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Entanglement of one-dimensional spin chains
10 June 2002
Physics Letters A 298 (2002) 219–224 www.elsevier.com/locate/pla
Entanglement of one-dimensional spin chains Tiefeng Xu ∗ , Yu Zhou, Yaxun Zhou, Qiuhua Nie Faculty of Information Science and Engineering, Ningbo University, Zhejiang 315211, PR China Received 2 October 2001; received in revised form 26 March 2002; accepted 27 March 2002 Communicated by P.R. Holland
Abstract Entanglement in one-dimensional Heisenberg model is investigated by Bethe ansatz method. By this approach, the reduced density matrix between any pair of spins in the chain can be obtained for ground state, and excited states as well. Especially, the reduced density matrix for nearest-neighbor spins has the block-diagonal form. Therefore, we can compute the concurrence of spin pairs in the chain steadily. 2002 Elsevier Science B.V. All rights reserved. PACS: 03.65.Ud; 75.10.Jm Keywords: Entanglement; Spin chain; Bethe ansatz
1. Introduction Recently quantum entanglement has become an active research field because of its key role played in quantum information science [1]. Most of the current studies focus on the following two aspects. The first one concerns with some more fundamental problems of entanglement, such as the conceptual foundation of entanglement and the definition of the measure of entanglement. These are hard problems and still widely debated. The second one is about the computation of the entanglement for a realistic physical model, for example, one-dimensional spin chains with nearest-neighbor interactions [2,3]. This is the foundation of every proposed experimental schemes to realize entanglement employed in quantum computation and quantum information processing. Due to its wide use and relative simplicity for mathematical treatment, entanglement of the one-dimensional Heisenberg model has been discussed by several authors [2,4–8]. In this Letter, we reconsider the problem by the Bethe ansatz [9] (and [10] for a pedagogical introduction) method. For one-dimensional quantum systems, Bethe ansatz is a very powerful tool which gives both wavefunctions and eigenvalues for ground state, and excited states as well. By partial tracing over the density matrix of the whole system, the reduced density matrix for any pair of spins in the chain can be obtained. We find that the reduced density matrix of the nearest-neighbor spins has the block diagonal form which is valid for both ground state and excited states. Therefore, by Wootters’ formulae [13], we can compute concurrence which is a reasonable measure for any pair of spins in a spin- 21 chain. * Corresponding author.
E-mail address: [email protected] (T. Xu). 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 2 ) 0 0 4 2 9 - 2
220
T. Xu et al. / Physics Letters A 298 (2002) 219–224
2. Reduced density matrix of spin pairs in the chain Consider a N -site Heisenberg spin- 21 chain with periodic boundary condition: H = −J
N
Sn · Sn+1 ,
(1)
n=1
where J > 0 and SN+1 = S1 . The Hilbert space of the chain has the dimension of 2N spanned by the orthogonal basis vectors |σ1 , σ2 , . . . , σN , where σn = h2¯ Sn which represents an up spin when σn =↑ and a down spin when z N σn =↓ at site n. Because of the conservation of the z-component of the total spins S z = N n=1 Sn , the 2 basis z vectors can be sorted according to the quantum number S = N/2 − r, where r is the number of down spins. The general form of the wavefunctions for the states with r spins flipped is supposed to be (Bethe ansatz) [9]: |Ψ = (2) a(n1 , . . . , nr )|n1 , . . . , nr , 1n1 <···
where the coefficients a(n1 , . . . , nr ) =
exp i
P ∈Sr
r j =1
kPj nj + i/2
θPj Pi ,
(3)
i
are characterized by r quasi-momentum kj (j = 1, 2, . . . , r) and one phase angle θij = −θj i for each (ki , kj ) pair. All these parameters are determined by the following equations (called the Bethe ansatz equations) coming from the eigenvalue equation and the translational invariance of the model: 2 cot θij /2 = cot ki /2 − cot kj /2, (4) Nki = 2πλi + θij , i, j = 1, . . . , r, j =i
where λi ∈ {0, 1, 2, . . . , N − 1}. The density matrix associated with the state |Ψ is then given by ρ= a(n1 , . . . , nr )a ∗ n1 , . . . , nr |n1 , . . . , nr n1 , . . . , nr .
