1. Introduction

The development of quantum communication [1–7] and quantum network [8,9] have attracted much attention from industries and researchers. Many important quantum secure communication protocols such as quantum key distribution (QKD) [1], quantum secret sharing (QSS) [2], quantum secure direct communication (QSDC) [3–5] and some quantum computation protocols [10–15] all employ quantum entanglement. Among these protocols, the communication parties should first distribute and share nonlocal high quality entanglement. However, photon loss during entanglement distribution scales exponentially with the length of quantum channel and limits the total distance and secrecy of quantum communication [16].

Quantum repeater (QR) is a promising method to realize long-distance quantum communication which was first proposed in 1998 [16] and has been widely developed during the last 20 years [17–32]. In general, the total distance is divided into several short-distance segments. Entanglement is distributed in each short-distance segments. During entanglement distribution, if the high quality entanglement is degraded because of environment noise, entanglement purification is exploited to distill high quality entangled states from low quality entangled states. Subsequently, the entanglement swapping is employed between two adjacent segments and finally the high quality entanglement is stored using quantum memory [16]. Much attention has been devoted not only in quantum repeater, but also in its building blocks such as entanglement distribution [33,34], entanglement purification [35–63], entanglement swapping [64] and quantum memory [65–68]. Although great efforts have been made and quantum repeaters were designed and optimized for three generations [27], practical quantum repeater is still a big challenge with current experimental technology, for all building blocks require sufficiently high accuracy, i.e., with errors of a few percent [69]. In recent years, Zwerger et al. introduced a new kind of quantum repeater, i.e., measurement-based quantum repeater (MBQR) [69,70]. Different from the conventional quantum repeaters, MBQR avoids performing accurate coherent quantum gates. Only resource states prepared off-line in a probabilistic way are coupled with incoming photons by simple Bell-state measurement (BSM) [71,72].

Entanglement purification is a key part of QR. Entanglement purification protocol (EPP) was first proposed to counteract the inherent noise aided with controlled-not (CNOT) gates in 1996 [35]. Since then, many recurrence EPPs [37–41,44–46,48–53,56–59] which require two noisy copies to yield one high-quality entanglement have been studied. However, this kind of EPPs often have low yield. Additionally, the deterministic EPPs [42,43] and robust EPPs using hashing protocol [36,47] were exploited. For example, Sheng et al. presented a deterministic EPP to obatin a maximally entangled state just with one step, which significantly reduces the consumption of resource states [43]. In EPPs using hashing protocol [36,47], arbitrary $n$ copies are selected from the total $N$ pairs and the bilateral local CNOT operations are performed on each of the $n$ pairs. After measuring the selected pairs, the information of the remaining state is revealed thereby the parties can purify the polluted states. In 2021, the authors in Refs. [60–63] only employed one noisy pair to distill a high-fidelity entanglement using hyperentanglement, which can efficiently improve the efficiency of existing EPPs. Moreover, the measurement-based entanglement purification protocols (MB-EPPs) [69,70] were introduced which are different from EPPs based on coherent gates [35,37]. Furthermore, the results indicated that the MB-EPPs can tolerate more noise than conventional EPPs [70,73,74]. The original MB-EPP proposed by Zwerger et al. [70,73] provided us a universal model for general mixed entangled states. However, the original MB-EPP not only requires the ideal entanglement source but also the ideal resource state, i.e., the clean multi-partite entangled state. Unfortunately, in optical system, the ideal entanglement source is unavailable. Current available entanglement source is the spontaneous parametric down conversion (SPDC) source. Due to the probabilistic nature of the SPDC source [51], the double-pair noise components, which make spurious contributions to the resultant entangled state in both entanglement purification and resource state preparation remain big challenges in the MB-EPP.

In this paper, we will propose a feasible MB-EPP in linear optical system, based on the original work in Ref. [73]. We first describe a simple example of the MB-EPP with two-copy of mixed states. Then, we extend it to the multi-copy case. Finally, we design a possible experimental realization and provide a detailed analysis of the MB-EPP with SPDC sources. We use SPDC sources to generate both the resource states and the incoming mixed states. Interestingly, in our MB-EPP, the double-pair emission noise caused by SPDC sources can be eliminated automatically. Our results show that it is possible to realize the MB-EPP in current experiment technology. Therefore, this work may become one of the key steps for MB-EPP from the original theoretical protocol to practical experiment implementation. Moreover, combined with suitable quantum memory and entanglement swapping, our MB-EPP may have application potential in the implementation of a practical MBQR.

