Measurement-based entanglement purification for entangled coherent states

Abstract

The entangled coherent states (ECSs) have been widely used to realize quantum information processing tasks. However, the ECSs may suffer from photon loss and decoherence due to the inherent noise in quantum channel, which may degrade the fidelity of ECSs. To overcome these obstacles, we present a measurement-based entanglement purification protocol (MBEPP) for ECSs to distill some high-quality ECSs from a large number of low-quality copies. We first show the principle of this MBEPP without considering the photon loss. After that, we prove that this MBEPP is feasible to correct the error resulted from the photon loss. Additionally, this MBEPP only requires to operate the Bell state measurement without performing local two-qubit gates on the noisy pairs and the purified high-quality ECSs can be preserved for other applications. This MBEPP may have application potential in the implementation of long-distance quantum communication.

Graphical abstract

Keywords

measurement-based entanglement purification / entangled coherent state / photon loss / decoherence

1 Introduction

Quantum entanglement is a counterintuitive phenomenon which leads the quantum mechanic to be different from classical one. There are two methods to encode the qubit such as discrete variables (DVs) and continuous variables (CVs). The DVs, i.e., polarization, time-bin and spatial modes, have been widely employed in quantum information processing, such as quantum dense coding [1, 2], quantum teleportation (QT) [3-6], quantum key distribution (QKD) [7-11], quantum secure direct communication (QSDC) [12-24], and some other important quantum information processing protocols [25, 26]. The CVs, i.e., coherent states, are quite important in quantum computation [27-30]. Entangled coherent state (ECS), which is the superposition of coherent states for different modes [31, 32], also plays a key role in the application of QT [3-37], QKD [38-43], and quantum metrology [44]. However, the ECSs are sensitive to photon loss in the noisy quantum channel, which makes the maximally entangled state become less-entangled state or even mixed state, thereby largely decreasing the security and distance of quantum communication.

To address this issue, the quantum repeater with ECSs [45] was proposed to establish entangled quantum channel with high fidelity. The research findings show that the photon-number-resolving detectors with extremely high efficiency are essential to perform entanglement swapping with high fidelity [45]. However, the realization of this kind of detector is beyond current technology. Thus, it is necessary for one to explore other methods to generate high-quality ECSs. For instance, Kuang et al. utilized technique of electromagnetically induced transparency to prepare ECSs between two distant atomic Bose–Einstein condensates, which is sensitive to detection inefficiency [46]. The scheme in Ref. [47] is robust to photon loss with inefficient detectors. In Ref. [48], the researchers presented a method to generate ECSs in a two-dimensional anisotropic trap. In 2019, Xiong et al. prepared a scheme to generate entangled cat states in a hybrid optomechanical system [49], which shows that scheme is feasible even in the presence of noise. Recently, Tian et al. demonstrated experimentally the simultaneous generation and detection of two types of continuous variable nonclassical states from one type-0 phase-matching optical parametric amplification (OPA) and subsequent two ring filter cavities (RFCs) [50].

Additionally, the entanglement concentration protocols (ECPs) [51-58] and entanglement purification protocols (EPPs) [59-75] can be separately employed to distill maximal entangled states and high-fidelity entangled states from corrupted copies. For example, the hyperentanglement concentration for polarization-spatial-time-bin was investigated in multi-photon systems in Ref. [54]. In Ref. [56], the entanglement concentration protocols for cluster-type ECSs utilizing single-mode and two-mode coherent states were proposed to obtain the maximal cluster-type ECSs. In 2002, Jeong et al. first proposed the entanglement purification for ECSs and it is applicable to Werner-type ECSs [74]. In Ref. [75], the authors considered the imperfect detection in the generation process of ECSs and employed a random mode of entanglement purification without the manual intervention on purifying lossy errors. After that, Andersen et al. experimentally demonstrated an EPP for coherent states in linear optics assisted with an ancillary vacuum state [76]. In 2013, Sheng et al. used linear optical elements to purify mixed hybrid entangled states. The research findings showed that the error caused by the dissipation can be transformed to bit-flip error and then be purified [64].

In this paper, we investigate a measurement-based entanglement purification protocol (MBEPP) for ECSs in linear optics, which bases on the original work in Ref. [65], to resist to the effects of noisy quantum channel on ECSs and pave the route of measurement-based quantum repeater (MBQR) [77, 78]. We first illustrate the principle of MBEPP for ECSs under the ideal case without photon loss. We show that our MBEPP has higher efficiency than previous EPP for discrete variables. Then we consider a practical scenario with the photon loss caused by the noisy environment. Our findings show that the error resulted from the photon loss can be converted to bit-flip error and be purified by the same method in the next step. Moreover, the purified high-quality ECSs can be well preserved for other applications. Consequently, this work may have application potential in the implementation of long-distance quantum communication.

The paper is organized as follows. In Section 2, we first recall the BSM for ECSs and present the principle of MBEPP to correct the bit-flip error without the photon loss. In Section 3, we consider a practical scenario that the ECSs may encounter the dissipation simultaneously suffering from the bit-flip error. In Section 4, we present a discussion and make a conclusion.

2 MBEPP for ECSs without photon loss

For this MBEPP, the Bell state measurement (BSM) is an indispensable part. Thus, we start to briefly review the BSM for the ECSs [79]. As discussed in Ref. [27], the four quasi-Bell states can be given by

\begin{array}{}\text{(1)}& \begin{array}{c}{|{\varphi }^{\pm }⟩}_{ab}={\displaystyle \frac{1}{N_{\pm }}}({|\alpha ⟩}_a{|\alpha ⟩}_b\pm {|-\alpha ⟩}_a{|-\alpha ⟩}_b),\hfill \\ {|{\psi }^{\pm }⟩}_{ab}={\displaystyle \frac{1}{N_{\pm }}}({|\alpha ⟩}_a{|-\alpha ⟩}_b\pm {|-\alpha ⟩}_a{|\alpha ⟩}_b),\hfill \end{array}\end{array}

