Abstract

The phase-matching quantum key distribution (PM-QKD), one of the variants of Twin-Field (TF) QKD protocol, was recently proposed to overcome the rate-distance limits of point to point protocol without quantum repeaters. In this paper, we propose a more practical PM-QKD protocol version with four-intensity decoy states and source errors, since neither the infinite decoy states nor the precise control of the light source is available in practice. We present the formulation of the secure key rate of the proposed protocol and analyze the performances of the protocol with and without source errors by numerical simulations.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Based on the laws of physics, quantum key distribution (QKD) protocol is able to guarantee the authorized users, Alice and Bob, to share unconditionally secure keys, even in the presence of a malicious eavesdropper, Eve. The first QKD (BB84-QKD) protocol was put forward in 1984, and its security was later proved [1,2]. Since then, many more secure and practical QKD protocols have been developed, such as decoy-state QKD protocols [35], measurement-device-independent quantum key distribution(MDI-QKD) protocols [68], and round-robin differential-phase-shift quantum key distribution(RRDPS-QKD) protocols [911]. However, the transmission distance is still a major obstacle in actual implementation of all these existing protocols due to the rate-distance limit, i.e., the secure key rate $R$ is bounded by the efficiency of quantum channel transmission $\eta$ without quantum repeaters ( $R \leq O(\eta )$) [12,13].

In 2018, Lucamarini et al. did a marvelous work by proposing a so-called twin-field quantum key distribution(TF-QKD), which can break the limit without quantum repeaters [14]. The secure key rate of the protocol scales with square root of the channel transmission efficiency, $R \leq O(\sqrt {\eta })$, which has beaten the PLOB bound [13]. Inspired by the amazing work, several variants of TF-QKD protocol turned out, such as the phase-matching quantum key distribution (PM-QKD) proposed by $Ma~et~al.$ [15,16], the sending-or-not-sending quantum key distribution (SNS-QKD) presented by $Wang~et~al.$ [17,18], and the twin-field quantum key distribution without phase post-selection by $Cui~et~al.$ [19] and by $Curty~et~al.$ [20].

The PM-QKD protocol is a more practical version of TF-QKD, where phase post-compensation is adopted. In the PM-QKD protocol, Alice (Bob) prepares their weak coherent states randomly and adds a random phase $\phi (A)(\phi (B))$ to each of his weak coherent states. Afterwards, both of them send the states to an untrusted party (Charlie) located in the middle of the channel. Depending on the measurement performed by Charlie, Alice and Bob are able to generate the raw key after a post-selection of the case satisfying $\phi (A)\approx \phi (B)$. After a sifting, parameter estimation and key distillation are necessary to be used to generate a final private and secure key. The infinite decoy-state method was used to estimate the performance in the original PM-QKD [15]. Moreover, it is assumed that the signal and decoy sources adopted in the protocol are accurately controlled in the photon-number space. But this assumption turns out to be unrealistic due to the existence of source errors caused by the fluctuations of source intensity, noises in the environment and others.

On the other hand, there are a lot of published works concerning the source errors in QKD protocols. For instance, $Wang~et~al.$ proposed a general theory of decoy-state BB84-QKD with source errors [21,22] to estimate the system performance. Then the secure key rate of BB84-QKD protocol with source errors adopting heralded single-photon source(HSPS) [23] or heralded pair coherent source(HPCS) [24] has been studied. In addition, the performance of other QKD protocols with source errors, such as reference-frame-independent quantum key distribution(RFI-QKD) protocol [25], MDI-QKD protocol [2628] and RRDPS-QKD protocol [29], have been demonstrated. However, up to now, the study of PM-QKD with source error has not yet been discussed.

In this paper, we first propose a four-intensity decoy-state PM-QKD protocol, and demonstrate that the four-intensity decoy-state method is enough in actual implementation. Then, we further study the four-intensity decoy-state PM-QKD protocol with source errors. We derive the formulations of the secure key rate of the proposed protocol with and without source errors. At last, we present some numerical simulations according to the analysis results.

The benefits of our work are listed as follows: (1) The proposed protocol is more practical, since the source errors has been included in the protocol. (2) Compared with PM-QKD protocol with infinite decoy-states, the four-intensity PM-QKD protocol is more feasible, because it is impossible to have infinite-intensity light source in real implementation. At the same time, the used four-intensity decoy-state is more universal for all the intensities are calculated and optimized. (3) The finite data-size analysis for the proposed protocol is presented, and a simple statistical fluctuation is analyzed. (4) The formula for the key generate rate with source errors is presented, and the simulation results verify the proposed protocol.

The organization of the paper is as follows. Firstly, we present the four-intensity decoy-state PM-QKD protocol, and give the derivation formula, a simple statistical fluctuation analysis and the simulation results on the performance of our protocol. Secondly, we study the effect of source errors on our protocol , and show the formulations of the secure key rate by deducing the lower or upper bound of count rate and quantum bit error of the $k$-photon state for the signal source. Thirdly, the numerical simulations of the protocol with source errors are presented, and at last, the conclusion is drawn.

2. Four-intensity decoy-state PM-QKD

2.1 Protocol

We adopt four-intensity decoy-state method to approach the performance of the PM-QKD protocol with infinite-intensity decoy-state method. The details of our four-intensity decoy-state PM-QKD protocol are as follows.

Step 1. Alice (Bob) prepares and sends a coherent state pulse. She(He) encodes a key bit $k_a(k_b)\in \{0,1\}$ into the phase of the coherent state and then adds a random phase $\varphi _A(\varphi _B)\in [0,2\pi )$ to the state. After that, she(he) sends the encoded state to a third party Charlie who can even be an eavesdropper.

Here, the pulse’s intensity $\mu _A(\mu _B)$ is randomly chosen from $\{\mu /2,v_1/2,v_2/2,v_3/2\}$ with the probabilities of $P_{\mu }$, $P_{v_{1}}$, $P_{v_{2}}$ and $P_{v_{3}}$, where $P_{\mu }+P_{v_{1}}+P_{v_{2}}+P_{v_{3}}=1$ [22,29]. $\mu /2$ represents the intensity of signal states, while $v_1/2$, $v_2/2$ and $v_3/2$ are intensities of decoy states, and they should satisfy the conditions: $\mu \ge v_1 \ge v_2 \ge v_3$ and $\mu > v_1+v_2+v_3$. Their density matrix can be written as $\rho _x=\begin {matrix} \sum _{k=0}^\infty p_{x}(k)\left |k\right \rangle \left \langle k \right | \end {matrix} (x=\mu /2,v_{1}/2,v_{2}/2,v_{3}/2)$.

Step 2. Charlie uses a beam splitter to carry out an interference measurements on the receiving pair of pulses. After the interference, he is supposed to announce the outcomes of the measurement when the interference measurement detection is a success, i.e., which detector clicks.

Step 3. The authorized users Alice and Bob, along with the third party Charlie, repeat Step 1. to Step 2. $N$ times.

Step 4. Charlie announces the results of all the measurements, and Alice(Bob) announces all the intensities and random phases she(he) chooses. Then Alice(Bob) keeps the results as the raw bits if

$$|\varphi_A-\varphi_B-k\pi|\leq\frac{2\pi}{M} (k=0,1),$$
where $M$ denotes the number of slices that Alice and Bob choose to divide $[0,2\pi )$ to the phase interval. Eq. (1) refers to the process called post-selection of phases. Note that, the raw bit is obtained in Z-basis if $\mu _A=\mu _B=\mu /2$ , or X-basis if $\mu _A=\mu _B \ne \mu /2$.

Step 5. Raw bits in X-basis are used for parameter estimations. A certain amount of bits in Z-basis are chosen for error evaluation and the others are used for key distillation.

During the protocol, some notes should be clarified [22]:

(1)For the $ith$ measurement, Charlie may have three outcomes: both detectors click, only one detector clicks or none of the detectors clicks. If only one detector clicks and the $ith$ measurement outcome satisfies the condition in Step 4, the $ith$ result is called as a success measurement event.

(2)The light source adopted in the protocol is a weak coherent source (WCS) [15], so the probability of the $k$-photon pulses follows a Poisson distribution, and is expressed by $p_{x}(k)=\frac {e^{-x}x^k}{k!}(x=\mu /2,v_1/2,v_2/2,v_3/2)$. Given $\mu _A=\mu _B=\mu /2$, the probability of a success measurement event for Charlie with $n$-photon pulse from Alice and $(n-m)$-photon pulse from Bob can be given by

$$\begin{aligned}\sum_{m=0}^n{p_{\mu_A}(m) {\cdot} p_{\mu_B}(n-m)} &= \sum_{m=0}^n{\frac{e^{-\mu/2}(\frac{\mu}{2})^m}{m!} {\cdot} \frac{e^{-\mu/2}(\frac{\mu}{2})^{n-m}}{(n-m)!}}\\ &=e^{-\mu}\mu^n \sum_{m=0}^n{\frac{1}{2^n}\frac{1}{m!(n-m)!}}\\ &=\frac{e^{-\mu}\mu^n}{n!} =p_{\mu}(n). \end{aligned}$$
The equation also stands when $\mu _A=\mu _B=x(x=v_1/2,v_2/2,v_3/2)$. Thus, it is said that Alice and Bob together send a weak coherent pulse with intensity $x(x=\mu ,v_1,v_2,v_3)$ [15]. And the density matrix can be written as $\rho _x=\begin {matrix} \sum _{k=0}^\infty p_{x}(k)\left |k\right \rangle \left \langle k \right | \end {matrix} (x=\mu ,v_1,v_2,v_3)$.

2.2 Secure key rate

After the protocol is executed, some measured values can be obtained, including the number of counts caused by states with different kinds of intensities $N_x\left (x=\mu ,v_{1},v_{2},v_{3}\right )$, the overall gain $Q_x(x=\mu ,v_{1},v_{2},v_{3})$ and the bit error rate $E_{\mu }^Z$. According to the original PM-QKD protocol [15], the secure key rate is given by

$$R=\frac{2}{M}Q_{\mu}[1-fH(E_{\mu}^Z)-H(E_{\mu}^X)],$$
where $f$ denotes the efficiency of the error correction, $H(x)=-xlog_2(x)-(1-x)log_2(1-x)$ is the binary Shannon information function. $E_{\mu }^X$ denotes the phase error rate, which can be calculated by
$$\begin{aligned} E_{\mu}^X &=\sum_{k=0}^{\infty}q_{2k+1}e_{2k+1}^Z+\sum_{k=0}^{\infty}q_{2k}(1-e_{2k}^Z)\\ &=\sum_{k=0}^{\infty}q_{2k+1}e_{2k+1}^Z+q_0e_0^Z+\sum_{k=1}^{\infty}q_{2k}-\sum_{k=1}^{\infty}q_{2k}e_{2k}\\ &=\sum_{k=0}^{\infty}q_{2k+1}e_{2k+1}^Z+q_0e_0^Z+(1-q_0-q_{odd})-\sum_{k=1}^{\infty}q_{2k}e_{2k}\\ &\leq \sum_{k=0}^{\infty}q_{2k+1}e_{2k+1}^Z+q_0e_0^Z+(1-q_0-q_{odd}). \end{aligned}$$

In Eq. (4), $q_k=\frac {Q_{k,\mu }}{Q_{\mu }}$ and $e_k^Z$ respectively represent the fractions and the bit error rate of different photon components $k$, and $e_0^Z=0.5$. According to the inequality [15]

$$\begin{aligned}E_{\mu}^X &=\sum_{k=0}^{\infty}q_{2k+1}e_{2k+1}^Z+\sum_{k=0}^{\infty}q_{2k}(1-e_{2k}^Z)\\ &\leq q_0e_0^Z+(q_1e_1^Z+q_3e_3^Z+q_5e_5^Z)+(1-q_0-q_1-q_3-q_5), \end{aligned}$$
$q_k$ and $e_k^Z$ can be calculated for $0\leq k\leq 5$.

As for our protocol, the upper and lower bounds of the yield of $k$-photon state $Y_k$, as well as the bit error rate of $k$-photon state $e_{k}^Z$, can be estimated using decoy state method. The phase error rate $E_{\mu }^X$ can be rewritten as

$$\begin{aligned} E_{\mu}^X &=\sum_{k=0}^{\infty}q_{2k+1}e_{2k+1}^Z+\sum_{k=0}^{\infty}q_{2k}(1-e_{2k}^Z)\\ &=\sum_{k=0}^{\infty}q_{2k+1}e_{2k+1}^Z+q_0e_0^Z+q_2(1-e_2^Z)+\sum_{k=2}^{\infty}q_{2k}-\sum_{k=2}^{\infty}q_{2k}e_{2k}\\ &\leq\sum_{k=0}^{\infty}q_{2k+1}e_{2k+1}^Z+q_0e_0^Z+q_2(1-e_2^Z)+(1-q_0-q_2-q_{odd})\\ &=\sum_{k=0}^{\infty}q_{2k+1}e_{2k+1}^Z+q_0e_0^Z-q_2e_2^Z+(1-q_0-q_{odd}). \end{aligned}$$

And with one signal state and three decoy states, $q_k$ and $e_k^Z$ can only be estimated for $0\leq k\leq 2$, the overall bit error rate can be calculated as

$$E_{\mu}^X = q_0e_0^Z+q_1e_1^Z-q_2e_2^Z+(1-q_0-q_1).$$

According to equations above, the upper bounds of $e_1^Z$ and the lower bounds of $Y_0$, $Y_1$, $Y_2$ and $e_2^Z$ are estimated as the follows.

