Gaussian entanglement in the turbulent atmosphere

We provide a rigorous treatment of the entanglement properties of two-mode Gaussian states in atmospheric channels by deriving and analyzing the input-output relations for the corresponding entanglement test. A key feature of such turbulent channels is a nontrivial dependence of the transmitted continuous-variable entanglement on coherent displacements of the quantum state of the input field. Remarkably, this allows one to optimize the entanglement certification by modifying local coherent amplitudes using a finite, but optimal amount of squeezing. In addition, we propose a protocol which, in principle, renders it possible to transfer the Gaussian entanglement through any turbulent channel over arbitrary distances. Therefore, our approach provides the theoretical foundation for advanced applications of Gaussian entanglement in free-space quantum communication.

Figure 1

The Simon entanglement test ${\mathcal{W}}_{\mathrm{atm}.}$ is shown for a TMSV state for two uncorrelated atmospheric channels with channel characteristics obtained from experiments. The solid and dashed lines correspond to a 144-km-long channel with $\langle T_a\rangle =\langle T_b\rangle =0.027$, $\langle T_a^2\rangle =\langle T_b^2\rangle =0.001$ ($\mathrm{\Gamma }\approx 0.75$) and to a 1.6-km-long channel with $\langle T_a\rangle =\langle T_b\rangle =0.398$, $\langle T_a^2\rangle =\langle T_b^2\rangle =0.163$ ($\mathrm{\Gamma }\approx 0.97$), respectively. ${\mathcal{W}}_{\mathrm{atm}.}$ is scaled by ${10}^6$ for the solid line. Surprisingly, if $\xi$ exceeds a certain value no Gaussian entanglement persists.

Figure 2

The roots of the Simon entanglement test, ${\mathcal{W}}_{\mathrm{atm}.}=0$, are displayed as a function of the coherent displacement $\alpha$. The circle and the ellipse correspond to the displaced symmetric and asymmetric TMSV states with squeezing parameter $\xi =1$. The interior (exterior) regions satisfy ${\mathcal{W}}_{\mathrm{atm}.}<0$ (${\mathcal{W}}_{\mathrm{atm}.}>0$). The channel parameters are $\langle T_a\rangle =\langle T_b\rangle =\langle T_a^2\rangle =\langle T_b^2\rangle =0.9$ ($\mathrm{\Gamma }=0.9$).

Figure 3

The contours illustrate the bounds of the regions of Gaussian entanglement of a displaced TMSV state. Entanglement is preserved in the gray shaded areas. Two cases of ideal correlations $T_a=T_b(\mathrm{\Delta }\mathrm{\Gamma }=\mathrm{\Gamma }=1)$ are considered, with the same two sets of experimental turbulence parameters as in Fig. 1: (a) $\langle T_a^2\rangle =0.163$ and $\langle T_a\rangle =0.398$ as well as (b) $\langle T_a^2\rangle =0.001$ and $\langle T_a\rangle =0.027$. The dependence on the square amplitude ${\left|\alpha \right|}^2={\left|\langle \widehat{a}\rangle \right|}^2$ and the sum of the phases of displacement $\varphi +\chi$ is shown. The squared displacement amplitude of the second mode is $|\langle \widehat{b}\rangle {|}^2=50-{\left|\langle \widehat{a}\rangle \right|}^2$. The squeezing parameter is $\xi =0.5$.