where N_{\pm }=\sqrt{2\pm 2{\text{e}}^{-4{\alpha }^2}} and the subscripts a and b represent two parties Alice and Bob, respectively. \pm \alpha represent the amplitudes of the coherent state |\pm \alpha ⟩, respectively. For simplicity, we consider α to be real throughout the paper. Obviously, the state {|{\varphi }^{+}⟩}_{ab} and {|{\psi }^{+}⟩}_{ab} are nonorthogonal due to the fact that the inner product is ⟨{\varphi }^{+}|{\psi }^{+}⟩=2{({\text{e}}^{2\alpha {}^2}+{\text{e}}^{-2\alpha {}^2})}^{-1}. However, if we choose a large α, i.e., \alpha =2, the overlap ⟨{\varphi }^{+}|{\psi }^{+}⟩ between them tends to 0. In this case, all the quasi-Bell states in Eq. (1) are nearly orthogonal. As shown in Fig. 1, the 50:50 beam-splitter (BS, take BS₁ as an example) makes the coherent state evolve to

\begin{array}{}\text{(2)}& \begin{array}{c}{|\alpha ⟩}_{a_1}{|\alpha ⟩}_{g_1}\to {|\sqrt{2}\alpha ⟩}_{c_1}{|0⟩}_{d_1},\hfill \\ {|\alpha ⟩}_{a_1}{|-\alpha ⟩}_{g_1}\to {|0⟩}_{c_1}{|\sqrt{2}\alpha ⟩}_{d_1},\hfill \\ {|-\alpha ⟩}_{a_1}{|\alpha ⟩}_{g_1}\to {|0⟩}_{c_1}{|-\sqrt{2}\alpha ⟩}_{d_1},\hfill \\ {|-\alpha ⟩}_{a_1}{|-\alpha ⟩}_{g_1}\to {|-\sqrt{2}\alpha ⟩}_{c_1}{|0⟩}_{d_1}.\hfill \end{array}\end{array}

After the photons in mode a₁ and mode g₁ passing through BS₁, the quasi-Bell states in Eq. (1) evolve to

\begin{array}{}\text{(3)}& \begin{array}{c}{|{\varphi }^{+(-)}⟩}_{a_1{g}_1}\to {|\mathrm{even}(\mathrm{odd})⟩}_{c_1}{|0⟩}_{d_1},\hfill \\ {|{\psi }^{+(-)}⟩}_{a_1{g}_1}\to {|0⟩}_{c_1}{|\mathrm{even}(\mathrm{odd})⟩}_{d_1},\hfill \end{array}\end{array}

where

\begin{array}{}\text{(4)}& \begin{array}{c}|\text{even}⟩=N_e\{{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{[{(\sqrt{2}\alpha )}^n+{(-\sqrt{2}\alpha )}^n]}{\sqrt{n!}}}\}|n⟩,\hfill \\ |\text{odd}⟩=N_o\{{\displaystyle \sum _{n=0}^{\mathrm{\infty }}}{\displaystyle \frac{[{(\sqrt{2}\alpha )}^n-{(-\sqrt{2}\alpha )}^n]}{\sqrt{n!}}}\}|n⟩.\hfill \end{array}\end{array}

The |\text{even}⟩ and |\text{odd}⟩ denote the even number and odd number of photons arriving at a photon detector, respectively. N_e and N_o are the normalization coefficients. Hence, we can unambiguously distinguish four quasi-Bell states according to the different responses of photon detectors. To be precise, if the photon detector in c₁ registers even (odd) number of photons and the photon detector in d₁ registers no photon, we can deterministically distinguish {|{\varphi }^{+}⟩}_{a_1{g}_1} ({|{\varphi }^{-}⟩}_{a_1{g}_1}). Similarly, we can discriminate {|{\psi }^{+}⟩}_{a_1{g}_1} ({|{\psi }^{-}⟩}_{a_1{g}_1}) if the photon detector in d₁ registers even (odd) photons while that in c₁ registers no photon. It needs to be pointed out that the state {|\sqrt{2}\alpha ⟩}_{c_1}{|0⟩}_{d_1} may be in {|0⟩}_{c_1}{|0⟩}_{d_1} with the probability of {\text{e}}^{-2{\alpha }^2}, which indicates a failure event for BSM. However, the failure probability tends to 0 when a large α is chosen. For example, if \alpha =2, the success probability of the BSM is 0.9997.

Similar to the polarization degree of freedom, the ECSs also suffer from the channel noise during the entanglement distribution, which makes the maximal ECSs become less entangled states or even mixed states. In detail, we consider the initial state is {|{\varphi }^{-}⟩}_{ab}. If {|{\varphi }^{-}⟩}_{ab} becomes {|{\psi }^{-}⟩}_{ab}, we call it the bit-flip error. If the phase-flip error happens, {|{\varphi }^{-}⟩}_{ab} will become {|{\varphi }^{+}⟩}_{ab}. And {|{\varphi }^{-}⟩}_{ab} will evolve to {|{\psi }^{+}⟩}_{ab} when the bit-flip and phase-flip errors occur simultaneously. The phase-flip error cannot be purified directly. It is usually transformed to bit-flip error after the Hadamard operations. We can obtain the transformation as

\begin{array}{}\text{(5)}& \begin{array}{c}|{\varphi }^{-}⟩\to |{\varphi }^{-}⟩,|{\varphi }^{+}⟩\to |{\psi }^{+}⟩,\hfill \\ |{\psi }^{-}⟩\to |{\psi }^{-}⟩,|{\psi }^{+}⟩\to |{\varphi }^{+}⟩,\hfill \end{array}\end{array}

with the assistance of operation U_x(\pi \mathrm{/}4) which means the rotation by \pi \mathrm{/}2 around the x axis [27]. This operation can be given by

\begin{array}{}\text{(6)}& \begin{array}{c}|\alpha ⟩\to {\displaystyle \frac{{\text{e}}^{-\text{i}\pi \mathrm{/}4}}{\sqrt{2}}}(|\alpha ⟩+\text{i}|-\alpha ⟩)\hfill \\ |-\alpha ⟩\to {\displaystyle \frac{{\text{e}}^{-\text{i}\pi \mathrm{/}4}}{\sqrt{2}}}(\text{i}|\alpha ⟩+|-\alpha ⟩).\hfill \end{array}\end{array}

