Entanglement in dimerized and frustrated spin-one Heisenberg chains

Entanglement in dimerized and frustrated spin-one Heisenberg chains Zhe Sun, XiaoGuang Wang and You-Quan Li Zhejiang Institute of Modern Physics, Department of Physics, Zhejiang University, HangZhou 310027, People’s Republic of China E-mail: [email protected] New Journal of Physics 7 (2005) 83

Received 10 January 2005 Published 24 March 2005 Online at http://www.njp.org/ doi:10.1088/1367-2630/7/1/083

Abstract. We investigate entanglement properties in dimerized and frustrated

spin-one models by applying the concept of negativity. On the basis of the relation between negativity and correlators, negativities are numerically calculated. We study the effects of dimerization and frustration on entanglement. We also consider the case of finite temperature, and determine the threshold temperature before which the thermal state is doomed to be entangled.

Contents

1. Introduction 2. Negativity and correlators 3. Dimerized model 3.1. Negativity at a low temperature . 3.2. Negativity at finite temperatures 4. Frustrated model 4.1. Negativity at a low temperature . 4.2. Negativity at finite temperatures 5. Conclusions Acknowledgment References

New Journal of Physics 7 (2005) 83 1367-2630/05/010083+14$30.00

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 3 4 4 4 5 6 9 12 12 12

PII: S1367-2630(05)92628-1 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

2

DEUTSCHE PHYSIKALISCHE GESELLSCHAFT

1. Introduction

Since Haldane predicted that the one-dimensional Heisenberg chain has a spin gap for integer spins [1], the physics of quantum spin chains has been the subject of many theoretical and experimental studies [2]–[4]. Moreover, the spin-one chain described by the bilinear–biquadratic model exhibits a very rich phase diagram [5]. Recently, the study of entanglement properties in Heisenberg chains received much attention [6]–[46]. Quantum entanglement lies at the heart of quantum mechanics and can be exploited to accomplish some physical tasks such as quantum teleportation [47]. Spin-half systems have been considered in most of these studies. However, due to the lack of entanglement measure for higher spin systems, the entanglement in these systems has been less studied. There are many ongoing works on entanglement in spin-one chains. Fan et al [32] and Verstraete et al [33] studied entanglement in the bilinear–biquadratic model with a special value of γ = 1/3, i.e., the AKLT model [2]. Zhou et al [34] studied entanglement with the bilinear–biquadratic Hamiltonian for the case of two spins. Wang et al [46] obtained some analytical results of entanglement in spin-one chains. Dimerized systems play an important role in condensed matter theory, and entanglement in dimerized systems has been considered [31, 37]. Frustrated systems with next-nearest-neighbour (NNN) interactions are also important in condensed matter physics. It is true that there exist some quasi-one-dimensional compounds offering us systems with NNN interactions. Bose and Chattopadhyay [42] and Gu et al [35] have investigated entanglement in frustrated spin-half Heisenberg chains. In this paper, by using the concept of negativity [48, 49], we study entanglement in both the dimerized and frustrated spin-one Heisenberg models. The dimerized system is described by the following Hamiltonian: Hd =

N/2

[J1 (S2i−1 · S2i ) + J2 (S2i · S2i+1 )],

(1)

i=1

where the Si is the ith spin-one operator and J1 and J2 denote the coupling strengths. The number of spins is chosen to be even, and the periodic boundary condition is adopted here. The frustrated model is described by the following Hamiltonian: Hf =

N

[J1 (Si · Si+1 ) + J2 (Si · Si+2 )],

(2)

i=1

where J1 and J2 are the nearest-neighbour (NN) exchange interaction and the NNN interaction respectively. Obviously, both Hamiltonians are SU(2)-invariant. Entanglement in a system with a few spins displays the general features of entanglement with more spins. For instance, in the anisotropic Heisenberg model with a large number of qubits, the pairwise entanglement shows a maximum in the isotropic point [35]. This feature was already shown in a small system with four or five qubits [36]. So, the study of small systems is meaningful in the study of entanglement as they may reflect the general features of larger or macroscopic systems. Also, due to limitations of our computation capability, we concentrate only on small systems such as 4-, 5- and 6-spin models. New Journal of Physics 7 (2005) 83 (http://www.njp.org/)

