Multipartite EntanglementinHeisenberg Model.doc

精品论文Multipartite Entanglement in Heisenberg ModelJie Ren and Shiqun ZhuSchool of Physical Science and Technology, Suzhou University, Suzhou, Jiangsu 215006, Peoples Republic of China AbstractThe effects of anisotropy and magnetic field on multipartite entanglement of ground state inHeisenberg XY model are investigated. The multipartite entanglement increases as a function of the inverse strength of the external field when the degree of anisotropy is finite. There are two peaks when the degree of anisotropy is = ±1. When the degree of anisotropy increases further, the multipartite entanglement will decrease and tend to a constant. The threshold of the inverse strength of the external field for generating multipartite entanglement generally decreases with the increasing of qubits.PACS numbers: 03.65.Ud, 03.67.Lx,75.10.Jm Corresponding author, E-mail: szhu91. IntroductionThe entanglement is an important resource in the fields of quantum computation and quantum information 1-3. Due to its potential applications, the pairwise entanglement of anisotropic Heisenberg model has been extensively studied in recent years 4, 5. The en- tanglement of thermal states is introduced. Its properties, including threshold temperature, magnetic field dependence and anisotropic effects, are studied. The entanglement properties of ground state are very important. Some properties of the ground state is studied 6. The pairwise entanglement in one-dimensional infinite-lattice anisotropic XY model is introduced 7. The bipartite entanglement is well understood, while the multipartite entanglement is still under intensive research. To understand the multipartite entanglement, the distributed entanglement has been presented 8. The residual entanglement is generalized to the mul- tipartite entanglement 9. The multipartite entanglement in Ising model is also studied 10.In this paper, the multipartite entanglement of ground state in a Heisenberg XY model with an external magnetic field is investigated. In section II, the basic measures of the multipartite entanglement are presented. In section III, the multipartite entanglement in Heisenberg XY model is studied when an external magnetic field is presented. A discussion concludes the paper.2. Measures of Multipartite EntanglementThe anisotropic Heisenberg XY model of a one-dimensional lattice with N sites in a trans- verse field can be described by the Hamiltonian 10 ofNH = X (1 + )xx+ (1 )y y + z (1)2i=1i i+1i i+1iwhere is the degree of anisotropy, is the inverse strength of the external magnetic field,i ( = x, y, z) are the Pauli matrices at qubit of i. The cyclic boundary conditions ofN +1 = 1 ( = x, y, z) is assumed.The quantity tangle 8 is introduced to measure the tripartite entanglement of a purestate |i. For a tripartite two-level system, the residual entanglement is referred to,A(BC ) C CABACABC = C 2 2 2(2)where CAB and CAC are the concurrence of the original pure state ABC with tracing overthe qubits C and B, respectively, CA(BC ) is the concurrence of A(BC ) with qubits B and Cregarded as a single object. It is shown that the residual entanglement of a three-qubit state|i = Pi,j,k aijk |ijki can be obtained 8,ABC = 2| X aijk ai0 j0 m anpk0 an0 p0 k0 ii0 jj0 kk0 mm0 nn0 pp0 |(3)where the sum is taking over all the indices, and = = .The residual entanglement can be generalized to the multipartite entanglement 9. The residual entanglement ABC D···N of N-particle system ABC D···N is defined as,ABC ···N = min | = 1, 2, 3.,n/2Xi=1NC i (4)Nwhere corresponds to all possible foci, C i= N and N/2 is N/2 when N is even, N/2i(N i)is (N-1)/2 when N is odd. When the foci is A, the residual entanglement is2 2 2 2A(BC ···N ) = CA(BC ···N ) CAB CAC · · · CAN (5)If the focus is changed, one will obtain the other N-1 equations. It is worth noting that AB,ABC and so on can be considered as focus. So there are PN/2 C ifocus. The multipartitei=1 N2entanglement of the well known Greenberger-Horne-Zeilinger (GHZ) state 1 (|00 · · · 0i +N|11 · · · 1i) and W state 1 (|0 · · · 01i + |0 · · · 10i + |1 · · · 00i) correspond to 1 and 0 respectively.3. Multipartite EntanglementThe generalized residual entanglement can be used to calculate the entanglement of an anisotropic Heisenberg XY model when there is an external magnetic field.3.1 Three and Four Qubits精品论文The ground state of the anisotropic Heisenberg XY model can be obtained by|gi = Nc(2 + p2 4 + 4 + 322)|000i + |110i + |011i + |101i(6) where Nc is a normalization constant. The multipartite entanglement of ground state in three qubits can be easily obtained by Eqs. (3) and (4). It is plotted as a functionof the magnetic field and the degree of anisotropy in Fig. 1.It is shown that the2multipartite entanglement of three qubits is symmetric about = 0. There is no multipartite entanglement of ground state in the isotropy Heisenberg XY model ( = 0). It is found that there are two peaks located at = ±1. The peak value will reach 1.0 with the increase of . The ground state can be described by |gi = 1 (|000i + |110i + |011i + |101i). When| < 1, the entanglement increases as a function of |. After reaches the peak at | = 1, the entanglement decreases and finally saturates to a constant value of about 0.77. The eigenvalues and eigenstates of isotropy Heisenberg XY model in a external field magnetic can be exactly solved by symmetric 11 or the Jordan-Wigner transformation 12. For the eigenstates of three qubits, there is on entanglement in two states of |000i and |111i. The other six states are just like W states. It means that all the states have no multipartite entanglement in the three-qubit isotropy Heisenberg XY model.The multipartite entanglement of the ground state in four qubits of an anisotropic Heisenberg model is plotted as a function of the magnetic field and the degree of anisotropy in Fig. 2. Similar result as that in Fig. 1 is obtained. The saturation value of is about0.414 when | is very large. When = 0 and > 2.414, the multipartite entanglement still exists with a value of about 0.414. For four qubits isotropy XY model, it can also be exactly solved by symmetric 11 or the Jordan-Wigner transformation 12. The eigenstates of the ground states can be given byE0 = 4, E1 = 2 2,E2 = 22,(7)and the corresponding eigenstates are|0i = |0000i,2|1i = 1 (|1110i + |1101i + |1011i + |0111i),|2i = 2 (|0011i + |0110i + |1100i + |1001i) + 1 (|0101i + |1010i) .(8)42When the inverse strength of the external field is small, the ground state is |0 i. There isno entanglement. When increases, the ground state changes to |1i. It is just the W state. There is no multipartite entanglement either. However, if increases further, the ground state changes to |2 i. The multipartite entanglement exists.3.2 General Case of N qubitsThe Hilbert space of N qubits in a one-dimensional Heisenberg chain is 2N -dimensional and the corresponding Hamiltonian HN has 2N eigenvectors and eigenvalues.When approaches zero, the ground state of the anisotropic Heisenberg XY model in aone-dimensional lattice with N sites in a transverse field becomes a product of spins pointing in the z direction,|gi = |00 · · · 00i(9) When = 1 the model will be reduced to Ising model. When approaches infinity, theground state becomes to a GHZ state and is given by 13,|gi = Nc Pi,j,.even|i, j, . . .i(10)where Nc is a normalization constant, i, j, . . .even means that the even state is selected forsumming up all the foci.For general case of N qubits, the multipartite entanglement is quite similar to that shown in Figs. 1 and 2. There are also two peaks located at the degree of anisotropy equal to = ±1. When the degree of anisotropy increases further, the multipartite entanglement will saturate to a constant value. Although the three and four qubits are simple, they share many features of the general case of a chain with arbitrary number of N qubits.