The paper is organized as follows. In Sec. 2, we first present a simple two-copy example of MB-EPP in the ideal optical system. In Sec. 3, we extend the MB-EPP to three-copy case and arbitrary $N$-copy case. In Sec. 4, we provide a detailed analysis of such MB-EPP in current experiment condition. We show that although the ideal entanglement source is unavailable, the current SPDC sources can also be employed to perform this MB-EPP and the double-pair emission noise caused by SPDC sources can be removed automatically during the BSMs. Finally, in Sec. 5, we present a discussion and make a conclusion.

2. MB-EPP with two noisy copies

In this section, we first consider the MB-EPP with two-copy mixed states as shown in Fig. 1. We let $|\phi ^{\pm }\rangle _{ab}$ and $|\psi ^{\pm }\rangle _{ab}$ are four Bell states of the form

Here $|H\rangle$ and $|V\rangle$ respectively denote horizontal polarization and vertical polarization of photon. The subscripts $a$ and $b$ represent the spatial modes. As shown in Fig. 1, suppose Alice and Bob want to share the maximally entangled state $|\phi ^{+}\rangle _{ab}$. However, a bit-flip error occurs with the probability of $1-F$ and $|\phi ^{+}\rangle _{ab}$ becomes $|\psi ^{+}\rangle _{ab}$. In this way, the whole state becomes

Hence, the whole system ${\rho _{{a_1}{b_1}}} \otimes {\rho _{{a_2}{b_2}}}$ can be described as follows. The system is in $|\phi ^+\rangle _{{a_1}{b_1}}|\phi ^+\rangle _{{a_2}{b_2}}$ with the probability of ${F^2}$. It is in the state $|\phi ^+\rangle _{{a_1}{b_1}}|\psi ^+\rangle _{{a_2}{b_2}}$ and $|\psi ^+\rangle _{{a_1}{b_1}}|\phi ^+\rangle _{{a_2}{b_2}}$ with an equal probability of $F(1 - F)$. With the probability of $(1 - F)^2$, it is in the state $|\psi ^+\rangle _{{a_1}{b_1}}|\psi ^+\rangle _{{a_2}{b_2}}$.

 

Fig. 1. The schematic diagram of MB-EPP in linear optics for two noisy copies. It needs two resource states R$_1$ and R$_2$. The photons in modes $a_1$ and $g_1$, $a_2$ and $g_2$, $b_1$ and $h_1$, $b_2$ and $h_2$ are directed to four 50:50 beam-splitters (BSs) to couple with resource states via BSMs, respectively. The polarizing beam-splitters (PBSs) totally transmit the photon in $|H\rangle$ and reflect the photon in $|V\rangle$. A successful BSM corresponds to a projection onto $|\psi ^\textrm{+}\rangle$ or $|\psi ^\textrm{-}\rangle$ [71,72]. We take the BSM1 for an example. If the detectors ${D_1}{D_{{1^\prime }}}$ or ${D_2}{D_{{2^\prime }}}$ each register one photon, it indicates a projection onto $|\psi ^+\rangle$. If the detectors ${D_1}{D_{{2^\prime }}}$ or ${D_2}{D_{{1^\prime }}}$ each register one photon, it corresponds to a projection onto $|\psi ^-\rangle$.

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In the MB-EPP, two pairs of resource states $R_{1}$ and $R_{2}$ are required in Alice’s and Bob’s locations, respectively. In this two-copy case, the resource state is a three-photon Greenberg-Horne-Zeilinger (GHZ) state of the form

The three-photon GHZ states $R_{1}$ and $R_{2}$ are in the spatial modes $g_{1}g_{2}g_{3}$ and $h_{1}h_{2}h_{3}$, respectively. Therefore, the two mixed states combined with two resource states can be described as ${\rho _{{a_1}{b_1}}} \otimes {\rho _{{a_2}{b_2}}}\otimes |\textrm{GHZ}\rangle _{g_{1}g_{2}g_{3}}\otimes |\textrm{GHZ}\rangle _{h_{1}h_{2}h_{3}}$. Alice (Bob) performs BSMs on ${a_i}$ (${b_i}$) and ${g_i}$ (${h_i}$) ($i = 1,2$). The success purification is that Alice and Bob pick up the same outcome of measurement [69,70,73]. Here, due to the fact that the standard BSM [71,72] can only distinguish two of the four Bell states $|\psi ^{\pm }\rangle$, we merely consider the case that all the measurement outcomes are $|\psi ^{+}\rangle$ or $|\psi ^{-}\rangle$. For example, with the probability ${F^2}$, state ${\rho _{{a_1}{b_1}}} \otimes {\rho _{{a_2}{b_2}}}\otimes |\textrm{GHZ}\rangle _{g_{1}g_{2}g_{3}}\otimes |\textrm{GHZ}\rangle _{h_{1}h_{2}h_{3}}$ can be described as

