Impurity entanglement in three-qubit Heisenberg XX chain

20 May 2002

Physics Letters A 297 (2002) 291–299 www.elsevier.com/locate/pla

Impurity entanglement in three-qubit Heisenberg XX chain Xiao-qiang Xi a,b,∗ , San-ru Hao a , Wen-xue Chen b , Rui-hong Yue a a Institute of Modern Physics, Northwest University, P.O. Box 105, Xi’an, 710069, China b Fundamental Department of Xi’an Institute of Posts and Telecommunications, Xi’an, 710061, China

Received 31 August 2001; accepted 12 December 2001 Communicated by P.R. Holland

Abstract We studied the influence of the impurity to the entanglement in three-qubit Heisenberg XX chain and got the analytical expression of the concurrence C, go a step further, for fixed J , we discussed the concrete √ effect of the impurity parameter J1 to the concurrence. At the condition T → 0, when |J1 |  |J |, the concurrence C = (2 2 − 1)/4; when J1 → −J and J → ∞, the concurrence C get its maximum value 1/2.  2002 Elsevier Science B.V. All rights reserved. PACS: 03.65.Ud; 03.65.Bz; 75.10.Jm

1. Introduction Many of the interesting properties of the quantum systems are attributable to the existence of entanglement. Entanglement is responsible for the non-local correlations which can exist between spatially separated quantum systems, as is revealed by the violation of Bell’s inequality [1]. Quantum entanglement has come to be viewed as a significant resource for quantum information processing, such as quantum teleportation [2] and super-dense cording [3], quantum computational speed-ups [4,5] and certain quantum cryptographic protocols [6,7]. For pure states, we know almost all the information about it. We can easily distinguish entangled and UN entangled states: a pure state is entangled if and only if its state vector cannot be expressed as a product of its parts. It has been shown that every entangled pure state violates some Bell-type inequality [8]. It is convenient to measure the pure state’s entanglement by calculating its entropy. Unfortunately, the pure state is only the special case of the mixed state, in practice, a pure state will be change into mixed state under non-unitary evolution. For further use, we must know all the properties of the mixed state. While the theory of mixed-state entanglement is more complicated and less well understood than that of pure-state entanglement. Though there have been much interesting work in the field of mixed-state entanglement—such as purification [9], concentration [10,11], and distillable entanglement [12,13], which are all belong to the manipulation of entanglement; measure of the entanglement, such as the entanglement of formation and the concurrence [9,14], negativity [15]; thermal entanglement in two-qubit quantum * Corresponding author.

E-mail address: [email protected] (X.-q. Xi). 0375-9601/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 8 4 3 - X

292

X.-q. Xi et al. / Physics Letters A 297 (2002) 291–299

spin models, such as Heisenberg XXX [16,17], XX [18], XXZ model [19] and Ising model in a magnetic field [20], more than two-qubit quantum spin models, see Refs. [21,22]—there still have plenty of work to be explored in mixed-state entanglement. The Heisenberg chain has been used to construct a quantum computer and quantum dots [23], nuclear spins [24], electronic spins [25] and optical lattices [26]. By suitable coding, the Heisenberg interaction alone can be used for quantum computation [27,28]. Heisenberg XY chain [29] is one of the most charming model: the interaction Hamiltonian between two quantum dots is just the XY Hamiltonian, the effective Hamiltonian can be used to construct the C-NOT gate [30] and swap gate [18]. The XY chain is also realized in the quantum-Hall system [31] and in cavity QED system [32]. The entanglement in the ground state of the Heisenberg model has been discussed in Ref. [21], Wang [18] studied the entanglement in the time evolution of the XX chain. Here we consider the entanglement of quantum Heisenberg XX chain [29] with impurity [33]. Impurity plays an important role in the 1D quantum system, even a small defect may destroy the whole properties of the quantum system. So it is meaningful to investigate the entanglement of the impurity. The state of the system at thermal equilibrium is represented by the density operator ρ(T ) = exp(−H /kT )/Z, where Z = tr[exp(−H /kT )/Z)] is the partition function, H is the system Hamiltonian, k is the Boltzmann’s constant, and T is the temperature. As ρ(T ) represents a thermal state, the entanglement in the state is called thermal entanglement [16,17]. In this Letter, we introduce impurity to the thermal state, so we call the entanglement impurity and thermal entanglement.