(5)
1n1 <···
The reduced density matrix for spins located at sites i and j (i < j ) is obtained by tracing out all of the spins in the chain except that at sites i and j : ρ ij = (6) σN , . . . , 0j , . . . , 0i , . . . , σ1 |ρ|σ1 , . . . , 0i , . . . , 0j , . . . , σN , σ1 ,...,0i ,...,0j ,...,σN
where 0i (0j ) means missing of σi (σj ). For every pair of nearest-neighbor spins, entanglement is same because of the translational invariance of the chain. For the simplicity of computation, let i = 1 and j = 2. The reduced density matrix for a pair of nearest-neighbor spins is thus given by ρ 12 = σN , . . . , σ3 |ρ|σ3 , . . . , σN σ3 ,...,σN
=
1n1 <···
a(n1 , . . . , nr )a n1 , . . . , nr σN , . . . , σ3 |n1 , . . . , nr nr , . . . , n1 |σ3 , . . . , σN . ∗
σ3 ,...,σN
(7)
The sum yields sixteen terms which correspond to sixteen matrix elements of the reduced two-spin density matrix. It should be emphasized that the symbols σN , . . . , σ3 |n1 , . . . , nr and nr , . . . , n1 |σ3 , . . . , σN do not represent
T. Xu et al. / Physics Letters A 298 (2002) 219–224
221
inner products because the lengths (sites) of the right and left state vectors are not equal. Explicitly, nr , . . . , n1 |σ3 , . . . , σN = 2 m 1 n b, σN , . . . , σ3 |n1 , . . . , nr = a|m1 |n2 ,
(8)
where the state vectors in the right-hand side are tensor products of the state vectors at site 1 and 2. Coefficients a and b are the products of a series of Kronecker delta symbols. If the number of indexes n1 , n2 , . . . , nr and n1 , n2 , . . . , nr locating at sites 1 and 2 (or equivalently, at sites 3, 4, . . . , N , are not equal, the sum (9) a(n1 , . . . , nr )a ∗ n1 , . . . , nr σN , . . . , σ3 |n1 , . . . , nr nr , . . . , n1 σ3 , . . . , σN , σ3 ,...,σN
is zero because of the summations over the Kronecker delta symbols with different indexes. Therefore, the summations are greatly simplified and can be rewritten as 12 a(n1 , . . . , nr )a ∗ n1 , . . . , nr ρ = n1 ,n2 ;n1 ,n2
n3 <···
σ3 ,...,σN
× σN , . . . , σ3 |n1 , . . . , nr nr , . . . , n1 σ3 , . . . , σN
.
(10)
The summation over n1 , n2 , n1 , n2 produces sixteen terms which correspond to the following sixteen combinations according to the location of indexes n1 , n2 , n1 and n2 : (1) (3) (5) (7) (9) (11) (13) (15)
n1 = 1, n1 = 1, n1 = 1, n1 = 1, n1 = 2, n1 = 2, n1 , n2 3, n1 , n2 3,
n1 = 1, n1 = 2, n1 = 1, n1 = 2, n1 = 1, n1 = 2, n1 = 1, n1 = 2,
n2 = 2, n2 = 2, n2 3, n2 3, n2 3, n2 3,
n2 = 2; n2 3; n2 = 2; n2 3; n2 = 2; n2 3; n2 = 2; n2 3;
(2) (4) (6) (8) (10) (12) (14) (16)
n1 = 1, n1 = 1, n1 = 1, n1 = 1, n1 = 2, n1 = 2, n1 , n2 3, n1 , n2 3,
n2 = 2, n2 = 2, n2 3, n2 3, n2 3, n2 3,
n1 = 1, n1 , n2 3; n1 = 1, n1 , n2 3; n1 = 1, n1 , n2 3; n1 = 1, n1 , n2 3.