Thus, we merely consider to purify the bit-flip error to describe the principle of this MBEPP. We assume that the mixed state can be written as

\begin{array}{}\text{(7)}& {\rho }_{ab}=F{|{\varphi }^{-}⟩}_{ab}{⟨{\varphi }^{-}|+(1-F)|{\psi }^{-}⟩}_{ab}⟨{\psi }^{-}|,\end{array}

where F=⟨{\varphi }^{-}\left|{\rho }_{ab}\right|{\varphi }^{-}⟩ is the initial fidelity of the mixed state. As a result, the whole system {\rho }_{a_1{b}_1}\otimes {\rho }_{a_2{b}_2} can be described as the mixture of four pure states. To be specific, the whole system is in the state {|{\varphi }^{-}⟩}_{a_1{b}_1}{|{\varphi }^{-}⟩}_{a_2{b}_2} with the probability of F². It is in the state {|{\varphi }^{-}⟩}_{a_1{b}_1}{|{\psi }^{-}⟩}_{a_2{b}_2} or {|{\psi }^{-}⟩}_{a_1{b}_1}{|{\varphi }^{-}⟩}_{a_2{b}_2} with an equal probability of F(1-F). The whole system is in the state {|{\psi }^{-}⟩}_{a_1{b}_1}{|{\psi }^{-}⟩}_{a_2{b}_2} with the probability of {(1-F)}^2. From Fig. 1, this MBEPP also needs two pairs of resource states which are entangled in modes g_1{g}_2{g}_3 and h_1{h}_2{h}_3, respectively. The resource state can be given by

\begin{array}{}\text{(8)}& |GHZ⟩={\displaystyle \frac{1}{N_1}}(|\alpha ⟩|\alpha ⟩|\alpha ⟩+|-\alpha ⟩|-\alpha ⟩|-\alpha ⟩),\end{array}

where N_1=\sqrt{2+2{\text{e}}^{-6{\alpha }^2}}. The Ref. [82] discussed the generation of arbitrary concatenated Greenberger-Horne-Zeilinger (C-GHZ) state encoded in coherent states with linear optics. Thus, we let N=1 and M=3 where N and M respectively represent the number of blocks and the physical qubits in each block. This resource state as Eq. (8) can be prepared off-line in a probabilistic way.

Then, Alice and Bob make BSMs for the photons in modes a_1{g}_1, a_2{g}_2, b_1{h}_1, and b_2{h}_2, respectively. Whether the purification is successful or not is determined by the outcomes of BSMs [77]. The success cases and the corresponding operations performed on the photon in one of the output modes are listed in Table 1. In detail, with the probability of F², the state {\rho }_{a_1{b}_1}\otimes {\rho }_{a_2{b}_2}\otimes {|GHZ⟩}_{g_1{g}_2{g}_3}\otimes {|GHZ⟩}_{h_1{h}_2{h}_3} is in

\begin{array}{}\text{(9)}& \begin{array}{c}{|{\varphi }^{-}⟩}_{a_1{b}_1}\otimes {|{\varphi }^{-}⟩}_{a_2{b}_2}\otimes {|GHZ⟩}_{g_1{g}_2{g}_3}\otimes {|GHZ⟩}_{h_1{h}_2{h}_3}\hfill \\ ={\displaystyle \frac{1}{N_{-}^2{N}_1^2}}({|\alpha ,\alpha ⟩}_{a_1{b}_1}-{|-\alpha ,-\alpha ⟩}_{a_1{b}_1})\hfill \\ \otimes ({|\alpha ,\alpha ⟩}_{a_2{b}_2}-{|-\alpha ,-\alpha ⟩}_{a_2{b}_2})\hfill \\ \otimes ({|\alpha ,\alpha ,\alpha ⟩}_{g_1{g}_2{g}_3}+{|-\alpha ,-\alpha ,-\alpha ⟩}_{g_1{g}_2{g}_3})\hfill \\ \otimes ({|\alpha ,\alpha ,\alpha ⟩}_{h_1{h}_2{h}_3}+{|-\alpha ,-\alpha ,-\alpha ⟩}_{h_1{h}_2{h}_3}).\hfill \end{array}\end{array}

From Eq. (9), the two modes in |{\varphi }^{-}⟩ have the same amplitudes. Therefore, if the results of BSMs on (g_1,a_1) and (h_1,b_1) are the same (different), the measurement outcomes on (g_2,a_2) and (h_2,b_2) must be the same (different) without considering the sign. Hence, the resultant state is {|{\varphi }^{+}⟩}_{g_3{h}_3} or {|{\varphi }^{-}⟩}_{g_3{h}_3}. We define that when above measurement results are obtained, our MBEPP is successful. The state {|{\psi }^{-}⟩}_{a_1{b}_1}\otimes {|{\psi }^{-}⟩}_{a_2{b}_2}\otimes {|GHZ⟩}_{g_1{g}_2{g}_3}\otimes {|GHZ⟩}_{h_1{h}_2{h}_3} can also lead to the successful measurement results. After the BSMs, {|{\psi }^{-}⟩}_{a_1{b}_1}\otimes {|{\psi }^{-}⟩}_{a_2{b}_2}\otimes {|GHZ⟩}_{g_1{g}_2{g}_3}\otimes {|GHZ⟩}_{h_1{h}_2{h}_3} will evolve to {|{\psi }^{+}⟩}_{g_3{h}_3} or {|{\psi }^{-}⟩}_{g_3{h}_3}.

