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Abstract
High quality time-bin entanglement is widely exploited to achieve the purposes of fundamental tests of physics and the implementation of quantum communication protocols both in free space and optical fiber propagation. However, the imperfect approaches of generating time-bin entangled state will degrade its quality and limit its practical application. Entanglement purification is to distill high quality entangled states from low quality entangled states. In this paper, we present the first entanglement purification protocol (EPP) for time-bin entanglement. We first explain this EPP for two-photon time-bin entangled state and then extend it to the system of multi-photon time-bin entangled state. We also design a possible realization of this EPP with practical spontaneous parametric down conversion (SPDC) source. Differ from the conventional EPPs, this EPP does not require the sophisticated controlled-not (CNOT) gate or similar operations, and it uses the feasible sum-frequency generation (SFG) to perform the purification. Moreover, the double-pair noise emitted from the SPDC source can be eliminated automatically which is the other advantage of this EPP. If we combine with the faithful entanglement swapping, this EPP may have potential to be a part of full quantum repeaters.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quantum entanglement is a strictly important element for quantum communication and quantum information processing. Some important quantum communication and quantum information processing protocols such as quantum key distribution [1,2], quantum secret sharing [3], quantum secure direct communication [4–6], distributed secure quantum machine learning [7], and quantum network [8,9] all require the communication parties to share the nonlocal maximally entangled state.
In quantum communication, photon is the best candidate for fast transmission. People usually employ the polarization degree of freedom (DOF) to encode the information. Besides the polarization, the time-bin DOF is also widely investigated, such as the generation of time-bin entanglement [10–18], entanglement distribution [19–22], and entanglement swapping [23–26]. Time-bin qubits and time-bin entanglement are also widely used in quantum communication [27–34]. Differ from the polarization, the time-bin entanglement is robust during entanglement distribution over the inherent noisy channel. The experiments showed that for 50 km distribution, the Clauser-Horne-Shimony-Holt Bell inequality can be violated by more than 15 standard deviations without removing the imperfect detectors [19]. For quantum communication which employs time-bin entanglement, it is critical to generate high fidelity entanglement [10–18]. For example, Jayakumar et al. utilized the biexciton-exciton cascade in a self-assembled quantum dot to emit time-bin entangled state [12]. To suppress the multi-pair emissions and further improve the fidelity of entanglement, a site-controlled quantum dot was used to be a source of time-bin entanglement [13]. Vedovato et al. designed the interferometers working as fast synchronized optical switches at the two measurement stations [15]. In this way, the genuine time-bin entanglement can be generated free of the post-selection and the resulting state can be further applied in quantum communication. However, the generation of time-bin entanglement still contains some imperfect factors, i.e., double excitations, environment-induced dephasing and excitation errors, which limit the fidelity of time-bin entanglement and its application [12,18]. In addition, after performing the entanglement swapping, one can only obtain a mixed state [23] or a poorly time-bin entangled state [24]. Such low quality entangled state becomes an obstacle in long-distance quantum communication.
Entanglement purification is a powerful tool to recover low quality entangled states to high quality entangled states, which has been widely investigated in recent years [35–65]. Entanglement purification is also indispensable for quantum repeaters to realize long-distance quantum communication [66]. The first entanglement purification protocol (EPP) was proposed by Bennett et al. in 1996 [35]. Specifically, two noisy copies are selected from the same ensemble. One pair is considered to be a control and the other was used as a target. After performing the controlled-not (CNOT) or similar operations, by measuring the target pair, the parties can pick out the control pairs if they obtain the same measurement results. Consequently, the fidelity of remaining control pair is higher than that of the initial states. After that, a larger number of EPPs were proposed to improve the quality of entanglement. For example, the EPP with polarization beam splitters (PBSs) rather than CNOT gate was proposed [36] and then experimentally demonstrated based on practical spontaneous parametric down conversion (SPDC) source [37]. Sheng et al. [39] proposed a higher yield EPP with nondestructive quantum nondemolition (QND) detectors using practical entanglement sources [35–37]. In 2011, Wang et al. purified the noisy hybrid entangled state employing quantum-dot and microcavity coupled system [43]. Subsequently, Sheng et al. exploited linear optical elements to purify the polluted hybrid entangled state and illustrated that the error resulted from photon dissipation can be transformed to bit-flip error [47]. In 2014, Ren et al. presented a high-efficiency EPP for hyperentanglement assisted by polarization-spatial phase-check QND detectors and the quantum-state joining method [48]. In 2016, Wang et al. constructed parity-check gate to purify the hyperentanglement in multiple DOFs [49]. Recently, the researchers in [63,64] employed high-dimensional auxiliary entanglement to learn about the information of noisy copies, thereby improving the fidelity of mixed entangled state. Recently, the first high-efficient and long-distance EPP was demonstrated, which shows powerful potential applications in future quantum repeaters, long-distance quantum communication and large-scale quantum networks [65].