n2 3; n2 3; n2 3; n2 3;
It can be checked that all terms except that corresponding to (1), (6), (7), (10), (11) and (16) are zero because for these terms the number of n1 and n2 locating at sites 1 and 2 are different from that of n1 and n2 locating at the same sites. The reduced density matrix finally appears as ρ 12 = ρ11 |↑1 |↑22 ↑|1 ↑| + ρ22 |↑1 |↓22 ↓|1 ↑| + ρ23 |↑1 |↓22 ↑|1 ↓| + ρ32 |↓1 |↑22 ↓|1 ↑| + ρ44 |↓1 |↓22 ↓|1 ↓|,
(11)
where the density matrix elements are given by a(n1 , . . . , nr )a ∗ n1 , . . . , nr σN , . . . , σ3 |n1 , . . . , nr nr , . . . , n1 σ3 , . . . , σN , ρ11 = 3n1 <···
ρ22 =
3n2 <···
a(2, n2, . . . , nr )a ∗ 2, n2 , . . . , nr σN , . . . , σ3 |n2 , . . . , nr × nr , . . . , n2 σ3 , . . . , σN ,
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T. Xu et al. / Physics Letters A 298 (2002) 219–224
ρ33 =
a(1, n2, . . . , nr )a ∗ 1, n2 , . . . , nr σN , . . . , σ3 |n2 , . . . , nr
3n2 <···
ρ44 =
× nr , . . . , n2 σ3 , . . . , σN , a(1, 2, n3, . . . , nr )a ∗ 1, 2, n3 , . . . , nr σN , . . . , σ3 |n3 , . . . , nr
3n3 <···
ρ23 =
× nr , . . . , n3 σ3 , . . . , σN , a(2, n2, . . . , nr )a ∗ 1, n2 , . . . , nr σN , . . . , σ3 |n2 , . . . , nr
3n2 <···
ρ32 =
× nr , . . . , n2 σ3 , . . . , σN , a(1, n2, . . . , nr )a ∗ 2, n2 , . . . , nr σN , . . . , σ3 |n2 , . . . , nr
3n2 <···
× nr , . . . , n2 σ3 , . . . , σN ,
(12)
where the vectors |ni , nj , . . . , nr (and their dual vectors) contain all the sites except site 1 and site 2. If we label the four orthogonal two-spin states as |↑1 |↑2 ≡ |1,
|↑1 |↓2 ≡ |2,
|↓1 |↑2 ≡ |3,
|↓1 |↑2 ≡ |4,
(13)
ρ 12
then, in this basis, the density matrix has the following block-diagonal form: ρ11 0 0 0 0 ρ ρ 0 22 23 ρ 12 = , 0 ρ32 ρ33 0 0 0 0 ρ44
(14)
where ρ22 = ρ33 and ρ23 = ρ32 .1 This is a very general result independent of the number of down spins r and also valid for antiferromagnetic chain.
3. Entanglement of the chain For a biparticle system in a pure quantum state |ψ, it’s reasonable to measure the entanglement of the system by the Von Neumann entropy of either of its two parts [11] E(ψ) = − Tr(ρ log2 ρ),
(15)
where ρ is the partial trace of the density matrix of the whole system |ψψ|. For mixed states, the entanglement can be defined by the concurrence introduced by Wootters [12,13] who demonstrated that there is a one-to-one correspondence between the concurrence and the entanglement of formation [14]. The value of the concurrence ranges from zero for an unentangled state to unity for a maximally entangled state. Let ρ 12 be the reduced twoparticle (spin) density matrix. The concurrence is defined as C = max{0, λ1 − λ2 − λ3 − λ4 }, 1 Indicated by the referee.
