performance of renormalized GQD and Bell nonlocality between two spins located on the sides of the block, we consider one of the degenerate ground state. The density matrix can be defined as ρ = |φ0 ⟩⟨φ0 |, where |φ0 ⟩ has been shown in Eq. (2). The results will be the same if we choose |φ0′ ⟩. Based on the above discussions, we acquire the reduced density matrix by tracing over site 2 as follows:
ρ13
(∆ + q)2 1 0 = 2q (∆ + q) 0 0
0
0
4 D2 + 1
−4 2iD + D2 + 1
0
−4 −2iD + D2 + 1 4 D2 + 1 0
0
0
. 0
0
(7)
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It is easy to obtain the concurrence of ρ13 [46]:
C (ρ13 ) =
4 1 + D2
q (q + ∆)
.
(8)
It is displayed that for a nonzero value of anisotropy, turning on the DM interaction can restore the spoiled quantum entanglement, while the quantum entanglement is a fixed value in the absence of the anisotropy for any values of D. Simultaneously, √ the concurrence between two blocks exhibits a quantum phase transition at D = 1 when ∆ = 2. But there is no study on the dynamical behaviors of GQD and Bell nonlocality in the XXZ model with DM interaction before. In the following section, the quantum critical traits of the two quantum correlations are investigated to compare with the previous works concerning quantum entanglement. 3.1. Geometric quantum discord The geometric measure of quantum discord based on the Hilbert–Schmidt norm is defined as the square-norm distance between the given state ρ and its closest classical state χρ , introduced by Dakić et al. [10]: 2 Dg (ρ) = ρ − χρ
(9)
where χρ takes the form χρ = i pi |i⟩⟨i| ⊗ ρiB , pi are probabilities, {|i⟩} is a completely orthogonal vector set, and ρiB is an arbitrary state of part B. ∥M ∥2 = Tr(MM + ) is the square-norm distance in the Hilbert–Schmidt space. For the two-qubit X state, normalized GQD has a straightforward form
Dg (ρ) =
1 2 ⃗x + ∥T ∥2 − kmax 2
(10)
where xi = tr (ρσi ⊗ I ) are components of the local Bloch vector, Tij = tr ρσi ⊗ σj are components of the correlation matrix, and kmax is the largest eigenvalue of the matrix K = ⃗ x⃗ xt + TT t (the superscript t stands for transpose). Using the explicit formula for arbitrary two-qubit state of GQD in Ref. [18], we can obtain:
Dg (ρ13 ) =
16 D2 + 1
2
q2 (∆ + q)2
= C 2 (ρ13 ) .
(11)
It is worth noticing that the GQD of the state for Eq. (7) is equivalent to the corresponding square concurrence. From Eq. (16), we find that the value of the function Dg (ρ13 ) is decided by the variables D and ∆. The change of GQD between the first and third sites of block versus ∆ for different values of D is plotted in Fig. 1. It is clearly observed that the GQD is insensitive to the DM value when ∆ = 0 or infinity, while for a nonzero value of anisotropy, GQD increases with the value of D increasing. Therefore, we can come to the conclusion that the suppression of the GQD can be compensated by tuning the DM interaction, i.e. the DM interaction enhances the GQD. Since the QRG method is introduced to explore the quantum properties of critical systems, and it has been proven that realization of QRG method is helpful in detecting the quantum critical points associated with quantum phase transition after enough iterations of the renormalization, it is significant for us to investigate the quantum phase transition of GQD for the spin-system via QRG method. For a fixed value of D = 1, the plots for the change of Dg (ρ13 ) are depicted in Fig. 2. We obtain that
√
the GQD cross each other at the critical point ∆ = 2. It is no wonder that in the thermodynamic √ limit of the model, GQD develops two different saturated values: Dg (ρ13 ) ≈ 0.25 for 0 ≤ ∆ < 2,
√
while Dg (ρ13 ) → 0 for ∆ > 2, which are associated with two different phases, spin-fluid and Néel phases. For the XXZ model of Ref. [40], the GQD exhibits a QPT at the critical point ∆ =√1. However, for the present model the GQD indicates the precise location of the critical point ∆ = 2. That is to say, the DM interaction has a great influence on the QPT of the system due to the quantum fluctuation. Having this in mind, people cannot help considering how the GQD versus D changes for a fixed √ value ∆. In Fig. 3, we illustrate the change of Dg (ρ13 ) versus D for different QRG step with ∆ = 2.
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Fig. 1. (Color online) The change of the geometric quantum discord between the first and third sites of a three-site model versus ∆ for different D.
Fig. 2. (Color online) The change of the geometric quantum discord versus ∆ in terms of QRG iterations for D = 1.
