Bipartite and Tripartite Entanglement in a Three-Qubit Heisenberg Model

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1 Commun. Theor. Phys. (Beijing, China) 46 (006) pp c International Academic Publishers Vol. 46, No. 6, December 5, 006 Bipartite and Tripartite Entanglement in a Three-Qubit Heisenberg Model REN ie and ZHU Shi-Qun School of Physical Science and Technology, Suzhou University, Suzhou 5006, China (Received anuary 6, 006) Abstract The bipartite and tripartite entanglement in a three-qubit Heisenberg XY model with a nonuniform magnetic field is studied. There are two or four peaks in the concurrence of the bipartite entanglement when the amplitudes of the magnetic fields are differently distributed between the three qubits. It is very interesting to note that there is no tangle of tripartite entanglement between the three qubits when the amplitudes of the magnetic fields are varied. However, the variation of the magnetic field direction can induce the tangle. The tangle is periodic about the angle between the magnetic field and the z axis of the spin. PACS numbers: Ud, Lx, 75.0.m Key words: bipartite entanglement, tripartite entanglement, Heisenberg XY model Introduction The entanglement is an important resource in the fields of quantum computation and quantum information. [ 3] Due to its potential applications, the thermal entanglement of Heisenberg model has been extensively studied in recent years. [4 6] The effects of anisotropy and magnetic field on the entanglement were investigated in a two-qubit Heisenberg XY chain. [5,6] In quantum teleportation, [7] dense coding, [8] and quantum cloning, [9] various kinds of entanglement states of three-qubit Heisenberg models were studied. [0 4] The thermal entanglement of pairwise qubits and three-qubit Heisenberg model with a magnetic impurity was investigated. [5,6] The bipartite entanglement was studied intensively while the multipartite entanglement still needs further investigation. To understand the multipartite entanglement, the distributed entanglement has been presented. [7] In most of the previous investigations, the magnetic field is uniformly distributed. However, the effect of a nonuniform magnetic field on the bipartite entanglement and the multipartite entanglement in a Heisenberg model also needs to be investigated. In this paper, the thermal entanglement in a Heisenberg XY model with a nonuniform magnetic field is studied. The three-qubit entanglement of the ground state is investigated. In Sec., the basic measures of the entanglement of two-qubit and three-qubit are presented. In Sec. 3, the thermal entanglement of the nearest neighboring qubits is analyzed in a nonuniform magnetic field. In Sec. 4, the effect of the magnetic field directions is explored. A discussion concludes the paper. Basic Measures of Entanglement The three-qubit Heisenberg model with a nonuniform magnetic field can be described by the Hamiltonian H = 3 (σ x nσn+ x + σnσ y y n+ ) + B (σz + σ3) z + B (σz cos α + σ x sin α), () n= where B and B are the external magnetic fields of different magnitudes, is the strength of the Heisenberg interaction, > 0 and < 0 correspond to the antiferromagnetic and ferromagnetic cases respectively, σ β i (β = x, y, z) are the Pauli matrices at qubit of i =,, 3, α is the angle between B and the z axis of the spin. The state of the system at thermal equilibrium of temperature T is represented by ρ(t ) = ( Z exp H ) = exp( βh), () kt Z where k is the Boltzmann constant, T is the temperature, β = /kt. The partition function of the system is Z = Tr(exp( βh)). Many measures of the multipartite entanglement were introduced, such as entanglement of information, [8] concurrence, [9] negativity, [0] and so on. The concurrence is chosen as a measurement of the bipartite entanglement. The concurrence C is defined as C = max{λ λ λ 3 λ 4, 0}, (3) The project supported by the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No Corresponding author, szhu@suda.edu.cn

