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1、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 mod
2、el 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 w
3、ill 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: 11. IntroductionThe entanglement is an i
4、mportant 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, incl
5、uding 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 introdu
6、ced 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.
7、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
8、 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 b
9、y 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
10、 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, C
11、A(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 indi
12、ces, and = = .The residual entanglement can be generalized to the multipartite entanglement 9. The residual entanglement ABC DN of N-particle system ABC DN 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
13、 (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 N2
14、entanglement 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 magnet