Therefore, by selecting the case that all BSM results are $|\psi ^{+}\rangle$ or $|\psi ^{-}\rangle$ (Appendix A), the photons in ${g_3}{h_3}$ will become

Similarly, after the parties performing the BSMs and picking up $|\psi ^+\rangle$ or $|\psi ^-\rangle$, the state $|\psi ^+\rangle _{{a_1}{b_1}}|\psi ^+\rangle _{{a_2}{b_2}}|\textrm{GHZ}\rangle _{{g_1}{g_2}{g_3}}|\textrm{GHZ}\rangle _{{h_1}{h_2}{h_3}}$ will evolve to Eq. (6) with the form of

The remaining items $|\psi ^+\rangle _{{a_1}{b_1}}|\phi ^+\rangle _{{a_2}{b_2}}$ and $|\phi ^+\rangle _{{a_1}{b_1}}|\psi ^+\rangle _{{a_2}{b_2}}$ of $\rho _{a_1b_1}\otimes \rho _{a_2b_2}$ combined with $|\textrm{GHZ}\rangle _{{g_1}{g_2}{g_3}}|\textrm{GHZ}\rangle _{{h_1}{h_2}{h_3}}$ fail to make all the BSM results be $|\psi ^+\rangle$ or $|\psi ^-\rangle$. This corresponds to a failure for MB-EPP. Therefore, by selecting the cases that all the BSM results are $|\psi ^+\rangle$ or $|\psi ^-\rangle$, state in ${g_3}{h_3}$ will become a new mixed state

As discussed in Ref. [35], if $F > 0.5$, the fidelity of resultant state is higher than that of initial one.

3. MB-EPP with multiple noisy copies

In this section, we will present the analysis of MB-EPP with multiple noisy copies. As shown in Fig. 2 ($N=3$), we need three copies of the mixed states ${\rho _{{a_1}{b_1}}}$, ${\rho _{{a_2}{b_2}}}$ and ${\rho _{{a_3}{b_3}}}$. The resource state is a four-photon GHZ state which can be given by

If a bit-flip error occurs, states in modes ${a_1}{b_1}$, ${a_2}{b_2}$ and ${a_3}{b_3}$ will become the mixed states shown in Eq. (2). The whole mixed system ${\rho _{{a_1}{b_1}}}\otimes {\rho _{{a_2}{b_2}}}\otimes {\rho _{{a_3}{b_3}}}$ can be described as follows. It is in the state $|\phi ^+\rangle _{{a_1}{b_1}}|\phi ^+\rangle _{{a_2}{b_2}}|\phi ^ +\rangle _{{a_3}{b_3}}$ with the probability of $F^3$. It is in the state $|\psi ^+\rangle _{{a_1}{b_1}}|\psi ^+\rangle _{{a_2}{b_2}}|\psi ^+\rangle _{{a_3}{b_3}}$ with the probability of $(1\!-\!F)^3$. With an equal probability of ${F^2}(1 \!-\! F)$, it is in the state $|\phi ^+\rangle _{{a_1}{b_1}}|\phi ^+\rangle _{{a_2}{b_2}}|\psi ^+\rangle _{{a_3}{b_3}}$, $|\phi ^+\rangle _{{a_1}{b_1}}|\psi ^+\rangle _{{a_2}{b_2}}|\phi ^+\rangle _{{a_3}{b_3}}$ and $|\psi ^+\rangle _{{a_1}{b_1}}|\phi ^ +\rangle _{{a_2}{b_2}}|\phi ^+\rangle _{{a_3}{b_3}}$. The whole mixed state is in $|\phi ^ +\rangle _{{a_1}{b_1}}|\psi ^+\rangle _{{a_2}{b_2}}|\psi ^ +\rangle _{{a_3}{b_3}}$, $|\psi ^ +\rangle _{{a_1}{b_1}}|\psi ^ +\rangle _{{a_2}{b_2}}|\phi ^ +\rangle _{{a_3}{b_3}}$ and $|\psi ^ +\rangle _{{a_1}{b_1}}|\phi ^ +\rangle _{{a_2}{b_2}}|\psi ^ +\rangle \!_{{a_3}{b_3}}$ with an equal probability of $F{(1 \!-\! F)^2}$.

 

Fig. 2. The schematic diagram of MB-EPP protocol for multiple noisy copies [73]. Two pairs of $N+1$-photon resource states are required. The boxes represent the standard BSMs [71,72] depicted as Fig. 1.