2.2.1 Estimation of the parameters

Using the decoy-state method, one can get the following overall gains and quantum bit error rates for the signal and decoy states

$$Q_{x}e^{x} = \sum_{k=0}^\infty Y_k \frac{(x)^k}{k!},\quad E_{x}Q_{x}e^{x} = \sum_{k=0}^\infty e_k^Z Y_k \frac{(x)^k}{k!} \quad (x=\mu,v_1,v_2,v_3).$$

From earlier work [30], the lower bounds of $Y_0$, $Y_1$, $Y_2$ and the upper bounds of $e_1$, $e_2$ can be obtained. With $(v_2-v_3)\times (E_{v_1}Q_{v_1}e^{v_1}-E_{v_2}Q_{v_2}e^{v_2})- (v_1-v_2)\times (E_{v_2}Q_{v_2}e^{v_2}-E_{v_3}Q_{v_3}e^{v_3})$, one can have

$$\begin{aligned} &e_2^ZY_2 \frac{(v_1-v_2)(v_2-v_3)(v_1-v_3)(\mu-v_1-v_2-v_3)}{2\mu}\\ &\ge (v_2-v_3)(E_{v_1}Q_{v_1}e^{v_1}-E_{v_2}Q_{v_2}e^{v_2})-(v_1-v_2)(E_{v_2}Q_{v_2}e^{v_2}-E_{v_3}Q_{v_3}e^{v_3})\\ &-\frac{(v_1-v_2)(v_2-v_3)(v_1-v_3)(v_1+v_2+v_3)}{\mu^3}(E_{\mu}Q_{\mu}e^{\mu}-e_0^ZY_0-e_1^ZY_1\mu). \end{aligned}$$

Therefore, the lower bound of $e_2^Z$ can be obtained as

$$\begin{aligned} e_2^Z&\ge \frac{2\mu}{Y_2^U(v_1-v_2)(v_1-v_3)(v_2-v_3)(\mu-v_1-v_2-v_3)}[(v_2-v_3)(E_{v_1}Q_{v_1}e^{v_1}\\ &-E_{v_2}Q_{v_2}e^{v_2})-(v_1-v_2)(E_{v_2}Q_{v_2}e^{v_2}-E_{v_3}Q_{v_3}e^{v_3})]-\frac{2(v_1+v_2+v_3)}{Y_2^U\mu^2(\mu-v_1-v_2-v_3)}\\ &{\cdot}[E_{\mu}Q_{\mu}e^{\mu}-e_0^ZY_0^L-(e_1^Z)^LY_1^L\mu]\\ &=(e_2^Z)^L. \end{aligned}$$

Obviously, the upper bound of $Y_2$ and lower bound of $e_1^Z$ are necessary for calculating $(e_2^Z)^L$. The upper bound of $Y_2$ can be evaluated as

$$Y_2 \leq \frac{2[(v_2-v_3)Q_{v_1}e^{v_1}-(v_1-v_3)Q_{v_2}e^{v_2}+(v_1-v_2)Q_{v_3}e^{v_3}]}{(v_1-v_2)(v_1-v_3)(v_2-v_3)} =Y_2^U.$$

It is known that $E_{v_1}Q_{v_1}e^{v_1}-E_{v_2}Q_{v_2}e^{v_2} \leq e_1^ZY_1\frac {(v_1-v_2)(\mu -v_1-v_2)}{\mu }+\frac {(v_1)^2-(v_2)^2}{\mu ^2}(E_{\mu }Q_{\mu }e^{\mu }-e_0^ZY_0)$, consequently, one can get the lower bound of $e_1^Z$ as

$$\begin{aligned} e_1^Z &\ge \frac{\mu(E_{v_1}Q_{v_1}e^{v_1}-E_{v_2}Q_{v_2}e^{v_2})}{(v_1-v_2)(\mu-v_1-v_2)Y_1^U}-\frac{v_1+v_2}{\mu(\mu-v_1-v_2)Y_1^U}(E_{\mu}Q_{\mu}e^{\mu}-e_0^ZY_0^L)\\ &=(e_1^Z)^L, \end{aligned}$$
where $Y_1^U=\frac {Q_{v_1}e^{v_1}-Q_{v_2}e^{v_2}}{v_1-v_2}$ is the upper bound of $Y_1$, calculated by the equation $Q_{v_1}e^{v_1}-Q_{v_2}e^{v_2}$.

2.2.2 Statistical fluctuation analysis

In this subsection, we discuss the effect of finite-size of data. Here we follow the Gaussian analysis method in Ref. [31], where the quantum channel is assumed to be fluctuated according to Gaussian distribution. Then, the equalities in Eq. (8) should turn out to inequalities as,

$$\begin{array}{r}(Q_{x})^L=Q_{x}(1-\frac{n_{\alpha}}{\sqrt{N_xQ_{x}}}) \leq \sum_{k=0}^\infty Y_k \frac{e^{-x}(x)^k}{k!}\\ \leq Q_{x}(1+\frac{n_{\alpha}}{\sqrt{N_xQ_{x}}})=(Q_{x})^U,\\ (E_{x}Q_{x})^L=E_{x}Q_{x}(1-\frac{n_{\alpha}}{\sqrt{N_xE_{x}Q_{x}}}) \leq \sum_{k=0}^\infty e_k^Z Y_k \frac{e^{-x}(x)^k}{k!}\\ \leq E_{x}Q_{x}(1+\frac{n_{\alpha}}{\sqrt{N_xE_{x}Q_{x}}})=(E_{x}Q_{x})^U. \end{array}$$

Here, $(Q_{x})^U$, $(Q_{x})^L$, $(E_{x}Q_{x})^U$ and $(E_{x}Q_{x})^L$ are thereafter used for obtaining the upper and lower bounds of $Y_k$ and $e_k$, $n_{\alpha }$ is the number of standard deviations choosing for the statistical fluctuation analysis, directly related to the failure probability $\varepsilon$, that is,

$$1-erf(\frac{n_{\alpha}}{\sqrt{2}})= \varepsilon$$
where $erf(x)=\frac {2}{\sqrt \pi }\int _0^x e^{-t^2}dt$, it is the error function.

2.2.3 Numerical simulation

In this section, we present the numerical simulation of the secret key rate(SKR), and compare it with that of the original PM-QKD protocol.

The overall gain $Q_x(x=\mu ,v_1,v_2,v_3)$ and the overall bit error rate $E_x(x=\mu ,v_1,v_2,v_3)$, can be directly measured in practical experiments. They are expressed as

$$Q_{x} = 1-e^{-\eta x}+2pde^{-\eta x},\qquad E_{x} = \frac{e^{-\eta x}(pd+\eta x e_{\delta})}{Q_{\mu}},$$
where, $e_{\delta }=\frac {\pi }{M}-\frac {M^2}{\pi ^2}sin^3(\frac {\pi }{M})$ can be regarded as the error probability caused by the misalignment error. $pd$ denotes the background count rate for the detector. And $\eta$ is the transmittance which includes detection efficiency $\eta _d$ and channel loss efficiency $\eta _t$ ($\eta _t=10^{-\alpha z/10}$), where $\alpha$ and $z$ are the channel transmission loss rate and the transmission distance, respectively. Utilizing the deductions Eq. (912), together with the expressions of the overall gain and bit error rate, the SKR of the proposed PM-QKD protocol can be evaluated.

In order to numerically compare the SKR performance of our proposed protocol and the original one, we setup $M=16$ and use the same parameters in Table 1. Furthermore, the values of $\mu$, $v_1$, $v_2$ and $v_3$ are first optimized in order to obtain the optimal SKR performance. Their optimal values are $0.16, 0.02, 0.01$ and $0$, respectively.

Tables Icon

Table 1. Parameters of the simulation

Figure 1 shows the performance of the proposed four-intensity decoy-state PM-QKD protocol, the protocol with infinite decoy states (original PM-QKD), new PM-QKD in [32] with the same parameter in Table 1, and new PM-QKD with the original parameters in [32]. The results show that the secure key rates decrease with the increase of transmission distance, the proposed protocol can exceed the SKR linear bound, and the performance of the proposed protocol is always close to that one with infinite decoy states. The new symmetry-based PM-QKD protocol [32] can greatly improve the SKR performance with the parameters in Ref. [32] ( $pd=1\times 10^{-8}$, $\eta _d=0.2$, $f=1.1$ and $e_d=0.1$). For the given parameters in Table 1, as it can be seen, the new PM-QKD only has a better performance than that of the original PM-QKD when the transmission distance is less than $353km$, and our proposed protocol has a better performance in comparison with the new PM-QKD in Ref. [32] when the transmission distance is greater than $360 km$.

 

Fig. 1. Secure key rates for the four-intensity decoy-state PM-QKD, original PM-QKD, new PM-QKD in [32] with the same parameter in Table 1, and new PM-QKD with the original parameters in [32].

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We further discuss the performance of the proposed protocol with finite data sizes. Statistical fluctuation would cause less accurate of the channel parameters estimation when a finite set of data is considered in the QKD system. Figure 2 demonstrates the four-intensity decoy-state PM-QKD’s performance with finite data sizes. The parameters are the same as those listed in Table 1 and the failure probability is set as $\varepsilon =5.733\times 10^{-7}$. The results show that the statistical fluctuation has some impact on the protocol’s performance. With the decrease of the data size, the maximum transmission distances are reduced. However, the key rate can beat the linear bound at 235 km under the condition where the data size $N \geq 1 \times 10^{15}$.

 

Fig. 2. Key rates for the four-intensity decoy-state PM-QKD under the data size $N=1\times 10^{15}$, $1\times 10^{16}$, $1\times 10^{17}$ and infinitely large

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3. Protocol with source errors

The assumption that the precise control of the light source in the photon number space is impractical. In this section, we discuss the performance of the four-intensity decoy-state PM-QKD protocol with source errors.

Assume that there exists some errors in the source state, each state deviates from the expectation at any time $\left (i\in \left \{1,2,\ldots ,N\right \}\right )$. The density matrix can be rewritten as $\rho _{xi}=\begin {matrix} \sum _{k=0}^\infty p_{xi}(k)\left |k\right \rangle \left \langle k \right | \end {matrix} \left (x=\mu ,v_1,v_2,v_3\right )$, and $p_{xi}(k)-p_{x}(k)$ is called source error. Without loss of generality, the probability $p_{xi}(k)$ is assumed to be bounded by $p_{x}^L(k)$ and $p_{x}^U(k)$, i.e., $p_{x}^L(k)\leq p_{xi}(k)\leq p_{x}^U(k)$.

3.1 Secure key rate with source errors

At first, we introduce the definition of set $S$ and $s_k$ [21]: Set $S$ contains all the effective events; set $s_k$ contains all the effective caused by pulses containing $k$ photons. Mathematically, the sufficient and necessary condition for $i\in$$S$ is that the outcome of the $ith$ measurement is an effective event. Equally, the sufficient and necessary condition for $i\in$$s_k$ is that the outcome of the $ith$ measurement caused by k-photon states is an effective event.

