1. Introduction

Due to the potential applications in quantum quantum information processing(QIP), entanglement plays a critical role in quantum information science [1], such as in quantum teleportation [2], quantum secret sharing [3–5 ] and quantum repeaters [6] where maximally entangled states service as the quantum channel to connect the two neighboring nodes in a network. Usually, these maximally entangled states required to be shared non-locally to complete these tasks efficiently. However, the particle loss and decoherence which caused by the interaction between a quantum system and the noise of quantum channel will inevitably occur during the practical entanglement distribution and long distance transmission. As a result, the maximally entangled states generally decay into partially entangled pure states or mixed states [7] leading to a decrement of the fidelity and the security of long-distance quantum communication.

To overcome a collective noise, there are some interesting methods of dealing with the photon systems actively before they are transmitted, such as decoherence-free-subspaces [8–10 ], faithful qubit distribution [11], quantum error correction [12], and so on. Traditionally, one can exploit some passive methods for depressing the noise effect on entangled photon systems, such as entanglement concentration(EC) [13–15 ] and entanglement purification(EP) [16–35 ] to eliminate the decoherence effect in entangled systems and to improve the entanglement of the quantum system [36]. Maximally entangled states could be achieved from partially entangled pure states by using EC. On the other hand, EP is used to distill high-fidelity entangled systems from a mixed ensemble which composed of different maximally entangled states. The application of both EC and EP has been widely studied in recent years, and these protocols work under the condition that, after the noisy channel the maximally entangled states will either decay into partially entangled pure states or decay into mixed states. In 1996, Bennett et al. [16] proposed the EP protocol by using controlled-not operations and postselection on Werner state to eliminate the channel errors. Later, Simon and Pan demonstrated a novel EP protocol to purify the entangled states with only bit-flip errors [19], and the available range for purifying two-photon mixed Bell-state ensemble is that the initial fidelity p > 1/2 against the bit-flip errors. However, the effect of the noisy channel is more complicated for the more general cases. Especially when the bit-flip error and phase-flip error both exist in a mixed entangled states ensemble and they are independent of each other, the iterating purification operations for one kind error will affect the fidelity against the other one, which even reduces the fidelity of the whole system. Therefore, the threshold value of entanglement purification is not p > 1/2 under this situation.

In this paper, we demonstrate that, when the bit-flip error and phase-flip error are coexistent, the available distillation ranges for a mixed two-photon ensemble is $0<p<1-\sqrt{0.5}$ and $\sqrt{0.5}<p<1$ where p is initial fidelity against the bit-flip error or the phase-flip error. Mean-while, we report a potentially practical entanglement distillation protocol (EDP) [37, 38] for a mixed photon-ensemble in which the vacuum error, the bit-flip error and the phase-flip error are coexistent. The key operations of our scheme are the parity-checking [39, 40] and qubit amplifying (PCQA) for photonic polarization degrees of freedom (DOF) by exploiting linear optics and entanglement resource locally (4-qubit GHZ states or 2-qubit Bell-states). With this operations, the vacuum error caused by the photon-transmission losses, phase-flip errors and bit-flip errors caused by the decoherence in noisy channels could all be corrected. Compared with the previous EC and EP methods, our protocol overcomes the difficulties posed by inherent channel losses during photon transmission. Especially, our protocol reduces the threshold value of the distilled mixed quantum system, and the available range for completely purifying bit-flip and phase-flip errors are much larger than the conventional EP schemes. All these advantages make this scheme more useful in the practical applications of long-distance quantum communication and scalable quantum computing.

This paper is organized as follows: In Sec. II, we first describe the approach for the PCQA setup assisted by W-state and linear optics. In Sec. III, we demonstrate the three steps of our EDP for a mixed photon-ensemble which composed of four kinds of partially entangled states and vacuum states. By iterating our EDP process, a more loose initialization requirement for entanglement distilling can be obtained. And the last part is the discussion and summary.