3


We study entanglement in the thermal state. The state of a system at thermal equilibrium described by the density operator ρ(T ) = exp(−βH )/Z, where β = 1/kB T , kB is the Boltzmann’s constant, which is assumed to be 1 throughout the paper, and Z = Tr{exp(−βH )} is the partition function. The entanglement in the thermal state is called thermal entanglement. We organize the paper as follows: in section 2, we give an introduction of negativity and the relation between negativity and two correlators for the SU(2)-invariant states. In sections 3 and 4, we study the effects of dimerization and frustration on entanglement, respectively. The conclusion is given in section 5. 2. Negativity and correlators

For the case of higher spins, a non-entangled state has necessarily a positive partial transpose (PPT) according to the Peres–Horodecki criterion [48]. In the case of two spin-halves, and the case of (1/2,1) mixed spins, a PPT is also sufficient. However, in the case of two spin-one particles, a PPT is not sufficient. Nevertheless, the negative partial transpose (NPT) gives a sufficient condition for entanglement and, due to the SU(2) symmetry in our systems, the NPT is expected to fully capture the entanglement properties. The Peres–Horodecki criterion gives an effective qualitative way for judging whether the state is entangled. The quantitative version of the criterion was developed by Vidal and Werner [49]. They presented a measure of entanglement called negativity that can be computed efficiently, and the negativity does not increase under local manipulations of the system. The negativity of a state ρ is defined as N (ρ) = |µi |, (3) i

where µi is the negative eigenvalue of ρT2 , and T2 denotes the partial transpose with respect to the second subsystem. The negativity N is related to the trace norm of ρT2 through the relation ρT2 1 − 1 , (4) 2 where the trace norm of ρT2 is equal to the sum of the absolute values of the eigenvalues of ρT2 . For an arbitrary pure state, we can write it in If N > 0, then the two-spin state is entangled. ⊗ c |e |e , where cα > 0. Then the negativity is given by [49] its Schmidt form | = M α α=1 α α   2 M 1 N (||) = cα − 1 . (5) 2 α=1 N (ρ) =

√ For the maximally entangled state cα = 1/ M, then N (||) = (M − 1)/2. Clearly, the maximal value of negativity for two qutrits is 1. Schliemann has analysed the Peres–Horodecki criterion in systems with SU(2) symmetry [44]. The SU(2)-invariant state of two spin-one particles takes the following form [44]: 1 H |S = 1, Sz S = 1, Sz | ρ = G|S = 0, Sz = 0S = 0, Sz = 0| + 3 S =−1 z

+

1−G−H 5

2

|S = 2, Sz S = 2, Sz |,

Sz =−2

New Journal of Physics 7 (2005) 83 (http://www.njp.org/)

(6)

4


where |S, Sz denotes a state of total spin S and z component Sz and G, H are two parameters. From this simple form, one can derive the negativity, which is given by [46] N (ij) =

1 2

max [0, (Si · Sj )2 − 2] + 13 max [0, 1 − (Si · Sj ) − (Si · Sj )2 ].

(7)

We see that, for the SU(2)-invariant state, the negativity is determined by two correlators (Si · Sj ) and (Si · Sj )2 . This expression greatly facilitates the following study of entanglement properties.

3. Dimerized model

In this section, we study entanglement in the dimerized model. From the expression given by equation (7), the two correlators completely determine the negativity. On the basis of this relation and the exact diagonalization method, we can numerically compute the negativities. When the negativity is larger than zero, the state is doomed to be entangled. 3.1. Negativity at a low temperature In figure 1, we plot the negativities N (12) and N (23) . Note that, in all the plots in this paper, we take J1 = 1. From figure 1(a), we see that, for the four-spin model, the negativity N (12) reaches its maximum 1 at J2 = 0. Clearly, when J2 = 0, the chain is reduced to two uncoupled spin-one pairs. In this situation, the negativity can be analytically obtained and N (12) = 1 at zero temperature [46], consistent with our numerical results. When the exchange interaction J2 increases after J2 = 0, N (12) decreases quickly and vanishes at about J2 = 2.48. The case of J2 < 0 is also considered, and we find that, with the increase in absolute value of J2 , N (12) decreases from the maximum to a steady value of about 0.4 without attaining zero. The negativity N (23) behaves quite differently. We can see that, in the region of J2 0, N (23) remains at zero; N (23) is a little larger than zero just when J2 = 0.35, and then becomes nonzero and increases gradually to the limit of 1. Now look at the six-spin case. The plot of negativity versus J2 is shown in figure 1(b). The behaviours of the negativity are very similar to those of the four-spin case. The visible difference is that, when J2 < 0, with the increase of |J2 |, N (12) goes to a steady value about N (12) = 0.037, which is much less than that in the four-spin model. Obviously, the increase of exchange interaction J2 suppresses the entanglement of spins coupled by J1 . When J2 > 0, the increase of J2 can enhance the entanglement between spins coupled by J2 , but when J2 < 0, the increase of |J2 | cannot enhance N (23) . 3.2. Negativity at finite temperatures At a larger finite temperature, because of the mixture of less entangled excited states to the ground state, entanglement of the system will be weakened. We plot the negativities N (12) and N (23) as a function of the exchange interaction coefficient J2 and the temperature T in figures 2 and 3, respectively. From figure 2, due to the thermal fluctuation and the exchange interaction J2 , N (12) is suppressed. The negativity N (12) has nonzero values in the region of J < 2.48 and T < 1.37. We may notice the curve on the J2 –T plane just lying on the boundary of the zero and nonzero New Journal of Physics 7 (2005) 83 (http://www.njp.org/)