Additionally, if the $ith$ measurement is an element of $s_k$, the probability that its intensity is $x(\mu ,v_1,v_2,v_3)$ should be [22,29]

$$P_{xi|k}=\frac{P_x{\cdot} p_{xi}(k)}{P_{\mu}{\cdot} p_{\mu i}(k)+P_{v_1}{\cdot} p_{v_1 i}(k)+P_{v_2}{\cdot} p_{v_2 i}(k)+P_{v_3}{\cdot} p_{v_3 i}(k)}.$$

Let $n_{k,x}$ denote the effective events caused by states containing $k$ photons whose intensity is $x(x=\mu ,v_1,v_2,v_3)$. And with the definition mentioned above, number of counts caused by source state $x(x=\mu ,v_1,v_2,v_3)$ can be formulated by

$$N_x=\sum_{k=0}^\infty{n_{k,x}}=\sum_{k=0}^\infty{\sum_{i\in s_k}{P_{xi|k}}}=\sum_{i\in s_0}{P_{xi|0}}+\sum_{i\in s_1}{P_{xi|1}}+\sum_{k=2}^\infty{\sum_{i\in s_k}{P_{xi|k}}}.$$

Define $d_{ki}=\frac {1}{P_{\mu }{\cdot} p_{\mu i}(k)+P_{v_1}{\cdot} p_{v_1 i}(k)+P_{v_2}{\cdot} p_{v_2 i}(k)+P_{v_3}{\cdot} p_{v_3 i}(k)}$, so $P_{xi|k}$ can be expressed by $P_{xi|k}=P_x{\cdot} p_{xi}(k){\cdot} d_{ki}$. Thus, $N_x$ can be rewritten as

$$N_x= \sum_{k=0}^\infty{\sum_{i\in s_k}{P_x{\cdot} p_{xi}(k){\cdot} d_{ki}}}=P_x{\cdot} \sum_{k=0}^\infty{\sum_{i\in s_k}{p_{xi}(k){\cdot} d_{ki}}}.$$

3.1.1 Bounds of $Q_{k,\mu }$

Since $Q_x=\frac {N_x}{P_x{\cdot} N}(x=\mu ,v_1,v_2,v_3)$, one can get

$$\begin{aligned}\frac{N_x}{P_x}&=Q_x{\cdot} N=N{\cdot} (Q_{0,x}+Q_{1,x}+\sum_{k=2}^\infty{Q_{k,x}})\\ &=\sum_{i\in s_0}{p_{xi}(0)d_{0i}}+\sum_{i\in s_1}{p_{xi}(1)d_{1i}}+\sum_{k=2}^\infty{\sum_{i\in s_k}{p_{xi}(k)d_{ki}}}, \end{aligned}$$
i.e.,
$$\begin{aligned} &\frac{N_{\mu}}{P_{\mu}}=\sum_{i\in s_0}{p_{\mu i}(0)d_{0i}}+\sum_{i\in s_1}{p_{\mu i}(1)d_{1i}}+\sum_{k=2}^\infty{\sum_{i\in s_k}{p_{\mu i}(k)d_{ki}}},\end{aligned}$$
$$\begin{aligned} &\frac{N_{v_1}}{P_{v_1}}=\sum_{i\in s_0}{p_{ v_1 i}(0)d_{0i}}+\sum_{i\in s_1}{p_{v_1i}(1)d_{1i}}+\sum_{k=2}^\infty{\sum_{i\in s_k}{p_{v_1 i}(k)d_{ki}}},\end{aligned}$$
$$\begin{aligned} &\frac{N_{v_2}}{P_{v_2}}=\sum_{i\in s_0}{p_{{v_2}i}(0)d_{0i}}+\sum_{i\in s_1}{p_{{v_2}i}(1)d_{1i}}+\sum_{k=2}^\infty{\sum_{i\in s_k}{p_{{v_2}i}(k)d_{ki}}},\end{aligned}$$
$$\begin{aligned} &\frac{N_{v_3}}{P_{v_3}}=\sum_{i\in s_0}{p_{{v_3}i}(0)d_{0i}}+\sum_{i\in s_1}{p_{{v_3}i}(1)d_{1i}}+\sum_{k=2}^\infty{\sum_{i\in s_k}{p_{{v_3}i}(k)d_{ki}}}.\end{aligned}$$

If $p_{xi}(k)$ is bounded by $p_{x}^L(k)\leq p_{xi}(k)\leq p_{x}^U(k)$, then $n_{k,x}$ is bounded by $\sum _{i\in s_k}{p_{x}^L(k)d_{ki}}\leq \frac {n_{k,x}}{P_x}=Q_{k,x}{\cdot} N=\sum _{i\in s_k}{p_{xi}(k)d_{ki}}\leq \sum _{i\in e_k}{p_{x}^U(k)d_{ki}}.$

Furthermore, we define $D_k=\begin {matrix} \sum _{i\in s_k}{d_{ki}} \end {matrix}$, hence $Q_{k,\mu }=\frac {p_{\mu }(k)D_k}{N}$. It is clear that $D_0$, $D_1$ and $D_2$ need to be bounded in order to get the bounds of $Q_{0,\mu }$, $Q_{1,\mu }$ and $Q_{2,\mu }$. From our previous work [29], the lower bounds of $D_0$, $D_1$ and $D_2$ can be got. So do the minimum value of $Q_{0,\mu }$, $Q_{1,\mu }$ and $Q_{2,\mu }$.

Based on Sect.2.2.1, the upper bounds of $Q_{1,\mu }$ and $Q_{2,\mu }$ is also necessary for calculating the lower bound of $e_2^Z$. So, $D_1$ and $D_2$ need to be maximized(details are in Appendix)

$$\begin{aligned}D_1^U&=\frac{p_{v_2}^U(0)\frac{N_{v_1}}{P_{v_1}}-p_{v_1}^L(0)\frac{N_{v_2}}{P_{v_2}}}{p_{v_2}^U(0)p_{v_1}^L(1)-p_{v_1}^L(0)p_{v_2}^U(1)}, \end{aligned}$$
$$\begin{aligned}D_2^U&=\frac{b_1[p_{v_2}^U(0)\frac{N_{v_1}}{P_{v_1}}-p_{v_1}^L(0)\frac{N_{v_2}}{P_{v_2}}]-b_2[p_{v_3}^L(0)\frac{N_{v_2}}{P_{v_2}}-p_{v_2}^U(0)\frac{N_{v_3}}{P_{v_3}}]} {b_1[p_{v_2}^U(0)p_{v_1}^L(2)-p_{v_1}^L(0)p_{v_2}^U(2)]-b_2[p_{v_3}^L(0)p_{v_2}^U(2)-p_{v_2}^U(0)p_{v_3}^L(2)]}, \end{aligned}$$
where $b_1=p_{v_3}^L(0)p_{v_2}^U(1)-p_{v_2}^U(0)p_{v_3}^L(1)$ and $b_2=p_{v_2}^U(0)p_{v_1}^L(1)-p_{v_1}^L(0)p_{v_2}^U(1)$.

Then the upper bounds of $Q_{1,\mu }$ and $Q_{2,\mu }$ can be obtained as

$$\begin{aligned}Q_{1,\mu}^U&=\frac{p_{\mu}^U(1)[p_{v_2}^U(0)Q_{v_1}-p_{v_1}^L(0)Q_{v_2}]}{p_{v_2}^U(0)p_{v_1}^L(1)-p_{v_1}^L(0)p_{v_2}^U(1)}, \end{aligned}$$
$$\begin{aligned}Q_{2,\mu}^U&=\frac{p_{\mu}^U(2)\big\{b_1[p_{v_2}^U(0)Q_{v_1}-p_{v_1}^L(0)Q_{v_2}]-b_2[p_{v_3}^L(0)Q_{v_2}-p_{v_2}^U(0)Q_{v_1}]\big\}} {b_1[p_{v_2}^U(0)p_{v_1}^L(2)-p_{v_1}^L(0)p_{v_2}^U(2)]-b_2[p_{v_3}^L(0)p_{v_2}^U(2)-p_{v_2}^U(0)p_{v_3}^L(2)]}. \end{aligned}$$

3.1.2 Bounds of $e_k^Z$

The overall quantum bit error rate is given by

$$N_x E_x=\sum_{k=0}^\infty{n_{k,x}e_k^Z}=\sum_{k=0}^\infty\sum_{i\in s_k}{P_x{\cdot} p_{xi}(k){\cdot} d_{ki}{\cdot} e_k^Z},$$
i.e.,
$$\begin{aligned} N_{\mu}E_{\mu}&=P_{\mu}[ \sum_{i \in s_0}p_{\mu i}(0)d_{0i}e_0^Z + \sum_{i \in s_1}p_{\mu i}(1)d_{1i}e_1^Z + \sum_{i \in s_2}p_{\mu i}(2)d_{2i}e_2^Z\\ &+\sum_{k=3}^\infty{\sum_{i \in s_k}p_{\mu i}(k)d_{ki}e_k^Z}], \end{aligned}$$
$$\begin{aligned} N_{v_1}E_{v_1}&=P_{v_1}[ \sum_{i \in s_0}p_{v_1i}(0)d_{0i}e_0^Z + \sum_{i \in s_1}p_{v_1i}(1)d_{1i}e_1^Z + \sum_{i \in s_2}p_{v_1i}(2)d_{2i}e_2^Z\\ &+\sum_{k=3}^\infty{\sum_{i \in s_k}p_{v_1i}(k)d_{ki}e_k^Z}], \end{aligned}$$
$$\begin{aligned} N_{v_2}E_{v_2}&=P_{v_2}[ \sum_{i \in s_0}p_{v_2i}(0)d_{0i}e_0^Z + \sum_{i \in s_1}p_{v_2i}(1)d_{1i}e_1^Z + \sum_{i \in s_2}p_{v_2i}(2)d_{2i}e_2^Z\\ &+\sum_{k=3}^\infty{\sum_{i \in s_k}p_{v_2i}(k)d_{ki}e_k^Z}], \end{aligned}$$
$$\begin{aligned} N_{v_3}E_{v_3}&=P_{v_3}[ \sum_{i \in s_0}p_{v_3i}(0)d_{0i}e_0^Z + \sum_{i \in s_1}p_{v_3i}(1)d_{1i}e_1^Z + \sum_{i \in s_2}p_{v_3i}(2)d_{2i}e_2^Z \\ &+\sum_{k=3}^\infty{\sum_{i \in s_k}p_{v_3i}(k)d_{ki}e_k^Z}].\end{aligned}$$

Previous work [29] has provided the upper bound of $e_1^Z$

$$(e_1^{Z})^U=\frac{[p_{v_2}^U(0)E_{v_1}Q_{v_1}-p_{v_1}^L(0)E_{v_2}Q_{v_2}]p_{\mu}^L(1)}{[p_{v_2}^U(0)p_{v_1}^L(1)-p_{v_1}^L(0)p_{v_2}^U(1)]Q_{1,\mu}^L},$$
but lower bounds of $e_1^Z$ and $e_2^Z$ are also needed. With Eq. (2932), we can minimize $e_1^Z$ and $e_2^Z$ (see the details in the Appendix):
$$(e_1^Z)^L=\frac{ \big\{[p_{v_2}^L(0)E_{v_1}Q_{v_1}-p_{v_1}^U(0)E_{v_2}Q_{v_2}]p_{\mu}^L(2)-a_1[E_{\mu}Q_{\mu}-Q_{0,\mu}^Le_0^Z]\big\}p_{\mu}^L(1) } { [a_3p_{\mu}^L(2)-a_1p_{\mu}^L(1)] Q_{1,\mu}^U},$$
$$\begin{aligned}(e_2^Z)^L&=\bigg\{ \Big\{ \big\{a_2[p_{v_2}^L(0)E_{v_1}Q_{v_1}-p_{v_1}^U(0)E_{v_2}Q_{v_2}] - a_3[p_{v_3}^U(0)E_{v_2}Q_{v_2}\\ &-p_{v_2}^L(0)E_{v_3}Q_{v_3}]\big\} p_{\mu}^L(3) - (a_2a_4-a_3a_5)(E_{\mu}Q_{\mu}-Q_{0,\mu}^Le_0^Z)\Big\}p_{\mu}^U(2)\bigg\}\\ &\bigg/ \bigg\{ \Big\{\big \{a_2a_1-a_3[p_{v_3}^U(0)p_{v_2}^L(2)-p_{v_2}^L(0)p_{v_3}^U(2)]\big\}p_{\mu}^L(3)-(a_2a_4\\ &-a_3a_5) p_{\mu}^L(2)\Big\}Q_{2,\mu}^U\bigg\}, \end{aligned}$$
where $a_1=p_{v_2}^L(0)p_{v_1}^U(2)-p_{v_1}^U(0)p_{v_2}^L(2)$, $a_2=p_{v_3}^U(0)p_{v_2}^L(1)-p_{v_2}^L(0)p_{v_3}^U(1)$, $a_3=p_{v_2}^L(0)p_{v_1}^U(1)-p_{v_1}^U(0)p_{v_2}^L(1)$,$a_4=p_{v_2}^L(0)p_{v_1}^U(3)-p_{v_1}^U(0)p_{v_2}^L(3)$ and $a_5=p_{v_3}^U(0)p_{v_2}^L(3)-p_{v_2}^L(0)p_{v_3}^U(3)$.

3.2 Numerical simulations

In this section, according to the above deductions, we present the SKR of our proposed protocol with source errors. The parameters in the simulation are the same listed in Table 1. Here we assume that the intensity fluctuation is the main factor of source error.