Similarly, we can discover that the GQD versus D can also effectively detect the quantum critical points associated √ with quantum phase transition after several iterations of the renormalization 2. But it is different from the behavior of GQD versus ∆ for D = 1. A comparison when ∆ = between Figs. 2 and 3 tells us that the anisotropy suppresses the GQD while the DM interaction can enhance it. 3.2. Bell nonlocality In the debate on the interpretation of the basic principles and concepts of quantum mechanics, Bell has proposed a Bell inequality to make the above-mentioned problems quantitatively and
X.-k. Song et al. / Annals of Physics 349 (2014) 220–231
Fig. 3. (Color online) The change of the geometric quantum discord versus D in terms of QRG iterations for ∆ =
225
√ 2.
specifically [47]. Bell inequality has many famous promotional forms. Among them, the famous one is the CHSH inequality (Clauser–Horne–Shimony–Holt) [48], which is more concise and symmetrical. Particularly, for two-qubit pure states, the presence of entanglement ensures the violation of a Bell–CHSH inequality. However, for the case of mixed states, the situation becomes more complicated [21,23]. Here, the Bell operator associated with the quantum CHSH inequality has the following general form BCHSH = a · σ ⊗ b + b′ · σ + a′ · σ ⊗ b − b′ · σ
(12)
where a, a′ , b, b′ are the unit vectors in R3 , and σ = σx i + σy j + σz k. Then the Bell–CHSH inequality can be written as:
|⟨BCHSH ⟩| ≤ 2
(13)
where |⟨BCHSH ⟩| = |tr (ρ BCHSH )|. For the two-qubit X states, the maximal value of Bell–CHSH which satisfy quantum CHSH inequality can be given by [19,20] Bmax CHSH = max tr(ρ BCHSH a,a′ ,b,b′
3 t2 − λ ) = 2 i
min
(14)
i=1
where ti = tr [ρ (σi ⊗ σi )] are the components of the correlation tensor in the Bloch representation, λmin = min{t12 , t22 , t32 }. The state ρ violates the Bell–CHSH inequality if and only if Bmax CHSH > 2. Thus, the maximal Bell–CHSH value of state (7) is
BCHSH
2 + 1 2 64 D2 + 1 2 + (∆ + q)2 − 8 D2 + 1 2 32 D = 2max , . q2 (∆ + q)2 4q2 (∆ + q)2
(15)
As mentioned above, the anisotropy induces limitation on the quantum entanglement and quantum discord due to favoring of alignment of spins, whereas the DM interaction restores the spoiled entanglement and quantum discord via the creation of quantum fluctuation. Specifically, it is confirmed that the values of concurrence and quantum discord become large with D increasing while
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Fig. 4. (Color online) The change of the Bell–CHSH value between the first and third sites of a three-site model versus ∆ for different D. In the illustration of diagram ∆ ranges from 0 to 5.
they decrease with ∆ increasing all along. Nevertheless, the influence of the DM interaction and anisotropy parameter on the Bell–CHSH inequality remains unknown. Motivated by this, we focus our eyes on how the Bell–CHSH nonlocality versus D and ∆ varies. The plots of Bell–CHSH value have been displayed in Fig. 4. Obviously, it can be seen that the block–block correlation in the XXZ model with DM interaction is not strong enough to violate the Bell–CHSH inequality, and the maximal value of the Bell–CHSH is 2. Moreover, we can observe that the Bell–CHSH value is a fixed nonzero constant for any values of D at ∆ = 0 or infinity. Interestingly, the Bell–CHSH value becomes large (i.e. the local correlation enlarges) with D increasing when ∆ is small. And its value decreases with D increasing when ∆ is large compared to D. In other words, for the value of Bell–CHSH inequality, the effect of DM interaction, which enhances quantum entanglement and quantum discord of system, is more important when ∆ is small, while the anisotropy parameter plays a prominent role when ∆ is large. Likewise, we also research the behavior of Bell–CHSH inequality in quantum phase transition employing the quantum renormalization group method. For the XXZ model with DM interaction, the change of the Bell–CHSH value versus ∆ in terms of QRG iteration steps at D = 1 has been plotted in Fig. values: √ we can gain that√the Bell–CHSH value also produces two saturated √ √5. From the picture, B = 2 for 0 ≤ ∆ < 2, B = 2 for ∆ > 2. In addition, at the critical point ∆ = 2, the minimal value of Bell–CHSH is a fixed nonzero constant Bc ≈ 0.94, irrespective of the iteration steps. But its change is in sharp contrast to situations of the above quantum correlations. As a comparison, we show √ the change of Bell–CHSH value versus D in terms of QRG iteration steps at ∆ = 2 in Fig. 6. It can be easily found that the Bell–CHSH value can obviously exhibit a QPT at the critical point D = 1. After some iteration steps, the Bell–CHSH value inequality can reach the maximal value 2. In the process of exploring the relationship between the Bell–CHSH nonlocality and geometric measure of quantum discord, a significant inequality which has been proven by Yao et al. [49] recently is very valuable for understanding the connection among various measures of quantum correlations. Now, we study the relation 4 Dg ≤ B ≤ 2 1 + 2Dg .
(16)
Noting that the GQD of Eq. (16) is not normalized, so we can rewrite it into the normalized form:
2 2DNg ≤ B ≤ 2 1 + DNg
(17)
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Fig. 5. (Color online) The change of Bell–CHSH value versus ∆ in terms of QRG iterations at D = 1.