2. The parity-checking and qubit amplifying (PCQA) setup

The designed PCQA setup as shown in Fig. 1 is used to eliminate the vacuum states and pick up odd-parity states in the polarized DOF of two-photon systems. As described in the figure, the polarizing beam splitter (PBS) is used to transmit horizontally polarized photon (|H〉) and reflect vertically polarized photon (|V〉). The H plate which is a half-wave plate with the angle of 22.5° to the horizontal direction can perform polarized Hadamard operation, that is $|H〉\to \frac{1}{\sqrt{2}}(|H〉+|V〉),|V〉\to \frac{1}{\sqrt{2}}(|H〉-|V〉)$. The F plate is a half-wave plate with the axis at 45° with respect to the horizontal direction rotates the photon polarization $|H〉\rightleftharpoons |V〉$. Four ancillary photons marked with E ₁ E ₂ E E ₄ are initially prepared in GHZ state ${|GHZ〉}_4=\frac{1}{\sqrt{2}}(|VHVH〉+|HVHV〉)E_1{E}_2{E}_3{E}_4$, and E ₁ E ₂ E ₃ E ₄ are sent to four input ports e ₁, e ₂, e ₃, and e ₄, respectively. With the PCQA setup shown in Fig. 1, which is composed of linear optical device and local entanglement resources, one can perform the nondemolition parity-checking measurement and the qubit amplifying on a polarized photon-pair in one step. Assume that the initial state of a photon-pair AB is ${|\phi 〉}_{AB}=\alpha {|HV〉}_{AB}+\beta {|VH〉}_{AB}+\gamma {|HH〉}_{AB}+\xi {|VV〉}_{AB}$ where $|\alpha {|}^2+|\beta {|}^2+|\gamma {|}^2+|\xi {|}^2=1$, and AB are sent to two input ports a and b, respectively. After passing through the whole circuit shown in Fig. 1, the total state of the system composed of the photon-pair AB and ancillary photons E ₁ E ₂ E ₃ E ₄ becomes

 

Fig. 1 The non-demolition parity-checking and qubit amplifying (PCQA) setup. The polarizing beam splitter (PBS) can transmit horizontally polarized photon (|H⟩) and reflect vertically polarized photon (|V⟩). The Hadamard operation could be performed using H plate which is a half-wave plate with the angle of 22.5° to the horizontal direction, that is $|H〉\to \frac{1}{\sqrt{2}}(|H〉+|V〉)$, $|V〉\to \frac{1}{\sqrt{2}}(|H〉+|V〉)$. The F mode plate is a half-wave plate with the axis at 45° with respect to the horizontal direction rotates the photon polarization as |H⟩⇌|V⟩. The eight detectors D ₁ D′₁ D ₂ D′₂ D ₃ D′₃ D ₄ D′₄ belong to four different groups ${\mathcal{D}}_1$ ${\mathcal{D}}_2$ ${\mathcal{D}}_3$ ${\mathcal{D}}_4$, i.e., {D_j,D′_j} ∈ ${\mathcal{D}}_j(j=1,2,3or4)$.

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Here the eight detectors D ₁ D′₁ D ₂ D′₂ D ₃ D′₃ D ₄ D′₄ belong to four different groups marked with ${\mathcal{D}}_1{\mathcal{D}}_2{\mathcal{D}}_3{\mathcal{D}}_4$, i.e., $\{D_j,D{\prime }_{\mathrm{j}}\}\in {\mathcal{D}}_j(j=1,2,3or4)$. If the polarization DOF of the two photons AB are initially in the even-parity, i.e., |HH⟩_AB or |VV⟩_AB, the photons ABE ₁ E ₂ E ₃ E ₄ will pass through the whole PCQA circuit and trigger only three detector groups of ${\mathcal{D}}_1$, ${\mathcal{D}}_2$, ${\mathcal{D}}_3$ and ${\mathcal{D}}_4$. This case can be removed by coincidence detection. If the state of two-photon system AB is in the odd-parity state as |odd⟩_AB = α′|HV⟩+β′|VH⟩_AB where ${|\alpha |}^{\prime }=\frac{\alpha }{{|\alpha |}^2+{|\beta |}^2}$ and ${|\beta |}^{\prime }=\frac{\beta }{{|\alpha |}^2+{|\beta |}^2}$, as the successful operation of the present scheme is conditioned by a four-photon coincidence detection by ${\mathcal{D}}_1$, ${\mathcal{D}}_2$, ${\mathcal{D}}_3$and ${\mathcal{D}}_4$, one can postselect only the following cases:

When the results of the detectors is |D ₁ D ₂ D ₃ D ₄⟩, |D′₁ D′₂ D′₃ D′₄⟩, |D′₁ D ₂ D ₃ D′₄⟩, or |D ₁ D′₂ D′₃ D ₄⟩, no operations are required at the output port. Meanwhile, if the result on detectors is either |D′₁ D′₂ D ₃ D ₄⟩, |D′₁ D ₂ D′₃ D ₄⟩, |D ₁ D′₂ D ₃ D′₄⟩, or |D ₁ D ₂ D′₃ D′₄⟩, a single-photon operation −I = −(|H⟩ ⟨H| + |V⟩ ⟨V|) could be performed on the photon in spatial-mode b′ at the output port. When the results on detectors reveals |D′₁ D ₂ D ₃ D ₄⟩, |D ₁ D ₂ D ₃ D′₄⟩, |D′₁ D′₂ D′₃ D ₄⟩, or |D ₁ D′₂ D′₃ D′₄⟩, a single qubit operation σ_Z = (|H⟩⟨H|−|V⟩⟨V|) on the photon in spatial-mode b′ is needed to recover the same state as |odd⟩_A′B′ at output port; otherwise, a single qubit operation −σ_Z is required on the photon in spatial-mode b′ at output port when the results of detectors is |D ₁ D ₂ D′₃ D ₄⟩, |D ₁ D′₂ D ₃ D ₄⟩, |D′₁ D′₂ D ₃ D′₄⟩, or |D′₁ D ₂ D′₃ D′₄⟩. From Eq. (1), one can see that by using the PCQA setup, the odd parity states of the photons AB can be picked up with a probability of 50%. Based on the above discussion, the process of the parity-check measurement is heralded by four detectors clicking simultaneously. The relation between the final results at the detectors D ₁ D ₂ D ₃ D ₄ D′₁ D′₂ D′₃ D′₄ and the single-photon operations at the output port b′ is shown in Table 1.

Table 1. Relation between the final states of D ₁ D ₂ D ₃ D ₄ D′₁ D′₂ D′₃ D′₄ and the corresponding single qubit operations on the photon at the output port b′.

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Moreover, if one of two photons A and B is in the vacuum state, these photons will pass through the whole circuit and trigger two or three detectors of the detectors’ groups ${\mathcal{D}}_1$, ${\mathcal{D}}_2$, ${\mathcal{D}}_3$ and ${\mathcal{D}}_4$. For example, if there is only the photon A existed, after passing through the whole circuit, the total state composed of the photon A and ancillary photons E ₁ E ₂ E ₃ E ₄ becomes

Obviously, at the outports, there are only two or three detectors triggered. Therefore, the case of the vacuum input can be removed as it would not trigger the succeeded index in the present scheme. According to the above discussion, exploiting the proposed setup, one can perform the parity-checking and qubit amplifying without disturbing the encoded information.

3. Three-step EDP for an unknown mixed polarized partially-entangled Bell-state ensemble

In this section, we introduce the EDP for a mixed two-photon systems which are composed of vacuum components and entangled states with both phase-flip errors and bit-flip errors in polarization DOF.