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The purification principle is similar to the two-copy case (Appendix B). After performing the BSMs, they pick up the case that all the BSM results are $|\psi ^{+}\rangle$ or $|\psi ^{-}\rangle$. In this way, the state $|\phi ^ +\rangle _{{a_1}{b_1}}|\phi ^ +\rangle _{{a_2}{b_2}}|\phi ^ +\rangle _{{a_3}{b_3}}$ combined with two resource states $|\textrm{GHZ}\rangle _{{g_1}{g_2}{g_3}{g_4}}$ and $| \textrm{GHZ}\rangle _{{h_1}{h_2}{h_3}{h_4}}$ will evolve to

On the other hand, with the same selection principle, the state $|\psi ^ +\rangle _{{a_1}{b_1}}|\psi ^ +\rangle _{{a_2}{b_2}}|\psi ^ +\rangle _{{a_3}{b_3}}$ combined with two resource states $|\textrm{GHZ}\rangle _{{g_1}{g_2}{g_3}{g_4}}$ and $|\textrm{GHZ}\rangle \!_{{h_1}{h_2}{h_3}{h_4}}$ will collapse to

Additionally, the other items of $\rho _{a_1b_1}\otimes \rho _{a_2b_2}\otimes \rho _{a_3b_3}$ such as $P\left [ {|{\psi ^ + }{\rangle ^{ \otimes 2}}|{\phi ^ + }\rangle } \right ]$ and $P\left [ {|{\phi ^ + }{\rangle ^{ \otimes 2}}|{\psi ^ + }\rangle } \right ]$ where $P\left [ \cdot \right ]$ denotes the permutation of three pairs of Bell states can be eliminated automatically according to the outcomes of BSMs. Finally, we can obtain a new mixed state

It is straightforward to extend this protocol to the MB-EPP with $N$ ($N>2$) noisy copies, as shown in Fig. 2 (Appendix C). After performing the MB-EPP, the new mixed state with $N$ noisy copies can be given by

In this way, we have completely explained the MB-EPP in ideal linear optics.

Fig. 3 plots the final fidelity $(F_{N})$ of the distilled new mixed state as a function of the initial fidelity of the input mixed states with $N=2, 3, 4, 5$. In addition, the research shows that the more noisy copies being purified, the higher fidelity of MB-EPP will be obtained. To be specific, if the initial fidelity is 0.65, the new fidelity is 0.775 for $N=2$, 0.865 for $N=3$, and 0.923 for $N=4$, as well as 0.956 for $N=5$. Refs. [48,57] also investigated the multi-copy EPP in linear optics. Our multi-copy MB-EPP has the same purified fidelity with Ref. [57]. Moreover, the previous multi-copy EPPs are based on the postselection, all purified entangled photons would be destroyed, similar to the two-copy case [38]. In our MB-EPP, the purified entangled photon pair in $g_{N+1}h_{N+1}$ can be remained for further application, which is an another advantage of our MB-EPP. In a practical experiment, the initial fidelity $F$ can be easily controlled like Refs. [40,51]. For example, if two parties want to prepare the mixed state as Eq. (2) with the fidelity of $F=0.75$, only one party should set a half-wave plate with the axis orienting at $\pm 14^{\circ }$. This will cause that the two-photon system is in the state $|\phi ^+\rangle$ with the probability $75{\%}$ and $|\psi ^+\rangle$ with the probability $25{\%}$.

 

Fig. 3. The fidelity $F_{N}$ altered with the initial fidelity $F$, where we control $N=2, 3, 4, 5$.

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4. MB-EPP in practical linear optics

In the previous section, we have completed the analysis for this MB-EPP with ideal linear optical elements, especially the ideal entanglement sources. Meanwhile, this MB-EPP also requires Alice and Bob to generate the ideal GHZ states. Unfortunately, the ideal entanglement source which can exactly generate the entangled photon pair is unavailable. The practical entanglement source is SPDC source, which generates entangled state in a probabilistic way. Moreover, the ideal GHZ state is also unavailable and the common method to generate GHZ state is still to exploit the SPDC sources. It seems that the inherent probability nature of SPDC sources may become an obstacle to realize the MB-EPP. Interestingly, we will show that the MB-EPP can also work with SPDC sources. Especially, the double-pair emission noise caused by the SPDC sources can be removed automatically by the BSMs [71,72]. In Fig. 4, we design an alternative approach to realize the MB-EPP in the two-copy case. The approach is composed of three parts. The first part is the entanglement generation. The second is the preparation of two resource states using the approach in Ref. [75]. The third is the BSM to perform the purification.