Considering the intensity fluctuation in practice, at any time $i$, the real intensity $x$ is not stable, that is, the intensity can be $x_i=x(1+\delta _{x_i})$ with $-\delta _x \leq \delta {x_i} \leq \delta _x$. For simplicity, we consider a situation in which the fluctuation bounds of different intensity $x(\mu ,v_1,v_2,v_3)$ are equal, i.e., $\delta _{\mu } = \delta _{v_1} = \delta _{v_2} = \delta _{v_3} =\delta$. This assumption is feasible if all pulses from Alice(Bob) are produced from a father pulse and randomly attenuates the pulse’s intensity to $x(\mu ,v_1,v_2,v_3)$. Since $p_{x}^L(k)\leq p_{xi}(k)\leq p_{x}^U(k)$, a series of bounds can be expressed by

$$\begin{aligned}p_{x}^U(0)&=p_{x(1-\delta)}(0)=e^{-x(1-\delta)},\\ p_{x}^L(0)&=p_{x(1+\delta)}(0)=e^{-x(1+\delta)},\end{aligned}$$
$$\begin{aligned} p_{x}^U(k)&=p_{x(1+\delta)}(k)=\frac{e^{-x(1+\delta)}(x(1+\delta))^k}{k!} \quad(k\ge 1),\\ p_{x}^L(k)&=p_{x(1-\delta)}(k)=\frac{e^{-x(1-\delta)}(x(1-\delta))^k}{k!} \quad(k\ge 1). \end{aligned}$$

Figure 3 depicts SKR of the proposed PM-QKD protocol with or without source errors, where the fluctuation of intensity is set as $\delta \in \{0.01,0.03,0.05\}$. The result shows that the proposed protocol without source errors can exceed the linear bound and perform better than that with source errors. The linear bound can obviously be exceeded when $\delta =0.01$ and $\delta =0.03$. And when the fluctuations of the source intensity $\delta$ is greater than $0.05$, the SKR performance is under the linear bound.

 

Fig. 3. Key rate comparison between MDI-QKD protocol, the proposed protocol with and without source error.

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Figure 4 demonstrates the key rate ratios $(R(\delta )/R(0))$ against transmission distance when $\delta =0.01, 0.03, 0.05$. Here, $R(\delta )$ represents the SKR with source errors while $R(0)$ means the SKR without source errors. From the figure, one can clearly see that the ratio decreases as transmission distance increases when $\delta$ is fixed. It is shown that source errors have a large impact on the protocol’s performance and the impact becomes heavier as $\delta$ grows.

Figure 5 shows that the key rate ratio against source error for different transmission distance. From Fig. 5, one can see the key rate ratios $(R(\delta )/R(0))$ against $\delta$ when the transmission distance $z$ is set up to 80km, 160km and 240km. The results also demonstrate that the longer transmission distance is, the smaller the key rate ratio is. It is indicated that there is a bigger influence on the SKR for a larger source error.

 

Fig. 4. Key rate ratio against transmission distance for different $\delta$.

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Fig. 5. Key rate ratio against $\delta$ for different transmission distance $z$.

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4. Conclusion

In this paper, we have proposed a four-intensity decoy-state PM-QKD protocol, and have demonstrated that its SKR performance has been close to that one with infinite-decoy-state. Our proposed protocol has a better performance in comparison with the new PM-QKD in [32] when the transmission distance is greater than 360km. At the same time, the statistical fluctuation analysis of the proposed protocol has been discussed. The key rate can beat the linear bound at 235 km under the condition where the data size $N \geq 1 \times 10^{15}$. We can use the four-intensity decoy-state one in practice. Furthermore, we have derived the formula of the bounds for the $k$-photon’s overall gain and the bit-error rate for signal state with some fluctuations in the state’s intensity, and have discussed the influence of source errors on the SKR performance of the four-intensity decoy-state PM-QKD protocol. The results have shown that the SKR performance of the proposed protocol can exceed the linear bound for a smaller source error, say $\delta \leq 0.03$, and there is a bigger influence on the SKR for a larger source error.

Appendix

Upper bounds of $D_1$,$D_2$

By applying $p_{v_2}^U(0)\times$Eq. (21)-$p_{v_1}^L(0)\times$Eq. (22), one can have

$$\begin{aligned} &p_{v_2}^U(0)\frac{N_{v_1}}{P_{v_1}}-p_{v_1}^L(0)\frac{N_{v_2}}{P_{v_2}}\\ &\ge \sum_{i\in s_0}[p_{v_2}^U(0)p_{v_1}^L(0)-p_{v_1}^L(0)p_{v_2}^U(0)]d_{0i} + \sum_{i\in s_1}[p_{v_2}^U(0)p_{v_1}^L(1)-p_{v_1}^L(0)p_{v_2}^U(1)]d_{1i}\\ &+ \sum_{k=2}^\infty\sum_{i\in s_k}[p_{v_2}^U(0)p_{v_1}^L(k)-p_{v_1}^L(0)p_{v_2}^U(k)]d_{ki}\\ &=[p_{v_2}^U(0)p_{v_1}^L(1)-p_{v_1}^L(0)p_{v_2}^U(1)]D_1 + \sum_{k=2}^\infty[p_{v_2}^U(0)p_{v_1}^L(k)-p_{v_1}^L(0)p_{v_2}^U(k)]D_k.\end{aligned}$$

And since the inequality $p_{v_2}^U(0)p_{v_1}^L(k)-p_{v_1}^L(0)p_{v_2}^U(k)\ge 0( k\ge 2)$ exists, Eq. (38) will become

$$p_{v_2}^U(0)\frac{N_{v_1}}{P_{v_1}}-p_{v_1}^l(0)\frac{N_{v_2}}{P_{v_2}} \ge [p_{v_2}^U(0)p_{v_1}^L(1)-p_{v_1}^L(0)p_{v_2}^U(1)]D_1,$$
then, the upper bound of $D_1$ can be obtained
$$D_1 \leq \frac{p_{v_2}^U(0)\frac{N_{v_1}}{P_{v_1}}-p_{v_1}^l(0)\frac{N_{v_2}}{P_{v_2}}}{p_{v_2}^U(0)p_{v_1}^L(1)-p_{v_1}^L(0)p_{v_2}^U(1)} = D_1^U.$$

Equation (38) can be further deduced as

$$\begin{aligned} &p_{v_2}^U(0)\frac{N_{v_1}}{P_{v_1}}-p_{v_1}^L(0)\frac{N_{v_2}}{P_{v_2}}\\ &\ge \sum_{i\in s_0}[p_{v_2}^U(0)p_{v_1}^L(0)-p_{v_1}^L(0)p_{v_2}^U(0)]d_{0i} + \sum_{i\in s_1}[p_{v_2}^U(0)p_{v_1}^L(1)-p_{v_1}^L(0)p_{v_2}^U(1)]d_{1i}\\ &+ \sum_{i\in s_2}[p_{v_2}^U(0)p_{v_1i}(2)-p_{v_1}^L(0)p_{v_2i}(2)]d_{2i} + \sum_{k=3}^\infty\sum_{i\in s_k}[p_{v_2}^U(0)p_{v_1}^L(k)-p_{v_1}^L(0)p_{v_2}^U(k)]d_{ki}\\ &=[p_{v_2}^U(0)p_{v_1}^L(1)-p_{v_1}^L(0)p_{v_2}^U(1)]D_1 + [p_{v_2}^U(0)p_{v_1}^L(2)-p_{v_1}^L(0)p_{v_2}^U(2)]D_2\\ &+ \sum_{k=3}^\infty\sum_{i\in s_k}[p_{v_2}^U(0)p_{v_1}^L(k)-p_{v_1}^L(0)p_{v_2}^U(k)]d_{ki}.\end{aligned}$$

Combine Eq. (41) with the equation below

$$\begin{aligned} &p_{v_3}^L(0) \times Eq.(22) - p_{v_2}^U(0) \times Eq.(23) = p_{v_3}^L(0) \frac{N_{v_2}}{P_{v_2}} - p_{v_2}^U(0) \frac{N_{v_3}}{P_{v_3}}\\ &\leq \sum_{i\in s_0}[p_{v_3}^L(0)p_{v_2}^U(0)- p_{v_2}^U(0)p_{v_3}^L(0)]d_{0i} + \sum_{i\in s_1}[p_{v_3}^L(0)p_{v_2}^U(1)- p_{v_2}^U(0)p_{v_3}^L(1)]d_{1i}\\ &+ \sum_{i\in s_2}[p_{v_3}^L(0)p_{v_2}^U(2)- p_{v_2}^U(0)p_{v_3}^L(2)]d_{2i}+ \sum_{k=3}^\infty\sum_{i\in s_k}[p_{v_3}^L(0)p_{v_2i}(k)- p_{v_2}^U(0)p_{v_3i}(k)]d_{ki}\\ &= [p_{v_3}^L(0)p_{v_2}^U(1)- p_{v_2}^U(0)p_{v_3}^L(1)]D_{1} + [p_{v_3}^L(0)p_{v_2}^U(2)- p_{v_2}^U(0)p_{v_3}^L(2)]D_{2}\\ &+\sum_{k=3}^\infty\sum_{i\in s_k}[p_{v_3}^L(0)p_{v_2i}(k)- p_{v_2}^U(0)p_{v_3i}(k)]d_{ki}.\end{aligned}$$
With $[p_{v_3}^L(0)p_{v_2}^U(1)-p_{v_2}^U(0)p_{v_3}^L(1)]\times$ Eq. (41)-$[p_{v_2}^U(0)p_{v_1}^L(1)-p_{v_1}^L(0)p_{v_2}^U(1)]\times$ Eq. (42)
$$\begin{aligned}&[p_{v_3}^L(0)p_{v_2}^U(1)-p_{v_2}^U(0)p_{v_3}^L(1)] [p_{v_2}^U(0)\frac{N_{v_1}}{P_{v_1}}-p_{v_1}^L(0)\frac{N_{v_2}}{P_{v_2}}]\\ &-[p_{v_2}^U(0)p_{v_1}^L(1)-p_{v_1}^L(0)p_{v_2}^U(1)][p_{v_3}^L(0)\frac{N_{v_2}}{P_{v_2}}-p_{v_2}^U(0) \frac{N_{v_3}}{P_{v_3}}]\\ &\ge\{[p_{v_3}^L(0)p_{v_2}^U(1)-p_{v_2}^U(0)p_{v_3}^L(1)][p_{v_2}^U(0)p_{v_1}^L(2)-p_{v_1}^L(0)p_{v_2}^U(2)]\\ &-[p_{v_2}^U(0)p_{v_1}^L(1)-p_{v_1}^L(0)p_{v_2}^U(1)][p_{v_3}^L(0)p_{v_2}^U(2)- p_{v_2}^U(0)p_{v_3}^L(2)]\}D_2\\ &+\sum_{k=3}^\infty \big\{[p_{v_3}^L(0)p_{v_2}^U(1)-p_{v_2}^U(0)p_{v_3}^L(1)][p_{v_2}^U(0)p_{v_1}^L(2)-p_{v_1}^L(0)p_{v_2}^U(2)]\\ &-[p_{v_2}^U(0)p_{v_1}^L(1)-p_{v_1}^L(0)p_{v_2}^U(1)][p_{v_3}^L(0)p_{v_2}^U(2)- p_{v_2}^U(0)p_{v_3}^L(2)]\big\}/p_{\mu}^U(k)\\ &{\cdot}\sum_{i\in s_k}p_{\mu i}(k), \end{aligned}$$
from which one can get the upper bound of $D_2$
$$\begin{aligned}D_2&\leq \bigg\{[p_{v_3}^L(0)p_{v_2}^U(1)-p_{v_2}^U(0)p_{v_3}^L(1)] [p_{v_2}^U(0)\frac{N_{v_1}}{P_{v_1}}-p_{v_1}^L(0)\frac{N_{v_2}}{P_{v_2}}]\\ &-[p_{v_2}^U(0)p_{v_1}^L(1)-p_{v_1}^L(0)p_{v_2}^U(1)] [p_{v_3}^L(0)\frac{N_{v_2}}{P_{v_2}}-p_{v_2}^U(0)\frac{N_{v_3}}{P_{v_3}}]\bigg\}\\ &\bigg/\bigg\{[p_{v_3}^L(0)p_{v_2}^U(1)-p_{v_2}^U(0)p_{v_3}^L(1)] [p_{v_2}^U(0)p_{v_1}^L(2)-p_{v_1}^L(0)p_{v_2}^U(2)]\\ &-[p_{v_2}^U(0)p_{v_1}^L(1)-p_{v_1}^L(0)p_{v_2}^U(1)] [p_{v_3}^L(0)p_{v_2}^U(2)-p_{v_2}^U(0)p_{v_3}^L(2)]\bigg\}\\ &=D_2^U. \end{aligned}$$