2. The solution of the XX chain with impurity The S = 1/2 XX chain H=

N


 y y  x Jn Snx Sn+1 + Sn Sn+1

(1)

n=1

is one of the simplest quantum many-body system conceivable. In this Letter, we do not consider the XX chain with homogeneous nearest-neighbor coupling (Ji ≡ J ), while consider the impurity spin locating at the boundary of the system J1 = JN+1 ≡ J1 ,

Ji ≡ J

for i (mod(N))
2.

(2)

Here we only consider the case N = 3, using the relations S α = σ α /2(α = x, y, z) and σ ± = (σ x ± σ y )/2, Hamiltonian (1) is rewritten as H=

3


  − + Jn σn+ σn+1 + σn+1 σn− .

(3)

n=1

In fact, the above Hamiltonian is a three-qubit Heisenberg ring. The first site is impurity and others are normal particles. For convenience, we write Eq. (3) more concrete as       H = J1 σ1+ σ2− + σ2+ σ1− + J σ2+ σ3− + σ3+ σ2− + J1 σ3+ σ1− + σ1+ σ3− . (4) The first step of studying the impurity and thermal entanglement is to get all the eigenvalues and eigenstates of Hamiltonian (4). Jordan–Wigner transformation [34] is the fundamental method to solve the eigenvalues problem of the XX model. In the three-qubit case, this problem can be solved easily. When J1 = J , the results are similar as in Ref. [22], when J1 = J , the eigenvalues are   J + J 2 + 8J12 J − J 2 + 8J12 , E2 = E5 = , E1 = E4 = E0 = E7 = 0, 2 2

X.-q. Xi et al. / Physics Letters A 297 (2002) 291–299

E3 = E6 = −J,

293

(5)

and the corresponding eigenstates are given by C1 B1 B1 |001 + |010 + |100, A1 A1 A1 √ √ 2 2 C2 B2 B2 |010 − |100, |Ψ2  = |001 + |010 + |100, |Ψ3  = A2 A2 A2 2 2 C1 B1 B1 C2 B2 B2 |Ψ4  = |110 + |101 + |011, |Ψ5  = |110 + |101 + |011, A1 A1 A1 A2 A2 A2 √ √ 2 2 |Ψ6  = |101 − |011, |Ψ7  = |111, 2 2

|Ψ0  = |000,

|Ψ1  =

where the parameters Ai , Bi , i = 1, 2, are    2  A1 = 2 J 2 + 2J12 + J J 2 + 8J12 + 3J1 J  B1 = J 2 + 2J12 + J J 2 + 8J12 , C1 = 3J1 J    2  A2 = 2 J 2 + 2J12 − J J 2 + 8J12 + 3J1 J  B2 = J 2 + 2J12 − J J 2 + 8J12 , C2 = 3J1 J

(6)

 2 + J1 J 2 + 8J12 ,  + J1 J 2 + 8J12 ,  2 − J1 J 2 + 8J12 ,  − J1 J 2 + 8J12 .

(7)

3. The impurity and thermal entanglement Before we talk about the impurity and thermal entanglement, let us have a brief review of EoF (entanglement of formation) and concurrence [14,35]. Concurrence C range from zero to one and it is monotonically relate to EoF, so that concurrence C is a kind of measure of entanglement. Let ρ12 (mixed or pure) be the density matrix of the pair. The concurrence corresponding to the density matrix is defined as C = max{λ1 − λ2 − λ3 − λ4 , 0},

(8)

where λi , i = 1, 2, 3, 4, are the square roots of the eigenvalues of the operator  y y ∗  y y ρ12 = ρ12 σ1 ⊗ σ2 ρ12 σ1 ⊗ σ2

(9)

in descending order. The eigenvalues of ρ12 are real and non-negative even though ρ12 is not necessarily Hermitian. When the concurrence C = 0, ρ12 is an UN entangled state; when C = 1, ρ12 is a maximally entangled state. Let Boltzmann’s constant be 1, we write the density matrix of the thermal equilibrium state as ρ(T ) =

  7 H 1
1 Ei exp − = |Ψi Ψi |. exp Z T Z T

(10)

i=0

From Eqs. (5), we get the partition function as Z = 2 + 2e

 −(J + J 2 +8J12 )/2T

+ 2e

 −(J − J 2 +8J12 )/2T

+ 2eJ /T .