 

Fig. 4. The schematic diagram of MB-EPP in linear optics. It has three parts which are entanglement generation, preparation of resource states and BSM. The UV laser passes through two BBO crystals to generate two entangled states $|\Phi ^+\rangle _{a_{1}b_{1}}$ and $|\Phi ^+\rangle _{a_{2}b_{2}}$. The photons in modes $a_1$, $b_1$, $a_2$ and $b_2$ are directed to four 50:50 beam splitters (BSs) to couple with resource states via BSMs, respectively. The polarizing beam splitters (PBSs) totally transmit the photon in $|H\rangle$ and reflect the photon in $|V\rangle$. A successful BSM corresponds to a projection into $|\psi ^\textrm{+}\rangle$ ($|\psi ^\textrm{-}\rangle$), providing a click on ${D_{H3}}{D_{V3}}$ (${D_{H3}}{D_{V4}}$) or ${D_{H4}}{D_{V4}}$ (${D_{H4}}{D_{V3}}$). The preparation of resource states can be described as follows. The UV laser passes through BBO$_{3}$, BBO$_{4}$, BBO$_{5}$, and BBO$_{6}$ to generate four entangled states $|\Phi ^+\rangle _{g_{1}g_{2}}$, $|\Phi ^+\rangle _{g_{3}g_{4}}$, $|\Phi ^+\rangle _{h_{1}h_{2}}$, and $|\Phi ^+\rangle _{h_{3}h_{4}}$. The photons in modes $g_{2}g_{3}$ and $h_{2}h_{3}$ are directed to two BSs, respectively. When a click on $D_3$ and $D_1$, the remaining particles can be utilized to help us to operate MB-EPP.

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In Fig. 4, the pump passes through a beta barium borate (BBO) crystal to probabilistically generate one entangled pair in modes $a_1$ and $b_1$ as [51]

Here, $|vac\rangle$ is the vacuum state and we omit the higher-order state for simplicity. Eq. (16) means that Alice and Bob share one pair of maximally entangled state $|\phi ^+\rangle _{a_{1}b_{1}}$ with the probability of $p$. However, a bit-flip error may occur during the photon transmission in the inherent noisy channels, which makes Eq. (16) become

Consequently, Alice and Bob share the mixed states with the form of

4.1 Preparation for resource states

In this subsection, we consider the preparation of resource states based on Ref. [75]. In this MB-EPP, we require two pairs of three-photon GHZ states. As shown in Fig. 4, the generation of three-photon GHZ state requires two pairs of two-photon Bell states. For the generation of the first GHZ state, we use BBO$_3$ and BBO$_4$ to generate two pairs of Bell states. BBO$_5$ and BBO$_6$ are employed to construct the second three-photon GHZ state. Here, we take the generation of the first GHZ state for example and the same principle can be carried out for the second resource state. We use a click in the detector $D_1$ of Fig. 4 to herald a success generation of the first resource state. In detail, the two entangled states generated by SPDC sources (BBO$_3$ and BBO$_4$) are $|\Phi ^+\rangle _{{h_1}{h_2}}$ and $|\Phi ^+\rangle _{{h_3}{h_4}}$. According to Eq. (16), the system $|\Phi ^+\rangle _{{h_1}{h_2}} \otimes |\Phi ^+\rangle _{{h_3}{h_4}}$ is in the state $\frac {1}{2}(|{HH}\rangle +|{VV}\rangle )\otimes (|{{h_1}{h_2}}\rangle +|{{h_3}{h_4}}\rangle )$ with the probability of $p$, which is essentially the hyperentanglement in polarization and spatial modes. With an equal probability of $\frac {{{p^2}}}{4}$, it is in the state $|{\phi ^ + }\rangle _{{h_1}{h_2}}^{ \otimes 2}$ and $|{\phi ^ + }\rangle _{{h_3}{h_4}}^{ \otimes 2}$. It is in the state $|\phi ^+\rangle _{{h_1}{h_2}}|\phi ^+\rangle _{{h_3}{h_4}}$ with the probability of ${{p^2}}$.