Lower bounds of $e_1^Z$,$e_2^Z$

With $p_{v_2}^L(0)\times$Eq. (30)- $p_{v_1}^U(0)\times$Eq. (31), one can get

$$\begin{aligned} &p_{v_2}^L(0) \frac{N_{v_1}E_{v_1}}{P_{v_1}} - p_{v_1}^U(0) \frac{N_{v_2}E_{v_2}}{P_{v_2}}\\ &\leq [p_{v_2}^L(0)p_{v_1}^U(0)-p_{v_1}^U(0)p_{v_2}^L(0)]D_0e_0^Z + [p_{v_2}^L(0)p_{v_1}^U(1)-p_{v_1}^U(0)p_{v_2}^L(1)]D_1e_1^Z\\ &+\sum_{k=2}^\infty \frac{p_{v_2}^L(0)p_{v_1}^U(k)-p_{v_1}^U(0)p_{v_2}^L(k)}{p_{\mu}^L(k)} \sum_{i\in s_k}p_{\mu i}(k)d_{ki}e_k^Z\\ &\leq [p_{v_2}^L(0)p_{v_1}^U(1)-p_{v_1}^U(0)p_{v_2}^L(1)]D_1e_1^Z\\ &+\frac{p_{v_2}^L(0)p_{v_1}^U(2)-p_{v_1}^U(0)p_{v_2}^L(2)}{p_{\mu}^L(2)}[\frac{N_{\mu}E_{\mu}}{P_{\mu}}-p_{\mu}(0)D_0e_0^Z-p_{\mu}(1)D_1e_1^Z].\end{aligned}$$

Thus, the lower bound of $e_1^Z$ can be got

$$\begin{aligned}e_1^Z&\ge \frac{p_{v_2}^L(0)\frac{N_{v_1}E_{v_1}}{P_{v_1}}-p_{v_1}^U(0)\frac{N_{v_2}E_{v_2}}{P_{v_2}} - \frac{p_{v_2}^L(0)p_{v_1}^U(2)-p_{v_1}^U(0)p_{v_2}^L(2)}{p_{\mu}^L(2)}[\frac{N_{\mu}E_{\mu}}{P_{\mu}}-p_{\mu}^L(0)D_0^Le_0^Z]} {p_{v_2}^L(0)p_{v_1}^U(1)-p_{v_1}^U(0)p_{v_2}^L(1)-\frac{[p_{v_2}^L(0)p_{v_1}^U(2)-p_{v_1}^U(0)p_{v_2}^L(2)]D_1^U}{p_{\mu}^L(2)}}\\ &=\bigg\{\big\{[p_{v_2}^L(0)\frac{N_{v_1}E_{v_1}}{P_{v_1}}-p_{v_1}^U(0)\frac{N_{v_2}E_{v_2}}{P_{v_2}}]p_{\mu}^L(2) - [p_{v_2}^L(0)p_{v_1}^U(2)-p_{v_1}^U(0)p_{v_2}^L(2)]\\ &{\cdot}[\frac{N_{\mu}E_{\mu}}{P_{\mu}}-p_{\mu}^L(0)D_0^Le_0^Z]\big\}p_{\mu}^U(1)\bigg\} \bigg/\bigg\{\big\{[p_{v_2}^L(0)p_{v_1}^U(1)-p_{v_1}^U(0)p_{v_2}^L(1)]p_{\mu}^L(2)\\ &-[p_{v_2}^L(0)p_{v_1}^U(2)-p_{v_1}^U(0)p_{v_2}^L(2)]p_{\mu}^L(1)\big\}D_1^Up_{\mu}^U(1)\bigg\}\\ &=\bigg\{ \big\{[p_{v_2}^L(0)E_{v_1}Q_{v_1}-p_{v_1}^U(0)E_{v_2}Q_{v_2}]p_{\mu}^L(2) - [p_{v_2}^L(0)p_{v_1}^U(2)-p_{v_1}^U(0)p_{v_2}^L(2)]\\ &{\cdot}[E_{\mu}Q_{\mu}-Q_{0,\mu}^Le_0^Z]\big\}p_{\mu}^U(1) \bigg\}\bigg/\bigg\{\big \{ [p_{v_2}^L(0)p_{v_1}^U(1)-p_{v_1}^U(0)p_{v_2}^L(1)]p_{\mu}^L(2)\\ &-[p_{v_2}^L(0)p_{v_1}^U(2)-p_{v_1}^U(0)p_{v_2}^L(2)]p_{\mu}^L(1)\big\}Q_{1,\mu}^U\bigg\}\\ &=(e_1^Z)^L. \end{aligned}$$

Equation (45) can be further deduced as

$$\begin{aligned} &p_{v_2}^L(0) \frac{N_{v_1}E_{v_1}}{P_{v_1}} - p_{v_1}^U(0) \frac{N_{v_2}E_{v_2}}{P_{v_2}}\\ &\leq[p_{v_2}^L(0)p_{v_1}^U(0)-p_{v_1}^U(0)p_{v_2}^L(0)]D_0e_0^Z + [p_{v_2}^L(0)p_{v_1}^U(1)-p_{v_1}^U(0)p_{v_2}^L(1)]D_1e_1^Z\\ &+[p_{v_2}^L(0)p_{v_1}^U(2)-p_{v_1}^U(0)p_{v_2}^L(2)]D_2e_2^Z +\sum_{k=3}\sum_{i\in s_k}[p_{v_2}^L(0)p_{v_1i}(k)-p_{v_1}^U(0)p_{v_2i}(k)]d_{ki}e_k^Z\\ &=[p_{v_2}^L(0)p_{v_1}^U(1)-p_{v_1}^U(0)p_{v_2}^L(1)]D_1e_1^Z + [p_{v_2}^L(0)p_{v_1}^U(2)-p_{v_1}^U(0)p_{v_2}^L(2)]D_2e_2^Z\\ &+\sum_{k=3}\sum_{i\in s_k}[p_{v_2}^L(0)p_{v_1i}(k)-p_{v_1}^U(0)p_{v_2i}(k)]d_{ki}e_k^Z.\end{aligned}$$

Combine Eq. (47) with the equation below:

$$\begin{aligned} &p_{v_3}^U(0) \frac{N_{v_2}E_{v_2}}{P_{v_2}} - p_{v_2}^L(0) \frac{N_{v_3}E_{v_3}}{P_{v_3}}\\ &\ge[p_{v_3}^U(0)p_{v_2}^L(0)-p_{v_1}^L(0)p_{v_3}^U(0)]D_0e_0^Z + [p_{v_3}^U(0)p_{v_2}^L(1)-p_{v_2}^L(0)p_{v_3}^U(1)]D_1e_1^Z\\ &+[p_{v_3}^U(0)p_{v_2}^L(2)-p_{v_2}^L(0)p_{v_3}^U(2)]D_2e_2^Z + \sum_{k=3}\sum_{i\in s_k}[p_{v_3}^U(0)p_{v_2i}(k)-p_{v_2}^L(0)p_{v_3i}(k)]d_{ki}e_k^Z\\ &=[p_{v_3}^U(0)p_{v_2}^L(1)-p_{v_2}^L(0)p_{v_3}^U(1)]D_1e_1^Z + [p_{v_3}^U(0)p_{v_2}^L(2)-p_{v_2}^L(0)p_{v_3}^U(2)]D_2e_2^Z\\ &+\sum_{k=3}\sum_{i\in s_k}[p_{v_3}^U(0)p_{v_2i}(k)-p_{v_2}^L(0)p_{v_3i}(k)]d_{ki}e_k^Z. \end{aligned}$$
With $[p_{v_3}^U(0)p_{v_2}^L(1)-p_{v_2}^L(0)p_{v_3}^U(1)]\times$Eq. (47) - $[p_{v_2}^L(0)p_{v_1}^U(1)-p_{v_1}^U(0)p_{v_2}^L(1)]\times$Eq. (48):
$$\begin{aligned}&[p_{v_3}^U(0)p_{v_2}^L(1)-p_{v_2}^L(0)p_{v_3}^U(1)][p_{v_2}^L(0)\frac{N_{v_1}E_{v_1}}{P_{v_1}}-p_{v_1}^U(0)\frac{N_{v_2}E_{v_2}}{P_{v_2}}]\\ &-[p_{v_2}^L(0)p_{v_1}^U(1)-p_{v_1}^U(0)p_{v_2}^L(1)] [p_{v_3}^U(0)\frac{N_{v_2}E_{v_2}}{P_{v_2}}-p_{v_2}^L(0)\frac{N_{v_3}E_{v_3}}{P_{v_3}}]\\ &\leq \{[p_{v_3}^U(0)p_{v_2}^L(1)-p_{v_2}^L(0)p_{v_3}^U(1)][p_{v_2}^L(0)p_{v_1}^U(2)-p_{v_1}^U(0)p_{v_2}^L(2)]\\ &-[p_{v_2}^L(0)p_{v_1}^U(1)-p_{v_1}^U(0)p_{v_2}^L(1)][p_{v_3}^U(0)p_{v_2}^L(2)-p_{v_2}^L(0)p_{v_3}^U(2)]\}D_2e_2^Z\\ &+ \Big\{[p_{v_3}^U(0)p_{v_2}^L(1)-p_{v_2}^L(0)p_{v_3}^U(1)][p_{v_2}^L(0)p_{v_1}^U(3)-p_{v_1}^U(0)p_{v_2}^L(3)]\\ &- [p_{v_2}^L(0)p_{v_1}^U(1)-p_{v_1}^U(0)p_{v_2}^L(1)][p_{v_3}^U(0)p_{v_2}^L(3)-p_{v_2}^L(0)p_{v_3}^U(3)]\Big\}\Big/p_{\mu}^L(3)\\ &{\cdot} [\frac{N_{\mu}E_{\mu}}{P_{\mu}} - p_{\mu}^L(0)D_0^Le_0^Z - p_{\mu}^L(1)D_1^L(e_1^Z)^L - p_{\mu}^L(2)D_2e_2^Z], \end{aligned}$$
one can finally calculate the lower bound of $e_2^Z$
$$\begin{aligned}e_2^Z&\ge \bigg\{\big\{a_2[p_{v_2}^L(0)\frac{N_{v_1}E_{v_1}}{P_{v_1}} - p_{v_1}^U(0)\frac{N_{v_2}E_{v_2}}{P_{v_2}}] - a_3[p_{v_3}^U(0)\frac{N_{v_2}E_{v_2}}{P_{v_2}}-p_{v_2}^L(0)\frac{N_{v_3}E_{v_3}}{P_{v_3}}]\big\}\\ &{\cdot}p_{\mu}^L(3) - (a_2a_4-a_3a_5){\cdot} [\frac{N_{\mu}E_{\mu}}{P_{\mu}} - p_{\mu}^L(0)D_0^Le_0^Z - p_{\mu}^L(1)D_1^L(e_1^Z)^L]\bigg\} \bigg/\bigg\{D_2^U\\ &{\cdot}\Big\{\big\{a_2a_1-a_3[p_{v_3}^U(0)p_{v_2}^L(2)-p_{v_2}^L(0)p_{v_3}^U(2)]\big\}p_{\mu}^L(3)-(a_2a_4-a_3a_5)p_{\mu}^L(3)\Big\}\bigg\}\\ &=\bigg\{p_{\mu}^U(2)\Big\{\big\{a_2[p_{v_2}^L(0)\frac{N_{v_1}E_{v_1}}{P_{v_1}} - p_{v_1}^U(0)\frac{N_{v_2}E_{v_2}}{P_{v_2}}] - a_3[p_{v_3}^U(0)\frac{N_{v_2}E_{v_2}}{P_{v_2}}-p_{v_2}^L(0)\\ &{\cdot}\frac{N_{v_3}E_{v_3}}{P_{v_3}}]\big\}p_{\mu}^L(3) - (a_2a_4-a_3a_5) [\frac{N_{\mu}E_{\mu}}{P_{\mu}} - p_{\mu}^L(0)D_0^Le_0^Z - p_{\mu}^L(1)D_1^L(e_1^Z)^L]\Big\}\bigg\}\\ &\bigg/\bigg\{p_{\mu}^U(2)D_2^U \Big\{\big\{a_2a_1-a_3[p_{v_3}^U(0)p_{v_2}^L(2)-p_{v_2}^L(0)p_{v_3}^U(2)]\big\}p_{\mu}^L(3)\\ &-(a_2a_4-a_3a_5)p_{\mu}^L(3)\Big\}\bigg\}\\ &=\bigg\{ \Big\{ \big\{a_2[p_{v_2}^L(0)E_{v_1}Q_{v_1}-p_{v_1}^U(0)E_{v_2}Q_{v_2}] - a_3[p_{v_3}^U(0)E_{v_2}Q_{v_2}-p_{v_2}^L(0)E_{v_3}Q_{v_3}]\big\}\\ &{\cdot}p_{\mu}^L(3) - (a_2a_4-a_3a_5)(E_{\mu}Q_{\mu}-Q_{0,\mu}^Le_0^Z)\Big\}p_{\mu}^U(2)\bigg\}\bigg/\bigg\{ \Big\{\big \{a_2a_1-a_3\\ &{\cdot}[p_{v_3}^U(0)p_{v_2}^L(2)-p_{v_2}^L(0)p_{v_3}^U(2)]\big\}p_{\mu}^L(3)-(a_2a_4-a_3a_5)p_{\mu}^L(2)\Big\}Q_{2,\mu}^U\bigg\}\\ &=(e_2^Z)^L, \end{aligned}$$
where $a_1=p_{v_2}^L(0)p_{v_1}^U(2)-p_{v_1}^U(0)p_{v_2}^L(2)$, $a_2=p_{v_3}^U(0)p_{v_2}^L(1)-p_{v_2}^L(0)p_{v_3}^U(1)$, $a_3=p_{v_2}^L(0)p_{v_1}^U(1)-p_{v_1}^U(0)p_{v_2}^L(1)$,$a_4=p_{v_2}^L(0)p_{v_1}^U(3)-p_{v_1}^U(0)p_{v_2}^L(3)$ and $a_5=p_{v_3}^U(0)p_{v_2}^L(3)-p_{v_2}^L(0)p_{v_3}^U(3)$.