Fig. 6. (Color online) The change of the Bell–CHSH value versus D in terms of QRG iterations at ∆ =
√ 2.
where DNg is normalized GQD. Although the exact lower and upper bounds of the Bell–CHSH inequality versus GQD are obtained for a particular class of states, Bell diagonal states, it is vital to see whether it can be verified really helpful in the further studies or not on the interplay between them. In Fig. 7, we show the change of difference for the Bell–CHSH inequality and normalizedGQD versus D for different QRG steps at ∆ =
√
2. We notice that both the difference B − 2 2DNg and 2 1 + DNg − B
can exhibit QPTs at the critical point D = 1 and each has a corresponding fixed nonzero constant. Both the upper bound and the lower bound of Bell–CHSH inequality versus normalized GQD is always
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Fig. 7. (Color online) The change of the difference for the Bell–CHSH inequality and corresponding normalized geometric √ quantum discord for Eq. (17) versus D in terms of QRG iterations at ∆ = 2.
established in the anisotropic XXZ model with DM interaction no matter how many QRG steps are carried out. Therefore, the results reveal that the Bell–CHSH nonlocality can show certain connection with quantum discord even if in the specific system. 4. Nonanalytic and scaling behavior In the above section, the block–block correlations and quantum phase transition of a large onedimensional XXZ model with DM interaction are discussed by using the QRG method. Indeed, a
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Fig. 8. (Color online) The logarithm of the absolute value of minimum ln(dB/dD|m ) versus the logarithm of the chain size, ln(N ), is linear and displays a scaling behavior.
general characteristic of the second-order quantum phase transitions is the appearance of nonanalytic behavior in some physical quantities. It is often accompanied by a scaling behavior since the correlation length diverges and there is no characteristic length scale in the system at the critical point. Recently, many schemes have demonstrated that the entanglement indicates the critical behavior such as diverging of its derivative as the phase transition. What is more, it has been illustrated that the entanglement in the vicinity of critical point shows a scaling behavior. Comparing with the previous works involved entanglement, we concentrate on disclosing the nonanalytic situation and scaling behavior of other block–block correlation witnesses. The GQD of the system is the corresponding square concurrence and the performance of quantum discord is more similar to the entanglement [45]; we explicitly present the behavior of the block–block Bell–CHSH inequality of this large one-dimensional spin XXZ model with DM interaction in this section. First, we √ analyze the scaling behavior of y = |dB/dD|Dm versus the size of the system N at ∆ = 2, where Dm is the position of the minimum of dB/dD. The plot of ln(y) versus ln(N ) is depicted in Fig. 8. Apparently, it shows a linear behavior. Meanwhile, we can get a more detailed comparison that the position of the minimum Dm of dB/dD slowly tends to the critical point Dc = 1, which has been discussed in Fig. 9. Through numerical computation, we can see that the exponent for this behavior is Dm = Dc − N −0.52 . These results convince us that the QRG implementation of Bell violation truly captures the critical behavior of the XXZ model with DM interaction. 5. Conclusion In this paper, we have investigated the dynamical behaviors of the two important quantum correlation witnesses, i.e. GQD and Bell–CHSH inequality, in the XXZ model with DM interaction by employing the QRG method. The DM interaction and anisotropy parameters have a crucial influence on the renormalization of the two quantum correlations in quantum phase transition. For a nonzero value of anisotropy, turning on the DM interaction can restore the spoiled entanglement and quantum discord, while the Bell–CHSH value becomes large with D increasing when ∆ is small, and the value decreases as D increases when ∆ is large. Additionally, the results have shown that after several iterations of the renormalization, the two quantum correlation measures can develop two saturated values, which are associated with two different phases: spin-fluid phase and the Néel phase. More importantly, we have obtained that the block–block correlation complies with the Bell–CHSH
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Fig. 9. (Color online) The scaling behavior of Dm in terms of system size N, where Dm is the position of the minimum derivative of geometric quantum discord in the Ising model with DM interaction.