On the other hand, we will show that the crossed combinations {|{\varphi }^{-}⟩}_{a_1{b}_1}{|{\psi }^{-}⟩}_{a_2{b}_2} and {|{\psi }^{-}⟩}_{a_1{b}_1}{|{\varphi }^{-}⟩}_{a_2{b}_2} can be eliminated automatically. For example,

\begin{array}{}\text{(10)}& \begin{array}{c}{|{\varphi }^{-}⟩}_{a_1{b}_1}\otimes {|{\psi }^{-}⟩}_{a_2{b}_2}\otimes {|GHZ⟩}_{g_1{g}_2{g}_3}\otimes {|GHZ⟩}_{h_1{h}_2{h}_3}\hfill \\ ={\displaystyle \frac{1}{N_{-}^2{N}_1^2}}({|\alpha ,\alpha ⟩}_{a_1{b}_1}-{|-\alpha ,-\alpha ⟩}_{a_1{b}_1})\hfill \\ \otimes ({|\alpha ,-\alpha ⟩}_{a_2{b}_2}-{|-\alpha ,\alpha ⟩}_{a_2{b}_2})\hfill \\ \otimes ({|\alpha ,\alpha ,\alpha ⟩}_{g_1{g}_2{g}_3}+{|-\alpha ,-\alpha ,-\alpha ⟩}_{g_1{g}_2{g}_3})\hfill \\ \otimes ({|\alpha ,\alpha ,\alpha ⟩}_{h_1{h}_2{h}_3}+{|-\alpha ,-\alpha ,-\alpha ⟩}_{h_1{h}_2{h}_3}).\hfill \end{array}\end{array}

It can be found that the measurement outcomes on (g_1,a_1) and (h_1,b_1) are the same (different) while the results on (g_2,a_2) and (h_2,b_2) are different (same). The similar measurement results can be obtained for {|{\psi }^{-}⟩}_{a_1{b}_1}{|{\varphi }^{-}⟩}_{a_2{b}_2}. In this way, the crossed combinations states cannot lead to the successful measurement results, so that they can be discarded automatically.

Consequently, if the number of sign “-” in the BSM outcomes is odd, i.e., {|{\varphi }^{+}⟩}_{g_1{a}_1}{|{\psi }^{-}⟩}_{g_2{a}_2}{|{\varphi }^{+}⟩}_{h_1{b}_1}{|{\psi }^{+}⟩}_{h_2{b}_2}, the output state can be described as

\begin{array}{}\text{(11)}& {\rho }_{g_3{h}_3}=F_1{|{\varphi }^{-}⟩}_{g_3{h}_3}{⟨{\varphi }^{-}|+(1-F_1)|{\psi }^{-}⟩}_{g_3{h}_3}⟨{\psi }^{-}|,\end{array}

where F_1=\frac{F^2}{F^2+{(1-F)}^2}. The fidelity of the new mixed state is larger than that of the initial one when F>0.5 [59]. While if the number of sign “-” is even, such as {|{\varphi }^{+}⟩}_{g_1{a}_1}{|{\varphi }^{-}⟩}_{g_2{a}_2}{|{\varphi }^{-}⟩}_{h_1{b}_1}{|{\varphi }^{+}⟩}_{h_2{b}_2}, the resultant state is

\begin{array}{}\text{(12)}& {\rho }_{g_3{h}_3}^{\prime }=F_1{|{\varphi }^{+}⟩}_{g_3{h}_3}{⟨{\varphi }^{+}|+(1-F_1)|{\psi }^{+}⟩}_{g_3{h}_3}⟨{\psi }^{+}|.\end{array}

After operating {\sigma }_z on the photon in one of the output modes, we can transform the mixed state {\rho }_{g_3{h}_3}^{\prime } in Eq. (12) to {\rho }_{g_3{h}_3} in Eq. (11) (see Appendix A). The operation {\sigma }_x corresponding to be a phase shift can be implemented by employing the giant Kerr nonlinearity [83] and the operation {\sigma }_z can be constructed by using a displacement operator D(\text{i}\pi /4\alpha \sqrt{1-T}), where T\to 1 is the transmission coefficient of beam splitter. The success probability of this MBEPP is {(1-{\text{e}}^{-2{\alpha }^2})}^4[F^2+{(1-F)}^2].

From the analysis beforehand, the fidelity of this MBEPP for ECSs is the same as that of the MBEPPs for discrete variables such as polarization in linear optics [77, 78] while the success probability of this MBEPP is larger. The BSM plays a key role in MBEPP. The success probability of those MBEPPs for polarization entanglement in linear optics [77, 78, 73] is [F^2+{(1-F)}^2]\mathrm{/}16, for only two of the four Bell states can be distinguished with the standard BSM in linear optics, so that the success probability for each BSM is only \frac{1}{2} [79]. However, the four Bell states in Eq. (1) can be completely discriminated with the probability 1-{\text{e}}^{-2{\alpha }^2}, which results in the total success probability of this MBEPP being {(1-{\text{e}}^{-2{\alpha }^2})}^4[F^2+{(1-F)}^2]. In this way, if \alpha >\sqrt{\ln 2\mathrm{/}2}, the success probability of this MBEPP is larger than that of the MBEPPs in Refs. [77, 78, 73]. Moreover, with the growth α, the success probability will tend to F^2+{(1-F)}^2 [59]. This MBEPP has another attractive advantage that the purified ECSs can be well remained for further application, i.e., quantum communication [38, 39, 45].

3 MBEPP for ECSs with photon loss

So far, we have completely discussed the principle of our MBEPP without considering the photon loss. However, in the practical scenario, the coherent states may suffer from photon loss which will degrade the fidelity of entanglement and threaten the security of quantum communication. To address this issue, it is essential to correct the error of photon loss caused by dissipation. Here, we utilize BS with the transmission coefficient of η and the reflection coefficient of 1-\eta to describe the loss model. Let’s denote |\beta ⟩=|\sqrt{\eta }\alpha ⟩ and |\gamma ⟩=|\sqrt{1-\eta }\alpha ⟩. In this case, the state |{\varphi }^{-}⟩ will not only face the bit-flip error but also suffer from the photon loss. Therefore, the mixed state {\rho }_{ab} in Eq. (7) will evolve to

\begin{array}{}\text{(13)}& {\rho }_{ab}^l=F{|{\varphi }_1^{-}⟩}_{ab}{⟨{\varphi }_1^{-}|+(1-F)|{\psi }_1^{-}⟩}_{ab}⟨{\psi }_1^{-}|,\end{array}

where

\begin{array}{}\text{(14)}& \begin{array}{cc}{|{\varphi }_1^{-}⟩}_{ab}=\hfill & {\displaystyle \frac{1}{N_{-}}}{(|\beta ⟩}_a{|\gamma ⟩}_{E_a}{|\beta ⟩}_b{|\gamma ⟩}_{E_b}\hfill \\ \hfill & {-|-\beta ⟩}_a{|-\gamma ⟩}_{E_a}{|-\beta ⟩}_b{|-\gamma ⟩}_{E_b}),\hfill \\ {|{\psi }_1^{-}⟩}_{ab}=\hfill & {\displaystyle \frac{1}{N_{-}}}{(|\beta ⟩}_a{|\gamma ⟩}_{E_a}{|-\beta ⟩}_b|-\gamma ⟩{}_{E_b}\hfill \\ \hfill & {-|-\beta ⟩}_a{|-\gamma ⟩}_{E_a}{|\beta ⟩}_b{|\gamma ⟩}_{E_b}).\hfill \end{array}\end{array}