(11)

294

X.-q. Xi et al. / Physics Letters A 297 (2002) 291–299

Take (5) into (10), we write the explicit expression of the density matrix of the state:  7 1
Ei ρ(T ) = exp |Ψi Ψi | Z T i=0



  1

−(J + J 2 +8J12 )/2T  = |Ψ1 Ψ1 | + |Ψ4 Ψ4 | |Ψ0 Ψ0 | + |Ψ7 Ψ7 | + e Z     −(J − J 2 +8J12 )/2T  +e |Ψ2 Ψ2 | + |Ψ5 Ψ5 | + eJ /T |Ψ3 Ψ3 | + |Ψ6 Ψ6 | .

(12)

We concern the pairwise entanglement between the impurity site 1 and the normal site 2, so we need to get the (xy) reduced density ρ12 = tr3 (ρ(T )). We denote the reduced matrix tr3 [|Ψx Ψx | + |Ψy Ψy |] as ρ12 . From Eqs. (6), we have  2    B1 0 0 0 1 0 0 0 1  0 B12 + C12 2B1 C1 0  0 0 0 0 (07) (14) , ρ12 = 2  ρ12 =  , 2 2  0 0 0 0 0 2B1 C1 B1 + C1 0 0  A1 0 0 0 1 0 0 0 B12  2    0 0 0 B2 1 0 0 0 2 2 1  0 B2 + C2 1 0 1 0 0 2B2 C2 0  (25) (36) , ρ12 (13) ρ12 = 2 =  . 2 + C20   C B 0 0 2B 2 0 0 1 0 A2 2 2 2 2 0 0 0 1 0 0 0 B22 From Eqs. (12) and (13), we get 

 J 2 +8J12 )/2T

1  (07) −(J + J 2 +8J12 )/2T (14) −(J − ρ +e ρ12 + e ρ12 (T ) = Z 12   v 0 0 0 1 0 w y 0 =  , Z 0 y w 0 0 0 0 v

(25)

(36)

ρ12 + eJ /T ρ12



(14)

where v, w, y are  J 2 +8J12 )/2T

B 2 −(J + 1 v = 1 + eJ /T + 12 e 2 A1

+

 J 2 +8J12 )/2T

B 2 + C 2 −(J + 1 w = eJ /T + 1 2 1 e 2 A1 y=

2B1 C1 A21

e

 −(J + J 2 +8J12 )/2T

+

2B2 C2 A22

e

B22 A22 +

e

 −(J − J 2 +8J12 )/2T

B22 + C22 A22

e

,

 −(J − J 2 +8J12 )/2T

 −(J − J 2 +8J12 )/2T

.

, (15)

∗ (σ ⊗ σ ) are The square roots of the eigenvalues of ρ12 = ρ12 (σ1 ⊗ σ2 )ρ12 1 2 y

λ1 =

w+y , Z

λ2 =

w−y , Z

λ3 = λ4 =

y

v . Z

From Eqs. (8), (11), (15) and (16), we get the concurrence    2 C = max |y| − v . Z

y

y

(16)

(17)

X.-q. Xi et al. / Physics Letters A 297 (2002) 291–299

295

Fig. 1. The concurrence of T → 0; J range from −100 to 100, J1 range from −200 to −102. The maximum concurrence is 1/2 when J1 → −|J |.

Fig. 2. The concurrence of finite temperature T = 25; J range from −100 to 100, J1 range from −200 to −102.