(16)
T. Xu et al. / Physics Letters A 298 (2002) 219–224
223
Table 1 Bethe ansatz solutions and concurrences for N = 5, r = 2 λ1
λ2
k
k1
k2
0 0 0 0 0
0 1 2 3 4
0 2π/5 4π/5 6π/5 8π/5
0 0 0 0 0
0 2π/5 4π/5 6π/5 8π/5
0 0 0 0 0
θ
0.25359 0 0 0 0
1 1 2 1 4
3 4 4 1 4
8π/5 0 2π/2 4π/5 6π/5
1.70533 π/2 2.96196 2π/5 + i1.19891 8π/5 + i1.19891
3.32122 3π/2 4.57786 2π/5 − i1.19891 2π/5 − i1.19891
2.24344 π/2 2.24344 5.99457 5.99457i
0.46420 0 0.46420 0 0
Concurrence
where the quantities λi (i = 1, 2, 3, 4) are the square roots of the eigenvalues in descending order of the product matrix 12 = ρ 12 (σ1y ⊗ σ2y )ρ ∗12 (σ1y ⊗ σ2y ),
(17)
and σiy (i = 1, 2) are Pauli matrices. Though 12 is not necessarily Hermitian, the eigenvalues of 12 are real and non-negative. Obviously, the ground state |↑ · · · ↑ is disentangled. For the excited states with one spin flipped (r = 1), the reduced density matrices are of the form N−2 0 0 0 N 1 ik 1 0 0 . N Ne ρ 12 = (18) 1 1 −ik 0 e 0 N N 0 0 0 0 Actually, for the excited states with r = 1, ρ 12 also represents the reduced density matrix between any pair of spins in the chain. This can be confirmed similar to the case of nearest-neighbor spins. The concurrence computed by ρ 12 is 2/N . This can also be understood in an alternative way. The Bethe ansatz wavefunction for r = 1 |ψ =
N 1 ikn e |n, N
(19)
n=1
is the generalization of the so-called W state [15] with N qubits. For such kind of states, it has been shown that the concurrence equals to 2/N [7] (and references therein). Furthermore, for a N -qubit symmetric state in which each pair out of N(N − 1)/2 possible combinations in the chain is entangled, the maximum concurrence is also equal to 2/N [16]. Therefore the excited state with r = 1 provides an example of such a symmetric state with maximum entanglement. For higher excited states r 2, the computation of concurrence becomes more complicated. Here we outline the main steps of computation procedure. They include: (1) Solving the Bethe ansatz equations (4) to get the quasi-momenta ki (i = 1, 2, . . . , r) and phases θij which are the parameters characterizing the wavefunctions or reduced density matrices; (2) Computing the matrix elements ρij by Eq. (13) (only sixteen possible non-zero terms); (3) Computing concurrence by Wootters’ formulae (17) and (18). Finally we give an example of r = 2 and N = 5 to end the discussion. The result is listed in the Table 1.
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4. Conclusions We have applied the Bethe ansatz method to discuss the entanglement of one-dimensional Heisenberg model measured by concurrence. By Bethe ansatz wavefunctions, the reduced density matrices for pairs of two spins in the chain are obtained. For a nearest-neighbor spin pair, the reduced density matrix has the block-diagonal form. These are general results which are valid for ground state, and excited states as well. We have also computed the concurrence of the Heisenberg chain by Wootters’ formulae. Therefore, by Bethe ansatz approach, the concurrence for any eigenstate of the Heisenberg chain can be computed steadily. The discussion of the present Letter focuses on the computational aspect of the entanglement of the Heisenberg chain. Further connections between entanglement and other physical properties of the model will be explored in subsequent papers.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
See, for example, M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000. K.M. O’Connor, W.K. Wootters, Phys. Rev. A 63 (2001) 052302. W.K. Wooters, quant-ph/0001114. A. Osterloh, L. Amico, G. Falci, R. Fazio, quant-ph/0202029. X. Wang, H. Fu, A.I. Solomon, J. Phys. A 34 (2001) 11307. G.L. Kamta, A.F. Starace, Phys. Rev. Lett. 88 (2002) 107901. X. Wang, Phys. Rev. A 64 (2001) 012313. M.C. Arnesen, S. Bose, V. Vedral, Phys. Rev. Lett. 87 (2001) 017901. H. Bethe, Z. Phys. 71 (1931) 205. M. Karbach, G. Müller, cond-mat/9809162. C.H. Bennett, H.J. Bernstein, S. Popescu, B. Schumacher, Phys. Rev. A 53 (1996) 2046. S. Hill, W.K. Wootters, Phys. Rev. Lett. 78 (1997) 5022. W.K. Wooters, Phys. Rev. Lett. 80 (1998) 2245. C.H. Bennett, D.P. DiVincenzo, J. Smolin, W.K. Wootters, Phys. Rev. A 54 (1996) 3824. W. Dür, G. Vidal, J.I. Cirac, Phys. Rev. A 62 (2000) 062314. M. Koashi, V. Buzek, N. Imoto, Phys. Rev. A 62 (2000) 050302.