Though many EPPs were proposed, they usually focus on the polarization or spatial entanglement in the optical system. None protocol discuss the entanglement purification for time-bin entangled state. In this paper, we will propose the first EPP for time-bin entanglement. We first describe the EPP in a simple example, i. e., the two-photon time-bin entangled state. We then extend this EPP to a more general case, i.e., the multi-photon entangled system. Subsequently, we design the EPP with current experiment technology. We show that the SPDC source can also be used to realize this EPP. Moreover, the double-pair emission can be eliminated with the working principle of the sum-frequency generation (SFG), which provides us the other advantage of this EPP. Combining with the faithful entanglement swapping [23–26], this EPP also has the potential to be a part of full quantum repeater.
This paper is organized as follows. In Sec. 2.1, we explain our EPP for bit-flip error. In Sec. 2.2, we describe our EPP for correcting phase-flip error. We show that phase-flip error can be transformed to bit-flip error and can be purified in a next step. In Sec. 3, we extend this protocol to the multi-photon entanglement system. In Sec. 4, we design a possible realization of this EPP with the SPDC source. In Sec. 5, we make a discussion. Finally, we draw a conclusion in Sec. 6.
2. Entanglement purification with sum-frequency generation
We assume that the two parties Alice and Bob share the time-bin entangled state described as [27]
where $L$ and $S$ are the pump photon’s arrival times corresponding to “Long” and “Short”. However, the imperfection and noise in the process of entanglement generation will make $|\phi ^{+}\rangle$ become $|\psi ^{+}\rangle$, which denotes that a bit-flip error occurs. Also the state $|\phi ^{+}\rangle$ will evolve to $|\phi ^{-}\rangle$ or $|\psi ^{-}\rangle$. We call that the original state becomes $|\phi ^{-}\rangle$ as a phase-flip error. The original state becomes $|\psi ^{-}\rangle$ when both bit-flip and phase-flip errors occur. Here $|\phi ^{-}\rangle$ and $|\psi ^{\pm }\rangle$ are the other three time-bin entangled Bell states with the form ofThus, if only a bit-flip error occurs with the probability of $1-F$, the initial state will become a mixed state described as
2.1 Purification of bit-flip error
Now we start to describe our protocol for bit-flip error correction. In Fig. 1, two copies of mixed states with the form of Eq. (3) are sent to Alice and Bob from S$_{1}$ and S$_{2}$. The whole system $\rho _{a_{1}b_{1}}\otimes \rho _{a_{2}b_{2}}$ can be viewed as a mixture of four pure states. To be precise, with the probability of $F^{2}$, it is in the state $|\phi ^{+}\rangle _{a_{1}b_{1}}\otimes |\phi ^{+}\rangle _{a_{2}b_{2}}$. With an equal probability of $F(1-F)$, it is in the state $|\phi ^{+}\rangle _{a_{1}b_{1}}\otimes |\psi ^{+}\rangle _{a_{2}b_{2}}$ and $|\psi ^{+}\rangle _{a_{1}b_{1}}\otimes |\phi ^{+}\rangle _{a_{2}b_{2}}$. With the probability of $(1-F)^{2}$, it is in the state $|\psi ^{+}\rangle _{a_{1}b_{1}}\otimes |\psi ^{+}\rangle _{a_{2}b_{2}}$. We first discuss the state $|\phi ^{+}\rangle _{a_{1}b_{1}}\otimes |\phi ^{+}\rangle _{a_{2}b_{2}}$, which can be written as
The photons in spatial modes $a_{1}$ ($b_{1}$) and $a_{2}$ ($b_{2}$) with different frequencies are combined into the SFG at the location of Alice (Bob), which creates a new photon in mode $D_{1}$ ($D_{2}$) with a frequency corresponding to the sum of photons in modes $a_{1}$ ($b_{1}$) and $a_{2}$ ($b_{2}$) with a small probability [24]. However, if two incoming photons come from the same spatial mode, they fail to create a new photon. In addition, the SFG produces a new photon in mode $D_{1}$ ($D_{2}$) only if both photons in modes $a_{1}$ ($b_{1}$) and $a_{2}$ ($b_{2}$) arrive at the same time. Therefore, one can clearly see from Eq. (4), the items $|S\rangle _{a_{1}}|L\rangle _{a_{2}}|S\rangle _{b_{1}}|L\rangle _{b_{2}}$ and $|L\rangle _{a_{1}}|S\rangle _{a_{2}}|L\rangle _{b_{1}}|S\rangle _{b_{2}}$ cannot create new photons because the photons are at the different arrival times. Thus, after passing through the SFGs, the state in Eq. (4) evolves to
