Fast and robust approach to long-distance quantum communication with atomic ensembles

Quantum repeaters create long-distance entanglement between quantum systems while overcoming difficulties such as the attenuation of single photons in a fiber. Recently, an implementation of a repeater protocol based on single qubits in atomic ensembles and linear optics has been proposed [Duan et al., Nature (London) 414, 413 (2001)]. Motivated by rapid experimental progress towards implementing that protocol, here we develop a more efficient scheme compatible with active purification of arbitrary errors. Using similar resources as the earlier protocol, our approach intrinsically purifies leakage out of the logical subspace and all errors within the logical subspace, leading to greatly improved performance in the presence of experimental inefficiencies. Our analysis indicates that our scheme could generate approximately one pair per $3\min$ over $1280\mathrm{km}$ distance with fidelity $(F⩾78\%)$ sufficient to violate Bell’s inequality.

Figure 1

(Color online) Repeater components: (a) entanglement generation (ENG) and (b) entanglement connection (ENC); indicated operations: retrieve $b_C$ and $a_C$ (additional 45° rotations only for the first level), join on polarizing beam splitter (PBS), detect in ± basis conditioned on one photon per output, and finally adjust the phase. (c) Entanglement purification (ENP); indicated operations: retrieve $a_1,b_1$ and $a_2,b_2$ (additional 45° rotations to purify phase error), interfere $a_1,a_2$ on PBS (same with $b_1,b_2$), restore $a_3,b_3$ conditioned on single photon at $a_4$ and $b_4$ respectively, and finally adjust the phase.

Figure 2

(Color online) Comparison between the DLCZ protocol and the new scheme (NS) [without active entanglement purification (ENP)]. For each distance, we optimize over the choice of the control parameters (the half distance between neighboring repeater stations, $L_0$, and the elementary pair generation probability, $p_c$). With efficiency $\eta =90\%$ and targeting fidelity $F=90\%$, we find the most efficient implementations to create the polarization entangled state [Eq. (2)] for both the DLCZ (circled black dashed line) and the new scheme (squared blue solid line). The fiber attenuation length is $L_{\mathit{att}}=20\mathrm{km}$, with no dynamical phase error. The main plot: we show the relationship between the (optimized) average creation time $t_{\mathit{avg}}$ and the final distance $L$ for both schemes, and the empirical estimate [Eq. (11)] of the time scaling for the new scheme (blue dotted line). Over long distances $(L⩾320\mathrm{km})$, the polynomial scaling of the new scheme is more favorable than the superpolynomial scaling of the DLCZ protocol. Inset: we plot the fidelities of the intermediate distances ($L_{\mathit{int}}=160$, 320, 640, and $1280\mathrm{km}$) to create polarization entangled states ($L=1280$, $F=90\%$), with the optimized choice of the control parameters $(L_0,p_c)=(80,0.0027)$ and (40,0.0081) for the DLCZ $(t_{\mathit{avg}}\approx 1900\mathrm{s})$ and the new scheme $(t_{\mathit{avg}}\approx 380\mathrm{s})$, respectively. The optimized choices of the control paramters are detailed in Tables .

Figure 3

(Color online) Average time versus final fidelity with no phase errors. For fixed final distance $L=1280$, we plot the (final) fidelity dependence of the (optimized) average pair creation time, for the DLCZ protocol (black dashed lines), new scheme without ENP (blue solid lines), and new scheme with ENP (red dashdotted lines), with efficiency $\eta$ to be 90% (circles), and 95% (no circles).

Figure 4

(Color online) Average time versus final fidelity with phase errors. This shows the same optimized approaches as Fig. 3 but with finite interferometric path length fluctuation characterized by the phase diffusion coefficient $D={10}^{-3}{\mathrm{rad}}^2∕\mathrm{km}$. Inclusion of ENP (red dash-dotted lines) yields a dramatic improvement in time for high fidelity operations.