Pair-wise Entanglement for Characterizing Quantum Phase Transition


Kaoru Shimizu and Akira Kawaguchi*
Optical Science Laboratory

 Quantum behaviors of a one-dimensional interacting spin system have attracted many research interests because that offers some important theoretical models for condensed matter physicist. Moreover, from the view of one-way quantum computation, study of the spin system may provide a variable knowledge for designing its operation scheme. In particular, it is most important for us to establish a physically-clear interpretation for the variety of the behaviors that are regulated by quantum uncertainty depending on the different values of spin-spin interaction coefficient J and external magnetic field h.
 By adapting some knowledge of quantum entanglement to the one-dimensional spin system, we here obtained an insight that the quantum behaviors of the system can be characterized in a quantitative way by small numbers of parameters; amplitudes and phases of four different types of quantum correlation (four Bell-states) between neighboring two spins, though the system is composed of many numbers of spins. We employed the one-dimensional anti-ferromagnetic Ising spin model represented by the Hamiltonian: H=J S SZiSZi+1+h xSSZi with the transverse magnetic field h x. Then we studied the behavior of pair-wise quantum entanglement with regarding the different values of h/J, where the system changes from the random phase of SZj ( for a large hx value) to the ordered phase (anti-ferromagnetic phase for a small hx value). From critical behaviors of the pair-wise entanglement observed around the phase transition point, we can conclude that the amplitudes and phases of different quantum correlation provide a quantitative description of quantum spin fluctuation[1].
 Our proposed method on the basis of the entanglement analysis is a useful tool for understanding the quantum behaviors of one or two dimensional spin systems.

[1] K. Shimizu and A. Kawaguchi, Phys. Lett. A 355 (2006) 176.
*Present address: Toyota Macs. Co. Ltd.
Reduced density matrix r for the neighboring spins is decomposed into the separable part (1 - L )rs and the inseparable part Lre , (Fig.1) where re is decomposed into the four Bell states so (Fig.2) that (1- L) is minimal. We employ concurrence C(r) as a quantitative measure of pair-wise entanglement.

Fig.1. h dependence of concurrence C(r)
(h=0.5: phase transition point).
Fig.2. Decomposition of re into four Bell states (spin correlation).

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