The superscript l means the mixed state undergoing the photon loss and the subscripts E_a and E_b denote the environment modes at Alice and Bob. Thus, the whole system {\rho }_{a_1{b}_1}^l\otimes {\rho }_{a_2{b}_2}^l can be described as follows. With the probability of F², the system is in the state {|{\varphi }_1^{-}⟩}_{a_1{b}_1}{|{\varphi }_1^{-}⟩}_{a_2{b}_2}. With an equal probability of F(1-F), the system is in the state {|{\varphi }_1^{-}⟩}_{a_1{b}_1}{|{\psi }_1^{-}⟩}_{a_2{b}_2} or {|{\psi }_1^{-}⟩}_{a_1{b}_1}{|{\varphi }_1^{-}⟩}_{a_2{b}_2}. It is in the state {|{\psi }_1^{-}⟩}_{a_1{b}_1}{|{\psi }_1^{-}⟩}_{a_2{b}_2} with the probability of {(1-F)}^2. In addition, the resource state entangled in modes g_1{g}_2{g}_3 and h_1{h}_2{h}_3 can be rewritten as

\begin{array}{}\text{(15)}& {|GHZ⟩}^n={\displaystyle \frac{1}{N_2}}(|\beta ⟩|\beta ⟩|\beta ⟩+|-\beta ⟩|-\beta ⟩|-\beta ⟩),\end{array}

where N_2=\sqrt{2+2{\text{e}}^{-6\eta {\alpha }^2}} and the superscript n means the new resource state regenerated corresponding to the transmission coefficient η. Therefore, with the probability of F², the state {|GHZ⟩}_{g_1{g}_2{g}_3}^n\otimes {|GHZ⟩}_{h_1{h}_2{h}_3}^n\otimes {\rho }_{a_1{b}_1}^l\otimes {\rho }_{a_2{b}_2}^l can be written as

\begin{array}{}\text{(16)}& \begin{array}{c}{|GHZ⟩}_{g_1{g}_2{g}_3}^n{|GHZ⟩}_{h_1{h}_2{h}_3}^n{|{\varphi }_1^{-}⟩}_{a_1{b}_1}{|{\varphi }_1^{-}⟩}_{a_2{b}_2}\hfill \\ ={\displaystyle \frac{1}{N_{-}^2{N}_2^2}}({|\beta ⟩}_{g_1}{|\beta ⟩}_{g_2}{|\beta ⟩}_{g_3}+{|-\beta ⟩}_{g_1}{|-\beta ⟩}_{g_2}{|-\beta ⟩}_{g_3})\hfill \\ \otimes ({|\beta ⟩}_{h_1}{|\beta ⟩}_{h_2}{|\beta ⟩}_{h_3}+{|-\beta ⟩}_{h_1}{|-\beta ⟩}_{h_2}{|-\beta ⟩}_{h_3})\hfill \\ \otimes {(|\beta ⟩}_{a_1}{|\beta ⟩}_{b_1}{|\gamma ⟩}_{E_{a_1}}{|\gamma ⟩}_{E_{b_1}}-{|-\beta ⟩}_{a_1}{|-\beta ⟩}_{b_1}\hfill \\ {\otimes |-\gamma ⟩}_{E_{a_1}}{|-\gamma ⟩}_{E_{b_1}})({|\beta ⟩}_{a_2}{|\beta ⟩}_{b_2}{|\gamma ⟩}_{E_{a_2}}{|\gamma ⟩}_{E_{b_2}}\hfill \\ {-|-\beta ⟩}_{a_2}{|-\beta ⟩}_{b_2}{|-\gamma ⟩}_{E_{a_2}}{|-\gamma ⟩}_{E_{b_2}}).\hfill \end{array}\end{array}

Due to the existence of photon loss, only the items

\begin{array}{}\text{(17)}& \begin{array}{c}{{{|GHZ⟩}_{g_1{g}_2{g}_3}^n|GHZ⟩}_{h_1{h}_2{h}_3}^n{(|-\beta ⟩}_{a_1}|-\beta ⟩}_{b_1}|-\gamma ⟩{}_{E_{a_1}}\hfill \\ \otimes {|-\gamma ⟩}_{E_{b_1}}{|-\beta ⟩}_{a_2}{|-\beta ⟩}_{b_2}{|-\gamma ⟩}_{E_{a_2}}{|-\gamma ⟩}_{E_{b_2}}\hfill \\ {+|\beta ⟩}_{a_1}{|\beta ⟩}_{b_1}{|\gamma ⟩}_{E_{a_1}}{|\gamma ⟩}_{E_{b_1}}{|\beta ⟩}_{a_2}{|\beta ⟩}_{b_2}{|\gamma ⟩}_{E_{a_2}}{|\gamma ⟩}_{E_{b_2}}),\hfill \end{array}\end{array}

can be employed to distill high-quality entanglement.