First, with the probability of $p$, the system is a hyperentanglement in polarization and spatial modes, which can be written as

Let the photons in spatial modes $h_2$ and $h_3$ pass through the PBSs which totally transmit the photons in $|H\rangle$ and reflect the photons in $|V\rangle$. Hence, the items ${|H\rangle _{{h_1}}}|H{\rangle _{{h_2}}}$ and ${|H\rangle _{{h_3}}}|H{\rangle _{{h_4}}}$ fail to make $D_1$ click, thereby can be eliminated. For the other two items ${|V\rangle _{{h_1}}}|V{\rangle _{{h_2}}}$ and ${|V\rangle _{{h_3}}}|V{\rangle _{{h_4}}}$, the photon in $|V\rangle$ in the spatial modes $h_2$ and $h_3$ separately result in a click in $D_1$. Thus, the system described as Eq. (19) becomes

The state $|{\phi ^ + }{\rangle _{{h_1}{h_2}}}|{\phi ^ + }{\rangle _{{h_3}{h_4}}}$ can be written as

Obviously, the item ${|H\rangle _{{h_1}}}|H{\rangle _{{h_2}}}|H{\rangle _{{h_3}}}|H{\rangle _{{h_4}}}$ can be eliminated because it makes no click in $D_1$. Additionally, the item $|H{\rangle _{{h_1}}}|H{\rangle _{{h_2}}}|V{\rangle _{{h_3}}}|V{\rangle _{{h_4}}}$ makes $D_1$ register one photon and will evolve to $|H{\rangle _{{h_1}}}|V{\rangle _{{h_4}}}|H{\rangle _{{h_5}}}$. With the same principle, $|V{\rangle _{{h_1}}}|V{\rangle _{{h_2}}}|H{\rangle _{{h_3}}}|H{\rangle _{{h_4}}}$ will collapse to $|V{\rangle _{{h_1}}}|H{\rangle _{{h_4}}}|V{\rangle _{{h_5}}}$. The item $|V{\rangle _{{h_1}}}|V{\rangle _{{h_2}}}|V{\rangle _{{h_3}}}|V{\rangle _{{h_4}}}$ can also make the detector $D_1$ click and will become $|V{\rangle _{{h_1}}}|V{\rangle _{{h_4}}}$. Therefore, after passing through the PBSs, BS and HWP, the state $|{\phi ^ + }{\rangle _{{h_1}{h_2}}}|{\phi ^ + }{\rangle _{{h_3}{h_4}}}$ will evolve to

The same analysis can be done for the double-pair emissions $|{\phi ^ + }\rangle _{{h_1}{h_2}}^{ \otimes 2}$ and $|{\phi ^ + }\rangle _{{h_3}{h_4}}^{ \otimes 2}$ from SPDC sources. After above operations, they will become

Then, after performing the operation ${\sigma _x} =|H\rangle \langle V| + |V\rangle \langle H|$ on the photon in spatial mode $h_4$, the remained state can be written as

The second resource state has the same form as Eq. (25), which can be given by

From Eq. (25) and Eq. (27), only the items $|\gamma \rangle _1$ and $|\gamma \rangle _2$ make contributions to the resource states. The other items are essentially the disturbed items which are caused by the SPDC sources. Interestingly, we will show that such disturbed items which have spurious contributions to this MB-EPP can be eliminated automatically by the coincidence measurements in spatial modes $h_1h_4h_5$ and $g_1g_4g_5$.

4.2 MB-EPP with SPDC sources

In order to realize this MB-EPP with current experimental technology, we should prepare two copies of clean input Bell states and two GHZ states. Unfortunately, the input state generated by SPDC sources can be described as

Let’s first discuss the item $|\phi ^+\rangle _{a_1b_1}\otimes |\phi ^+\rangle _{a_2b_2}\otimes |\textrm{Res}\rangle _1\otimes |\textrm{Res}\rangle _2$. The whole state $|\Gamma _1\rangle$ generated by six SPDC sources from Fig. 4 can be written as

The preconditions to realize this MB-EPP are the successful operation of four BSMs and the preparation of resource states. In this way, the basic principle of this MB-EPP is to pick up the "twelve-mode" case, i.e., all the BSM results are the same, such as $|\psi ^{+}\rangle$. As a consequent, we select the coincidence detection on $D_{H1}D_{V1}$, $D_{H3}D_{V3}$, $D_{H5}D_{V5}$ and $D_{H7}D_{V7}$. On the other hand, according to the mentioned above, the success preparation of resource states are heralded by one of the detectors $D_{1}D_{2}$ and $D_{3}D_{4}$ clicking one photon. Here, we select the case that $D_{1}$ and $D_{3}$ register one photon, respectively. Finally, in order to verify the successful purification, we should detect the purified photon pair in spatial modes $h_{5}$ and $g_{5}$. Hence, the new mixed entanglement with higher fidelity can be obtained by selecting the "twelve-mode" case.

