Thermal entanglement in three-qubit Heisenberg models

Over the past few years much effort has been put into studying the entanglement of multipartite systems both qualitatively and quantitatively. Entangled states constitute a valuable resource in quantum information processing [1]. Quite recently, entanglement in quantum operations [2–4] and entanglement in indistinguishable fermionic and bosonic systems [5–7] have been considered. Entanglement in two-qubit states has been well studied in the literature, as have various kinds of threequbit entangled states [8–10]. The three-qubit entangled states have been shown to possess advantages over the two-qubit states in quantum teleportation [11], dense coding [12] and quantum cloning [13]. An interesting and natural type of entanglement, thermal entanglement, was introduced and analysed within the HeisenbergXXX [14], XX [15], andXXZ [16] models as well as the Ising model in a magnetic field [17]. The state of the system at thermal equilibrium is represented by the density operator ρ(T ) = exp (− H kT ) /Z,