Dynamics of non-classically-reproducible entanglement

We investigate when the quantum correlations of a bipartite system, under the influence of environments with memory, are not reproducible with certainty by a classical local hidden variable model. For this purpose, we compare the dynamics of a Bell inequality with that of entanglement, as measured by concurrence. We find time regions when the Bell inequality is not violated even in correspondence with high values of concurrence (up to $\approx 0.8$). We also suggest that these results may be observed by adopting a modification of a recent experimental optical setup. These findings indicate that even highly entangled systems cannot be exploited with certainty in contexts where the nonclassical reproducibility of quantum correlations is required.

Figure 1

(Color online) Maximum of Bell function, $B_{\max }$, in terms of the initial degree of entanglement (represented by ${\alpha }^2$) and of the dimensionless time $\Gamma t$ for the initial Bell-like states $\mid \Phi ⟩=\alpha \mid 01⟩+\beta \mid 10⟩$ (left plot) and $\mid \Psi ⟩=\alpha \mid 00⟩+\beta \mid 11⟩$ (right plot) at $\lambda ∕\Gamma ={10}^{-3}$. The parts of the graphs emerging after the first collapse under the classical threshold value 2, “Bell islands,” correspond to a return of INLCs.

Figure 2

(Color online) Maximum of Bell function, $B_{\max }-2$ (solid line), and concurrence $C$ (dashed line) as a function of the dimensionless time $\Gamma t$ for initial Bell-like states $\mid \Phi ⟩=(\mid 01⟩+\sqrt{2}\mid 10⟩)\sqrt{3}$ (left plot) and $\mid \Psi ⟩=(\mid 00⟩+\sqrt{2}\mid 11⟩)∕\sqrt{3}$ (right plot). The value of ${\alpha }^2$ is fixed at $1∕3$ so that dark intervals of entanglement $(C=0)$ occur for the state $\mid \Psi ⟩$. The coupling parameter is fixed so that $\lambda ∕\Gamma ={10}^{-4}$, representing strong non-Markovian effects.

Figure 3

(Color online) Comparison between the maximum of Bell function $B_{\max }$ and the fixed angle function $B_{\mathrm{fix}}$ in terms of the decay probability $p$, for initial Bell states $\mid \Phi ⟩=(\mid 01⟩\pm \mid 10⟩)∕\sqrt{2}$ (left plot) and $\mid \Psi ⟩=(\mid 00⟩\pm \mid 11⟩)∕\sqrt{2}$ (right plot).