Suppose a polarization-entangled photon source is located at Alices side and the entangled state can be written as ${|{\varphi }^{+}〉}_{ab}=\frac{1}{\sqrt{2}}{(|HH〉+|VV〉)}_{ab}$. The photon b is sent to Bob through a lossy channel with a transmission coefficient T (1 > T > 0), and the channel would deteriorate the photon pair ab to state ${\rho }_{ab}=(1-T){\rho }_{{\chi }_a}\otimes {|vac〉}_b〈vac|+T|{{\varphi }^{+}〉}_{ab}〈{\varphi }^{+}|$, where ${\rho }_{{\chi }_a}={|\chi 〉}_a〈\chi |=(|{H〉}_a〈H|+{|V〉}_a〈V|)/2$ is a mixed state of the photon a. Under the channel loss, the state of the photon b becomes a mixture of a vacuum component with probability 1 − T and a qubit component with probability T. In a practical transmission, besides the photon loss, the inevitable decoherence will also degrade the entanglement of the quantum system. The photon loss model was introduced for the first time in [41] for the effect of the loss in teleportation and cloning, and a more detailed model where all types of channel behavior is clearly introduced and studied is discussed in [42,43] where the imperfections of detectors and photon sources in experiments would affect quantum protocols.

For the more general case of the quantum system transmitted through a noisy channel, it becomes a mixed state. Therefore, after the channel transmitting, we suppose that the two-photon systems in a mixed ensemble which composed of vacuum states and four kinds of four maximally entangled states as follows:

There are three generic error modes on polarized photons, which could be described as the following forms: |ϕ ⁻⟩ represents the phase-flip error mode with the probabilities of p ₁(1 − p ₂), |ψ ⁺⟩ represents the bit-flip error mode with the probabilities of p ₂(1 − p ₁) and |ψ⁻⟩ represents both the two error mode with the probabilities of (1 − p ₁)(1 – p ₂). From Eq. (6), one can see that in the state ρ^p, the probabilities p ₁ and p ₂ are independent, and the bit-flip error which would occur with the probability (1− p ₁) and phase-flip error with the probability (1 − p ₂) are also independent of each other.

After transmission, Alice and Bob pick up two identical two-photon systems marked with AB and CD from the mixed ensemble ρ ⁰, and the states of the composite system can be described as

Here the subscripts AB and CD represent two photon pairs shared by Alice and Bob, respectively. The two photons A and C belong to Alice, and the other two photons B and D belong to Bob. With probability T ², the two pairs AB and CD are in the state ${\rho }_{AB}^p\otimes {\rho }_{CD}^p$, and it can be viewed as the mixture of 16 maximally entangled pure states:

With probability 1 − T ², there is a vacuum component existed in the 4-photon system ABCD. The aim of the entanglement distillation is to correct vacuum errors, bit-flip errors and phase-flip errors, and make the two parties share a high-fidelity maximally entangled state.

The principle of the polarized EDP includes three steps, and they are discussed in detail as follows. The PCQA setup shown in Sec. II is used in this EDP.

Step 1: Alice and Bob perform the polarized PCQA operations on the photons AC and BD, respectively. And with PCQA setups, they can remove the case that there is a vacuum component existed in the 4-photon system ABCD, and pick up the events that the two photon-pairs (AC and BD) are both in the odd-parity mode, the polarization state of the four-photon system is projected to

Then Alice and Bob perform Hardmard operations on the polarization DOF of the two photons A and B, respectively, and measure the states of the photons AB in the polarization basis {|H⟩,|V⟩}, respectively. If the final states of photons A and B are in the even-parity(|HH⟩ or |VV⟩), the two-photon system CD is projected to mixed maximally entangled state

Here, the bit-flip error which would occur with the probability (1 p′₁) and phase-flip error with the probability (1 – p′₂) are independent of each other. If the results of the photons AB is in the odd-parity(|HV〉 or |VH〉), phase-flip operation ${\sigma }_Z^D$ on the photon D is required to obtain the state ${\rho }_1^p$. By performing this step on the whole mixed quantum ensemble ρ^p, a new mixed ensemble ${\rho }_1^p$ is shared between Alice and Bob which composed of polarized maximally entangled states with phase-flip errors, the bit-flip errors, and both the two errors modes.