inequality all along in the whole iteration steps, and obviously exhibit a QPT. Coincidentally, although the given density matrix of our scheme is a general X state, we also have acquired the exact lower and upper bounds of Bell–CHSH inequality versus GQD for this specific system. We expect that the results presented here will be helpful in the further work to clarify that the quantum discord and Bell–CHSH inequality can be used as the indicators of the quantum phase transition in the spin models. To gain further insight, the nonanalytic and scaling behavior of Bell–CHSH inequality have also been analyzed in detail. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants No. 11074002 and No. 61275119, the Doctoral Foundation of the Ministry of Education of China under Grant No. 20103401110003, and also by the Personal Development Foundation of Anhui Province (2008Z018). References [1] M.A. Nilsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2000. [2] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Modern Phys. 81 (2009) 865. [3] C.H. Bennett, D.P. DiVincenzo, C.A. Fuchs, T. Mor, E. Rains, P.W. Shor, J.A. Smolin, W.K. Wootters, Phys. Rev. A 59 (1999) 1070. [4] M. Horodecki, P. Horodecki, R. Horodecki, J. Oppenheim, A. Sen, U. Sen, B. Synak-Radtke, Phys. Rev. A 71 (2005) 062307. [5] J. Niset, N.J. Cerf, Phys. Rev. A 74 (2006) 052103. [6] H. Ollivier, W.H. Zurek, Phys. Rev. Lett. 88 (2001) 017901. [7] M. Piani, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 100 (2008) 090502. [8] A. Ferraro, L. Aolito, D. Cavalcanti, F.M. Cucchietti, A. Acín, Phys. Rev. A 81 (2010) 052318. [9] S. Luo, Phys. Rev. A 77 (2008) 022301. [10] [11] [12] [13] [14] [15] [16]
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Renormalization of quantum discord and Bell nonlocality in the XXZ model with Dzyaloshinskii–Moriya interaction Xue-ke Song a , Tao Wu b , Shuai Xu a , Juan He a , Liu Ye a,∗ a
School of Physics & Material Science, Anhui University, Hefei, 230039, China
b
School of Physics & Electronics Science, Fuyang Normal College, Fuyang, 236037, China
article
info
Article history: Received 23 January 2013 Accepted 10 June 2014 Available online 14 June 2014 Keywords: Quantum phase transition Geometric quantum discord Bell–CHSH inequality
abstract In this paper, we have investigated the dynamical behaviors of the two important quantum correlation witnesses, i.e. geometric quantum discord (GQD) and Bell–CHSH inequality in the XXZ model with DM interaction by employing the quantum renormalization group (QRG) method. The results have shown that the anisotropy suppresses the quantum correlations while the DM interaction can enhance them. Meanwhile, using the QRG method we have studied the quantum phase transition of GQD and obtained two saturated values, which are associated with two different phases: spin-fluid phase and the Néel phase. It is worth mentioning that the block–block correlation is not strong enough to violate the Bell–CHSH inequality in the whole iteration steps. Moreover, the nonanalytic phenomenon and scaling behavior of Bell inequality are discussed in detail. As a byproduct, the conjecture that the exact lower and upper bounds of Bell inequality versus GQD can always be established for this spin system although the given density matrix is a general X state. © 2014 Elsevier Inc. All rights reserved.
1. Introduction Quantum entanglement, as one of the most intriguing traits of quantum information theory, plays a crucial role in quantum information processing [1,2]. However, it has been demonstrated that
∗
Corresponding author. Tel.: +86 13956016055. E-mail address: [email protected] (L. Ye).
http://dx.doi.org/10.1016/j.aop.2014.06.006 0003-4916/© 2014 Elsevier Inc. All rights reserved.
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quantum entanglement fails to be considered as the unique measure of quantum correlations since there exist other types of nonclassical correlations even in some mixed separable (not entangled) states [3–8]. As a result, the general entanglement–separability framework seems to be not suitable in the sense of quantifying and characterizing nonclassical correlations. Therefore, it is natural for us to search for new methods to quantify them. Quantum discord, introduced as a novel method to measure the nonclassical correlations by Ollivier and Zurek [6], is nonzero in almost all quantum states. Nevertheless, owing to the drastic difficulties of computation, quantum discord of arbitrary two-qubit states cannot be obtained [9]; Dakić et al. [10] propose an alternative measure of quantum correlation, called ‘‘geometric quantum discord’’, which can be used to get an analytical formula of universal two-qubit states in contrast to the original one. Subsequently, the dynamics of quantum discord attracts a great deal of attention [11–18]. On the other hand, quantum nonlocality which is generally captured by the violation of Bell-type inequality also stands for correlation measures of many systems [19,20]. Intuitively, Bell nonlocality can display some connection with other quantum correlation measures since they are used to characterize different aspects of nonclassical correlations. Great efforts have been intensively made to investigate their relations in recent years [21–27]. In particular, quantum entanglement can be regarded as an effective indicator of quantum phase transition (QPT) which is induced by the change of an external parameter or coupling constant [28] in several critical systems. This change occurs at absolute zero temperature where the quantum fluctuations play a dominant role while all the thermal fluctuations become frozen [29–33]. On the other hand, it