Obviously, item $|S\rangle _{a_{1}}|S\rangle _{b_{1}}|S\rangle _{a_{2}}|L\rangle _{b_{2}}$ can only lead the SFG in Alice’s location to create a new photon. In this way, it can be eliminated automatically. The other items are analogy with the item $|S\rangle _{a_{1}}|S\rangle _{b_{1}}|S\rangle _{a_{2}}|L\rangle _{b_{2}}$, thereby can be eliminated automatically.
Fig. 1. The EPP for purifying bit-flip error with two noisy copies entangled in modes $a_1,b_1$ and $a_2,b_2$ using the sum-frequency generations (SFGs). After the photons passing through the SFGs, the resultant state will be entangled in modes $D_1,D_2$ with a frequency corresponding to the sum of photons in modes $a_1(b_1)$ and $a_2(b_2)$ [24].
This is similar to the EPP of Ref. [36]. In Ref. [36], they choose the four-mode case and can automatically eliminate the cross-combination items, for the cross-combination items always leading two photons to be in the same output mode. Similar to Ref. [36], the selection principle of our purification can be named as two-mode case, which means that two SFGs in Alice’s and Bob’s sites simultaneously create new photons. Finally, with the probability of $(1-F)^{2}$, by selecting the two-mode case, the item $|\psi ^{+}\rangle _{a_{1}b_{1}}\otimes |\psi ^{+}\rangle _{a_{2}b_{2}}$ evolves to
As a result, we can obtain a new mixed state
If $F>\frac {1}{2}$, we can obtain $F'>F$ [35].
2.2 Purification of phase-flip error
In this section, we will discuss this EPP for purifying phase-flip error. In the conventional EPPs [35,39,40], phase-flip error should be transformed to bit-flip error first and then be purified. Here, we describe an approach to transform phase-flip error to bit-flip error for time-bin entanglement. From Fig. 2, the entanglement source generates a pair of time-bin entanglement such as ${|{{\phi ^\pm }}\rangle _{{a_1}{b_1}}}$ which has the same polarization, i.e., $|H\rangle$. Thus, we can obtain
The superscript “$H$” denotes the polarization $|H\rangle$. PC$_1$ is the fast Pockels cell, which can be employed to control the polarization mode of each state corresponding to different time-bins [67–69]. As discussed in Refs. [67,68], when the $|S\rangle$ component arrives, the PC$_1$ performs the transformation $|H{\rangle _S} \Leftrightarrow |V{\rangle _S}$. As a result, after the photons passing through the PC$_1$, the states ${| {{\phi ^ \pm }}\rangle _{{a_1}{b_1}}}$ evolve to
Subsequently, the photons are directed into PBSs, which can totally transmit the photon in $|H\rangle$ and reflect the photon in $|V\rangle$. After the PBSs, the photons in $|H\rangle$ passes through the half wave plates (HWPs), so that the photons in $|H\rangle$ are transformed to $|V\rangle$. Then, we set the path difference between spatial mode $c_1$($c_2$) and spatial mode $d_1$($d_2$) to be $c(L - S)$, where $c$ is the speed of ligh. Consequently, the time-bin entanglement can be changed into the spatial mode entanglement with the form of
The subscript “$s$” means the entanglement in spatial mode. As shown in Fig. 2, the 50:50 beam splitters (BSs) can make
Here, the superscript “$S$” and “$L$” mean the short arm and long arm which make the states in different spatial modes be in different time-bins. To erase the information on the spatial mode, the photons in four spatial modes are sent to the optical switches (OSs). The OS is used to put the photons from different spatial modes into one mode, but temporally multiplexed [70]. Then, the initial states $|\phi ^{+}\rangle$ and $|\phi ^{-}\rangle$ finally become
Consequently, one can correct the phase-flip error with the same method as discussed in Sec. II A once it is transformed to the bit-flip error.