Funding

National Natural Science Foundation of China (11847062, 61871234).

Acknowledgments

The work is supported by the National Natural Science Foundation of China (61871234), and Postgraduate Research & Practice Innovation Program of Jiangsu Province (SJCX19_0251).

Disclosures

The authors declare that there are no conflicts of interest related to this paper.

References

1. H. K. Lo and H. F. Chau, “Unconditional Security of Quantum Key Distribution over Arbitrarily Long Distances,” Science 283(5410), 2050–2056 (1999). [CrossRef]  

2. D. Mayers, “Unconditional security in quantum cryptography,” J. Assoc. Comput. Mach. 48(3), 351–406 (2001). [CrossRef]  

3. W. Y. Hwang, “Quantum Key Distribution with High Loss: Toward Global Secure Communication,” Phys. Rev. Lett. 91(5), 057901 (2003). [CrossRef]  

4. X. B. Wang, “Beating the Photon-Number-Splitting Attack in Practical Quantum Cryptography,” Phys. Rev. Lett. 94(23), 230503 (2005). [CrossRef]  

5. X. B. Wang, “Decoy-state protocol for quantum cryptography with four different intensities of coherent light,” Phys. Rev. A 72(1), 012322 (2005). [CrossRef]  

6. H. K. Lo, M. Curty, and B. Qi, “Measurement-Device-Independent Quantum Key Distribution,” Phys. Rev. Lett. 108(13), 130503 (2012). [CrossRef]  

7. L. Wang, S. M. Zhao, L. Y. Gong, and W. W. Cheng, “Free-space measurement-device-independent quantum-key-distribution protocol using decoy states with orbital angular momentum,” Chin. Phys. B 24(12), 120307 (2015). [CrossRef]  

8. H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016). [CrossRef]  

9. T. Sasaki, Y. Yamamoto, and M. Koashi, “Practical quantum key distribution protocol without monitoring signal disturbance,” Nature 509(7501), 475–478 (2014). [CrossRef]  

10. L. Wang and S. Zhao, “Round-robin differential-phase-shift quantum key distribution with heralded pair-coherent sources,” Quantum Inf. Process. 16(4), 100 (2017). [CrossRef]  

11. Q. P. Mao, L. Wang, and S. M. Zhao, “Plug-and-play round-robin differential phase-shift quantum key distribution,” Sci. Rep. 7(1), 15435 (2017). [CrossRef]  

12. M. Takeoka, S. Guha, and M. M. Wilde, “Fundamental rate-loss tradeoff for optical quantum key distribution,” Nat. Commun. 5(1), 5235 (2014). [CrossRef]  

13. S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental limits of repeaterless quantum communications,” Nat. Commun. 8(1), 15043 (2017). [CrossRef]  

14. M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate-distance limit of quantum key distribution without quantum repeaters,” Nature 557(7705), 400–403 (2018). [CrossRef]  

15. X. Ma, P. Zeng, and H. Zhou, “Phase-Matching Quantum Key Distribution,” Phys. Rev. X 8(3), 031043 (2018). [CrossRef]  

16. W. Li, L. Wang, and S. Zhao, “Phase-Matching Quantum Key Distribution based on Single-Photon Entanglement,” Sci. Rep. 9(1), 15466 (2019). [CrossRef]  

17. X. B. Wang, Z. W. Yu, and X. L. Hu, “Twin-field quantum key distribution with large misalignment error,” Phys. Rev. A 98(6), 062323 (2018). [CrossRef]  

18. Z. W. Yu, X. L. Hu, C. Jiang, H. Xu, and X. B. Wang, “Sending-or-not-sending twin-field quantum key distribution in practice,” Sci. Rep. 9(1), 3080 (2019). [CrossRef]  

19. C. Cui, Z. Q. Yin, R. Wang, W. Chen, S. Wang, G. C. Guo, and Z. F. Han, “Twin-Field Quantum Key Distribution without Phase Postselection,” Phys. Rev. Appl. 11(3), 034053 (2019). [CrossRef]  

20. M. Curty, K. Azuma, and H. K. Lo, “Simple security proof of twin-field type quantum key distribution protocol,” npj Quantum Inf. 5(1), 64 (2019). [CrossRef]  

21. X. B. Wang, C. Z. Peng, J. Zhang, L. Yang, and J. W. Pan, “General theory of decoy-state quantum cryptography with source errors,” Phys. Rev. A 77(4), 042311 (2008). [CrossRef]  

22. X. B. Wang, L. Yang, C. Z. Peng, and J. W. Pan, “Decoy-state quantum key distribution with both source errors and statistical fluctuations,” New J. Phys. 11(7), 075006 (2009). [CrossRef]  

23. S. Wang, S. L. Zhang, H. W. Li, Z. Q. Yin, Y. B. Zhao, W. Chen, Z. F. Han, and G. C. Guo, “Decoy-state theory for the heralded single-photon source with intensity fluctuations,” Phys. Rev. A 79(6), 062309 (2009). [CrossRef]  

24. C. Zhou, W. S. Bao, and X. Q. Fu, “Decoy-state quantum key distribution for the heralded pair coherent state photon source with intensity fluctuations,” Sci. China Inf. Sci. 53(12), 2485–2494 (2010). [CrossRef]  

25. K. Liu, J. Li, J. R. Zhu, C. M. Zhang, and Q. Wang, “Decoy-state reference-frame-independent quantum key distribution with both source errors and statistical fluctuations,” Chin. Phys. B 26(12), 120302 (2017). [CrossRef]  

26. Q. Wang and X. B. Wang, “Simulating of the measurement-device independent quantum key distribution with phase randomized general sources,” Sci. Rep. 4(1), 4612 (2015). [CrossRef]  

27. C. Jiang, Z. W. Yu, and X. B. Wang, “Measurement-device-independent quantum key distribution with source state errors in photon number space,” Phys. Rev. A 94(6), 062323 (2016). [CrossRef]  

28. C. Jiang, Z. W. Yu, and X. B. Wang, “Measurement-device-independent quantum key distribution with source state errors in photon number space,” Phys. Rev. A 94(6), 062323 (2016). [CrossRef]  

29. Q. P. Mao, L. Wang, and S. M. Zhao, “Decoy-state round-robin differential-phase-shift quantum key distribution with source errors,” Quantum Inf. Process. 19(2), 56 (2020). [CrossRef]  

30. Y. Y. Zhang, W. S. Bao, C. Zhou, H. W. Li, Y. Wang, and M. S. Jiang, “Practical round-robin differential phase-shift quantum key distribution,” Opt. Express 24(18), 20763–20773 (2016). [CrossRef]  

31. X. Ma, B. Qi, Y. Zhao, and H. K. Lo, “Practical decoy state for quantum key distribution,” Phys. Rev. A 72(1), 012326 (2005). [CrossRef]  

32. P. Zeng, W. Wu, and X. Ma, “Symmetry-Protected Privacy: Beating the Rate-Distance Linear Bound Over a Noisy Channel,” Phys. Rev. Appl. 13(6), 064013 (2020). [CrossRef]  

References

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  • |

  1. H. K. Lo and H. F. Chau, “Unconditional Security of Quantum Key Distribution over Arbitrarily Long Distances,” Science 283(5410), 2050–2056 (1999).
    [Crossref]
  2. D. Mayers, “Unconditional security in quantum cryptography,” J. Assoc. Comput. Mach. 48(3), 351–406 (2001).
    [Crossref]
  3. W. Y. Hwang, “Quantum Key Distribution with High Loss: Toward Global Secure Communication,” Phys. Rev. Lett. 91(5), 057901 (2003).
    [Crossref]
  4. X. B. Wang, “Beating the Photon-Number-Splitting Attack in Practical Quantum Cryptography,” Phys. Rev. Lett. 94(23), 230503 (2005).
    [Crossref]
  5. X. B. Wang, “Decoy-state protocol for quantum cryptography with four different intensities of coherent light,” Phys. Rev. A 72(1), 012322 (2005).
    [Crossref]
  6. H. K. Lo, M. Curty, and B. Qi, “Measurement-Device-Independent Quantum Key Distribution,” Phys. Rev. Lett. 108(13), 130503 (2012).
    [Crossref]
  7. L. Wang, S. M. Zhao, L. Y. Gong, and W. W. Cheng, “Free-space measurement-device-independent quantum-key-distribution protocol using decoy states with orbital angular momentum,” Chin. Phys. B 24(12), 120307 (2015).
    [Crossref]
  8. H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
    [Crossref]
  9. T. Sasaki, Y. Yamamoto, and M. Koashi, “Practical quantum key distribution protocol without monitoring signal disturbance,” Nature 509(7501), 475–478 (2014).
    [Crossref]
  10. L. Wang and S. Zhao, “Round-robin differential-phase-shift quantum key distribution with heralded pair-coherent sources,” Quantum Inf. Process. 16(4), 100 (2017).
    [Crossref]
  11. Q. P. Mao, L. Wang, and S. M. Zhao, “Plug-and-play round-robin differential phase-shift quantum key distribution,” Sci. Rep. 7(1), 15435 (2017).
    [Crossref]
  12. M. Takeoka, S. Guha, and M. M. Wilde, “Fundamental rate-loss tradeoff for optical quantum key distribution,” Nat. Commun. 5(1), 5235 (2014).
    [Crossref]
  13. S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental limits of repeaterless quantum communications,” Nat. Commun. 8(1), 15043 (2017).
    [Crossref]
  14. M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate-distance limit of quantum key distribution without quantum repeaters,” Nature 557(7705), 400–403 (2018).
    [Crossref]
  15. X. Ma, P. Zeng, and H. Zhou, “Phase-Matching Quantum Key Distribution,” Phys. Rev. X 8(3), 031043 (2018).
    [Crossref]
  16. W. Li, L. Wang, and S. Zhao, “Phase-Matching Quantum Key Distribution based on Single-Photon Entanglement,” Sci. Rep. 9(1), 15466 (2019).
    [Crossref]
  17. X. B. Wang, Z. W. Yu, and X. L. Hu, “Twin-field quantum key distribution with large misalignment error,” Phys. Rev. A 98(6), 062323 (2018).
    [Crossref]
  18. Z. W. Yu, X. L. Hu, C. Jiang, H. Xu, and X. B. Wang, “Sending-or-not-sending twin-field quantum key distribution in practice,” Sci. Rep. 9(1), 3080 (2019).
    [Crossref]
  19. C. Cui, Z. Q. Yin, R. Wang, W. Chen, S. Wang, G. C. Guo, and Z. F. Han, “Twin-Field Quantum Key Distribution without Phase Postselection,” Phys. Rev. Appl. 11(3), 034053 (2019).
    [Crossref]
  20. M. Curty, K. Azuma, and H. K. Lo, “Simple security proof of twin-field type quantum key distribution protocol,” npj Quantum Inf. 5(1), 64 (2019).
    [Crossref]
  21. X. B. Wang, C. Z. Peng, J. Zhang, L. Yang, and J. W. Pan, “General theory of decoy-state quantum cryptography with source errors,” Phys. Rev. A 77(4), 042311 (2008).
    [Crossref]
  22. X. B. Wang, L. Yang, C. Z. Peng, and J. W. Pan, “Decoy-state quantum key distribution with both source errors and statistical fluctuations,” New J. Phys. 11(7), 075006 (2009).
    [Crossref]
  23. S. Wang, S. L. Zhang, H. W. Li, Z. Q. Yin, Y. B. Zhao, W. Chen, Z. F. Han, and G. C. Guo, “Decoy-state theory for the heralded single-photon source with intensity fluctuations,” Phys. Rev. A 79(6), 062309 (2009).
    [Crossref]
  24. C. Zhou, W. S. Bao, and X. Q. Fu, “Decoy-state quantum key distribution for the heralded pair coherent state photon source with intensity fluctuations,” Sci. China Inf. Sci. 53(12), 2485–2494 (2010).
    [Crossref]
  25. K. Liu, J. Li, J. R. Zhu, C. M. Zhang, and Q. Wang, “Decoy-state reference-frame-independent quantum key distribution with both source errors and statistical fluctuations,” Chin. Phys. B 26(12), 120302 (2017).
    [Crossref]
  26. Q. Wang and X. B. Wang, “Simulating of the measurement-device independent quantum key distribution with phase randomized general sources,” Sci. Rep. 4(1), 4612 (2015).
    [Crossref]
  27. C. Jiang, Z. W. Yu, and X. B. Wang, “Measurement-device-independent quantum key distribution with source state errors in photon number space,” Phys. Rev. A 94(6), 062323 (2016).
    [Crossref]
  28. C. Jiang, Z. W. Yu, and X. B. Wang, “Measurement-device-independent quantum key distribution with source state errors in photon number space,” Phys. Rev. A 94(6), 062323 (2016).
    [Crossref]
  29. Q. P. Mao, L. Wang, and S. M. Zhao, “Decoy-state round-robin differential-phase-shift quantum key distribution with source errors,” Quantum Inf. Process. 19(2), 56 (2020).
    [Crossref]
  30. Y. Y. Zhang, W. S. Bao, C. Zhou, H. W. Li, Y. Wang, and M. S. Jiang, “Practical round-robin differential phase-shift quantum key distribution,” Opt. Express 24(18), 20763–20773 (2016).
    [Crossref]
  31. X. Ma, B. Qi, Y. Zhao, and H. K. Lo, “Practical decoy state for quantum key distribution,” Phys. Rev. A 72(1), 012326 (2005).
    [Crossref]
  32. P. Zeng, W. Wu, and X. Ma, “Symmetry-Protected Privacy: Beating the Rate-Distance Linear Bound Over a Noisy Channel,” Phys. Rev. Appl. 13(6), 064013 (2020).
    [Crossref]