2 970 REN ie and ZHU Shi-Qun Vol. 46 where the quantities λ i (i =,, 3, 4) are the square roots of the eigenvalues of the operator ϱ = ρ (σ y σy )ρ (σ y σy ). They are in descending order. The case of C = corresponds to the maximum entanglement between the two qubit, while C = 0 means that there is no entanglement between the two qubits. The quantity tangle τ [7] is introduced to measure the tripartite entanglement of a pure state ψ. For a tripartite two-level system, the residual entanglement is referred to τ ABC = C A(BC) C AB C AC, (4) where C AB and C AC are the concurrence of the original pure state ρ ABC with tracing over the qubits C and B, respectively, C A(BC) is the concurrence of ρ A(BC) with qubits B and C regarded as a single object. It is shown that the residual entanglement of a three-qubit state ψ = i,j,k a ijk ijk can be obtained by [7] τ ABC = a ijk a i j ma npk a n p k ɛ ii ɛ jj ɛ kk ɛ mm ɛ nn ɛ pp, (5) where the sum is taken over all the indices, and ɛ αβ = ɛ βα = δ αβ. Another method of the calculation of the tangle can be obtained by τ ABC = (λ AB λ AB + λ AC λ AC ), (6) where λ i, (i = AB, AC) is the eigenvalues of the matrix ρ i (σ y σy )ρ i (σ y σy ). It is known that the global entanglement is a measure of the entanglement of a many-qubit pure state ψ. [] It can be written as Q(ψ) = [ /nσ n k= Tr(ρ k )]. The relation between the global entanglement and the tangle can be expressed by [] τ ABC = Q ( C 3 AB + CAC + CBC ), (7) where C AB, C AC, and C BC are the concurrence of the original pure state ρ ABC with tracing over qubits C, B, and A, respectively. The tangles marked by τ ABC is the three-qubit entanglement. The tangles of the well-known three-qubit Greenberger Horne Zeilinger (GHZ) state (/ )( ) and three-qubit W state (/ 3)( ) correspond to and 0 respectively. 3 Effect of Nonuniform Magnetic Field When the nonuniform magnetic fields B and B are along the z direction with α = 0, the analytic expressions of the eigenvalues and eigenvectors of H can be obtained. These eigenvalues and eigenvectors are listed in Table. Here, only the effect of the magnitude of the nonuniform magnetic fields is investigated. Table Eigenvalues E 0 = B + B Eigenvalues and eigenvectors of the three-qubit Heisenberg model. E = + B Eigenvectors ψ 0 = 000 E = B + + M ψ = ψ = [ ] M + [ 00 + M ] E 3 = B + M ψ 3 = [ 00 + M ] M + E 4 = B + + N ψ 4 = [ 0 + N ] N + E 5 = B + N ψ 5 = [ 0 + N ] N + E 6 = B E 7 = B B ψ 6 = [ 0 0 ] ψ 7 = with B B + 9 B B + B + B M =, M = E B B N = + B + 9 B B B + B, N = E 4 + B, M = E 3 B,, N = E 5 + B

3 No. 6 Bipartite and Tripartite Entanglement in a Three-Qubit Heisenberg Model 97 In the basis of { 00, 0, 0, }, the reduced density matrix of qubits and 3 is given by u ρ 3 = tr (ρ) = 0 w 3 y 3 0 Z 0 y 3 w 3 0. (8) v 3 The concurrence can be calculated by where u 3, v 3, y 3 are given by u 3 = exp( βe 0 ) + v 3 = exp( βe 7 ) + C 3 = Z max ( y 3 u 3 v 3, 0), (9) M + M exp( βe ) + M + M exp( E 3 ), N + N exp( βe 4 ) + N + N exp( βe 5 ), y 3 = exp( βe ) exp( βe 6) + + M + exp( βe ) + M + exp( βe 3) N + exp( βe 4) + N + exp( βe 5). (0) Similarly, the reduced density matrix of qubits and is given by u ρ = tr 3 (ρ) = 0 w y 0 Z 0 y w 0. () v The concurrence of qubits and can also be obtained, where u, v, y are given by u = exp( βe 0 ) + exp( βe ) + v = exp( βe 7 ) + exp( βe 6) + y = C = Z max ( y u v, 0), () M + M exp( βe ) + M + M exp( βe 3 ) + M + exp( βe ) + M + exp( βe 3), N + exp( βe 4) + N + exp( βe 5), N + N exp( βe 4 ) + N + N exp( βe 5 ). (3) The concurrence is plotted as a function of the magnetic fields B and B in Fig.. Figure (a) is a plot of the