Long-Distance Entanglement Purification for Quantum Communication

High-quality long-distance entanglement is essential for both quantum communication and scalable quantum networks. Entanglement purification is to distill high-quality entanglement from low-quality entanglement in a noisy environment and it plays a key role in quantum repeaters. The previous significant entanglement purification experiments require two pairs of low-quality entangled states and were demonstrated in tabletop. Here we propose and report a high-efficiency and long-distance entanglement purification using only one pair of hyperentangled state. We also demonstrate its practical application in entanglement-based quantum key distribution (QKD). One pair of polarization spatial-mode hyperentanglement was distributed over 11 km multicore fiber (noisy channel). After purification, the fidelity of polarization entanglement arises from 0.771 to 0.887 and the effective key rate in entanglement-based QKD increases from 0 to 0.332. The values of Clauser-Horne-Shimony-Holt inequality of polarization entanglement arises from 1.829 to 2.128. Moreover, by using one pair of hyperentanglement and deterministic controlled-NOT gates, the total purification efficiency can be estimated as $6.6\times {10}^3$ times than the experiment using two pairs of entangled states with spontaneous parametric down-conversion sources. Our results offer the potential to be implemented as part of a full quantum repeater and large-scale quantum network.

Figure 1

Protocol of entanglement purification using hyperentanglement. (a) General protocol of long-distance entanglement purification using hyperentanglement. After long-distance distribution, deterministic CNOT operations are performed between different degrees of freedom of single particles. Then the target qubits (encoded in 1 degree of freedom) are measured. If we only keep the same measurement results, then the entanglement encoded in the other degree of freedom (control qubits) is purified. (b) Purification protocol based on polarization-spatial-mode hyperentanglement. The hyperentangled state $|{\mathrm{\Phi }}^{+}{⟩}_{ab}\otimes |{\varphi }^{+}{⟩}_{ab}$ is first generated by the entanglement source ($S$) and then distributed to Alice and Bob in the channel ($a_1{b}_1$, $a_2{b}_2$). After suffering from the channel noise, the entanglement purification is performed. The entanglement purification operation uses a half-wave plate (HWP, set at 45°) and a polarizing beam splitter (PBS, which transmits the $H$-polarized photon and reflects the $V$-polarized photon). This essentially acts as a CNOT gate with a success probability of 100% between the polarization (target qubits) and spatial-mode (control qubits) degree of freedom. After PBS1 and PBS2, we convert the spatial mode to polarization. Thus, we can verify its success by selecting the cases in which the two photons are in the output-mode state $D_1{D}_2$ or $D_3{D}_4$. The beam displacer (BD) can couple $H$- and $V$-polarized components from different spatial modes.

Figure 2

Experimental setup. (a) Preparation of hyperentangled state. The pump light beam is separated into two spatial modes by the BD. These two beams are injected into a Sagnac interferometer to pump a type-II cut periodically poled potassium titanyl phosphate (PPKTP) crystal ($1\times 7\times 10$) mm and generate the two-photon polarization entanglement $1/\sqrt{2}(|HV⟩+|VH⟩)$ in each spatial mode. After HWP1 (set at 45°), the hyperentangled state $|{\mathrm{\Phi }}^{+}{⟩}_{ab}\otimes |{\varphi }^{+}{⟩}_{ab}=1/\sqrt{2}(|HH⟩+|VV⟩)\otimes 1/\sqrt{2}(|a_1{b}_1⟩+|a_2{b}_2⟩)$ is generated. (b) Quantum noisy channel. This channel is divided into two parts. The first part is a controllable spatial mode and polarization BF or PF loading noise setup. The other part is an 11 km MCF. (c) Purification process. This process is also divided into two steps. The first step is a CNOT gate between the spatial mode and polarization d.o.f. The setup consists of a HWP (set at 45°) placed on spatial modes $a_2$ and $b_2$. The PBS is used for postselection of the polarization qubit, and the spatial-mode states of the same polarization are preserved, while different polarization states are ignored. Hadamard operations are needed to convert the PF noise to BF noise before purification. (d) Quantum state tomography setup. DM, dichroic mirror; QWP, quarter-wave plate.

Figure 3

Experimental results. (a) Results before and after BF noise purification. On the left are the density matrices of polarization and path-entangled states before purification, and on the right are the density matrices of path states after purification. The yellow column represents a value greater than 0, and the blue column represents a value less than 0. The transparent column represents the value of the ideal maximally entangled state. The fidelity of the purified quantum state has obviously been improved significantly. (b) Results before and after PF noise purification. The first half is the result of loading 20% PF noise. The second half is the result of noise introduced only by the MCF.

Figure 4

QKD based on entanglement purification. (a)–(c) The correlations for mutually unbiased bases required for QKD [computational bases ($|0{⟩}_Z$, $|1{⟩}_Z$) and Fourier bases $|0{⟩}_F$, $|1{⟩}_F$] before and after purification in the case of loading 20% BF noise, 20% PF noise, and using MCF only. The corresponding QBER and quantum key rate ($R$) are also presented.