Step 2: Alice and Bob perform the local single-photon Hadamard operations in the polarization DOF on the photons C and D of the ${\rho }_1^p$, respectively. Then, by ignoring the overall phase of the photon-pair system, the state of ${\rho }_{1,CD}^p$ is transformed to

Different from Eq. (10), the probability of the bit-flip error occurring becomes $(1-p_2^{{}^{\prime }})$ and the probability of phase-flip error occurring is changed into $(1-p_1^{{}^{\prime }})$ which are also independent of each other [44].

Step 3: Alice and Bob pick up two identical photon pairs, CD and C′D′ from the new ensemble ρ′ ₁. The state of photon pairs CD and C′D′ are denoted as ${\rho }_{1,CD}^{\prime }$ and ${\rho }_{1,C^{\prime }{D}^{\prime }}^{\prime }$, respectively. Here photons C, C′ belong to Alice, and photons D, D′ are on Bob’s side. The state of ${\rho }_{1,CD}^p$ and ${\rho }_{1,C^{\prime }{D}^{\prime }}^p$ are

Alice and Bob performs the nondemolition parity-checking operations on the polarized DOF between the photons C and C′ and between the photons D and D′, respectively, as in the first step. The successful operation of the our polarized EDP is heralded on condition that both Alice’s and Bob’s detection are odd-parity events.

When Alice and Bob postselect successful EDP operation cases, only odd-parity state are preserved as

Then Alice and Bob perform Hardmard operations on the polarization DOF of the two photons C′ and D′, and measure the state in the polarization basis {|H⟩, |V⟩}, respectively. If the two clicked photon detectors of photons C′ and D′ are in the even-parity polarization mode, the two-photon system CD is projected into mixed maximally entangled state

If the outcome of two clicked detectors is in the odd-parity polarization state, a phase-flip operation ${\sigma }_Z^D$ on the photon D is required to obtain the state ${\rho }_2^p$. After one round EDP process, the state of the remaining photon-pair CD is transformed into ${\rho }_2^p$ with the total fidelity of the photon-pair CD is $F_p=〈{\varphi }^{+}|{\rho }_2^p|{\varphi }^{+}〉=p_1^{″}{p}_2^{″}$.

As the PCQA measurement is the nondemolition parity-check process, by iterating our EDP several times, and the fidelity $F_{p,n}=〈{\varphi }^{+}|{\rho }_n^p|{\varphi }^{+}〉=p_{1,n}\times p_{2,n}$ [41–43 , 45–47 ] with $p_{1,n}=\frac{{[p_{2,n-1}^2+{(1-p_{2,n-1})}^2]}^2}{{[p_{2,n-1}^2+{(1-p_{2,n-1})}^2]}^2+{[1-p_{2,n-1}^2-{(1-p_{2,n-1})}^2]}^2}$ and $p_{2,n}={[\frac{p_{1,n-1}^2}{p_{1,n-1}^2+{(1-p_{1,n-1})}^2}]}^2+{[1-\frac{p_{1,n-1}^2}{p_{1,n-1}{1}^2+{(1-p_{1,n-1})}^2}]}^2$. Here n is the iteration number of the EC processes, and ${\rho }_n^p$ is the state of the photon system after n times iteration. $p_{1(2),1}=p_{1(2)}^{″}$. Here we numerically simulated the relation between the fidelity F_p,n and the parameter p ₁ = p ₂ = p with the different iteration time n, and the results is shown in Fig. 2. From Fig. 2, one can see that there are two areas at $0.5\le p\le \sqrt{0.5}$ and $p=1-\sqrt{0.5}$ where the fidelity cannot be improved. In our EDP, with the iteration time n = 1, in the range of $0\le p<1-\sqrt{0.5,}1-\sqrt{0.5}<p<0.5$ and $\sqrt{0.5}<p<1$, the fidelity F_{p, 1} can be improved. When n = 3, in the case of 0 ≤ p < 0.18 and 0.82 ≤ p < 1, the fidelity F_{p, 3} can be improved to ∼ 1. When n = 4, the fidelity F_{p, 4} can be improved to 1 in the case of 0 ≤ p < 0.22 and 0.78 ≤ p < 1. Theoretically, within the range $0\le p<1-\sqrt{0.5}$ and $\sqrt{0.5}<p<1$, the fidelity of an arbitrary mixed partially-entangled spatial-mode states can be improved to the maximal value through enough iterating operations of our EDP.