Similarly, with the probability of {(1-F)}^2, the state {|GHZ⟩}_{g_1{g}_2{g}_3}^n\otimes {|GHZ⟩}_{h_1{h}_2{h}_3}^n\otimes {\rho }_{a_1{b}_1}^l\otimes {\rho }_{a_2{b}_2}^l is in

\begin{array}{}\text{(18)}& \begin{array}{c}{|GHZ⟩}_{g_1{g}_2{g}_3}^n{|GHZ⟩}_{h_1{h}_2{h}_3}^n{|{\psi }_1^{-}⟩}_{a_1{b}_1}{|{\psi }_1^{-}⟩}_{a_2{b}_2}\hfill \\ ={\displaystyle \frac{1}{N_{-}^2{N}_2^2}}({|\beta ⟩}_{g_1}{|\beta ⟩}_{g_2}{|\beta ⟩}_{g_3}+{|-\beta ⟩}_{g_1}{|-\beta ⟩}_{g_2}{|-\beta ⟩}_{g_3})\hfill \\ \otimes ({|\beta ⟩}_{h_1}{|\beta ⟩}_{h_2}{|\beta ⟩}_{h_3}+{|-\beta ⟩}_{h_1}{|-\beta ⟩}_{h_2}{|-\beta ⟩}_{h_3})\hfill \\ {\otimes {(|\beta ⟩}_{a_1}|-\beta ⟩}_{b_1}{|\gamma ⟩}_{E_{a_1}}{|-\gamma ⟩}_{E_{b_1}}-{|-\beta ⟩}_{a_1}|\beta ⟩{}_{b_1}\hfill \\ {\otimes |-\gamma ⟩}_{E_{a_1}}{|\gamma ⟩}_{E_{b_1}}){(|\beta ⟩}_{a_2}{|-\beta ⟩}_{b_2}{|\gamma ⟩}_{E_{a_2}}|-\gamma ⟩{}_{E_{b_2}}\hfill \\ {-|-\beta ⟩}_{a_2}{|\beta ⟩}_{b_2}{|-\gamma ⟩}_{E_{a_2}}{|\gamma ⟩}_{E_{b_2}}),\hfill \end{array}\end{array}

in which only the components

\begin{array}{}\text{(19)}& \begin{array}{c}{{{|GHZ⟩}_{g_1{g}_2{g}_3}^n|GHZ⟩}_{h_1{h}_2{h}_3}^n{(|\beta ⟩}_{a_1}|-\beta ⟩}_{b_1}|\gamma ⟩{}_{E_{a_1}}\hfill \\ \otimes {|-\gamma ⟩}_{E_{b_1}}{|\beta ⟩}_{a_2}{|-\beta ⟩}_{b_2}{|\gamma ⟩}_{E_{a_2}}{|-\gamma ⟩}_{E_{b_2}}+{|-\beta ⟩}_{a_1}\hfill \\ {\otimes |\beta ⟩}_{b_1}{|-\gamma ⟩}_{E_{a_1}}{|\gamma ⟩}_{E_{b_1}}{|-\beta ⟩}_{a_2}{|\beta ⟩}_{b_2}{|-\gamma ⟩}_{E_{a_2}}{|\gamma ⟩}_{E_{b_2}}),\hfill \end{array}\end{array}

make contributions to this MBEPP.

Moreover, with the similar principle as described in Section 2, the crossed combinations {|{\varphi }_1^{-}⟩}_{a_1{b}_1}{|{\psi }_1^{-}⟩}_{a_2{b}_2} and {|{\psi }_1^{-}⟩}_{a_1{b}_1}{|{\varphi }_1^{-}⟩}_{a_2{b}_2} can be eliminated automatically. All the cases corresponding to the successful purification and the corresponding operations are listed in Table 2 in detail.

In this way, when the MBEPP is successful, we can obtain the resultant state as (see Appendix B)

\begin{array}{}\text{(20)}& {\rho }_{g_3{h}_3}^l=F_1{|{\varphi }_1^{-}⟩}_{g_3{h}_3}{⟨{\varphi }_1^{-}|+(1-F_1)|{\psi }_1^{-}⟩}_{g_3{h}_3}⟨{\psi }_1^{-}|,\end{array}

where F_1=\frac{F^2}{F^2+{(1-F)}^2} and

\begin{array}{}\text{(21)}& \begin{array}{cc}{|{\varphi }_1^{-}⟩}_{g_3{h}_3}=\hfill & {\displaystyle \frac{1}{N_3}}{(|\beta ⟩}_{g_3}{|\beta ⟩}_{h_3}{|\gamma ⟩}_{E_{a_1}}{|\gamma ⟩}_{E_{a_2}}{|\gamma ⟩}_{E_{b_1}}|\gamma ⟩{}_{E_{b_2}}\hfill \\ \hfill & -{|-\beta ⟩}_{g_3}{|-\beta ⟩}_{h_3}{|-\gamma ⟩}_{E_{a_1}}{|-\gamma ⟩}_{E_{a_2}}{|-\gamma ⟩}_{E_{b_1}}\hfill \\ \hfill & {|-\gamma ⟩}_{E_{b_2}}),\hfill \end{array}\end{array}

\begin{array}{}\text{(22)}& \begin{array}{cc}{|{\psi }_1^{-}⟩}_{g_3{h}_3}=\hfill & {\displaystyle \frac{1}{N_3}}{(|\beta ⟩}_{g_3}{|-\beta ⟩}_{h_3}{|\gamma ⟩}_{E_{a_1}}{|\gamma ⟩}_{E_{a_2}}{|-\gamma ⟩}_{E_{b_1}}|-\gamma ⟩{}_{E_{b_2}}\hfill \\ \hfill & {-|-\beta ⟩}_{g_3}{|\beta ⟩}_{h_3}{|-\gamma ⟩}_{E_{a_1}}{|-\gamma ⟩}_{E_{a_2}}{|\gamma ⟩}_{E_{b_1}}{|\gamma ⟩}_{E_{b_2}}),\hfill \end{array}\end{array}

where 1\mathrm{/}{N}_3 is a normalization coefficient. For the whole system, the loss modes {|\pm \gamma ⟩}_{E_{a_1}}{|\pm \gamma ⟩}_{E_{a_2}} and {|\pm \gamma ⟩}_{E_{b_1}}{|\pm \gamma ⟩}_{E_{b_2}} can be respectively viewed as a large environment as {|\pm \sqrt{2}\gamma ⟩}_{E_a} and {|\pm \sqrt{2}\gamma ⟩}_{E_b} for Alice and Bob. As a result, the Eqs. (21) and (22) can be rewritten as

