Decoherence and Multipartite Entanglement

We study the dynamics of multipartite entanglement under the influence of decoherence. A suitable generalization of concurrence reveals distinct scaling of the entanglement decay rate of Greenberger-Horne-Zeilinger and $W$ states, for various environments.

Figure 1

Time dependence of the multipartite concurrence $C_N(t)$, for a $N=3$-partite system initially prepared in a GHZ (dashed lines) or a $W$ (solid lines) state. Only for an infinite temperature reservoir (squares) does the state turn separable after a finite time $t_{\mathrm{s}\mathrm{e}\mathrm{p}}$. In contrast, the zero temperature (circles) as well as the dephasing environment (triangles) induce separability only in the limit $t\to \infty$. In all cases, the numerical results are very well fitted by an exponential (solid and dashed lines).

Figure 2

Decay rates $\gamma$ (in units of the reservoir rate $\Gamma$) for GHZ (top) and $W$ (bottom) states, as a function of the system size (i.e., the particle number) $N$. The different environment models are represented by circles connected by a dotted line (zero temperature), by squares connected by a solid line (infinite temperature), and by triangles connected by a long-dashed line (dephasing), respectively. While for GHZ states the decay rates increase roughly linearly with $N$, independently of the specific environment, the $W$ states exhibit increasing decay with system size only for the infinite temperature environment. Remarkably, the decay rate of the $W$ states is size independent for dephasing and zero temperature environments.