Physics Letters A 298 (2002) 219–224 www.elsevier.com/locate/pla
Entanglement of one-dimensional spin chains Tiefeng Xu ∗ , Yu Zhou, Yaxun Zhou, Qiuhua Nie Faculty of Information Science and Engineering, Ningbo University, Zhejiang 315211, PR China Received 2 October 2001; received in revised form 26 March 2002; accepted 27 March 2002 Communicated by P.R. Holland
Abstract Entanglement in one-dimensional Heisenberg model is investigated by Bethe ansatz method. By this approach, the reduced density matrix between any pair of spins in the chain can be obtained for ground state, and excited states as well. Especially, the reduced density matrix for nearest-neighbor spins has the block-diagonal form. Therefore, we can compute the concurrence of spin pairs in the chain steadily. 2002 Elsevier Science B.V. All rights reserved. PACS: 03.65.Ud; 75.10.Jm Keywords: Entanglement; Spin chain; Bethe ansatz
1. Introduction Recently quantum entanglement has become an active research field because of its key role played in quantum information science [1]. Most of the current studies focus on the following two aspects. The first one concerns with some more fundamental problems of entanglement, such as the conceptual foundation of entanglement and the definition of the measure of entanglement. These are hard problems and still widely debated. The second one is about the computation of the entanglement for a realistic physical model, for example, one-dimensional spin chains with nearest-neighbor interactions [2,3]. This is the foundation of every proposed experimental schemes to realize entanglement employed in quantum computation and quantum information processing. Due to its wide use and relative simplicity for mathematical treatment, entanglement of the one-dimensional Heisenberg model has been discussed by several authors [2,4–8]. In this Letter, we reconsider the problem by the Bethe ansatz [9] (and [10] for a pedagogical introduction) method. For one-dimensional quantum systems, Bethe ansatz is a very powerful tool which gives both wavefunctions and eigenvalues for ground state, and excited states as well. By partial tracing over the density matrix of the whole system, the reduced density matrix for any pair of spins in the chain can be obtained. We find that the reduced density matrix of the nearest-neighbor spins has the block diagonal form which is valid for both ground state and excited states. Therefore, by Wootters’ formulae [13], we can compute concurrence which is a reasonable measure for any pair of spins in a spin- 21 chain. * Corresponding author.
E-mail address: [email protected] (T. Xu). 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 2 ) 0 0 4 2 9 - 2
220
T. Xu et al. / Physics Letters A 298 (2002) 219–224
2. Reduced density matrix of spin pairs in the chain Consider a N -site Heisenberg spin- 21 chain with periodic boundary condition: H = −J
N
Sn · Sn+1 ,
(1)
n=1
where J > 0 and SN+1 = S1 . The Hilbert space of the chain has the dimension of 2N spanned by the orthogonal basis vectors |σ1 , σ2 , . . . , σN , where σn = h2¯ Sn which represents an up spin when σn =↑ and a down spin when z N σn =↓ at site n. Because of the conservation of the z-component of the total spins S z = N n=1 Sn , the 2 basis z vectors can be sorted according to the quantum number S = N/2 − r, where r is the number of down spins. The general form of the wavefunctions for the states with r spins flipped is supposed to be (Bethe ansatz) [9]: |Ψ = (2) a(n1 , . . . , nr )|n1 , . . . , nr , 1n1 <···
where the coefficients a(n1 , . . . , nr ) =
exp i
P ∈Sr
r j =1
kPj nj + i/2
θPj Pi ,
(3)
i
are characterized by r quasi-momentum kj (j = 1, 2, . . . , r) and one phase angle θij = −θj i for each (ki , kj ) pair. All these parameters are determined by the following equations (called the Bethe ansatz equations) coming from the eigenvalue equation and the translational invariance of the model: 2 cot θij /2 = cot ki /2 − cot kj /2, (4) Nki = 2πλi + θij , i, j = 1, . . . , r, j =i
where λi ∈ {0, 1, 2, . . . , N − 1}. The density matrix associated with the state |Ψ is then given by ρ= a(n1 , . . . , nr )a ∗ n1 , . . . , nr |n1 , . . . , nr n1 , . . . , nr .