\begin{array}{}\text{(23)}& \begin{array}{cc}{|{\varphi }_2^{-}⟩}_{g_3{h}_3}=\hfill & {\displaystyle \frac{1}{N_3}}{(|\beta ⟩}_{g_3}{|\beta ⟩}_{h_3}{|\sqrt{2}\gamma ⟩}_{E_a}|\sqrt{2}\gamma ⟩{}_{E_b}\hfill \\ \hfill & {-|-\beta ⟩}_{g_3}{|-\beta ⟩}_{h_3}{|-\sqrt{2}\gamma ⟩}_{E_a}{|-\sqrt{2}\gamma ⟩}_{E_b}),\hfill \end{array}\end{array}

and

\begin{array}{}\text{(24)}& \begin{array}{cc}{|{\psi }_2^{-}⟩}_{g_3{h}_3}=\hfill & {\displaystyle \frac{1}{N_3}}{(|\beta ⟩}_{g_3}{|-\beta ⟩}_{h_3}{|\sqrt{2}\gamma ⟩}_{E_a}|-\sqrt{2}\gamma ⟩{}_{E_b}\hfill \\ \hfill & {-|-\beta ⟩}_{g_3}{|\beta ⟩}_{h_3}{|-\sqrt{2}\gamma ⟩}_{E_a}{|\sqrt{2}\gamma ⟩}_{E_b}).\hfill \end{array}\end{array}

With the method described in Ref. [64], we use an orthogonal two dimensional basis \{|u⟩,|v⟩\} to represent {|\pm \sqrt{2}\gamma ⟩}_y where y=E_a or E_b, yielding

\begin{array}{}\text{(25)}& \begin{array}{c}{|\sqrt{2}\gamma ⟩}_y={\mu }_y{|u⟩}_y+{\nu }_y{|v⟩}_y,\hfill \\ {|-\sqrt{2}\gamma ⟩}_y={\mu }_y{|u⟩}_y-{\nu }_y{|v⟩}_y,\hfill \end{array}\end{array}

in which {\mu }_y^2=\frac{1+{\text{e}}^{-4(1-\eta ){\alpha }^2}}{2} and {\nu }_y^2=\frac{1-{\text{e}}^{-4(1-\eta ){\alpha }^2}}{2}.

Then, when we trace out the loss modes, the states {|{\varphi }_2^{-}⟩}_{g_3{h}_3} and {|{\psi }_2^{-}⟩}_{g_3{h}_3} will become

\begin{array}{}\text{(26)}& \begin{array}{cc}{\rho }_{l1}=\hfill & N_4^{-1}{[N_{1-}^2(1+{\text{e}}^{-8(1-\eta ){\alpha }^2})|{\varphi }_3^{-}⟩}_{g_3{h}_3}⟨{\varphi }_3^{-}|\hfill \\ \hfill & +N_{1+}^2(1-{\text{e}}^{-8(1-\eta ){\alpha }^2})|{\psi }_3^{+}⟩{}_{g_3{h}_3}⟨{\psi }_3^{+}|],\hfill \end{array}\end{array}

and

\begin{array}{}\text{(27)}& \begin{array}{cc}{\rho }_{l2}=\hfill & N_4^{-1}{[N_{1-}^2(1+{\text{e}}^{-8(1-\eta ){\alpha }^2})|{\psi }_3^{-}⟩}_{g_3{h}_3}⟨{\psi }_3^{-}|\hfill \\ \hfill & +N_{1+}^2(1-{\text{e}}^{-8(1-\eta ){\alpha }^2})|{\varphi }_3^{+}⟩{}_{g_3{h}_3}⟨{\varphi }_3^{+}|],\hfill \end{array}\end{array}

where N_4^{-1} is a normalization factor and

\begin{array}{}\text{(28)}& \begin{array}{c}{|{\varphi }_3^{\pm }⟩}_{g_3{h}_3}={\displaystyle \frac{1}{N_{1\pm }}}({|\beta ⟩}_{g_3}{|\beta ⟩}_{h_3}\pm {|-\beta ⟩}_{g_3}{|-\beta ⟩}_{h_3}),\hfill \\ {|{\psi }_3^{\pm }⟩}_{g_3{h}_3}={\displaystyle \frac{1}{N_{1\pm }}}({|\beta ⟩}_{g_3}{|-\beta ⟩}_{h_3}\pm {|-\beta ⟩}_{g_3}{|\beta ⟩}_{h_3}).\hfill \end{array}\end{array}

In Eq. (28), N_{1\pm }=\sqrt{2\pm 2{\text{e}}^{-4\eta {\alpha }^2}}. Subsequently, we can rewrite Eq. (20) as

\begin{array}{}\text{(29)}& \begin{array}{cc}{\rho }_1^l=\hfill & N_5{[F_1{F}_2|{\varphi }_3^{-}⟩}_{g_3{h}_3}⟨{\varphi }_3^{-}|+F_1(1-F_2)\hfill \\ \hfill & \times {|{\psi }_3^{+}⟩}_{g_3{h}_3}{⟨{\psi }_3^{+}|+(1-F_1)F_2|{\psi }_3^{-}⟩}_{g_3{h}_3}⟨{\psi }_3^{-}|\hfill \\ \hfill & +(1-F_1)(1-F_2)|{\varphi }_3^{+}⟩{}_{g_3{h}_3}⟨{\varphi }_3^{+}|],\hfill \end{array}\end{array}

where N₅ is the normalization factor and

\begin{array}{}\text{(30)}& F_2={\displaystyle \frac{N_{1-}^2[1+{\text{e}}^{-8(1-\eta ){\alpha }^2}]}{N_{1-}^2[1+{\text{e}}^{-8(1-\eta ){\alpha }^2}]+N_{1+}^2[1-{\text{e}}^{-8(1-\eta ){\alpha }^2}]}}.\end{array}