concurrence C 3 between qubits and 3 and figure (c) is the contour lines of C 3. From Figs. (a) and (c), it is seen that C 3 is symmetric about B and B. There are four peaks located on the B and B plane. The four peaks are divided by two groups of the same height. One group of the peaks is located at B = 0 and increased as B is increased. Another group is located at B =.3, B = 8.06, and B =.3, B = 8.06 and decreased when both B and B are increased. There is no entanglement between qubits and 3 in the area between the contour lines of C 3 = 0. If either B = 0 or B = 0, there are only two peaks in C 3. That is, the nonuniform magnetic field can induce more entanglement in C 3. Figure (b) is a plot of the concurrence C between qubits and and figure (d) is the contour lines of C. From Figs. (b) and (d), it is seen that C is symmetric about the diagonal line of B = B. There are two peaks located on the B and B plane. The two peaks are located at B = 5.3, B = 3.09 and B = 5.3, B = From the left peak in Fig. (b), it is seen that the height of the peak is decreased rapidly when B is increased and B is decreased, while the peak is decreased slowly when B is decreased and B is increased. The right peak has the similar decrease tendency with opposite values of B and B. From Fig. (d), it is seen that there is no entanglement between qubits and in the area between the contour lines of C = 0. From Figs. (a) and (b), it is clear that the maximum value of C can reach 0.6 while that of C 3 is only

4 97 REN ie and ZHU Shi-Qun Vol. 46 The tangle labelled by τ 3 is the measure of the three-qubit entanglement of the pure state. The tangle in the ground state of Heisenberg XY model is calculated. It is very interesting to note that there is no three-qubit entanglement since τ 3 = 0. From Table, it is seen that there is no entanglement in eigenvectors ψ 0 and ψ 7. There is only bipartite entanglement in eigenvectors ψ i (i =, 6). It is similar to the W state since the tangle τ 3 = 0 in the eigenvectors ψ i (i =, 3, 4, 5). So there is also no three-qubit entanglement. Fig. The concurrence C is plotted as a function of the amplitudes B and B of the magnetic fields when =, β = /kt =, and α = 0. (a) The concurrence C 3 of qubits and 3; (b) The concurrence C of qubits and ; (c) The contour lines of C 3; (d) The contour lines of C. 4 Effect of Magnetic Field Directions If the magnitude of the magnetic field is fixed with B = B = B, the effect of the direction of the magnetic field can be investigated. It is assumed that the direction of the magnetic field B of qubit is varied with α 0. Since the analytic expressions of the eigenvalues and eigenvectors of the Hamiltonian in Eq. () are difficult to obtain, the numerical simulation of Eq. () is performed. The pairwise entanglement as a function of the angle α and the magnetic field B is plotted in Fig.. Since the concurrence is symmetric about B = 0, only the part of B > 0 is plotted. Figure (a) is a plot of the concurrence C 3 between qubits and 3 and figure (c) is the contour lines of C 3. From Figs. (a) and (c), it is seen that the maximum value of C 3 is located at α = π and B = 8.67 and equals to When B < 5.8, C 3 is always equal to zero no matter what the value of α is. When B is increased greater than 5.8, C 3 is increased rapidly to a peak. Then C 3 is gradually decreased to zero when B is further increased. There is no entanglement between qubits and 3 in the area between the contour lines of C 3 = 0. Fig. (b) is a plot of the concurrence C between qubits and and Fig. (d) is the contour lines of C. Similar results as that in Figs. (a) and (c) are obtained. The maximum value of C equals 0.55 and located at α = π, B = From Fig., it is seen that the maximum height of C is much larger than that of C 3. The value of C starts to reach the peak at B =.0 while that of C 3 starts at B = 5.8. The value of C reaches the peak at much smaller value of B. The tangle τ 3 in the ground state is calculated and is plotted as a function of the angle α in Fig. 3 when the magnetic field B is varied. It is seen that the tangle τ 3 is symmetrically located at two sides of B = 0 and is a periodic function of α. The period is α = π. When α = nπ, the tangle arrives to the minimum value of zero. The maximum value of the tangle is symmetrically located at two sides of α = π. When B = 0, the tangle τ 3 = 0. When