2020 (2)

Q. P. Mao, L. Wang, and S. M. Zhao, “Decoy-state round-robin differential-phase-shift quantum key distribution with source errors,” Quantum Inf. Process. 19(2), 56 (2020).
[Crossref]

P. Zeng, W. Wu, and X. Ma, “Symmetry-Protected Privacy: Beating the Rate-Distance Linear Bound Over a Noisy Channel,” Phys. Rev. Appl. 13(6), 064013 (2020).
[Crossref]

2019 (4)

Z. W. Yu, X. L. Hu, C. Jiang, H. Xu, and X. B. Wang, “Sending-or-not-sending twin-field quantum key distribution in practice,” Sci. Rep. 9(1), 3080 (2019).
[Crossref]

C. Cui, Z. Q. Yin, R. Wang, W. Chen, S. Wang, G. C. Guo, and Z. F. Han, “Twin-Field Quantum Key Distribution without Phase Postselection,” Phys. Rev. Appl. 11(3), 034053 (2019).
[Crossref]

M. Curty, K. Azuma, and H. K. Lo, “Simple security proof of twin-field type quantum key distribution protocol,” npj Quantum Inf. 5(1), 64 (2019).
[Crossref]

W. Li, L. Wang, and S. Zhao, “Phase-Matching Quantum Key Distribution based on Single-Photon Entanglement,” Sci. Rep. 9(1), 15466 (2019).
[Crossref]

2018 (3)

X. B. Wang, Z. W. Yu, and X. L. Hu, “Twin-field quantum key distribution with large misalignment error,” Phys. Rev. A 98(6), 062323 (2018).
[Crossref]

M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate-distance limit of quantum key distribution without quantum repeaters,” Nature 557(7705), 400–403 (2018).
[Crossref]

X. Ma, P. Zeng, and H. Zhou, “Phase-Matching Quantum Key Distribution,” Phys. Rev. X 8(3), 031043 (2018).
[Crossref]

2017 (4)

L. Wang and S. Zhao, “Round-robin differential-phase-shift quantum key distribution with heralded pair-coherent sources,” Quantum Inf. Process. 16(4), 100 (2017).
[Crossref]

Q. P. Mao, L. Wang, and S. M. Zhao, “Plug-and-play round-robin differential phase-shift quantum key distribution,” Sci. Rep. 7(1), 15435 (2017).
[Crossref]

S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental limits of repeaterless quantum communications,” Nat. Commun. 8(1), 15043 (2017).
[Crossref]

K. Liu, J. Li, J. R. Zhu, C. M. Zhang, and Q. Wang, “Decoy-state reference-frame-independent quantum key distribution with both source errors and statistical fluctuations,” Chin. Phys. B 26(12), 120302 (2017).
[Crossref]

2016 (4)

Y. Y. Zhang, W. S. Bao, C. Zhou, H. W. Li, Y. Wang, and M. S. Jiang, “Practical round-robin differential phase-shift quantum key distribution,” Opt. Express 24(18), 20763–20773 (2016).
[Crossref]

C. Jiang, Z. W. Yu, and X. B. Wang, “Measurement-device-independent quantum key distribution with source state errors in photon number space,” Phys. Rev. A 94(6), 062323 (2016).
[Crossref]

C. Jiang, Z. W. Yu, and X. B. Wang, “Measurement-device-independent quantum key distribution with source state errors in photon number space,” Phys. Rev. A 94(6), 062323 (2016).
[Crossref]

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

2015 (2)

L. Wang, S. M. Zhao, L. Y. Gong, and W. W. Cheng, “Free-space measurement-device-independent quantum-key-distribution protocol using decoy states with orbital angular momentum,” Chin. Phys. B 24(12), 120307 (2015).
[Crossref]

Q. Wang and X. B. Wang, “Simulating of the measurement-device independent quantum key distribution with phase randomized general sources,” Sci. Rep. 4(1), 4612 (2015).
[Crossref]

2014 (2)

M. Takeoka, S. Guha, and M. M. Wilde, “Fundamental rate-loss tradeoff for optical quantum key distribution,” Nat. Commun. 5(1), 5235 (2014).
[Crossref]

T. Sasaki, Y. Yamamoto, and M. Koashi, “Practical quantum key distribution protocol without monitoring signal disturbance,” Nature 509(7501), 475–478 (2014).
[Crossref]

2012 (1)

H. K. Lo, M. Curty, and B. Qi, “Measurement-Device-Independent Quantum Key Distribution,” Phys. Rev. Lett. 108(13), 130503 (2012).
[Crossref]

2010 (1)

C. Zhou, W. S. Bao, and X. Q. Fu, “Decoy-state quantum key distribution for the heralded pair coherent state photon source with intensity fluctuations,” Sci. China Inf. Sci. 53(12), 2485–2494 (2010).
[Crossref]

2009 (2)

X. B. Wang, L. Yang, C. Z. Peng, and J. W. Pan, “Decoy-state quantum key distribution with both source errors and statistical fluctuations,” New J. Phys. 11(7), 075006 (2009).
[Crossref]

S. Wang, S. L. Zhang, H. W. Li, Z. Q. Yin, Y. B. Zhao, W. Chen, Z. F. Han, and G. C. Guo, “Decoy-state theory for the heralded single-photon source with intensity fluctuations,” Phys. Rev. A 79(6), 062309 (2009).
[Crossref]

2008 (1)

X. B. Wang, C. Z. Peng, J. Zhang, L. Yang, and J. W. Pan, “General theory of decoy-state quantum cryptography with source errors,” Phys. Rev. A 77(4), 042311 (2008).
[Crossref]

2005 (3)

X. Ma, B. Qi, Y. Zhao, and H. K. Lo, “Practical decoy state for quantum key distribution,” Phys. Rev. A 72(1), 012326 (2005).
[Crossref]

X. B. Wang, “Beating the Photon-Number-Splitting Attack in Practical Quantum Cryptography,” Phys. Rev. Lett. 94(23), 230503 (2005).
[Crossref]

X. B. Wang, “Decoy-state protocol for quantum cryptography with four different intensities of coherent light,” Phys. Rev. A 72(1), 012322 (2005).
[Crossref]

2003 (1)

W. Y. Hwang, “Quantum Key Distribution with High Loss: Toward Global Secure Communication,” Phys. Rev. Lett. 91(5), 057901 (2003).
[Crossref]

2001 (1)

D. Mayers, “Unconditional security in quantum cryptography,” J. Assoc. Comput. Mach. 48(3), 351–406 (2001).
[Crossref]

1999 (1)

H. K. Lo and H. F. Chau, “Unconditional Security of Quantum Key Distribution over Arbitrarily Long Distances,” Science 283(5410), 2050–2056 (1999).
[Crossref]

Azuma, K.

M. Curty, K. Azuma, and H. K. Lo, “Simple security proof of twin-field type quantum key distribution protocol,” npj Quantum Inf. 5(1), 64 (2019).
[Crossref]

Banchi, L.

S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental limits of repeaterless quantum communications,” Nat. Commun. 8(1), 15043 (2017).
[Crossref]

Bao, W. S.

Y. Y. Zhang, W. S. Bao, C. Zhou, H. W. Li, Y. Wang, and M. S. Jiang, “Practical round-robin differential phase-shift quantum key distribution,” Opt. Express 24(18), 20763–20773 (2016).
[Crossref]

C. Zhou, W. S. Bao, and X. Q. Fu, “Decoy-state quantum key distribution for the heralded pair coherent state photon source with intensity fluctuations,” Sci. China Inf. Sci. 53(12), 2485–2494 (2010).
[Crossref]

Chau, H. F.

H. K. Lo and H. F. Chau, “Unconditional Security of Quantum Key Distribution over Arbitrarily Long Distances,” Science 283(5410), 2050–2056 (1999).
[Crossref]

Chen, H.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

Chen, S. J.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

Chen, T. Y.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

Chen, W.

C. Cui, Z. Q. Yin, R. Wang, W. Chen, S. Wang, G. C. Guo, and Z. F. Han, “Twin-Field Quantum Key Distribution without Phase Postselection,” Phys. Rev. Appl. 11(3), 034053 (2019).
[Crossref]

S. Wang, S. L. Zhang, H. W. Li, Z. Q. Yin, Y. B. Zhao, W. Chen, Z. F. Han, and G. C. Guo, “Decoy-state theory for the heralded single-photon source with intensity fluctuations,” Phys. Rev. A 79(6), 062309 (2009).
[Crossref]

Cheng, W. W.

L. Wang, S. M. Zhao, L. Y. Gong, and W. W. Cheng, “Free-space measurement-device-independent quantum-key-distribution protocol using decoy states with orbital angular momentum,” Chin. Phys. B 24(12), 120307 (2015).
[Crossref]

Cui, C.

C. Cui, Z. Q. Yin, R. Wang, W. Chen, S. Wang, G. C. Guo, and Z. F. Han, “Twin-Field Quantum Key Distribution without Phase Postselection,” Phys. Rev. Appl. 11(3), 034053 (2019).
[Crossref]

Curty, M.

M. Curty, K. Azuma, and H. K. Lo, “Simple security proof of twin-field type quantum key distribution protocol,” npj Quantum Inf. 5(1), 64 (2019).
[Crossref]

H. K. Lo, M. Curty, and B. Qi, “Measurement-Device-Independent Quantum Key Distribution,” Phys. Rev. Lett. 108(13), 130503 (2012).
[Crossref]

Dynes, J. F.

M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate-distance limit of quantum key distribution without quantum repeaters,” Nature 557(7705), 400–403 (2018).
[Crossref]

Fu, X. Q.

C. Zhou, W. S. Bao, and X. Q. Fu, “Decoy-state quantum key distribution for the heralded pair coherent state photon source with intensity fluctuations,” Sci. China Inf. Sci. 53(12), 2485–2494 (2010).
[Crossref]

Gong, L. Y.

L. Wang, S. M. Zhao, L. Y. Gong, and W. W. Cheng, “Free-space measurement-device-independent quantum-key-distribution protocol using decoy states with orbital angular momentum,” Chin. Phys. B 24(12), 120307 (2015).
[Crossref]

Guha, S.

M. Takeoka, S. Guha, and M. M. Wilde, “Fundamental rate-loss tradeoff for optical quantum key distribution,” Nat. Commun. 5(1), 5235 (2014).
[Crossref]

Guo, G. C.

C. Cui, Z. Q. Yin, R. Wang, W. Chen, S. Wang, G. C. Guo, and Z. F. Han, “Twin-Field Quantum Key Distribution without Phase Postselection,” Phys. Rev. Appl. 11(3), 034053 (2019).
[Crossref]

S. Wang, S. L. Zhang, H. W. Li, Z. Q. Yin, Y. B. Zhao, W. Chen, Z. F. Han, and G. C. Guo, “Decoy-state theory for the heralded single-photon source with intensity fluctuations,” Phys. Rev. A 79(6), 062309 (2009).
[Crossref]

Han, Z. F.