(5)
1n1 <···
The reduced density matrix for spins located at sites i and j (i < j ) is obtained by tracing out all of the spins in the chain except that at sites i and j : ρ ij = (6) σN , . . . , 0j , . . . , 0i , . . . , σ1 |ρ|σ1 , . . . , 0i , . . . , 0j , . . . , σN , σ1 ,...,0i ,...,0j ,...,σN
where 0i (0j ) means missing of σi (σj ). For every pair of nearest-neighbor spins, entanglement is same because of the translational invariance of the chain. For the simplicity of computation, let i = 1 and j = 2. The reduced density matrix for a pair of nearest-neighbor spins is thus given by ρ 12 = σN , . . . , σ3 |ρ|σ3 , . . . , σN σ3 ,...,σN
=
1n1 <···
a(n1 , . . . , nr )a n1 , . . . , nr σN , . . . , σ3 |n1 , . . . , nr nr , . . . , n1 |σ3 , . . . , σN . ∗
σ3 ,...,σN
(7)
The sum yields sixteen terms which correspond to sixteen matrix elements of the reduced two-spin density matrix. It should be emphasized that the symbols σN , . . . , σ3 |n1 , . . . , nr and nr , . . . , n1 |σ3 , . . . , σN do not represent
T. Xu et al. / Physics Letters A 298 (2002) 219–224
221
inner products because the lengths (sites) of the right and left state vectors are not equal. Explicitly, nr , . . . , n1 |σ3 , . . . , σN = 2 m 1 n b, σN , . . . , σ3 |n1 , . . . , nr = a|m1 |n2 ,
(8)
where the state vectors in the right-hand side are tensor products of the state vectors at site 1 and 2. Coefficients a and b are the products of a series of Kronecker delta symbols. If the number of indexes n1 , n2 , . . . , nr and n1 , n2 , . . . , nr locating at sites 1 and 2 (or equivalently, at sites 3, 4, . . . , N , are not equal, the sum (9) a(n1 , . . . , nr )a ∗ n1 , . . . , nr σN , . . . , σ3 |n1 , . . . , nr nr , . . . , n1 σ3 , . . . , σN , σ3 ,...,σN
is zero because of the summations over the Kronecker delta symbols with different indexes. Therefore, the summations are greatly simplified and can be rewritten as 12 a(n1 , . . . , nr )a ∗ n1 , . . . , nr ρ = n1 ,n2 ;n1 ,n2
n3 <···
σ3 ,...,σN
× σN , . . . , σ3 |n1 , . . . , nr nr , . . . , n1 σ3 , . . . , σN
.