From Eq. (30), it is clear that the items which contain the single photon state $|\alpha \rangle _1$ or $|\alpha \rangle _2$, two-photon state $|\beta \rangle _1$ or $|\beta \rangle _2$ cannot lead the "twelve-mode" case, for the total photon number is less than twelve. In this way, such items can be eliminated automatically. Similarly, the items which contain three-photon states $|\delta \rangle _1$ or $|\delta \rangle _2$ cannot satisfy the "twelve-mode" case, for the two photons are in the same spatial modes, i. e., $h_{1}$ or $h_{4}$ and $g_{1}$ or $g_{4}$.

For example, we discuss the item that the single-photon state is $|\alpha \rangle _1$ in $|\textrm{Res}\rangle _1$. We can obtain

This state indicates that there is no photon either in spatial modes $h_4$ or in $h_1$ regardless of the resource state $|\textrm{Res}\rangle _2$. Meanwhile, both the spatial modes $h_5$ and $g_5$ are lack of photons, which fails to satisfy the "twelve-mode" case.

For the two-photon states such as $|VH{\rangle _{{h_1}{h_4}}}$, $|HH{\rangle _{{h_1}{h_1}}}$ and $|VV{\rangle _{{h_4}{h_4}}}$ in $|\textrm{Res}\rangle _{1}$, they can also be removed automatically according to the coincidence detection. In detail, if we combine $|\textrm{Res}{\rangle _1}$, $|\textrm{Res}{\rangle _2}$ with the input Bell states ${|{{\phi ^+}}\rangle _{{a_1}{b_1}}}$ and ${|{{\phi ^+}}\rangle _{{a_2}{b_2}}}$, we obtain

For the three-photon states such as $|HHV{\rangle _{{h_5}{h_1}{h_1}}}$ and $|VHV{\rangle _{{h_5}{h_4}{h_4}}}$ in $|\textrm{Res}\rangle _1$, they also fail to satisfy the "twelve-mode" case. For example,

One can clearly observe from Eq. (33) that no matter what is the second resource state $|\textrm{Res}\rangle _2$, the absence of photons in spatial modes $h_4$ or $h_1$ will result in a failure in BSMs. Therefore, such three-photon items can also be eliminated automatically.

If the first resource state is $|\gamma _1\rangle$, we can obtain

Similarly, the items in Eq. (34) which contain single-photon state $|\alpha _2\rangle$, two-photon state $|\beta _2\rangle$ or three-photon state $|\delta _2\rangle$ can be removed automatically with the same principle as Eqs. (31), (32), and (33). Only the item $|{\phi ^ + }{\rangle _{{a_1}{b_1}}} \otimes |{\phi ^ + }{\rangle _{{a_2}{b_2}}}\otimes |\gamma _1\rangle \otimes \!|\gamma _2\rangle \!$, i.e., the perfect Bell states combined with GHZ states can lead the "twelve-mode" case, which will collapse to

Similarly, item $|{\psi ^ + }{\rangle _{{a_1}{b_1}}} \otimes |{\psi ^ + }{\rangle _{{a_2}{b_2}}} \otimes |\textrm{Res}{\rangle _1} \otimes |\textrm{Res}{\rangle _2}$ can be discussed with the same principle. Only the item $|{\psi ^ + }{\rangle _{{a_1}{b_1}}} \otimes |{\psi ^ + }{\rangle _{{a_2}{b_2}}} \otimes |\textrm{GHZ}{\rangle _{h_1h_4h_5}} \otimes |\textrm{GHZ}{\rangle _{g_1g_4g_5}}$ can satisfy the "twelve-mode" case and will collapse to

In addition, the items $|\phi ^+\rangle _{a_1b_1}|\psi ^+\rangle _{a_2b_2}\otimes |\textrm{Res}{\rangle _1} \otimes |\textrm{Res}{\rangle _2}$ and $|\psi ^+\rangle _{a_1b_1}|\phi ^+\rangle _{a_2b_2}\otimes |\textrm{Res}{\rangle _1} \otimes |\textrm{Res}{\rangle _2}$ in input state $\rho _{in}$ can be automatically eliminated because of the basic purification principle.

