Optimal Dynamical Characterization of Entanglement

We show that, for experimentally relevant systems, there is an optimal measurement strategy to monitor the time evolution of entanglement under open system dynamics. This suggests an efficient, dynamical characterization of the entanglement of composite, open quantum systems.

Figure 1

Time evolution of the bipartite mixed state concurrence for initial states $|{\Psi }_0⟩=(|00⟩+|11⟩)/\sqrt{2}$ [$\overline{c}(t=0)=1$] and $|{\Psi }_0⟩=\sqrt{1/8}|00⟩+\sqrt{7/8}|11⟩$ [$\overline{c}(t=0)\simeq 0.7$], under incoherent coupling to a zero temperature environment. Continuous lines represent exact solutions; filled squares stem from a randomly chosen unraveling. Symbols show the results for improving unravelings with increasing $|{\mu }_1|=|{\mu }_2|=0.8$ (filled diamonds), 1.0 (filled pyramids), 3.0 (filled circles), 4.0 (open squares), 7.0 (open circles), and 15.0 (open diamonds). The dashed line shows the time evolution of $|\Lambda (t)|=|{\lambda }_1-\sum _{i=2}^4{\lambda }_i|$ beyond the disentanglement time $t_d$. 1000 realizations were used to generate the unraveling data, in all cases.

Figure 2

Concurrence of two qubits initially prepared in $|{\Psi }_0⟩=(|00⟩+|10⟩)/\sqrt{2}$, under the action of the CNOT gate Hamiltonian (see text) with dephasing. Open circles show the evolution for an optimal unraveling, in perfect agreement with the exact solution (continuous line), and in pronounced contrast to the result obtained from the least favorable unraveling (open squares).

Figure 3

Concurrence of a tripartite GHZ state under a dephasing environment. Filled circles represent the tripartite concurrence $c_3(t)$ obtained upon averaging over 1000 realisations of the optimal unraveling defined by the jump operators $J_k=\sqrt{\Gamma }{\sigma }_{+}^{(k)}{\sigma }_{-}^{(k)}$ ($k=1$, 2, 3), in perfect agreement with the exact solution (solid line). A nonoptimal unraveling (pyramids) is shown for comparison.