(10)
The summation over n1 , n2 , n1 , n2 produces sixteen terms which correspond to the following sixteen combinations according to the location of indexes n1 , n2 , n1 and n2 : (1) (3) (5) (7) (9) (11) (13) (15)
n1 = 1, n1 = 1, n1 = 1, n1 = 1, n1 = 2, n1 = 2, n1 , n2 3, n1 , n2 3,
n1 = 1, n1 = 2, n1 = 1, n1 = 2, n1 = 1, n1 = 2, n1 = 1, n1 = 2,
n2 = 2, n2 = 2, n2 3, n2 3, n2 3, n2 3,
n2 = 2; n2 3; n2 = 2; n2 3; n2 = 2; n2 3; n2 = 2; n2 3;
(2) (4) (6) (8) (10) (12) (14) (16)
n1 = 1, n1 = 1, n1 = 1, n1 = 1, n1 = 2, n1 = 2, n1 , n2 3, n1 , n2 3,
n2 = 2, n2 = 2, n2 3, n2 3, n2 3, n2 3,
n1 = 1, n1 , n2 3; n1 = 1, n1 , n2 3; n1 = 1, n1 , n2 3; n1 = 1, n1 , n2 3.
n2 3; n2 3; n2 3; n2 3;
It can be checked that all terms except that corresponding to (1), (6), (7), (10), (11) and (16) are zero because for these terms the number of n1 and n2 locating at sites 1 and 2 are different from that of n1 and n2 locating at the same sites. The reduced density matrix finally appears as ρ 12 = ρ11 |↑1 |↑22 ↑|1 ↑| + ρ22 |↑1 |↓22 ↓|1 ↑| + ρ23 |↑1 |↓22 ↑|1 ↓| + ρ32 |↓1 |↑22 ↓|1 ↑| + ρ44 |↓1 |↓22 ↓|1 ↓|,
(11)
where the density matrix elements are given by a(n1 , . . . , nr )a ∗ n1 , . . . , nr σN , . . . , σ3 |n1 , . . . , nr nr , . . . , n1 σ3 , . . . , σN , ρ11 = 3n1 <···
ρ22 =
3n2 <···
a(2, n2, . . . , nr )a ∗ 2, n2 , . . . , nr σN , . . . , σ3 |n2 , . . . , nr × nr , . . . , n2 σ3 , . . . , σN ,
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ρ33 =
a(1, n2, . . . , nr )a ∗ 1, n2 , . . . , nr σN , . . . , σ3 |n2 , . . . , nr
3n2 <···
ρ44 =
× nr , . . . , n2 σ3 , . . . , σN , a(1, 2, n3, . . . , nr )a ∗ 1, 2, n3 , . . . , nr σN , . . . , σ3 |n3 , . . . , nr
3n3 <···
ρ23 =
× nr , . . . , n3 σ3 , . . . , σN , a(2, n2, . . . , nr )a ∗ 1, n2 , . . . , nr σN , . . . , σ3 |n2 , . . . , nr
3n2 <···
ρ32 =
× nr , . . . , n2 σ3 , . . . , σN , a(1, n2, . . . , nr )a ∗ 2, n2 , . . . , nr σN , . . . , σ3 |n2 , . . . , nr
3n2 <···
× nr , . . . , n2 σ3 , . . . , σN ,
(12)
where the vectors |ni , nj , . . . , nr (and their dual vectors) contain all the sites except site 1 and site 2. If we label the four orthogonal two-spin states as |↑1 |↑2 ≡ |1,
|↑1 |↓2 ≡ |2,
|↓1 |↑2 ≡ |3,
|↓1 |↑2 ≡ |4,
(13)
ρ 12
then, in this basis, the density matrix has the following block-diagonal form: ρ11 0 0 0 0 ρ ρ 0 22 23 ρ 12 = , 0 ρ32 ρ33 0 0 0 0 ρ44
(14)
where ρ22 = ρ33 and ρ23 = ρ32 .1 This is a very general result independent of the number of down spins r and also valid for antiferromagnetic chain.
3. Entanglement of the chain For a biparticle system in a pure quantum state |ψ, it’s reasonable to measure the entanglement of the system by the Von Neumann entropy of either of its two parts [11] E(ψ) = − Tr(ρ log2 ρ),
(15)
where ρ is the partial trace of the density matrix of the whole system |ψψ|. For mixed states, the entanglement can be defined by the concurrence introduced by Wootters [12,13] who demonstrated that there is a one-to-one correspondence between the concurrence and the entanglement of formation [14]. The value of the concurrence ranges from zero for an unentangled state to unity for a maximally entangled state. Let ρ 12 be the reduced twoparticle (spin) density matrix. The concurrence is defined as C = max{0, λ1 − λ2 − λ3 − λ4 }, 1 Indicated by the referee.