has been proven that QPT is closely related to the entanglement of the ground state in the critical system, and the property of the ground state will be changed qualitatively at the critical point. Generally, performance of pairwise quantum entanglement in QPT is studied by using the quantum renormalization group method. Invoked by the method, it has been shown that implementation of the QRG method is helpful for detecting the nonanalytic phenomenon and the scaling behavior near the critical points in some physical systems. Furthermore, since quantum discord is introduced as a new crucial resource for quantum computation and Bell nonlocality is found to display certain connection with other quantum correlation witnesses, it is indispensable for us to clarify the roles played by them in quantum phase transition. To the best of our knowledge, many surveys have already illustrated that quantum discord and Bell nonlocality as well as quantum entanglement can observe the quantum critical points associated with QPT after enough iterations of the renormalization of coupling constants in one-dimensional Ising and Heisenberg models [34–40]. However, the dynamical properties of quantum correlations in the XXZ model with Dzyaloshinskii–Moriya (DM) interaction seem to have been seldom studied before. The DM interaction [41–43], both the Ising model in the transverse field and anisotropic XXZ models [44] can be supplemented with a magnetic term, arising from the spin–orbit coupling. It can restore the spoiled quantum entanglement via creation of the quantum fluctuation and influence the quantum phase transitions of many systems. Motivated by this, the main aim of this paper is to investigate quantum correlation properties of the ground state in the XXZ model with DM interaction by employing the quantum renormalization group method. In this paper, the effect of the DM interaction and anisotropy parameter, respectively, on renormalized geometric quantum discord and Bell nonlocality in quantum phase transition are discussed in detail. To serve as a further comparison, the relationship between the violation of Bell–CHSH inequality and geometric measure of quantum discord in this spin system is also taken into consideration. The paper is organized as follows. In Section 2 we will first briefly review the quantum renormalization group method in the XXZ model with DM interaction. In Section 3 the dynamical behaviors of the renormalized GQD and Bell nonlocality in QPT are investigated in detail. In Section 4, we are devoted to studying the nonanalytic and the scaling behavior of the system. Finally conclusion is given in Section 5. 2. Quantum renormalization of the XXZ model with Dzyaloshinskii–Moriya interaction In this section, we are committed to recalling the quantum renormalization group method and its application in the XXZ model with Dzyaloshinskii–Moriya interaction. Actually, the main idea of the QRG method is the mode elimination or the thinning of the degrees of freedom followed by exploring
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a recursive procedure, and purpose of the procedure is to reduce a certain amount of variables step by step until reaching a fixed point. In Kadanoff’s approach, the lattice is split into blocks in which the Hamiltonian is exactly diagonalized. Each block is treated independently to obtain the lower energy renormalized Hilbert subspace. Consequently, an effective Hamiltonian H eff is achieved by the full Hamiltonian being mapped into the renormalized space. The Hamiltonian of the XXZ model with DM interaction in the z direction on a periodic chain of N site is N J
H ( J , ∆) =
σix σix+1 + σiy σiy+1 + 1σiz σiz+1 + D σix σiy+1 − σiy σix+1
4
(1)
i
where J , ∆, D > 0, J is the exchange coupling constant, ∆ is the anisotropy parameter, and D is the χ strength of z component of DM interaction. σi (χ = x, y, z ) are the standard Pauli operators at site i. In the case of D = 0, the model becomes the known one-dimensional anisotropic XXZ model. To get a self-similar Hamiltonian after each QRG step, it is essential to divide the spin chain into three-site blocks. The corresponding block Hamiltonian has two degenerate ground states as follows [45]: 1 2 |φ0 ⟩ = {2(D + 1)|↓↓↑⟩ − (1 − iD)(q + ∆)| ↓↑↓⟩ 2q (q + ∆) 1 + D2
− 2[2iD + (D2 + 1)]|↑↓↓⟩} |φ0′ ⟩ =
1 2q (q + ∆) 1 +
D2
(2)
2 {2(D + 1)|↓↑↑⟩ − (1 − iD)(q + ∆)|↑↓↑⟩
− 2[2iD + (D2 + 1)]|↑↑↓⟩} where |↑⟩, |↓⟩ are the eigenstates of σ z Pauli matrix and q = ∆2 + 8 D2 + 1 .
(3)
(4)
Then the effective Hamiltonian of the renormalized XXZ chain with DM interaction can be expressed as H eff =
N J′
4
σix σix+1 + σiy σiy+1 + ∆′ σiz σiz+1 + D′ σix σiy+1 − σiy σix+1
(5)
i
where the renormalized couplings are ′
2
J =J
2 q
2
1+D
∆ ∆ = 1 + D2
,
′
∆+q
2
4
,
D′ = D.
′ By considering ∆c = √ ∆ = ∆, we can obtain a phase boundary √ phase ∆ < 1 + D2 from the Néel phase ∆ > 1 + D2 .
(6)
√
1 + D2 that separates the spin-fluid
3. Dynamical properties of renormalized geometric quantum discord and Bell nonlocality In order to study the performance of renormalized GQD and Bell nonlocality between two spins located on the sides of the block, we consider one of the degenerate ground state. The density matrix can be defined as ρ = |φ0 ⟩⟨φ0 |, where |φ0 ⟩ has been shown in Eq. (2). The results will be the same if we choose |φ0′ ⟩. Based on the above discussions, we acquire the reduced density matrix by tracing over site 2 as follows:
ρ13
(∆ + q)2 1 0 = 2q (∆ + q) 0 0
0
0
4 D2 + 1
−4 2iD + D2 + 1
0
−4 −2iD + D2 + 1 4 D2 + 1 0
0
0
. 0
0
(7)
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It is easy to obtain the concurrence of ρ13 [46]:
C (ρ13 ) =
4 1 + D2
q (q + ∆)
.
