We solve the eigenvalue problem of the five-qubit anisotropic Heisenberg model, without use of Bethe's Ansatz, and give analytical results for entanglement and mixedness of two nearest-neighbor qubits. The entanglement takes its maximum at ∆ = 1 (∆ > 1) for the case of zero (finite) temperature with ∆ being the anisotropic parameter. In contrast, the mixedness takes its minimum at ∆ = 1 (∆ > 1) for the case of zero (finite) temperature.PACS numbers: 03.65. Ud, Recently, the study of entanglement properties of many-body systems has received much attention [1]- [22]. To obtain analytical results for entanglement, one may consider the case of infinite lattice or a small lattice with a few qubits. It is hard to get some analytical results between these two extreme cases.It was well-known that the anisotropic Heisenberg model can be solved formally by Bethe's Ansatz method [23,24] for arbitrary number of qubits N , however, we have to solve a set of transcendental equations. For N ≤ 7, the isotropic Heisenberg Hamiltonian can be analytically solved [25,26]. Here, we give the analytical results of the eigenvalues of the anisotropic Heisenberg model with N = 5, without use of Bethe's Ansatz, from which the analytical expressions for entanglement and mixedness of two nearest-neighbor qubits are readily obtained.It is interesting to see that the entanglement properties of a pair of nearest-neighbor qubits at a finite temperature is completely determined by the partition function. The entanglement, quantified by the concurrence [27], relates to the partition function Z via [28,29,30] being the internal energy, andbeing the correlation function. Here, β = 1/T and the Boltzmann's constant k = 1. Thus, once we know the eigenenergies versus the temperature and the anisotropic parameter, we can completely determine the entanglement.There exists another concept, the mixedness of a state, is central in quantum information theory [31]. For instance, Bose and Vedral have shown that entangled states become useless for quantum teleportation on exceeding a certain degree of mixedness [32]. Mixedness is also related to quantum entanglement. We will study both the entanglement and mixedness properties.
Eigenvalue problem.The anisotropic Heisenberg Hamiltonian is given bywhereis the swap operator between qubit i and j, σ i = (σ ix , σ iy , σ iz ) is the vector of Pauli matrices, and J is the exchange constant. We have assumed the periodic boundary condition, i.e., N +1 ≡ 1. In the following discussions, we also assume J = 1 (antiferromagnetic case) and ∆ ≥ 0.Since we impose the periodic boundary condition, the Hamiltonian is translational invariant, i.e., [H, T ] = 0, where T is the cyclic right shift operator defined asThe translational invariant symmetry can be used to reduce the Hamiltonian matrix to smaller submatrices by a factor of N [33]. Now we focus our attention to five-qubit settings, and solve the eigenvalue problem of the anisotropic Heisenberg model. Since [H, J z ] = 0, the whole 32-dimentional Hilbert space can be divide...