Fig. 2. The schematic diagram of transforming phase-flip error to bit-flip error. The source generates one pair of time-bin entanglement. PC$_1$ represents the Pockels cells, which can realize the transformation from $|H_S\rangle$ to $|V_S\rangle$, when the photon in $|S\rangle$ arrives. PBS means the polarization beam splitter, which can totally transmit the photon in $|H\rangle$ and reflect the photon in $|V\rangle$. BS represents the 50:50 beam splitter and HWP represents the half-wave plate, which can perform $|H\rangle \Leftrightarrow |V\rangle$.
3. Entanglement purification for multi-photon entangled system
It is straightforward to extend our EPP to the multi-photon entanglement system. An $N$-photon time-bin entangled state can be described as
Such multi-photon time-bin entangled state can be generated in quantum dot by having the pump pulses to be in resonance with the triexciton or even higher excitonic states instead of the biexciton [10].
Here, we first take three-particle system as an example to show the basic principle of this multi-photon EPP and then extend it to the $N$-photon system. The three-photon Greenberger-Horne-Zeilinger (GHZ) states can be described as
We first describe the purification for bit-flip error. Without loss of generality, we assume that a bit-flip error occurs on the first qubit with the probability of $1-F$ and the same analysis can be carried out for the other qubits. Certainly, if the bit-flip error occurs on more than one qubit of the multi-photon system, the bit-flip errors can be corrected with the same method. The whole mixed state can be written as
In Fig. 3, two copies of mixed states with the form of Eq. (18) are sent to Alice, Bob, and Charlie, respectively. We denote the photons in the spatial modes $a_1$, $b_1$ and $c_1$ as $A_{1}$, $B_{1}$, and $C_{1}$, respectively, and other in $a_2$, $b_2$, and $c_2$, as $A_{2}$, $B_{2}$, and $C_{2}$, respectively. The whole system can be described as follows. With the probability of $F^{2}$, it is in the state $|\Phi ^{+}\rangle _{A_{1}B_{1}C_{1}}\otimes |\Phi ^{+}\rangle _{A_{2}B_{2}C_{2}}$. With the probability of $(1-F)^{2}$, it is in the state $|\Phi _{1}^{+}\rangle _{A_{1}B_{1}C_{1}}\otimes |\Phi _{1}^{+}\rangle _{A_{2}B_{2}C_{2}}$. With the same probability of $F(1-F)$, it is in the state $|\Phi ^{+}\rangle _{A_{1}B_{1}C_{1}}\otimes |\Phi ^{+}_{1}\rangle _{A_{2}B_{2}C_{2}}$ or $|\Phi _{1}^{+}\rangle _{A_{1}B_{1}C_{1}}\otimes |\Phi ^{+}\rangle _{A_{2}B_{2}C_{2}}$. After the six photons being sent through the SFGs, we find that only $|\Phi ^{+}\rangle _{A_{1}B_{1}C_{1}}\otimes |\Phi ^{+}\rangle _{A_{2}B_{2}C_{2}}$ and $|\Phi ^{+}_{1}\rangle _{A_{1}B_{1}C_{1}}\otimes |\Phi ^{+}_{1}\rangle _{A_{2}B_{2}C_{2}}$ make contributions to this EPP. For example,
Similarly, the state $|\Phi _{1}^{+}\rangle _{A_{1}B_{1}C_{1}}\otimes |\Phi _{1}^{+}\rangle _{A_{2}B_{2}C_{2}}$ becomes
In addition, the cross-combinations $|\Phi ^{+}\rangle _{A_{1}B_{1}C_{1}}\otimes |\Phi ^{+}_{1}\rangle _{A_{2}B_{2}C_{2}}$ and $|\Phi _{1}^{+}\rangle _{A_{1}B_{1}C_{1}}\otimes |\Phi ^{+}\rangle _{A_{2}B_{2}C_{2}}$ do not fulfill the principle of the SFGs and fail to create new photons, thereby can be eliminated automatically.