(8)
It is displayed that for a nonzero value of anisotropy, turning on the DM interaction can restore the spoiled quantum entanglement, while the quantum entanglement is a fixed value in the absence of the anisotropy for any values of D. Simultaneously, √ the concurrence between two blocks exhibits a quantum phase transition at D = 1 when ∆ = 2. But there is no study on the dynamical behaviors of GQD and Bell nonlocality in the XXZ model with DM interaction before. In the following section, the quantum critical traits of the two quantum correlations are investigated to compare with the previous works concerning quantum entanglement. 3.1. Geometric quantum discord The geometric measure of quantum discord based on the Hilbert–Schmidt norm is defined as the square-norm distance between the given state ρ and its closest classical state χρ , introduced by Dakić et al. [10]: 2 Dg (ρ) = ρ − χρ
(9)
where χρ takes the form χρ = i pi |i⟩⟨i| ⊗ ρiB , pi are probabilities, {|i⟩} is a completely orthogonal vector set, and ρiB is an arbitrary state of part B. ∥M ∥2 = Tr(MM + ) is the square-norm distance in the Hilbert–Schmidt space. For the two-qubit X state, normalized GQD has a straightforward form
Dg (ρ) =
1 2 ⃗x + ∥T ∥2 − kmax 2
(10)
where xi = tr (ρσi ⊗ I ) are components of the local Bloch vector, Tij = tr ρσi ⊗ σj are components of the correlation matrix, and kmax is the largest eigenvalue of the matrix K = ⃗ x⃗ xt + TT t (the superscript t stands for transpose). Using the explicit formula for arbitrary two-qubit state of GQD in Ref. [18], we can obtain:
Dg (ρ13 ) =
16 D2 + 1
2
q2 (∆ + q)2
= C 2 (ρ13 ) .
(11)
It is worth noticing that the GQD of the state for Eq. (7) is equivalent to the corresponding square concurrence. From Eq. (16), we find that the value of the function Dg (ρ13 ) is decided by the variables D and ∆. The change of GQD between the first and third sites of block versus ∆ for different values of D is plotted in Fig. 1. It is clearly observed that the GQD is insensitive to the DM value when ∆ = 0 or infinity, while for a nonzero value of anisotropy, GQD increases with the value of D increasing. Therefore, we can come to the conclusion that the suppression of the GQD can be compensated by tuning the DM interaction, i.e. the DM interaction enhances the GQD. Since the QRG method is introduced to explore the quantum properties of critical systems, and it has been proven that realization of QRG method is helpful in detecting the quantum critical points associated with quantum phase transition after enough iterations of the renormalization, it is significant for us to investigate the quantum phase transition of GQD for the spin-system via QRG method. For a fixed value of D = 1, the plots for the change of Dg (ρ13 ) are depicted in Fig. 2. We obtain that
√
the GQD cross each other at the critical point ∆ = 2. It is no wonder that in the thermodynamic √ limit of the model, GQD develops two different saturated values: Dg (ρ13 ) ≈ 0.25 for 0 ≤ ∆ < 2,
√
while Dg (ρ13 ) → 0 for ∆ > 2, which are associated with two different phases, spin-fluid and Néel phases. For the XXZ model of Ref. [40], the GQD exhibits a QPT at the critical point ∆ =√1. However, for the present model the GQD indicates the precise location of the critical point ∆ = 2. That is to say, the DM interaction has a great influence on the QPT of the system due to the quantum fluctuation. Having this in mind, people cannot help considering how the GQD versus D changes for a fixed √ value ∆. In Fig. 3, we illustrate the change of Dg (ρ13 ) versus D for different QRG step with ∆ = 2.
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Fig. 1. (Color online) The change of the geometric quantum discord between the first and third sites of a three-site model versus ∆ for different D.
Fig. 2. (Color online) The change of the geometric quantum discord versus ∆ in terms of QRG iterations for D = 1.
Similarly, we can discover that the GQD versus D can also effectively detect the quantum critical points associated √ with quantum phase transition after several iterations of the renormalization 2. But it is different from the behavior of GQD versus ∆ for D = 1. A comparison when ∆ = between Figs. 2 and 3 tells us that the anisotropy suppresses the GQD while the DM interaction can enhance it. 3.2. Bell nonlocality In the debate on the interpretation of the basic principles and concepts of quantum mechanics, Bell has proposed a Bell inequality to make the above-mentioned problems quantitatively and
X.-k. Song et al. / Annals of Physics 349 (2014) 220–231
Fig. 3. (Color online) The change of the geometric quantum discord versus D in terms of QRG iterations for ∆ =
225
√ 2.