It is clear for one to observe from Eq. (29) that the photon loss resulted from the noisy environment can be transformed to the bit-flip error. Thus, the mixed state given by Eq. (20) can be rewritten as the general case containing both bit-flip error and phase-flip error. In this case, the high-quality ECSs can be obtained from Eq. (29) if F_1{F}_2>\frac{1}{2}. As a result, after performing one round of purification, one can obtain the new mixed state as

\begin{array}{}\text{(31)}& \begin{array}{cc}{\rho }_n^l=\hfill & A{|{\varphi }_3^{-}⟩}_{g_3{h}_3}{⟨{\varphi }_3^{-}|+B|{\psi }_3^{+}⟩}_{g_3{h}_3}⟨{\psi }_3^{+}|\hfill \\ \hfill & +C{|{\psi }_3^{-}⟩}_{g_3{h}_3}{⟨{\psi }_3^{-}|+D|{\varphi }_3^{+}⟩}_{g_3{h}_3}⟨{\varphi }_3^{+}|,\hfill \end{array}\end{array}

with

\begin{array}{}\text{(32)}& \begin{array}{c}A={\displaystyle \frac{[F_1^2+{(1-F_1)}^2]F_2^2}{F_2^2+{(1-F_2)}^2}},B={\displaystyle \frac{[F_1^2+{(1-F_1)}^2]{(1-F_2)}^2}{F_2^2+{(1-F_2)}^2}},\hfill \\ C={\displaystyle \frac{2F_1(1-F_1)F_2^2}{F_2^2+{(1-F_2)}^2}},D={\displaystyle \frac{2F_1(1-F_1){(1-F_2)}^2}{F_2^2+{(1-F_2)}^2}}.\hfill \end{array}\end{array}

The fidelity of {\rho }_n^l is ⟨{\varphi }_3^{-}|{\rho }_n^l|{\varphi }_3^{-}⟩=A. To ensure that the fidelity of Eq. (31) is larger than that of Eq. (13) after tracing out the loss modes, we require

\begin{array}{}\text{(33)}& {\displaystyle \frac{[F_1^2+{(1-F_1)}^2]F_2^2}{[F_2^2+{(1-F_2)}^2]F}}>{\displaystyle \frac{(1-{\text{e}}^{-4\eta {\alpha }^2})(1+{\text{e}}^{-4(1-\eta ){\alpha }^2})}{2(1-{\text{e}}^{-4{\alpha }^2})}}.\end{array}

Further, it is necessary for one to perform the operation U_x(\pi \mathrm{/}4) to transform phase-flip error to bit-flip error, i.e., |{\varphi }_3^{+}⟩\iff |{\psi }_3^{+}⟩ before each round of purification which is similar to that in Ref. [60]. Thus, the fidelity of the state {|{\varphi }^{-}⟩}_{g_3{h}_3} increases with an increasing round of purification, accordingly. In addition, we assume that F_1 tends to unity after purifying the mixed state as the form of Eq. (20) after a large number of purification rounds. Under this circumstance, we can obtain

\begin{array}{}\text{(34)}& N_{1-}^2[1+{\text{e}}^{-8(1-\eta ){\alpha }^2}]>N_{1+}^2[1-{\text{e}}^{-8(1-\eta ){\alpha }^2}],\end{array}

when \eta >\frac{2}{3}.

It is clear to observe from Eq. (34) that the basic requirement of this MBEPP just relies on the transmission efficiency without depending on the photon number. So far, we have completely carried out the analysis for the MBEPP under the dissipation combined with the bit-flip error. Interestingly, the error caused by dissipation can be transformed to bit-flip error which is similar as Ref. [64].

4 Discussion and conclusion

We have discussed the MBEPP for ECSs under the photon loss combined with the bit-flip error. The phase-flip error can be transformed to bit-flip error assisted by the operation U_x(\frac{\pi }{4}) [27] and can be purified in the subsequent purification step. It is interesting for us to compare this MBEPP with the conventional EPP using controlled-not (CNOT) gate [59], in which one pair is considered to be a control and the other pair as a target. After passing through the CNOT gate, the target pair is measured by z-basis and the source pair is retained if the measurement results are the same. In this MBEPP, let us denote |\varphi ⟩\equiv |0⟩ and |\psi ⟩\equiv |1⟩. For Alice’s side, the outcomes of BSMs on g_1{a}_1 and g_2{a}_2 have four cases, i.e., {|\varphi ⟩}_{g_1{a}_1}{|\varphi ⟩}_{g_2{a}_2}, {|\varphi ⟩}_{g_1{a}_1}{|\psi ⟩}_{g_2{a}_2}, {|\psi ⟩}_{g_1{a}_1}{|\varphi ⟩}_{g_2{a}_2} and {|\psi ⟩}_{g_1{a}_1}{|\psi ⟩}_{g_2{a}_2}, which correspond to |0⟩|0⟩, |0⟩|1⟩, |1⟩|0⟩ and |1⟩|1⟩. Subsequently, we can obtain |0⟩ and |1⟩ by applying additional modulo 2 to these two measurement outcomes. The same case happens at Bob’s side. We retain the states entangled in modes g₃ and h₃ when the results of addition modulo 2 at Alice’s and Bob’s sides are the same. In addition, the corresponding operations should be performed on one of the output modes based on the number of sign “-” among four outcomes of BSMs. In this view, it plays a similar role as a CNOT gate. Moreover, this MBEPP is able to tolerate photon loss while the MBEPPs in Refs. [77, 78, 73] become invalid once any photon loses. In this MBEPP, the photon-number-resolving detector is pivotal. Fortunately, the photon counting detector working at visible and near-infrared wavelengths have been constructed [80], which employs superconducting transition-edge sensors. The efficiency of this detector is 95\% at 1556 nmwavelengths and it can be further improved by utilizing some appropriate methods.

In conclusion, we propose the first MBEPP for ECSs to correct the bit-flip error. Subsequently, we consider the practical scenario that the coherent states suffer from the dissipation and the bit-flip error. Surprisingly, we show that the error resulted from the dissipation can be converted to bit-flip error and then be corrected with the same method for correcting the bit-flip error. Additionally, if one combines our MBEPP with the entanglement generation [46-49] and entanglement swapping [45] as well as quantum memory [84] for ECSs, the long-distance quantum communication based on ECSs may be realized in the future.

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