5


1

1

0.9

0.9 (23)

0.8

0.8

N

(23)

N

N

0.6

0.6 0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

−4

N

(12)

0.5

0 −6

N

(12)

0.7

0.7

−2

0

2

4

6

8

0 −5 −4 −3 −2 −1

J2 (a)

0 1 J2

2

3

4

5

(b)

Figure 1. Negativity versus the exchange interaction coefficient J2 at the low

temperature T = 0.05 in the four-spin dimerized system (a) and a six-spin model (b). N (12) and N (23) denote the negativities between the spins coupled by J1 and between the spins coupled by J2 , respectively. negativities. It just shows Tth changes when J2 varies in the regions of J2 < 2.48, where Tth is a threshold temperature at which the negativity just vanishes. From another point of view, there exists a threshold J2th and, from the curve on the J2 –T plane, we can also find that J2th is not affected greatly by varying the temperature. In figure 3, we show how N (23) behaves at finite temperatures. Also, the thermal fluctuation suppresses the negativity to zero finally. But just because the exchange interaction J2 helps to enhance the N (23) , there should exist a competition between the thermal fluctuation and the interaction J2 . So, the threshold Tth increases nearly linearly when J2 increases, which is in sharp contrast with the behaviours of N (12) . 4. Frustrated model

For the frustrated model, we may also use equation (7) to compute the negativity as there also exists an SU(2) symmetry in the model. In the frustrated model, there is a competition between the NN and NNN interactions. So, the model displays some special features of entanglement. New Journal of Physics 7 (2005) 83 (http://www.njp.org/)

6


1 0.9 0.8 0.7

N

0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 T

2

3

2

1

0

−1 J2

−2

−4

−3

−5

−6

Figure 2. Negativity N (12) versus the exchange interaction coefficient J2 and the

temperature T in the four-spin dimerized model.

1 0.8

N

0.6 0.4 0.2 0 8 7 6 5 4 3 2 1 0 −1 12 J2

10

6

8

4

2

0

T

Figure 3. Negativity N (23) versus the exchange interaction coefficient J2 and the

temperature T in the four-spin dimerized model.

4.1. Negativity at a low temperature In figure 4, we plot the negativities versus J2 in the four-spin system at a lower temperature. The negativity N (12) keeps a steady value 1/3 when J2 increases from J2 < 0 to J2 = 0.5 and, then, New Journal of Physics 7 (2005) 83 (http://www.njp.org/)

7


1

0.8 0.7

N

0.6 0.5 0.4 0.3

−3 −4 −5 −6 −7 −8 −9 −10 −1

N(13)

E

0.9

−0.5

0 J2

0.5

1

N(12)

0.2 0.1 0 −1

−0.8 −0.6 −0.4 −0.2

0 J2

0.2

0.4

0.6

0.8

1

Figure 4. N (12) and N (13) versus the NNN interaction coefficient J2 at the

temperature T = 0.02 in the four-spin system. N (12) and N (13) denote the negativities of NN and NNN spins, respectively. The inset shows the groundstate energy and a few nearest excited energies versus J2 .