specifically [47]. Bell inequality has many famous promotional forms. Among them, the famous one is the CHSH inequality (Clauser–Horne–Shimony–Holt) [48], which is more concise and symmetrical. Particularly, for two-qubit pure states, the presence of entanglement ensures the violation of a Bell–CHSH inequality. However, for the case of mixed states, the situation becomes more complicated [21,23]. Here, the Bell operator associated with the quantum CHSH inequality has the following general form BCHSH = a · σ ⊗ b + b′ · σ + a′ · σ ⊗ b − b′ · σ
(12)
where a, a′ , b, b′ are the unit vectors in R3 , and σ = σx i + σy j + σz k. Then the Bell–CHSH inequality can be written as:
|⟨BCHSH ⟩| ≤ 2
(13)
where |⟨BCHSH ⟩| = |tr (ρ BCHSH )|. For the two-qubit X states, the maximal value of Bell–CHSH which satisfy quantum CHSH inequality can be given by [19,20] Bmax CHSH = max tr(ρ BCHSH a,a′ ,b,b′
3 t2 − λ ) = 2 i
min
(14)
i=1
where ti = tr [ρ (σi ⊗ σi )] are the components of the correlation tensor in the Bloch representation, λmin = min{t12 , t22 , t32 }. The state ρ violates the Bell–CHSH inequality if and only if Bmax CHSH > 2. Thus, the maximal Bell–CHSH value of state (7) is
BCHSH
2 + 1 2 64 D2 + 1 2 + (∆ + q)2 − 8 D2 + 1 2 32 D = 2max , . q2 (∆ + q)2 4q2 (∆ + q)2
(15)
As mentioned above, the anisotropy induces limitation on the quantum entanglement and quantum discord due to favoring of alignment of spins, whereas the DM interaction restores the spoiled entanglement and quantum discord via the creation of quantum fluctuation. Specifically, it is confirmed that the values of concurrence and quantum discord become large with D increasing while
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Fig. 4. (Color online) The change of the Bell–CHSH value between the first and third sites of a three-site model versus ∆ for different D. In the illustration of diagram ∆ ranges from 0 to 5.
they decrease with ∆ increasing all along. Nevertheless, the influence of the DM interaction and anisotropy parameter on the Bell–CHSH inequality remains unknown. Motivated by this, we focus our eyes on how the Bell–CHSH nonlocality versus D and ∆ varies. The plots of Bell–CHSH value have been displayed in Fig. 4. Obviously, it can be seen that the block–block correlation in the XXZ model with DM interaction is not strong enough to violate the Bell–CHSH inequality, and the maximal value of the Bell–CHSH is 2. Moreover, we can observe that the Bell–CHSH value is a fixed nonzero constant for any values of D at ∆ = 0 or infinity. Interestingly, the Bell–CHSH value becomes large (i.e. the local correlation enlarges) with D increasing when ∆ is small. And its value decreases with D increasing when ∆ is large compared to D. In other words, for the value of Bell–CHSH inequality, the effect of DM interaction, which enhances quantum entanglement and quantum discord of system, is more important when ∆ is small, while the anisotropy parameter plays a prominent role when ∆ is large. Likewise, we also research the behavior of Bell–CHSH inequality in quantum phase transition employing the quantum renormalization group method. For the XXZ model with DM interaction, the change of the Bell–CHSH value versus ∆ in terms of QRG iteration steps at D = 1 has been plotted in Fig. values: √ we can gain that√the Bell–CHSH value also produces two saturated √ √5. From the picture, B = 2 for 0 ≤ ∆ < 2, B = 2 for ∆ > 2. In addition, at the critical point ∆ = 2, the minimal value of Bell–CHSH is a fixed nonzero constant Bc ≈ 0.94, irrespective of the iteration steps. But its change is in sharp contrast to situations of the above quantum correlations. As a comparison, we show √ the change of Bell–CHSH value versus D in terms of QRG iteration steps at ∆ = 2 in Fig. 6. It can be easily found that the Bell–CHSH value can obviously exhibit a QPT at the critical point D = 1. After some iteration steps, the Bell–CHSH value inequality can reach the maximal value 2. In the process of exploring the relationship between the Bell–CHSH nonlocality and geometric measure of quantum discord, a significant inequality which has been proven by Yao et al. [49] recently is very valuable for understanding the connection among various measures of quantum correlations. Now, we study the relation 4 Dg ≤ B ≤ 2 1 + 2Dg .
(16)
Noting that the GQD of Eq. (16) is not normalized, so we can rewrite it into the normalized form:
2 2DNg ≤ B ≤ 2 1 + DNg
(17)
X.-k. Song et al. / Annals of Physics 349 (2014) 220–231
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Fig. 5. (Color online) The change of Bell–CHSH value versus ∆ in terms of QRG iterations at D = 1.
Fig. 6. (Color online) The change of the Bell–CHSH value versus D in terms of QRG iterations at ∆ =
√ 2.
where DNg is normalized GQD. Although the exact lower and upper bounds of the Bell–CHSH inequality versus GQD are obtained for a particular class of states, Bell diagonal states, it is vital to see whether it can be verified really helpful in the further studies or not on the interplay between them. In Fig. 7, we show the change of difference for the Bell–CHSH inequality and normalizedGQD versus D for different QRG steps at ∆ =
√
2. We notice that both the difference B − 2 2DNg and 2 1 + DNg − B
can exhibit QPTs at the critical point D = 1 and each has a corresponding fixed nonzero constant. Both the upper bound and the lower bound of Bell–CHSH inequality versus normalized GQD is always
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X.-k. Song et al. / Annals of Physics 349 (2014) 220–231
Fig. 7. (Color online) The change of the difference for the Bell–CHSH inequality and corresponding normalized geometric √ quantum discord for Eq. (17) versus D in terms of QRG iterations at ∆ = 2.
