Disentanglement in bipartite continuous-variable systems

Entanglement in bipartite continuous-variable systems is investigated in the presence of partial losses such as those introduced by a realistic quantum communication channel, e.g., by propagation in an optical fiber. We find that entanglement can vanish completely for partial losses, in a situation reminiscent of so-called entanglement sudden death. Even states with extreme squeezing may become separable after propagation in lossy channels. Having in mind the potential applications of such entangled light beams to optical communications, we investigate the conditions under which entanglement can survive for all partial losses. Different loss scenarios are examined, and we derive criteria to test the robustness of entangled states. These criteria are necessary and sufficient for Gaussian states. Our study provides a framework to investigate the robustness of continuous-variable entanglement in more complex multipartite systems.

Figure 1

(a) Dual-channel losses: An entangled quantum state is distributed to two parties, Alice and Bob, over two lossy quantum channels. (b) Single-channel losses: Alice holds the quantum-state generator and only distributes one entangled mode to Bob over a single lossy quantum channel.

Figure 2

Possible behaviors of the PPT entanglement witness $W_{\text{PPT}}^{\prime }$ under attenuation, as a function of the transmittances $T_1$ and $T_2$. (a) Fully robust entanglement. (b) Fragility for any combination of beam attenuations. (c) Separable state. (d) Single-channel partial robustness: either mode, i.e., the state is robust for any individual attenuation, but not for a combination of attenuations, such as equal attenuations. (e) Single-channel partial robustness: specific mode, i.e., the state is robust when one mode is attenuated but presents ESD upon attenuation of the other mode.

Figure 3

Entanglement as a function of losses for the covariance matrix given by Eq. (29). Disentanglement may occur only for combined losses on both modes. In this example, the symplectic eigenvalue [6] is only 0.22 for the initial state.

Figure 4

The space of states with covariance matrices of the form of Eq. (33) is plotted as a function of the normalized correlations ${\overline{C}}_p$ and ${\overline{C}}_q$. Separable states lie in the region IV; fully robust states are comprised within the region I; partially robust states are in the region II, and fragile states are in the region III. Points outside of these regions do not correspond to physical states. Here, we use ${\Delta }^{2}p=1.80$ and ${\Delta }^{2}q=2.55$. The points included represent the states in Figs. 2a, 2b, 2c, 2d.

Figure 5

The space of symmetric two-mode states is plotted as a function of the EPR variances ${\Delta }^2{\widehat{q}}_{+}$ and ${\Delta }^2{\widehat{p}}_{-}$, normalized to the standard quantum limit (SQL). Separable states lie in the region IV; fully robust entangled states are within the region I; partially robust states are in the region II, and fragile states are in the region III. The partial purities are ${\mu }_{-}=0.7267$ and ${\mu }_{+}=0.4529$.