N (12) falls to zero. In contrast, the negativity N (13) goes up to a nonzero value of 1 at the point J2 = 0.5. This behaviour is due to the energy level cross at the point J2 = 0.5, as seen clearly from the inset to figure 4. Before and after the critical point, the entanglements display distinct behaviours. For the five-spin system, numerical results of negativities versus J2 are plotted in figure 5. From the inset, we observe that there are two level-crossing points J2 = −3.32 and J2 = 1, which is a particular feature of the five-spin system. These two critical points greatly affect the behaviours of entanglement. Over the region between the two points, N (12) stays at a steady value of about 0.124, and out of the region, N (12) falls to zero. For the negativity N (13) , it only obtains a nonzero value after J2 = 1, and before that, it always remains at zero. Moreover, the nonzero values of N (12) and N (13) are the same. For larger systems such as the six-spin chain, as shown in figure 6, the negativities show very different behaviours from the former cases. When J2 < 0, as |J2 | increases, N (12) tends to a steady value of about 0.223, and it gets the maximum value of 0.263 near J2 = −0.3. At J2 ≈ 0.366, N (12) displays a decrease. With N (12) decreasing, N (13) gets a little jump from zero and then increases gradually to 0.312. N (12) disappears after about J2 = 2.22. The complex rich behaviours of N (12) may be due to the very close energy levels as shown in the inset. There is a level crossing at about J2 = 0.366. The approximate degenerate energy levels can remarkably change the occupation probability of a eigenstate in the thermal state even at a very low temperature, thereby greatly affecting the negativity.


8


−5

0.2

0.16 0.14

E

0.18

−10 −15 −20 −25 −4

−3

−2

−1

0 J2

1

2

3

4

N

0.12 N(12)

0.1

N(13)

0.08 0.06 0.04 0.02 0 −4

−3

−2

−1

0 J2

1

2

3

4


temperature T = 0.02 in the five-spin system. N (12) and N (13) denote the negativities of NN and NNN spins, respectively. The inset shows the groundstate energy and a few nearest excited energies versus J2 . 0.45 −10

0.35

−20

E

0.4

−30

0.3

−4

−3

−2

−1

0.25 N

0.366 1

2

3

J2 N

0.2

(13)

N

(12)

0.15 0.1 0.05 0 −0.05 −4

−3

−2

−1

0

1 J2

2

3

4

5


temperature T = 0.05 in the four-spin system. N (12) and N (13) denote the negativities of NN and NNN spins, respectively. The inset shows the groundstate energy and a few nearest excited energies versus J2 . New Journal of Physics 7 (2005) 83 (http://www.njp.org/)

9


0.35 0.3 0.25

N

0.2 0.15 0.1 0.05 0 0

0.5

1

1.5 T

2

2.5 1

0.5

−0.5

0

−1

J2

Figure 7. Negativity N (12) versus the NNN interaction J2 and the temperature T

in a four-spin system.

1.2 1

N

0.8 0.6 0.4 0.2 0

1

0 0.8

0.6 J2

1 0.4

2 0.2 3

T


in the four-spin system. 4.2. Negativity at finite temperatures We consider the effects from both the thermal fluctuation and the NNN interaction. Similar to the dimerized model, thermal fluctuation suppresses the entanglement, while the NNN interaction has different effects on the N (12) and N (13) . In figure 7, we show the negativity N (12) versus J2 and T in the four-spin system. With increasing temperature, the negativity is suppressed until it vanishes. The NNN interaction has New Journal of Physics 7 (2005) 83 (http://www.njp.org/)

10


0.14 0.12 0.1

N

0.08 0.06 0.04 0.02 0 0 0.5

T

1 1.5

−4

−2

0

2

4

J2


in the five-spin system. 0.14 0.12 0.1

N

0.08 0.06 0.04 0.02 0

4

3

2

1 J2

0

−1 4

3

2 T

1

0

Figure 10. Negativity N (13) versus the NNN interaction J2 and the temperature

T in the five-spin system. a frustration effect on N (12) . So, only in the region J2 < 0.5 and T < 2.04, can the negativity survive. On the J2 –T plane, from the curve just detaching the nonzero and zero negativities, we can see that the threshold temperature Tth is a decreasing function of J2 . On the other hand, N (13) (see figure 8) does not appear before J2 = 0.5. The thermal fluctuation suppresses and finally completely destroys the negativity. But the NNN interaction enhances the entanglement of NNN spins, so the threshold temperature Tth is an increasing function of J2 . New Journal of Physics 7 (2005) 83 (http://www.njp.org/)

11


0.35 0.3 0.25

N

0.2 0.15 0.1 0.05 0 0 1 2 3

T

4

3

2

−1

0

1

−2

−3

−4

J2


T in the six-spin system.