5 No. 6 Bipartite and Tripartite Entanglement in a Three-Qubit Heisenberg Model 973 B is increased from zero, two peaks appear in τ 3. The two peaks are increased and broadened rapidly when B is increased. When B is increased further, the two peaks are decreased and finally disappeared. Fig. The concurrence C is plotted as a function of the angle α and the magnetic field B when =, β = /kt =, and B = B. (a) The concurrence C 3 of qubit and 3; (b) The concurrence C of qubit and ; (c) The contour lines of C 3; (d) The contour lines of C. Fig. 3 The tangle τ 3 of the three-qubit entanglement is plotted as a function of the angle α when the magnetic field B is varied and =, β = /kt. 5 Discussion The effect of the nonuniform magnetic field on the entanglement in a three-qubit Heisenberg model is investigated when the amplitude and the direction of the magnetic field are varied. If the amplitude of the magnetic field of qubits and 3 is different from that of qubit, there are two peaks in C and four peaks in C 3. There is no thermal entanglement when B = B. The maximum value of C is much larger than that of C 3. If the direction of the

6 974 REN ie and ZHU Shi-Qun Vol. 46 magnetic field is changed, there are two peaks in C 3 and C and the two peaks are located at α = π and symmetric about B = 0. It is very interesting to note that there is no tangle τ 3 when the amplitude of the magnetic field is varied. However, the variation of the magnetic field direction can induce the tangle τ 3 between the three qubits. The tangle τ 3 is symmetric about B = 0 and a periodic function of the angle α. Acknowledgments It is a pleasure to thank Xiang Hao, ian-xing Fang and Yin-Sheng Ling for their useful and extensive discussions about the topic. References [] C.H. Bennett, G. Brassard, C. Crepeau, R. ozsa, A. Peres, and W.K. Wootters, Phys. Rev. Lett. 70 (993) 895. [] M. Murao, D. onathan, M.B. Plenio, and V. Vedral, Phys. Rev. A 59 (999) 56. [3] M.A. Nielson and I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, England (000). [4] X.G. Wang, Phys. Rev. A 64 (00) 033. [5] G.L. Kamta and A.F. Starace, Phys. Rev. Lett. 88 (00) [6] Y. Sun, Y.G. Chen, and H. Chen, Phys. Rev. A 68 (003) [7] A.K. Rajagopal and M. Bourennane, Phys. Rev. A 58 (998) 439. [8].C. Hao, C.F. Li, and G.C. Guo, Phys. Rev. A 57 (998) [9] D. Bruβ, D.P. DiVincenzo, A. Ekert, C.A. Fuchs, C. Macchiavello, and.a. Smolin, Phys. Rev. A 57 (998) 368. [0] W. Dür, G. Vidal, and.i. Cirac, Phys. Rev. A 6 (000) [] H.A. Carteret and A. Sudbery,. Phys. A 33 (000) 498. [] T.A. Brun and O. Cohen, Phys. Lett. A 8 (00) 567. [3] A. Ac in, A. Andrianov, L. Costa, E. ane,.i. Latorre, and R. Tarrach, Phys. Rev. Lett. 85 (000) 560. [4] A.K. Rajagopal and R.W. Rendell, Phys. Rev. A 65 (00) [5] X. Wang, H. Fu, and A.I. Solomon,. Phys. A 34 (00) 307. [6] H. Fu, A.I. Solomon, and X.G. Wang, arxiv:quantph/0005 vo. 8, an. (00). [7] V. Coffman,. Kundu, and W.K. Wootters, Phys. Rev. A 6 (000) [8] C.H. Bennett, D.P. DiVincenzo,.A. Smolin, and W.K. Wootters, Phys. Rev. A 54 (996) 386. [9] W.K. Wootters, Phys. Rev. Lett. 80 (998) 45. [0] G. Vidal and R.F. Werner Phys. Rev. A 65 (00) [] D. Meyer and N. Wallach,. Math. Phys. 43 (00) 473. []. Endrejat and H. Buttner, Phys. Rev. A 7 (005) 0305.