精品论文Table 1. The threshold of the inverse strength of the external field when there exists multipartiteentanglement of the ground state for N qubits when = 0.N3456782.421.621.371.251.18Table 2. The multipartite entanglement of ground state of N qubits when and = 0, .N357468 0.76980.77400.77250.4140.54630.5272 = 0, 00.56830.62450.4140.54630.5272The limiting cases of = 0 and need to be investigated when N is very large.When the degree of anisotropy = 0 and the inverse strength of the external field , the Hamiltonian HN has the following formNH1 = CI Pxx+ y y(11)i=1i i+1i i+1While the degree of anisotropy , the Hamiltonian HN has the following formNH2 = CI Pxx y y(12)i=1i i+1i i+1The ground state is degenerate. The magnetic field can eliminate the degeneracy and the perturbation theory can be used. When N is even, it is easy to findH1, H2 = 0(13) The result can not be extended to odd qubits. So it is not strange that the multipartite entanglement for the cases of even and odd number of qubits are different.For an isotropic Heisenberg model with = 0, the multipartite entanglement exists forfinite values of when the number of qubits increases. The threshold of that the multi- partite entanglement exists is shown in Table 1 when the number of qubits is varied. Small value of can induce multipartite entanglement if the number of qubits increases. When the精品论文inverse strength of the external field , the multipartite entanglement will be stable.The stable value is shown in Table 2. Meanwhile, the stable value when is also shown. It is found that the stable values are different for odd and even qubits respectively when the qubits increases. For even number of qubits, the value of = 0, equals to the value of . While for odd qubits, these values are different.4. DiscussionIn the paper, the multipartite entanglement of the ground state in an anisotropic Heisen- berg model of three and four qubits is investigated. The effects of anisotropy and magnetic field are discussed. Some properties can be extended to the general case of N qubits. The multipartite entanglement is a increasing function of the inverse strength of the external field when the degree of anisotropy is not equal to zero. It is found that there are two peaks located at = ±1. When = 1, the model reduces to Ising model. When the degree of anisotropy increases further, the multipartite entanglement will saturate to a constant value. It is found that the constant value for the multipartite entanglement generally decreases with the increase of qubits. While the inverse strength of the external field approaches infinity, it is found that the value of = 0, equals to the value of for even number of qubits. While for odd qubits, these values are different.AcknowledgementIt is a pleasure to thank Xiang Hao, Jianxing Fang and Yinsheng Ling for their many useful and extensive discussions about the topic. The financial support from the Special- ized Research Fund for the Doctoral Program of Higher Education of China (Grant No.20050285002) is gratefully acknowledged.Reference1. C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895(1993).2. M. Murao, D. Jonathan, M. B. Plenio, and V. Vedral, Phys. Rev. A59, 156 (1999).3. M. A. Nielson and I. L. Chuang, Quantum Computation and QuantumInformation (Cambridge University Press, Cambridge, England, 2000).4. G. L. Kamta, A. F. Starace, Phys. Rev. Lett. 88, 107901 (2002).5. N. Canosa and R. Rossignoli, Phys. Rev. A69 052306 (2004).6. X. G. Wang, Phys. Rev. A66, 034301 (2002).7. T. J. Osborne and M. A. Nielsen, Phys. Rev. A66, 032110 (2002).8. V. Coffman, J. Kundu, and W. K. Wootters, Phys. Rev. A61, 052306 (2000).9. C. S. Yu and H. S. Song, Phys. Rev. A71, 002331 (2005).10. P. W. Anderson, Phys. Rev. 112, 1900 (1958).11 Y. S. Ling, private communication.12 P. Jordan and E. Wigner, Z. Phys.47, 631 (1928).13. P. Stelmachovic and V. Buzek, Phys. Rev. A70, 032314 (2004).FIG. 1: The multipartite entanglement of ground state in three qubits of is plotted as functions ofthe magnetic fields and degree of anisotropy .FIG. 2: The multipartite entanglement of ground state in four qubits of is plotted as functions of the magnetic fields and degree of anisotropy .