 

Fig. 2 The fidelity of the two-photon entangled Bell state in our polarized EDP. F_p,n alters with the iteration number n of EDP processes and the initial the coefficient of the mixed less-entangled Bell state p = p ₁ = p ₂. Here p ₁(p ₂) is the initial fidelity against the bit-(phase-) flip error.

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If one knows in advance, after the communication channel, the transmitted state become a pure state with only a flip error, such as, p ₁ = p ₂ = 0 or p ₁ = 0 and p ₂ = 1, the best strategy is to add proper unitary transformation after the channel (in these simulation, our EDP also can work) [48]. However, when the initial maximally entangled states decay into mixed states, only with the unitary transformation, the errors can not be corrected. Therefore, the available distillation ranges of our EDP for a mixed two-photon ensemble is $0<p<1-\sqrt{0.5}$ and $\sqrt{0.5}<p<1$. Compared with the the conventional EP schemes, we add the Hadamard operation on the reserved photons after the step 1. The reason is that, in the case of p ₁ ≠ 0, p ₂ ≠ 0, and p ₁ and p ₂ are independent, the iterating parity-checking operations for one error will affect the other one, and even reduces the fidelity of the whole system. Here, we will give a discussion about this reason in detail in Section 4.

4. Discussion and summary

In the conventional EP schemes, especially, in the non-linear optical EP schemes which can get a high-fidelity yield by iterating the EP processes, if the bit-flip error and phase-flip error are coexistent, Alice and Bob should firstly perform the bilateral parity-checking operations on the mixed photon-ensemble for several times to purify the bit-flip error, and then they transform the polarization phase-flip error into a polarization bit-flip error with local operation so that the phase-flip error can also be purified by iterating the bilateral parity-checking operations [35]. However, when the bit-flip error and phase-flip error are coexistent independently, the iterating purification operations for one error will affect the other one, and even reduces the fidelity of the whole system. For example, from Eq. (8), one can conclude that after the bilateral parity-checking operation for purifying the bit-flip error which can improve p ₁ to $p_1^{{}^{\prime }}=\frac{p_1^2}{p_1^2+{(1-p_1)}^2}$ and the whole probability for a bit-flip erroroccurring $1-p_1^{{}^{\prime }}$ becomes smaller in the case of p ₁ > 1/2. But, the whole probability for a phase-flip error occurring becomes $1-p_2^{{}^{\prime }}=1-P_2^2-{(1-P_2)}^2$ which is bigger than 1−p ₂ in the case of p ₂ > 1/2. If Alice and Bob both iterate the bilateral parity-checking operation for purifying the bit-flip error for m times, the fidelity of the whole photon-pair ensemble becomes $F_{s,m}^0=f_{b,m}\times f_{p,m}$, where $f_{b,m}=\frac{f_{b,m-1}^2}{f_{b,m-1}^2+{(1-f_{b,m-1})}^2}$ is the fidelity against the bit-flip error [shown in Fig. 3(a)] and $f_{p,m}=f_{p,m-1}^2+{(1-f_{p,m-1})}^2$ is the fidelity against the phase-flip error[shown in Fig. 3(b)] with f_{b, 1} = p ₁ and f_{p, 1} = p ₂. Then, by using local operations, the polarization phase-flip error can be transformed into polarization bit-flip error, and the users iterate the purification operations for phase-flip error for m times by using the same bilateral parity-checking operations. Finally, the fidelity of the whole photon-pair ensemble becomes $F_{s,m}=f_{p,m}^{{}^{\prime }}\times f_{b,m}^{{}^{\prime }}$, where $f_{p,m}^{{}^{\prime }}=\frac{f_{p,m-1}^2}{f_{p,m-1}^2+{(1-f_{p,m-1}^{{}^{\prime }})}^2}$ [shown in Fig. 3(c)] and $f_{b,m}^{{}^{\prime }}=f_{b,m-1}^{{}^{\prime }2}-{(1-f_{b,m-1}^{{}^{\prime }})}^2$ [shown in Fig. 3(d)] with $f_{p,1}^{{}^{\prime }}=\frac{f_{b,m}^2}{f_{b,m}^2+{(1-f_{b,m})}^2}$ and $f_{b,1}^{{}^{\prime }}=f_{p,m}^2+{(1-f_{p,m}^{{}^{\prime }})}^2$. From the Figs. 3(a) and 3(b), One can see that in the case $p_1>\frac{1}{2}$ and $p_2>\frac{1}{2}$ the coefficient f_b,m can be improved largely with the iteration of the bit-flip error purifying process. When m = 4, in the case of 0.6 ≤ p ₁ < 1, the coefficient f_b,m can be improved to ~1. However, the coefficient f_p,m rapidly degrades to 1/2 simultaneously. When m = 4, in the case of 0.2 ≤ p ₂ < 0.8, the coefficient f_p,m is reduced to 1/2. Although the polarization phase-flip error can be transformed into the polarization bit-flip error with local operations, once the f_p,m = 1/2, the coefficient $f_{p,m}^{{}^{\prime }}$ can’t be improved any more and it always equals to 1/2 [as shown in Fig. 3(d)].