(16)
T. Xu et al. / Physics Letters A 298 (2002) 219–224
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Table 1 Bethe ansatz solutions and concurrences for N = 5, r = 2 λ1
λ2
k
k1
k2
0 0 0 0 0
0 1 2 3 4
0 2π/5 4π/5 6π/5 8π/5
0 0 0 0 0
0 2π/5 4π/5 6π/5 8π/5
0 0 0 0 0
θ
0.25359 0 0 0 0
1 1 2 1 4
3 4 4 1 4
8π/5 0 2π/2 4π/5 6π/5
1.70533 π/2 2.96196 2π/5 + i1.19891 8π/5 + i1.19891
3.32122 3π/2 4.57786 2π/5 − i1.19891 2π/5 − i1.19891
2.24344 π/2 2.24344 5.99457 5.99457i
0.46420 0 0.46420 0 0
Concurrence
where the quantities λi (i = 1, 2, 3, 4) are the square roots of the eigenvalues in descending order of the product matrix 12 = ρ 12 (σ1y ⊗ σ2y )ρ ∗12 (σ1y ⊗ σ2y ),
(17)
and σiy (i = 1, 2) are Pauli matrices. Though 12 is not necessarily Hermitian, the eigenvalues of 12 are real and non-negative. Obviously, the ground state |↑ · · · ↑ is disentangled. For the excited states with one spin flipped (r = 1), the reduced density matrices are of the form N−2 0 0 0 N 1 ik 1 0 0 . N Ne ρ 12 = (18) 1 1 −ik 0 e 0 N N 0 0 0 0 Actually, for the excited states with r = 1, ρ 12 also represents the reduced density matrix between any pair of spins in the chain. This can be confirmed similar to the case of nearest-neighbor spins. The concurrence computed by ρ 12 is 2/N . This can also be understood in an alternative way. The Bethe ansatz wavefunction for r = 1 |ψ =
N 1 ikn e |n, N
(19)
n=1
is the generalization of the so-called W state [15] with N qubits. For such kind of states, it has been shown that the concurrence equals to 2/N [7] (and references therein). Furthermore, for a N -qubit symmetric state in which each pair out of N(N − 1)/2 possible combinations in the chain is entangled, the maximum concurrence is also equal to 2/N [16]. Therefore the excited state with r = 1 provides an example of such a symmetric state with maximum entanglement. For higher excited states r 2, the computation of concurrence becomes more complicated. Here we outline the main steps of computation procedure. They include: (1) Solving the Bethe ansatz equations (4) to get the quasi-momenta ki (i = 1, 2, . . . , r) and phases θij which are the parameters characterizing the wavefunctions or reduced density matrices; (2) Computing the matrix elements ρij by Eq. (13) (only sixteen possible non-zero terms); (3) Computing concurrence by Wootters’ formulae (17) and (18). Finally we give an example of r = 2 and N = 5 to end the discussion. The result is listed in the Table 1.
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T. Xu et al. / Physics Letters A 298 (2002) 219–224
4. Conclusions We have applied the Bethe ansatz method to discuss the entanglement of one-dimensional Heisenberg model measured by concurrence. By Bethe ansatz wavefunctions, the reduced density matrices for pairs of two spins in the chain are obtained. For a nearest-neighbor spin pair, the reduced density matrix has the block-diagonal form. These are general results which are valid for ground state, and excited states as well. We have also computed the concurrence of the Heisenberg chain by Wootters’ formulae. Therefore, by Bethe ansatz approach, the concurrence for any eigenstate of the Heisenberg chain can be computed steadily. The discussion of the present Letter focuses on the computational aspect of the entanglement of the Heisenberg chain. Further connections between entanglement and other physical properties of the model will be explored in subsequent papers.
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