established in the anisotropic XXZ model with DM interaction no matter how many QRG steps are carried out. Therefore, the results reveal that the Bell–CHSH nonlocality can show certain connection with quantum discord even if in the specific system. 4. Nonanalytic and scaling behavior In the above section, the block–block correlations and quantum phase transition of a large onedimensional XXZ model with DM interaction are discussed by using the QRG method. Indeed, a
X.-k. Song et al. / Annals of Physics 349 (2014) 220–231
229
Fig. 8. (Color online) The logarithm of the absolute value of minimum ln(dB/dD|m ) versus the logarithm of the chain size, ln(N ), is linear and displays a scaling behavior.
general characteristic of the second-order quantum phase transitions is the appearance of nonanalytic behavior in some physical quantities. It is often accompanied by a scaling behavior since the correlation length diverges and there is no characteristic length scale in the system at the critical point. Recently, many schemes have demonstrated that the entanglement indicates the critical behavior such as diverging of its derivative as the phase transition. What is more, it has been illustrated that the entanglement in the vicinity of critical point shows a scaling behavior. Comparing with the previous works involved entanglement, we concentrate on disclosing the nonanalytic situation and scaling behavior of other block–block correlation witnesses. The GQD of the system is the corresponding square concurrence and the performance of quantum discord is more similar to the entanglement [45]; we explicitly present the behavior of the block–block Bell–CHSH inequality of this large one-dimensional spin XXZ model with DM interaction in this section. First, we √ analyze the scaling behavior of y = |dB/dD|Dm versus the size of the system N at ∆ = 2, where Dm is the position of the minimum of dB/dD. The plot of ln(y) versus ln(N ) is depicted in Fig. 8. Apparently, it shows a linear behavior. Meanwhile, we can get a more detailed comparison that the position of the minimum Dm of dB/dD slowly tends to the critical point Dc = 1, which has been discussed in Fig. 9. Through numerical computation, we can see that the exponent for this behavior is Dm = Dc − N −0.52 . These results convince us that the QRG implementation of Bell violation truly captures the critical behavior of the XXZ model with DM interaction. 5. Conclusion In this paper, we have investigated the dynamical behaviors of the two important quantum correlation witnesses, i.e. GQD and Bell–CHSH inequality, in the XXZ model with DM interaction by employing the QRG method. The DM interaction and anisotropy parameters have a crucial influence on the renormalization of the two quantum correlations in quantum phase transition. For a nonzero value of anisotropy, turning on the DM interaction can restore the spoiled entanglement and quantum discord, while the Bell–CHSH value becomes large with D increasing when ∆ is small, and the value decreases as D increases when ∆ is large. Additionally, the results have shown that after several iterations of the renormalization, the two quantum correlation measures can develop two saturated values, which are associated with two different phases: spin-fluid phase and the Néel phase. More importantly, we have obtained that the block–block correlation complies with the Bell–CHSH
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Fig. 9. (Color online) The scaling behavior of Dm in terms of system size N, where Dm is the position of the minimum derivative of geometric quantum discord in the Ising model with DM interaction.
inequality all along in the whole iteration steps, and obviously exhibit a QPT. Coincidentally, although the given density matrix of our scheme is a general X state, we also have acquired the exact lower and upper bounds of Bell–CHSH inequality versus GQD for this specific system. We expect that the results presented here will be helpful in the further work to clarify that the quantum discord and Bell–CHSH inequality can be used as the indicators of the quantum phase transition in the spin models. To gain further insight, the nonanalytic and scaling behavior of Bell–CHSH inequality have also been analyzed in detail. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants No. 11074002 and No. 61275119, the Doctoral Foundation of the Ministry of Education of China under Grant No. 20103401110003, and also by the Personal Development Foundation of Anhui Province (2008Z018). References [1] M.A. Nilsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, UK, 2000. [2] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Modern Phys. 81 (2009) 865. [3] C.H. Bennett, D.P. DiVincenzo, C.A. Fuchs, T. Mor, E. Rains, P.W. Shor, J.A. Smolin, W.K. Wootters, Phys. Rev. A 59 (1999) 1070. [4] M. Horodecki, P. Horodecki, R. Horodecki, J. Oppenheim, A. Sen, U. Sen, B. Synak-Radtke, Phys. Rev. A 71 (2005) 062307. [5] J. Niset, N.J. Cerf, Phys. Rev. A 74 (2006) 052103. [6] H. Ollivier, W.H. Zurek, Phys. Rev. Lett. 88 (2001) 017901. [7] M. Piani, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 100 (2008) 090502. [8] A. Ferraro, L. Aolito, D. Cavalcanti, F.M. Cucchietti, A. Acín, Phys. Rev. A 81 (2010) 052318. [9] S. Luo, Phys. Rev. A 77 (2008) 022301. [10] [11] [12] [13] [14] [15] [16]
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