Fig. 3. The schematic diagram of the EPP for the multi-photon time-bin entanglement system with multiple SFGs. Each photon is directed into the corresponding SFG to create a high quality entangled state.
For an $N$-photon entanglement system, if a bit-flip error occurs, the parties correct the error with the same method as mentioned above. Let’s assume that a bit-flip error happens on the first photon, we also choose the cases where each SFG creates a new photon in the output mode. Thus, we will get
Consequently, we will get a new mixed state with a higher fidelity $F'=\frac {F^{2}}{F^{2}+(1-F)^{2}}$ compared with the initial state provided $F > \frac {1}{2}$ [35].
Finally, we briefly explain the principle of phase-flip error purification. We also take three-photon system as an example. If a phase-flip error occurs, the mixed state becomes
Each of photons converts the phase-flip error to bit-flip error with the same setup shown in Fig. 2. After the transformation, the mixed state in Eq. (23) evolves to
whereThus, in the EPP, we prepare two pairs of mixed state with the form of $\rho _{b}$. The whole system can be described as the mixture of four pure states. With the probability of $F^2$, it is in the state $|\Psi ^{+}\rangle _{{A_1}{B_1}{C_1}}|\Psi ^{+}\rangle _{{A_2}{B_2}{C_2}}$. With an equal probability of $F(1-F)$, the system is in $|\Psi ^{+}\rangle _{{A_1}{B_1}{C_1}}|\Psi ^{-}\rangle _{{A_2}{B_2}{C_2}}$ and $|\Psi ^{-}\rangle _{{A_1}{B_1}{C_1}}|\Psi ^{+}\rangle _{{A_2}{B_2}{C_2}}$. With the probability of $(1-F)^2$, the system is in $|\Psi ^{-}\rangle _{{A_1}{B_1}{C_1}}|\Psi ^{-}\rangle _{{A_2}{B_2}{C_2}}$. According to the principle of the SFG, only the states $|\Psi ^{+}\rangle _{{A_1}{B_1}{C_1}}|\Psi ^{+}\rangle _{{A_2}{B_2}{C_2}}$ and $|\Psi ^{-}\rangle _{{A_1}{B_1}{C_1}}|\Psi ^{-}\rangle _{{A_2}{B_2}{C_2}}$ make contributions to this EPP and the other items such as $|\Psi ^{+}\rangle |\Psi ^{-}\rangle$ and $|\Psi ^{-}\rangle |\Psi ^{+}\rangle$ can be eliminated automatically. Therefore, after directing the photons into SFGs, the remained state is $|\Psi ^{+}\rangle _{{D_1}{D_2}{D_3}}$ with the probability of $\frac {{{F^2}}}{4}$ and $|\Psi ^{-}\rangle _{{D_1}{D_2}{D_3}}$ with the probability of $\frac {{{(1-F)^2}}}{4}$.
As discussed in Ref. [71], we can convert $|\Psi ^{+}\rangle _{{D_1}{D_2}{D_3}}$ to $|\Phi ^{+}\rangle _{{D_1}{D_2}{D_3}}$, and $|\Psi ^{-}\rangle _{{D_1}{D_2}{D_3}}$ to $|\Phi ^{-}\rangle _{{D_1}{D_2}{D_3}}$, yielding a new mixed state $\rho '_{p}$ as
4. Entanglement purification with practical SPDC source
So far, we have discussed this EPP for purifying an arbitrary mixed time-bin entangled state based on the ideal source. However, the ideal source is not available within the reach of current technique. As pointed out in Ref. [24], one method to generate the time-bin entanglement is to use the SPDC source. As shown in Fig. 4, a pump pulse of ultraviolet light is sent through a beta barium borate crystal named BBO$_2$ to produce polarization entangled pairs of photons in the spatial modes $a_{2}$ and $b_{2}$ with a small probability of $p$. The state can be written as
Then, the light pulse is sent through a second nonlinear crystal BBO$_1$ to create the correlated photons in the spatial modes $a_{1}$ and $b_{1}$ with the same form as Eq. (27). Similarly, let photons pass through the PBSs and PCs, the time-bin entangled state can be generated as the form of Eq. (28). Therefore, we can write the whole system ${\rho _{{c_1}{d_1}}} \otimes {\rho _{{c_2}{d_2}}}$ as
From Eq. (29), the system is in the state $\frac {1}{2}(|SS\rangle + |LL\rangle ) \otimes (|{c_1}{d_1}\rangle + |{c_2}{d_2}\rangle )$ with the probability of $p$. Actually, it is the hyperentanglement between the time-bin and spatial mode DOFs. With an equal probability of $p^2$, the system is in the state $|\phi ^+\rangle _{{c_1}{d_1}}^{ \otimes 2}$, $|\phi ^+\rangle _{{c_2}{d_2}}^{ \otimes 2}$, or $|\phi ^+\rangle _{{c_1}{d_1}}|\phi ^+\rangle _{c_{2}d_{2}}$. As pointed out in Refs. [24,50], the double-pair emission becomes a big challenge in the realization of quantum information processing. For example, in Ref. [24], after performing the entanglement swapping, the remained state is a poorly entangled state and the fidelity is upper bounded by $1/2$. Interestingly, we will show that the double-pair emission problem can be overcome by using the SFG.