0.35 0.3 0.25

N

0.2 0.15 0.1 0.05 0 5

4

3 J2

2

1

0

−1 6

5

4

2

3

1

0

T


T in the six-spin system.

In figure 9, we present the negativity N (12) versus J2 and T in the five-spin system. The negativity N (12) here exhibits some strange phenomena. It has only a nonzero value approximately in the region of −3.32 < J2 < 1 and T < 1.15. N (12) is suppressed by the thermal fluctuation, but is not as smooth as in the four-spin case. We see that it exhibits some ridges. The threshold temperature Tth does not behave as a monotonic function of J2 and takes its maximum at about J2 = − 1. The NNN negativity N (13) is shown in figure 10. The main property is very New Journal of Physics 7 (2005) 83 (http://www.njp.org/)

12


similar to the case of the four-spin case, except that N (13) exhibits a dent and is larger than zero when J2 > 1. Figure 11 shows N (12) versus J2 and T in a six-spin frustrated model. When J2 < 2.22 and T < 2.32, N (12) exists. Obviously, the region of nonzero negativity displays two distinct parts, which are due to a special energy-level structure as shown in figure 6. The threshold temperature Tth behaves as a decreasing function of J2 . The NNN negativity N (13) versus J2 and T is shown in figure 12. The negativity vanishes when J2 < 0.42 at any temperature. The threshold temperature Tth is increased by increasing J2 , similar to the cases of four and five spins. 5. Conclusions

In this paper, we have studied the properties of entanglement in both the dimerized and frustrated spin-one models. On the basis of the relation between negativity and correlators, we compute the negativities by the exact diagonalization method. In dimerized models, the maximal negativity is 1, and the interaction J2 suppresses N (12) but enhances N (23) . When we increase the size of the dimerized model to six spin, the behaviours of negativity are not changed significantly. At different finite temperatures, it is true that both N (12) and N (23) are suppressed by the thermal fluctuation. We have numerically determined the threshold temperature Tth , before which the state is doomed to be entangled. For the frustrated models, we have considered the cases from four to six spins. At low temperature, the NNN interaction frustrates the NN negativity N (12) and enhances the NNN negativity N (13) . The negativities display a strong jump up or down at some critical points of J2 , which has close relations with the energy level crossing. Specifically, for the case of the five-spin system, there exist two critical points: J2 = −3.32 and J2 = 1, at which the negativity N (12) falls to zero. In the six-spin frustrated model, N (12) has a particular sharp decrease near J2 = 0.336 just due to its approximate degenerate energy-level structure. In our numerical studies, although we cannot confirm whether the state is entangled or not in the region of zero negativity, we believe that the behaviour of negativities fully captures the key properties of entanglement in our dimerized and frustrated spin-one models as these models display an SU(2) symmetry. It is more interesting to study the effects of dimerization and frustration in higher spin systems and to study spin models without SU(2) symmetry, which is under consideration. Acknowledgment

This work was supported by the National Natural Science Foundation of China under grant no 10405019. References [1] Haldane F D M 1983 Phys. Lett. 93A 464 Haldane F D M 1983 Phys. Rev. Lett. 50 1153 [2] Affleck I, Kennedy T, Lieb E H and Tasaki H 1987 Phys. Rev. Lett. 59 799 [3] Millet P, Mila F, Zhang F C, Mambrini M, Van Oosten A B, Pashchenko V A, Sulpice A and Stepanov A 1999 Phys. Rev. Lett. 83 4176 New Journal of Physics 7 (2005) 83 (http://www.njp.org/)

13 [4] [5] [6] [7] [8] [9]

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]

[37] [38] [39] [40] [41] [42] [43] [44]


