Robust Asymptotic Entanglement under Multipartite Collective Dephasing

We derive an analytic solution for the ensemble-averaged collective dephasing dynamics of $N$ noninteracting atoms in a fluctuating homogeneous external field. The obtained Kraus map is used to specify families of states whose entanglement properties are preserved at all times under arbitrary field orientations, even for states undergoing incoherent evolution. Our results apply to arbitrary spectral distributions of the field fluctuations.

Figure 1

Dynamics of a bipartite system with the initial state ${\rho }_0={\mathbb{I}}_4/20+4|{\mathrm{\Phi }}^{-}⟩⟨{\mathrm{\Phi }}^{-}|/5$. (a) The evolution of the state under Lorentzian- (green) and box-distributed (red) noise, $C_{0,1}(\omega )$ and $B_{0,1}(\omega )$ (see the text for a definition), respectively, with a magnetic field direction $\mathit{n}=(2,1,1{)}^T/\sqrt{6}$, is depicted inside the tetrahedron of Bell-diagonal states. The inner octahedron marks the set of separable states. (b) Depending on the orientation of the magnetic field, the concurrence remains constant [for $\mathit{n}=(1,0,0{)}^T$, orange], decays to a finite value [for $\mathit{n}=(2,1,1{)}^T/\sqrt{6}$, blue], or decays to zero [for $\mathit{n}=(0,1,1{)}^T/\sqrt{2}$, red]. The decay is monotonic for Lorentzian (dashed lines), or nonmonotonic for box-distributed (solid lines) noise.

Figure 2

Influence of collective dephasing (5) on $N$-partite entangled $W$ states. Here we show the upper bound $k_{\mathrm{eff}}$ to the asymptotic state’s separability vs the polar angles $0\le \theta \le \pi /2$ (measured from the $z$ axis to the $x,y$ plane) of the fluctuating magnetic field’s direction. In blue and orange we show, respectively, the dependence of the critical angles ${\theta }_{\mathrm{E}}$ and ${\theta }_{\mathrm{NPE}}$, Eqs. (6) and (7), on $N$. The dots correspond to the numerical estimation of the smallest angle where $k_{\mathrm{eff}}$ changes from 1 to 2 (defining ${\theta }_{\mathrm{NPE}}$) or from $N-1$ to $N$ (defining ${\theta }_{\mathrm{E}}$). The uncertainty on this angle, due to the finite bin width of our sampling, is covered by the dot size. The lines represent the expressions (6) and (7).

Figure 3

Evolution of the trace distance $D_{\mathrm{tr}}$ between $\rho (t)$ and the asymptotic state ${\rho }_s$ (solid lines), and of the state’s separability bound $k_{\mathrm{eff}}$ (connected dots), respectively, for eight-partite $|W⟩$ (green) and $|\tilde{W}⟩$ (blue) initial states (see the text for a definition), box-distributed noise fluctuations $B_{0,1}(\omega )$, and a polar angle $\theta =\pi /8$. The values where $k_{\mathrm{eff}}>N$, marked in a lighter green shade, indicate that the state is compatible with full separability, and hence they do not provide additional information than the case $k_{\mathrm{eff}}=N$. Note that the $|\tilde{W}⟩$ state exhibits time-invariant genuine multipartite entanglement outside of a time-invariant subspace.