Entanglement and spin squeezing in the three-qubit transverse Ising model

We study entanglement and spin squeezing in the ground state of three qubits interacting via the transverse Ising model. We give analytical results for the entanglement and spin squeezing, and a quantitative relation between the concurrence, quantifying the entanglement of two spins, and the spin squeezing parameter, measuring the degree of squeezing. Finally, by appropriately choosing the exchange interaction and strengths of the transverse field, we propose a scheme for generating entangled W state from an unentangled initial state with all spins down.  2004 Elsevier B.V. All rights reserved. PACS: 03.67.-a; 03.65.Ud; 75.10.Jm Entanglement in many-body systems has attracted much attention [1–15]. Spin squeezing, resulting from quantum correlations between spins, has been a subject of much studies in recent years [16–37]. Interestingly, there exist close relations between the entanglement and spin squeezing [23,38], which enhance the importance of spin squeezing. Spin squeezing in quantum systems exhibiting a quantum phase transition [12] has been studied [13,14]. In this Letter, we study ground-state entanglement and spin squeezing in the three-qubit transverse Ising E-mail address: [email protected] (X. Wang). 0375-9601/$ – see front matter  2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.08.019 model (TIM), which can be exactly solved. For the two-qubit case, analytical results of the entanglement in the TIM have been given by Gunlycke et al. [3]. Here, we give analytical results of pairwise entanglement and spin squeezing for the three-qubit case. It was found that the spin squeezing is equivalent to bipartite entanglement for an arbitrary two-qubit pure state [27]. However, for the case of three qubits or more, in general the entanglement is not equivalent to spin squeezing, and there could exist some relations between them [38]. We study relations between entanglement and spin squeezing for three-qubit states. We also propose a scheme for dynamically producing an entangled W state [4,39–43] in the TIM. X. Wang / Physics Letters A 331 (2004) 164–169 165 Transverse Ising model. We first consider the N qubit TIM, and the corresponding Hamiltonian is given by