Case I. J1 < 0. In this region, y > 0, we take Eqs. (15) into Eq. (17) and get  C = max

  −(J + J 2 +8J12 ) −(J − J 2 +8J12 ) 2 2 2 2 (2B1 C1 − B1 )/A1 z + (2B2 C2 − B2 )/A2 z   2 −(J + J +8J12 ) −(J − J 2 +8J12 ) 2J

1+z

+z

− (1/2)z2J − 1

 ,0 ,

+z

(18) where z = e1/2T . After calculating Eq. (18), we get: In the ferromagnetic (J > 0) and antiferromagnetic (J < 0) case, when T → 0, the entanglement exist at the condition |J1 | > |J |. If J1 → −|J |, the concurrence attains its maximum value 1/2, so is the entanglement (entanglement is a monotonic increasing function of the concurrence C). If |J1 |  |J |, the concurrence arrive at √ (2 2 − 1)/4, see Fig. 1. At the finite temperature, there also exist entanglement, see Fig. 2.

296

X.-q. Xi et al. / Physics Letters A 297 (2002) 291–299

Fig. 3. The concurrence of T → 0; J range from −100 to 100, J1 range from 102 to 200. The maximum concurrence is 1/2 when J1 → −|J |.

Fig. 4. The concurrence of finite temperature T = 25; J range from −100 to 100, J1 range from 102 to 200.

Case II. J1 > 0. In this region, y < 0, we take Eqs. (15) into Eq. (17) and get  C = max

(−2B1 C1 − B12 )/A21 z

 −(J + J 2 +8J12 )

1+z

+ (−2B2 C2 − B22 )/A22 z

 −(J + J 2 +8J12 )

+z

 −(J − J 2 +8J12 )

 −(J − J 2 +8J12 )

− (1/2)z2J − 1

 ,0 ,

+ z2J (19)

where z = e1/2T . After calculating Eq. (19), we get: In the ferromagnetic (J > 0) and antiferromagnetic (J < 0) case, when T → √ 0, the entanglement exist at the condition J1 > |J |. If J1  |J |, the concurrence arrive at its maximum value (2 2 − 1)/4, see Fig. 3. At the finite temperature, there also exist entanglement, see Fig. 4.

4. Conclusions In this Letter, we get the analytical expression of the concurrence in the three-qubit Heisenberg XX chain with impurity.

X.-q. Xi et al. / Physics Letters A 297 (2002) 291–299

297

Fig. 5. T → 0, J = 1, J1 < 0, where J1 ranges from −1000 to 0 (left) and from −2 to −1 (right).

Fig. 6. T = 25, J = 1, J1 < 0, where J1 ranges from −500 to 0 (left) and from −100 to −20 (right).

Fig. 7. T → 0, J = 1, J1 > 0, where J1 ranges from 0 to 2000 (left) and from 0 to 2 (right).

If J1 = J , the impurity site is a normal site, the conclusion has been shown in Ref. [22], we list the conclusion here: the XX model is thermally entanglement if and only if J < −0.7866T ; maximum entanglement is attended when T → 0 or J → −∞, pairwise thermal entanglement occurs only in the ferromagnetic case [22]. If J1 = −J , the pairwise entanglement of ρ12 is the same as J1 = J . If J1 = ±J , the pairwise entanglement between impurity site and normal site is richful. Firstly, J1 < 0, at the condition T → 0, when |J1 | > |J |, the thermal entanglement exist at ferromagnetic and antiferromagnetic case: when J1 → −J and J → ∞, the entanglement gets its maximum value, concurrence C = 1/2; when |J1 |  |J |,

298

X.-q. Xi et al. / Physics Letters A 297 (2002) 291–299

Fig. 8. T = 25, J = 1, J1 > 0, where J1 ranges from 0 to 2000 (left) and from 0 to 2 (right).