Lou J Z, Xiang T and Su Z B 2000 Phys. Rev. Lett. 85 2380 Schollwöck, Jolicoeur T and Garel T 1996 Phys. Rev. B 53 3304 Nielsen M A 1998 PhD Thesis University of Mexico (Preprint quant-ph/0011036) Arnesen M C, Bose S and Vedral V 2001 Phys. Rev. Lett. 87 017901 Gunlycke D, Kendon V M, Vedral V and Bose S 2001 Phys. Rev. A 64 042302 Wang X 2001 Phys. Rev. A 64 012313 Wang X 2001 Phys. Lett. A 281 101 Wang X and Zanardi P 2002 Phys. Lett. A 301 1 Wang X 2002 Phys. Rev. A 66 044305 Wang X, Fu H and Solomon A I 2001 J. Phys. A: Math. Gen. 34 11307 Wang X and Mølmer K 2002 Eur. Phys. J. D 18 385 Jaeger G, Sergienko A V, Saleh B E A and Teich M C 2003 Phys. Rev. A 68 022318 Bose S and Vedral V 2000 Phys. Rev. A 61 040101 Kamta G L and Starace A F 2002 Phys. Rev. Lett. 88 107901 O’Connor K M and Wootters W K 2001 Phys. Rev. A 63 0520302 Meyer D A and Wallach N R 2001 Preprint quant-ph/0108104 Osborne T J and Nielsen M A 2002 Phys. Rev. A 66 032110 Osterloh A, Amico L, Falci G and Fazio R 2002 Nature 416 608 Sun Y, Chen Y G and Chen H 2003 Phys. Rev. A 68 044301 Glaser U, Büttner H and Fehske H 2003 Phys. Rev. A 68 032318 Santos L F 2003 Phys. Rev. A 67 062306 Yeo Y 2002 Phys. Rev. A 66 062312 Khveshchenko D V 2003 Phys. Rev. B 68 193307 Zhou L, Song H S, Guo Y Q and Li C 2003 Phys. Rev. A 68 024301 Brennen G K and Bullock S S 2004 Phys. Rev. A 70 52303 Xin R, Song Z and Sun C P 2004 Preprint quant-ph/0411177 T´oth G 2005 Phys. Rev. A 71 010301 Verstraete F, Popp M and Cirac J I 2004 Phys. Rev. Lett. 92 027901 Verstraete F, Martin-Delgado M A and Cirac J I 2004 Phys. Rev. Lett. 92 087201 Vidal J, Palacios G and Mosseri R 2004 Phys. Rev. A 69 022107 Lambert N, Emary C and Brandes T 2004 Phys. Rev. Lett. 92 073602 Ghose S, Rosenbaum T F, Aeppli G and Coppersmith S N 2003 Nature 425 48 Brukner C, Vedral V and Zeilinger A 2004 Preprint quant-ph/0410138 Brukner C and Vedral V 2004 Preprint quant-ph/0406040 Fan H, Korepin V and Roychowdhury V 2004 Phys. Rev. Lett. 93 227203 Verstraete F, Mart´ın-Delgado M A and Cirac J I 2004 Phys. Rev. Lett. 92 087201 Zhou L, Yi X X, Song H S and Quo Y Q 2003 Preprint quant-ph/0310169 Gu S J, Lin H Q and Li Y Q 2003 Phys. Rev. A 68 042330 Wang X 2004 Phys. Rev. E 69 066118 Wang X 2004 Phys. Lett. A 329 439 Wang X 2005 Phys. Lett. A 334 352 Chen Y, Zanardi P, Wang Z D and Zhang F C 2004 Preprint quant-ph/0407228 Vidal G, Latorre J I, Rico E and Kitaev A 2003 Phys. Rev. Lett. 90 227902 Wootters W K 1998 Phys. Rev. Lett. 80 2245 Kouzoudis D 1997 J. Magn. Magn. Mater. 173 259 Kouzoudis D 1998 J. Magn. Magn. Mater. 189 366 Bärwinkel K, Schmidt H-J and Schnack J 2000 J. Magn. Magn. Mater. 220 227 Bose I and Chattopadhyay E 2002 Phys. Rev. A 66 062320 Lin H Q 1990 Phys. Rev. B 42 6561 Schliemann J 2003 Phys. Rev. A 68 012309


14 [45] [46] [47] [48]


Kay A, Lee D K K, Pachos J K, Plenio M B, Reuter M E and Rico E 2004 Preprint quant-ph/0407121 Wang X, Li H B, Sun Z and Li Y Q 2005 Preprint quant-ph/0501032 Bennett C H, Brassard G, Crepeau C, Jozsa R, Peres A and Wootters W K 1993 Phys. Rev. Lett. 70 1895 Peres A 1996 Phys. Rev. Lett. 77 1413 Horodecki M, Horodecki P and Horodecki R 1996 Phys. Lett. A 223 1 [49] Vidal G and Werner R F 2002 Phys. Rev. A 65 032314
