Pair-wise Entanglement for Characterizing Quantum Phase Transition

Kaoru Shimizu and Akira Kawaguchi

^{*}

Optical Science LaboratoryQuantum 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

Jand external magnetic fieldh.

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=JSS+^{Z}_{i}S^{Z}_{i+1}hS^{x}Swith the transverse magnetic field^{Z}_{i}h. Then we studied the behavior of pair-wise quantum entanglement^{ x}^{†}with regarding the different values ofh/J, where the system changes from the random phase ofS( for a large^{Z}_{j}hvalue) to the ordered phase (anti-ferromagnetic phase for a small^{x}hvalue). 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].^{x}

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 matrixrfor the neighboring spins is decomposed into the separable part (1 -L)rand the inseparable part_{s}Lr, (Fig.1) where_{e}ris decomposed into the four Bell states so (Fig.2) that (1-_{e}L) is minimal. We employ concurrenceC(r) as a quantitative measure of pair-wise entanglement.

Fig.1. hdependence of concurrenceC(r)

(h=0.5: phase transition point).

Fig.2. Decomposition of rinto four Bell states (spin correlation)._{e}

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