Finally, let us discuss the double-pair emissions $|{{\phi ^+}}\rangle _{a_{1}b_{1}}^{\otimes 2}(|{{\psi ^+}}\rangle _{a_{1}b_{1}}^{\otimes 2})$ and $|{{\phi ^+}}\rangle _{a_{2}b_{2}}^{\otimes 2}(|{{\psi ^+}}\rangle _{a_{1}b_{1}}^{\otimes 2})$ in input state $\rho _{in}$. For example, the state $|{{\phi ^+}}\rangle _{a_{1}b_{1}}^{\otimes 2}$ combined with the two resource states can be described as

Obviously, the state in Eq. (37) shows that both the spatial modes $a_2$ and $b_2$ are lack of photons, which will be a failure in the BSMs. In this way, such double-pair emissions of noisy inputs can be automatically removed according to the final coincidence detection. Here, if the spatial modes $g_{1}$ and $g_{4}$ have extra photons, they can still lead successful BSMs. Such successful cases with disturbed items will cause error items for final purification. Fortunately, from Eqs. (25) and (27), states $|\textrm{Res}\rangle _{1}$ and $|\textrm{Res}\rangle _{2}$ cannot contribute to such extra photons.

According to above detailed analysis, only the items $|{\phi ^ + }{\rangle _{{a_1}{b_1}}} \otimes |{\phi ^ + }{\rangle _{{a_2}{b_2}}} \otimes |\textrm{GHZ}{\rangle _{h_1h_4h_5}} \otimes |\textrm{GHZ}{\rangle _{g_1g_4g_5}}$ and $|{\psi ^ + }{\rangle _{{a_1}{b_1}}} \otimes |{\psi ^ + }{\rangle _{{a_2}{b_2}}} \otimes |\textrm{GHZ}{\rangle _{h_1h_4h_5}} \otimes |\textrm{GHZ}{\rangle _{g_1g_4g_5}}$ in the input state $\rho _{in}$ can lead to the "twelve-mode" case which lead to an output state ${\rho _{{g_5}{h_5}}}$ as

5. Discussion and conclusion

So far, we have completely explained this MB-EPP for correcting the bit-flip error in linear optics. Similar to existing EPPs [35,37,38], this MB-EPP cannot directly purify phase-flip error. The phase-flip error should be firstly transformed to the bit-flip error with the help of the Hadamard operations, and then be corrected by the above method in the next step. In a practical scenario, bit-flip and phase-flip errors may occur simultaneously, which makes the state $|\phi ^+\rangle$ degrade to

If $A>0.5$, the fidelity of $|{\phi ^ + }\rangle$ in Eq. (40) is larger than that of Eq. (39).

The key element in this MB-EPP is the BSM coupling the noisy pairs with the resource states. However, the probability of BSM in linear optics cannot exceed $0.5$ [71,72], which limits the efficiency of this MB-EPP. Actually, if we use the complete BSM [76] where the parity-check gates were respectively constructed in spatial and polarization degrees of freedom, the efficiency of MB-EPP can be improved but the fidelity will not change. Recently, some parity-check gates with cross-Kerr nonlinearity [77,78] and Rydberg atoms [79–83] were constructed. In 2020, the Ref. [81] achieved two-qubit nondestructive Rydberg parity meter (NRPM) using Rydberg atoms and applied the complete BSM to the quantum teleportation using the NRPM. Furthermore, the complete and nondestructive distinguishment of many-body Rydberg entanglement was realized using Rydberg atoms [82]. Consequently, the recent experimental progress would promote the experimental realization of our MB-EPP.

In practical experiment, the main challenge comes from the multi-photon entanglement. We require to generate two copies of mixed states and two copies of GHZ states. In this way, six SPDC sources are required to generate six pairs of two-photon Bell states simultaneously. Fortunately, the technology of multi-photon entanglement have been well developed in the last decade. For example, in Ref. [84], they reported the 12-photon entanglement generation using six SPDC sources. Their entangled-photon source shows simultaneously 97% heralding efficiency and 96% indistinguishability between independent single photons without narrow-band filtering. Moreover, two-dimensional metasurfaces have shown great potential in quantum-optical technologies. Li et al. reported a 100-path SPDC sources in a 10$\times$10 array by integrating a metalens array with a nonlinear crystal [85]. Such compact, stable and controllable metalens-array-based quantum photon source may also provide the potential to realize this MB-EPP.

In conclusion, we have proposed a feasible MB-EPP with linear optical elements based on the original works in Refs. [70,73]. We first take a simple example to describe the basic principle of MB-EPP in two-copy case. We show that the original MB-EPP is feasible in linear optics. Subsequently, we extend it to the multi-copy case. Finally, we design a possible experiment realization of such MB-EPP in current experiment condition. Especially, we provide the analysis on the realization of this MB-EPP with practical SPDC sources. We show that the SPDC source is not an obstacle and the double-pair emission noise caused by the SPDC sources can be eliminated according to the coincidence detection. Combined with the suitable quantum memory and entanglement swapping, this MB-EPP may have the potential to become a building block in measurement-based quantum repeater.