We investigate the creation of highly entangled ground states in a system of three exchangecoupled qubits arranged in a ring geometry. Suitable magnetic field configurations yielding approximate GHZ and exact W ground states are identified. The entanglement in the system is studied at finite temperature in terms of the mixed-state tangle τ . By generalizing a conjugate gradient optimization algorithm originally developed to evaluate the entanglement of formation, we demonstrate that τ can be calculated efficiently and with high precision. We identify the parameter regime for which the equilibrium entanglement of the tripartite system reaches its maximum. Entangled quantum systems have been the focus of numerous theoretical and experimental investigations [1, 2, 3] . In particular, entanglement has been identified as the primary resource for quantum computation and communication [4] . Compared to the case of a bipartite system, multipartite entanglement exhibits various new features. Notably, there are two different equivalence classes of genuine three-qubit entanglement [3], the representatives being any one of the two maximally entangled Greenberger-Horne-Zeilinger (GHZ) states [2] |GHZ ± = (|↑↑↑ ± |↓↓↓ )/ √ 2 on the one hand, and the W state [3] |W = (|↑↑↓ +|↑↓↑ +|↓↑↑ )/ √ 3 on the other. The ability to realize both representatives in real physical systems is thus of high importance in the study of genuine tripartite entanglement. Particularly interesting is the GHZ state, as it represents the strongest quantum correlations possible in a system of three qubits. Furthermore, it is equivalent to the three-qubit cluster state used in one-way quantum computation [5] . It is favorable to obtain the GHZ and W states as the eigenstate of a suitable system, rather than by engineering them using quantum gates. In this Letter, we demonstrate the possibility of obtaining approximate GHZ and exact W states as the ground state (g.s.) of three spin-qubits in a ring geometry coupled via an anisotropic Heisenberg interaction. The use of quantum gates is therefore not required. Rather, the desired states are achieved merely by cooling down to sufficiently low temperatures. We state all our results in terms of the exchange coupling strengths in order to keep our proposal open to a broad set of possible implementations of the qubits. We remark that, while Heisenberg models have been studied frequently in the context of entanglement [6] (also with respect to entangled eigenstates [7] ), this is the first time that highly entangled states are reported as the non-degenerate g.s. of three exchange-coupled qubits. Our study inevitably involves the issue of quantifying entanglement [8, 9, 10, 11, 12] : At finite temperatures, the mixing of the g.s. with excited states forces us to evaluate a mixed-state entanglement measure (EM) in order to study the entanglement in the system meaningfully. Computationally, this is a rather formidable task. We generalize a numerical scheme that has originally been developed to compute the entanglement of formation (EOF) [10, 13] . Our scheme can be used to evaluate any mixed-state EM defined as a socalled convex roof [14] . Model.-We assume that three spins S i , with S = 1/2, are located at the corners i = 1, 2, 3 of an equilateral triangle lying in the xy-plane. Their interaction is described by the anisotropic Heisenberg Hamiltonian (1) where S 4 = S 1 . Here, J xy and J z are the in-and outof-plane exchange coupling constants, respectively, and denotes the Zeeman coupling of the spins S i to the externally applied magnetic fields b i at the sites i [15] . We now seek a configuration of b i 's yielding a highly entangled GHZ-or W-type ground state. Finitetemperature effects will then be studied in a second step. Ground-state properties.-We first consider isotropic exchange couplings, i.e., J xy = J z ≡ J. For b i = 0, we naturally find two fourfold-degenerate eigenspaces due to the high symmetry of the system. For J > 0, i.e., ferromagnetic coupling, the ground-state quadruplet is spanned by the two GHZ states |GHZ ± , the W and the spin-flipped W state. Appropriately chosen magnetic fields allow one, however, to split off an approximate GHZ state from this degenerate eigenspace. To identify the optimal field geometry, we first observe that the two states |GHZ ± have the form of a tunnel doublet. If we thus find a set of b i 's, which, in the classical spin system, results in precisely two degenerate minima for the configurations ↑↑↑ and ↓↓↓ with an energy barrier in between, quantum tunneling will yield the desired states. In order to single out exactly the two directions perpendicular to the xy-plane, the magnetic fields must be in-plane, be of the same strength, and sum to zero. This immediately implies that successive directions of the fields must differ by an angle of 2π/3 from each other. We choose the fields to point radially outwards, although any other config-
Related work has shown similar properties for a ring of exchange-coupled qubits REF even in the ground state.
Highly Entangled Ground States in Tripartite Qubit Systems
Phys. Rev. Lett. 100, 100502 (2008).
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