Fig. 4. The schematic diagram of the EPP for purifying mixed states encoded in time-bin DOF under the practical SPDC source.
For two-photon cases which mean that one SPDC source generates a pair of entangled state while the other generates vacumm state, the mixed system is in the state $|\phi ^+\rangle _{c_{1}d_{1}}$ ($|\phi ^+\rangle _{c_{2}d_{2}}$) or $|\psi ^+\rangle _{c_{1}d_{1}}$ ($|\psi ^+\rangle _{c_{2}d_{2}}$) with a probability of $pF$ and $p(1-F)$, respectively. For four-photon cases where one SPDC source generates two pairs entangled states and the other emits none, the whole system is in the state $|{\phi ^ + }\rangle _{{c_1}{d_1}}^{ \otimes 2}$ or $|{\phi ^ + }\rangle _{{c_2}{d_2}}^{ \otimes 2}$ with an equal probability of $p^2F$. It is in the state $|{\psi ^ + }\rangle _{{c_1}{d_1}}^{ \otimes 2}$ or $|{\psi ^ + }\rangle _{{c_2}{d_2}}^{ \otimes 2}$ with an equal probability of $p^2(1-F)$. In addition, if each of SPDC sources generates one pair of entanglement, the system will be in the state $|\phi ^+\rangle _{c_{1}d_{1}}|\phi ^+\rangle _{c_{2}d_{2}}$ with the probability of $p^2F^2$. The system is in the state $|\phi ^+\rangle _{c_{1}d_{1}}|\psi ^+\rangle _{c_{2}d_{2}}$ or $|\psi ^+\rangle _{c_{1}d_{1}}|\phi ^+\rangle _{c_{2}d_{2}}$ with an equal probability of $p^2F(1-F)$. The whole system is in $|\psi ^+\rangle _{c_{1}d_{1}}|\psi ^+\rangle _{c_{2}d_{2}}$ with the probability of $p^2(1-F)^2$.
First, we discuss the two-photon cases. There is only one photon either in $c_1(d_1)$ or in $c_2(d_2)$. Consequently, it fails to create a new photon [24] and has no contribution to the purification, which can be eliminated automatically.
Similar to the two-photon cases, the four-photon cases also have no contribution to the EPP and they can be eliminated automatically. To elaborate, with the probability of $p^2F$, the whole mixed system is in $|\phi ^+\rangle _{{c_1}{d_1}}^{ \otimes 2}$ which can be written as
From Eq. (30), each of spatial modes $c_{1}$ and $d_{1}$ contain two photons which unsuccessfully creates new photons due to the violation of the principle of SFGs [24]. As a result, the item $|\phi ^+\rangle _{{c_1}{d_1}}^{ \otimes 2}$ has no contribution to the purification and can be eliminated automatically. Moreover, the same analysis can be carried out for $|\phi ^+\rangle _{{a_2}{b_2}}^{ \otimes 2}$, $|\psi ^+\rangle _{{a_1}{b_1}}^{ \otimes 2}$ and $|\psi ^+\rangle _{{a_2}{b_2}}^{ \otimes 2}$ and they can also be automatically eliminated.