 

Fig. 3 The fidelity f_b,m (a), f_p,m (b), $f_{b,m}^{{}^{\prime }}$ (c) and $f_{p,m}^{{}^{\prime }}$ (d) are altered with the iteration number of entanglement concentration processes m and the coefficients p ₁ and p ₂.

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Figure 4 illustrates the fidelities of the conventional EP schemes F_s,m and the fidelities of our proposed EDP F_p,n versus the initial coefficient of the mixed entangled Bell-state p in the case p ₁ = p ₂ = p for different iteration times of EP processes. As described in the figure, when m = n = 1, we have F_{s, 1} = F_{p, 1}. By iterated three times as m = n = 3, the fidelity F_{s, 3} can be improved to ∼ 1 in the case of 0 ≤ p < 0.08 and 0.92 ≤ p < 1, but the available ranges for F_{p, 3} ∼ 1, in which both the two errors can be corrected, are 0 ≤ p < 0.18 and 0.82 ≤ p < 1. If we continuing increasing the iteration number to m = n = 4, the available ranges for F_{s, 4} ~ 1 are 0 ≤ p < 0.06 and 0.94 ≤ p < 1, but the available ranges for F_{p, 4} ~ 1 are 0 ≤ p < 0.22 and 0.78 ≤ p < 1. From Fig. 4, one can see that the available ranges of our EDP is much larger than the one of the conventional EP schemes. When both the bit-flip error and phase-flip error are existed independently, the available range of the conventional EP strategies in which both the two errors can be corrected becomes smaller with the increment of the iteration number m, but the available purification range for our EDP becomes larger with the increment of the iteration number n. So our EDP shows a wide distillation range compared to the conventional EP schemes.

 

Fig. 4 The fidelities of the two-photon entangled Bell states in the conventional EP schemes F_s,m and in our polarized EDP F_p,n in the case of p ₁ = p ₂ = p

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In our scheme, we consider the influence of the photon loss in the channel which generally is ignored in the conventional EP and EC schemes. Such a heralding scheme can overcome the difficulties posed by the inherent channel losses in photon transmission which not only reduces the efficiency of quantum communication but also the security of the quantum communication [49–51 ]. For example, photon losses can cause a low detection efficiency in quantum communications (the overall detection efficiency is below a certain threshold value), and lead to the unsatisfying of the optical loophole-free violation of Bell inequalities [49]. Therefore, the low signal detection can open the so-called detection efficiency loophole [52] for the attacks. Eliminating the vacuum component and improving the detection efficiency are the main requirements for the practical quantum commination.