√ C = (2 2 − 1)/4. At finite temperature, we can get the corresponding conclusion through plot the 3D figure, as Figs. 2 and 4. Secondly, J1 > 0, at the condition T → 0, √ the entanglement exist when J1 > |J |, and the entanglement attains its maximum value, concurrence C = (2 2 − 1)/4, if J1  |J |. For finite temperature we can plot the 3D figure. In order to express our results more concretely, we plot the 2D figures between J1 and concurrence C for fixed J = 1 (Figs. 5–8). The difference between the following two figures lies in the range of J1 . Through Figs. 5–8, for fixed J , we can control the entanglement by changing the impurity’s coupling constant J1 to satisfy our need. It is also interesting to discuss the entanglement between the normal sites.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

J.S. Bell, Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy, 1987. C.H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70 (1993) 1895. C.H. Bennett, S.J. Wiesner, Phys. Rev. Lett. 69 (1992) 2881. P.W. Shor, SIAM J. Comput. 26 (5) (1997) 1484. L.K. Grover, Phys. Rev. Lett. 79 (1997) 325. A.K. Ekert, Phys. Rev. Lett. 67 (1991) 661. D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, A. Sanpera, Phys. Rev. Lett. 77 (1996) 2818. N. Gisin, Phys. Lett. A 154 (1991) 201. C.H. Bennett, D.P. Vincenzo, J.A. Smolin, W.K. Wootters, Phys. Rev. A 54 (1996) 3824. C.H. Bennett, H.J. Bernstein, S. Popescu, B. Schumacher, Phys. Rev. A 53 (1996) 2046. R.T. Thew, W.J. Munro, quant-ph/0012049. E.M. Rains, Phys. Rev. A 60 (1999) 173. V. Vedral, M.B. Plenio, M.A. Rippen, P.L. Knight, Phys. Rev. Lett. 78 (1997) 2275. W. Wootters, Phys. Rev. Lett. 80 (1998) 2245. K. Zyczkowski, P. Horodecki, A. Sanpera, M. Lewenstion, Phys. Rev. A 58 (1998) 883. M.C. Arnesen, S. Bose, V. Vedral, quant-ph/0009060. M.A. Nielsen, Ph.D. thesis, University of Mexico (1998); M.A. Nielsen, quant-ph/0011036. X.G. Wang, Phys. Rev. A 64 (2001) 012313. X.G. Wang, Phys. Rev. A 281 (2001) 101. D. Gunlycke, S. Bose, V.M. Kendon, V. Vedral, quant-ph/0102137. E.M. O’Connor, W.K. Wootters, quant-ph/0009041. X.G. Wang, H.C. Fu, A.I. Solomon, quant-ph/0105057; X.G. Wang, K. Molmer, quant-ph/0106145. D. Loss, D.P. Divincenzo, Phys. Rev. A 57 (1998) 120; G. Burkard, D. Loss, D.P. Divincenzo, Phys. Rev. B 59 (1999) 2070.

X.-q. Xi et al. / Physics Letters A 297 (2002) 291–299

[24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]

B.E. Kane, Nature 393 (1998) 133. R. Vrijen et al., quant-ph/9905096. A. Sorensen, K. Molmer, Phys. Rev. Lett. 83 (1999) 2274. D.A. Lidar, D. Bacon, K.B. Whaley, Phys. Rev. Lett. 82 (1999) 4556. D.P. Divincenzo, D. Bacon, J. Kempe, G. Burkard, K.B. Whaley, Nature 408 (2000) 339. E. Lieb, T. Schultz, D. Mattis, Ann. Phys. (NY) 16 (1961) 407. A. Imamoglu, D.D. Awshalom, G. Burkard, D.P. Divincenzo, D. Loss, M. Sherwin, A. Small, Phys. Rev. Lett. 83 (1999) 4204. V. Privman, I.D. Vagner, G. Kventsel, Phys. Lett. A 239 (1998) 141. S.B. Zheng, G.C. Guo, Phys. Rev. Lett. 85 (2000) 2392. J. Stolze, M. Vogel, Phys. Rev. B 61 (2000) 4026. P. Jordan, E. Wigner, Z. Phys. 47 (1928) 631. S. Hill, W.K. Wootters, Phys. Rev. Lett. 78 (1997) 5022; V. Coffman, J. Kundu, W.K. Wootters, Phys. Rev. A 61 (2000) 052306.

299