Finally, we discuss the cases where one pair of entangled state is in spatial modes $c_{1}d_{1}$ and the other is in $c_{2}d_{2}$ as $|\phi ^+\rangle _{c_{1}d_{1}}|\phi ^+\rangle _{c_{2}d_{2}}$, $|\phi ^+\rangle _{c_{1}d_{1}}|\psi ^+\rangle _{c_{2}d_{2}}$ and $|\psi ^+\rangle _{c_{1}d_{1}}|\phi ^+\rangle _{c_{2}d_{2}}$ as well as $|\psi ^+\rangle _{c_{1}d_{1}}|\psi ^+\rangle _{c_{2}d_{2}}$. These items are essentially the ideal source. Thus, they make contributions to the EPP. Consequently, we can get a new mixed state with the fidelity of $\frac {{{F^2}}}{{{F^2} + {{(1 - F)}^2}}}$, which illustrates that the SPDC source is not an obstacle for EPP.
5. Discussion
For the conventional EPPs in an optical system, the PBS essentially acts as the role of CNOT gate between polarization and spatial mode. By classical communication, Alice and Bob choose the same measurement results, which corresponds to the even parity of the mixed states. Surprisingly, the SFG in this EPP plays the similar role as the PBS in Ref. [36]. Here, we choose the case that the photons in Alice’s and Bob’s location are in the same time-bin, i.e., $|SS\rangle$ or $|LL\rangle$. We call this case two-mode case, which is an even parity-check. In this way, the total success probability is the same as the protocol of Ref. [36], and half as the protocol of Ref. [35] without considering the efficiency of SFG. Meanwhile, the SFG is quite different from the PBS in Ref. [36], for in Ref. [36], the photon number is conversed, but only collapsed to the four-photon state. Therefore, they need other measurement operations to get the two-photon state. However, the photon number conversation in this EPP is substituted to energy conversation and the four photons directly become two photons after passing through the SFGs without additional measurement as Ref. [36].
The key element in this protocol is the nonlinear element SFG which has been widely used in quantum information processing [24,72–76]. The Hamiltonian of the SFG to generate a photon can be described as $H = i\alpha (a_S^\dag {b_S}{c_S}-a_L^\dag {b_L}{c_L}) + h.c.$ [24] where $a_{S}^{\dag }$ and $a_{L}^{\dag }$ are associated to the photons created by different times $|S\rangle$ and $|L\rangle$. The success probability of SFG to create a new photon is $\eta _{SFG}=\alpha ^2\tau ^{2}$ [24] where $\tau$ is the interaction time between the incoming photon and the nonlinear medium. Thus, the whole success probability of purifying a bit-flip error is $\frac {1}{2}[F^{2}+(1-F)^{2}]\alpha ^{2N}\tau ^{2N}$ where $N$ is the number of the particles. As discussed in Ref. [74], the efficiency of SFG can reach $1.06 \times 1{0^{ - 7}}$ which is sufficient to provide a simple method based on linear optics for the implementation device-independent quantum key distribution (DI-QKD). If one employs 10 cm nonlinear waveguide, the efficiency of SFG can be greater than $6 \times 1{0^{ - 7}}$, which may significantly increase the success probability of this EPP for a large $N$. To further increase the efficiency of SFG, one can adopt highly nonlinear organic materials [75] or tighter field confinement [76].
6. Conclusion
In conclusion, we have presented the first EPP to purify arbitrary mixed entangled states encoded in time-bin assisted by the SFGs. We first explain this EPP with two-photon entangled states, and then we extend this EPP to a multi-photon quantum system. Subsequently, we consider this EPP under the practical SPDC source. We show that the double-pair emission in the conventional EPPs can be eliminated automatically. This EPP does not require the sophisticated CNOT gate or similar operations and the whole purification is based on the common element SFG. This EPP is feasible in current experiment condition. Moreover, since entanglement swapping and entanglement purification are two key elements in a quantum repeater, our work may be combined with the faithful entanglement swapping [23–26] to realize a full quantum repeater. It may also be used in DI-QSDC [77] to improve the security and communication quality.
Funding
National Natural Science Foundation of China (12005106, 11974189); Priority Academic Program Development of Jiangsu Higher Education Institutions; the Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX20-0731).
Disclosures
The authors declare no conflicts of interest.
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