The core device of our scheme is the PCQA setup, and the similar setups, which are used for nondemolition detection of the presence of a single photon and for qubit amplification using entangled two-photon Bell-state ancilla, are discussed in [53, 54]. In these papers, the probability of successful the single photon QND is 50%, and so the probability of successful two-photon presence is 25% by using two single-photon QND devices with two Bell-state photon-pairs. However, in our scheme, not only the entangled GHZ-state four-photon but also the Bell-state photon-pairs can be used to perform the PCQA operation. For example, if the four ancillary photons E ₁ E ₂ and E ₃ E ₄ are initially in the maximally entangled Bell state ${|{\psi }^{+}〉}_{E_1{E}_2}$ and ${|{\psi }^{+}〉}_{E_3{E}_4}$, respectively, after passing through the whole circuit shown in Fig. 1, the total state of the system composed of the photon-pair AB and ancillary photons E ₁ E ₂ E ₃ E ₄ becomes

As the successful operation of the present scheme is conditioned by a four-photon coincidence detection by ${\mathcal{D}}_1$, ${\mathcal{D}}_2$, ${\mathcal{D}}_3$ and ${\mathcal{D}}_4$, from Eq. (15), one can see that, only the cases in which the states of the two photons at the outports a′b′ are odd parity can be postselected, and the probability of successful operation is 25%. Therefore, comparing with two single-photon QNDs in [53, 54], the PCQA setup in our scheme can perform not only the QND but also the parity-checking of the photon-pairs with the same entanglement resources and the same the success probability.

To mirror a likely experimental scenario, we numerically simulated the dark error rate P_vac of the PCQA setup at output port a with the initial spontaneous-parametric down-conversion entanglement source with the single pair generator rate of η_p = 0.058, a collection and the detector efficiency of η_d = 0.265 (an average two-photon coincidence count rate of M = 1 ∼ 3 MHz) [56], and the dark count probability ∼ 1 dark count/s [53]. From Eq. (1), one can see that the output photons are come from the local entangled photon pairs, and the rate of dark error could be described as $P_e=\frac{D}{\frac{T^{2}M[p\frac{2}{1}+{(1-p_1)}^2]}{2}+D}$, where D denotes the error key rate (per second) caused by the dark count of the detectors [50,53]. We assume that the channel loss is 0.2dB/km, after 100km transmission and T ∼ 0.01, we found that the error rate after one time parity check P_e is below 0.23. By increasing T=0.1(the transmission length ∼ 50km), P_e ~ 0.0035 could be solved. Moreover, the success probability of the PCQA setup P_s relies on the transmission efficiency T and the initial fidelity p against both bit-flip error and phase-flip error which could be expressed as T ²[p ² + (1 − p)²]/2.

In summary, we proposed an efficient linear-optical entanglement distillation scheme for achieving high-fidelity Bell states from photon losses and decoherence in a heralded strategy. Exploiting our strategy, a mixed photonic ensemble which composed of different partially entangled states and vacuum states can be distilled to a high-fidelity mixed Bell-state ensemble in the polarization DOF, and in the available distillation ranges for our EDP ( $0<p<1-\sqrt{0.5}$ and $\sqrt{0.5}<p<1$, that is, the whole initial fidelity F = p ² is in the range $0<F<1.5-\sqrt{2}$ and 0.5 < F < 1), the fidelity of this ensemble can be improved to the maximal value through iterated operations. Compared to the first EP scheme in [16], the initial state of our scheme is a more universal entangled mixed state ensemble in which the bit-flip error and phase-flip error are coexistent independently, and it is different from the Werner state [55]. Compared to the conventional EP schemes, our scheme considers both the bit-flip error, phase-flip error and vacuum noise state, which is more general entanglement distillation scheme for practical channel noise. Also the distillation range of our scheme is enlarged compared with conventional schemes. Moreover, our scheme largely reduces the requirement of the distilled mixed quantum system with a lower threshold, and overcomes the difficulties posed by inherent channel losses during photon transmission. With these advantages, our scheme may be further applied in long-distance quantum communication and quantum networks.