C. Cui, Z. Q. Yin, R. Wang, W. Chen, S. Wang, G. C. Guo, and Z. F. Han, “Twin-Field Quantum Key Distribution without Phase Postselection,” Phys. Rev. Appl. 11(3), 034053 (2019).
[Crossref]

S. Wang, S. L. Zhang, H. W. Li, Z. Q. Yin, Y. B. Zhao, W. Chen, Z. F. Han, and G. C. Guo, “Decoy-state theory for the heralded single-photon source with intensity fluctuations,” Phys. Rev. A 79(6), 062309 (2009).
[Crossref]

Hu, X. L.

Z. W. Yu, X. L. Hu, C. Jiang, H. Xu, and X. B. Wang, “Sending-or-not-sending twin-field quantum key distribution in practice,” Sci. Rep. 9(1), 3080 (2019).
[Crossref]

X. B. Wang, Z. W. Yu, and X. L. Hu, “Twin-field quantum key distribution with large misalignment error,” Phys. Rev. A 98(6), 062323 (2018).
[Crossref]

Huang, M. Q.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

Hwang, W. Y.

W. Y. Hwang, “Quantum Key Distribution with High Loss: Toward Global Secure Communication,” Phys. Rev. Lett. 91(5), 057901 (2003).
[Crossref]

Jiang, C.

Z. W. Yu, X. L. Hu, C. Jiang, H. Xu, and X. B. Wang, “Sending-or-not-sending twin-field quantum key distribution in practice,” Sci. Rep. 9(1), 3080 (2019).
[Crossref]

C. Jiang, Z. W. Yu, and X. B. Wang, “Measurement-device-independent quantum key distribution with source state errors in photon number space,” Phys. Rev. A 94(6), 062323 (2016).
[Crossref]

C. Jiang, Z. W. Yu, and X. B. Wang, “Measurement-device-independent quantum key distribution with source state errors in photon number space,” Phys. Rev. A 94(6), 062323 (2016).
[Crossref]

Jiang, M. S.

Jiang, X.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

Koashi, M.

T. Sasaki, Y. Yamamoto, and M. Koashi, “Practical quantum key distribution protocol without monitoring signal disturbance,” Nature 509(7501), 475–478 (2014).
[Crossref]

Laurenza, R.

S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental limits of repeaterless quantum communications,” Nat. Commun. 8(1), 15043 (2017).
[Crossref]

Li, H. W.

Y. Y. Zhang, W. S. Bao, C. Zhou, H. W. Li, Y. Wang, and M. S. Jiang, “Practical round-robin differential phase-shift quantum key distribution,” Opt. Express 24(18), 20763–20773 (2016).
[Crossref]

S. Wang, S. L. Zhang, H. W. Li, Z. Q. Yin, Y. B. Zhao, W. Chen, Z. F. Han, and G. C. Guo, “Decoy-state theory for the heralded single-photon source with intensity fluctuations,” Phys. Rev. A 79(6), 062309 (2009).
[Crossref]

Li, J.

K. Liu, J. Li, J. R. Zhu, C. M. Zhang, and Q. Wang, “Decoy-state reference-frame-independent quantum key distribution with both source errors and statistical fluctuations,” Chin. Phys. B 26(12), 120302 (2017).
[Crossref]

Li, M. J.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

Li, W.

W. Li, L. Wang, and S. Zhao, “Phase-Matching Quantum Key Distribution based on Single-Photon Entanglement,” Sci. Rep. 9(1), 15466 (2019).
[Crossref]

Liu, H.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

Liu, K.

K. Liu, J. Li, J. R. Zhu, C. M. Zhang, and Q. Wang, “Decoy-state reference-frame-independent quantum key distribution with both source errors and statistical fluctuations,” Chin. Phys. B 26(12), 120302 (2017).
[Crossref]

Lo, H. K.

M. Curty, K. Azuma, and H. K. Lo, “Simple security proof of twin-field type quantum key distribution protocol,” npj Quantum Inf. 5(1), 64 (2019).
[Crossref]

H. K. Lo, M. Curty, and B. Qi, “Measurement-Device-Independent Quantum Key Distribution,” Phys. Rev. Lett. 108(13), 130503 (2012).
[Crossref]

X. Ma, B. Qi, Y. Zhao, and H. K. Lo, “Practical decoy state for quantum key distribution,” Phys. Rev. A 72(1), 012326 (2005).
[Crossref]

H. K. Lo and H. F. Chau, “Unconditional Security of Quantum Key Distribution over Arbitrarily Long Distances,” Science 283(5410), 2050–2056 (1999).
[Crossref]

Lucamarini, M.

M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate-distance limit of quantum key distribution without quantum repeaters,” Nature 557(7705), 400–403 (2018).
[Crossref]

Ma, X.

P. Zeng, W. Wu, and X. Ma, “Symmetry-Protected Privacy: Beating the Rate-Distance Linear Bound Over a Noisy Channel,” Phys. Rev. Appl. 13(6), 064013 (2020).
[Crossref]

X. Ma, P. Zeng, and H. Zhou, “Phase-Matching Quantum Key Distribution,” Phys. Rev. X 8(3), 031043 (2018).
[Crossref]

X. Ma, B. Qi, Y. Zhao, and H. K. Lo, “Practical decoy state for quantum key distribution,” Phys. Rev. A 72(1), 012326 (2005).
[Crossref]

Mao, Q. P.

Q. P. Mao, L. Wang, and S. M. Zhao, “Decoy-state round-robin differential-phase-shift quantum key distribution with source errors,” Quantum Inf. Process. 19(2), 56 (2020).
[Crossref]

Q. P. Mao, L. Wang, and S. M. Zhao, “Plug-and-play round-robin differential phase-shift quantum key distribution,” Sci. Rep. 7(1), 15435 (2017).
[Crossref]

Mao, Y.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

Mayers, D.

D. Mayers, “Unconditional security in quantum cryptography,” J. Assoc. Comput. Mach. 48(3), 351–406 (2001).
[Crossref]

Nolan, D.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

Ottaviani, C.

S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental limits of repeaterless quantum communications,” Nat. Commun. 8(1), 15043 (2017).
[Crossref]

Pan, J. W.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

X. B. Wang, L. Yang, C. Z. Peng, and J. W. Pan, “Decoy-state quantum key distribution with both source errors and statistical fluctuations,” New J. Phys. 11(7), 075006 (2009).
[Crossref]

X. B. Wang, C. Z. Peng, J. Zhang, L. Yang, and J. W. Pan, “General theory of decoy-state quantum cryptography with source errors,” Phys. Rev. A 77(4), 042311 (2008).
[Crossref]

Peng, C. Z.

X. B. Wang, L. Yang, C. Z. Peng, and J. W. Pan, “Decoy-state quantum key distribution with both source errors and statistical fluctuations,” New J. Phys. 11(7), 075006 (2009).
[Crossref]

X. B. Wang, C. Z. Peng, J. Zhang, L. Yang, and J. W. Pan, “General theory of decoy-state quantum cryptography with source errors,” Phys. Rev. A 77(4), 042311 (2008).
[Crossref]

Pirandola, S.

S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental limits of repeaterless quantum communications,” Nat. Commun. 8(1), 15043 (2017).
[Crossref]

Qi, B.

H. K. Lo, M. Curty, and B. Qi, “Measurement-Device-Independent Quantum Key Distribution,” Phys. Rev. Lett. 108(13), 130503 (2012).
[Crossref]

X. Ma, B. Qi, Y. Zhao, and H. K. Lo, “Practical decoy state for quantum key distribution,” Phys. Rev. A 72(1), 012326 (2005).
[Crossref]

Sasaki, T.

T. Sasaki, Y. Yamamoto, and M. Koashi, “Practical quantum key distribution protocol without monitoring signal disturbance,” Nature 509(7501), 475–478 (2014).
[Crossref]

Shields, A. J.

M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate-distance limit of quantum key distribution without quantum repeaters,” Nature 557(7705), 400–403 (2018).
[Crossref]

Takeoka, M.

M. Takeoka, S. Guha, and M. M. Wilde, “Fundamental rate-loss tradeoff for optical quantum key distribution,” Nat. Commun. 5(1), 5235 (2014).
[Crossref]

Wang, L.

Q. P. Mao, L. Wang, and S. M. Zhao, “Decoy-state round-robin differential-phase-shift quantum key distribution with source errors,” Quantum Inf. Process. 19(2), 56 (2020).
[Crossref]

W. Li, L. Wang, and S. Zhao, “Phase-Matching Quantum Key Distribution based on Single-Photon Entanglement,” Sci. Rep. 9(1), 15466 (2019).
[Crossref]

Q. P. Mao, L. Wang, and S. M. Zhao, “Plug-and-play round-robin differential phase-shift quantum key distribution,” Sci. Rep. 7(1), 15435 (2017).
[Crossref]

L. Wang and S. Zhao, “Round-robin differential-phase-shift quantum key distribution with heralded pair-coherent sources,” Quantum Inf. Process. 16(4), 100 (2017).
[Crossref]

L. Wang, S. M. Zhao, L. Y. Gong, and W. W. Cheng, “Free-space measurement-device-independent quantum-key-distribution protocol using decoy states with orbital angular momentum,” Chin. Phys. B 24(12), 120307 (2015).
[Crossref]

Wang, Q.

K. Liu, J. Li, J. R. Zhu, C. M. Zhang, and Q. Wang, “Decoy-state reference-frame-independent quantum key distribution with both source errors and statistical fluctuations,” Chin. Phys. B 26(12), 120302 (2017).
[Crossref]

Q. Wang and X. B. Wang, “Simulating of the measurement-device independent quantum key distribution with phase randomized general sources,” Sci. Rep. 4(1), 4612 (2015).
[Crossref]

Wang, R.

C. Cui, Z. Q. Yin, R. Wang, W. Chen, S. Wang, G. C. Guo, and Z. F. Han, “Twin-Field Quantum Key Distribution without Phase Postselection,” Phys. Rev. Appl. 11(3), 034053 (2019).
[Crossref]

Wang, S.

C. Cui, Z. Q. Yin, R. Wang, W. Chen, S. Wang, G. C. Guo, and Z. F. Han, “Twin-Field Quantum Key Distribution without Phase Postselection,” Phys. Rev. Appl. 11(3), 034053 (2019).
[Crossref]

S. Wang, S. L. Zhang, H. W. Li, Z. Q. Yin, Y. B. Zhao, W. Chen, Z. F. Han, and G. C. Guo, “Decoy-state theory for the heralded single-photon source with intensity fluctuations,” Phys. Rev. A 79(6), 062309 (2009).
[Crossref]

Wang, X. B.

Z. W. Yu, X. L. Hu, C. Jiang, H. Xu, and X. B. Wang, “Sending-or-not-sending twin-field quantum key distribution in practice,” Sci. Rep. 9(1), 3080 (2019).
[Crossref]

X. B. Wang, Z. W. Yu, and X. L. Hu, “Twin-field quantum key distribution with large misalignment error,” Phys. Rev. A 98(6), 062323 (2018).
[Crossref]

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

C. Jiang, Z. W. Yu, and X. B. Wang, “Measurement-device-independent quantum key distribution with source state errors in photon number space,” Phys. Rev. A 94(6), 062323 (2016).
[Crossref]

C. Jiang, Z. W. Yu, and X. B. Wang, “Measurement-device-independent quantum key distribution with source state errors in photon number space,” Phys. Rev. A 94(6), 062323 (2016).
[Crossref]

Q. Wang and X. B. Wang, “Simulating of the measurement-device independent quantum key distribution with phase randomized general sources,” Sci. Rep. 4(1), 4612 (2015).
[Crossref]

X. B. Wang, L. Yang, C. Z. Peng, and J. W. Pan, “Decoy-state quantum key distribution with both source errors and statistical fluctuations,” New J. Phys. 11(7), 075006 (2009).
[Crossref]

X. B. Wang, C. Z. Peng, J. Zhang, L. Yang, and J. W. Pan, “General theory of decoy-state quantum cryptography with source errors,” Phys. Rev. A 77(4), 042311 (2008).
[Crossref]

X. B. Wang, “Beating the Photon-Number-Splitting Attack in Practical Quantum Cryptography,” Phys. Rev. Lett. 94(23), 230503 (2005).
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X. B. Wang, “Decoy-state protocol for quantum cryptography with four different intensities of coherent light,” Phys. Rev. A 72(1), 012322 (2005).
[Crossref]

Wang, Y.

Wang, Z.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

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M. Takeoka, S. Guha, and M. M. Wilde, “Fundamental rate-loss tradeoff for optical quantum key distribution,” Nat. Commun. 5(1), 5235 (2014).
[Crossref]

Wu, W.

P. Zeng, W. Wu, and X. Ma, “Symmetry-Protected Privacy: Beating the Rate-Distance Linear Bound Over a Noisy Channel,” Phys. Rev. Appl. 13(6), 064013 (2020).
[Crossref]

Xu, H.

Z. W. Yu, X. L. Hu, C. Jiang, H. Xu, and X. B. Wang, “Sending-or-not-sending twin-field quantum key distribution in practice,” Sci. Rep. 9(1), 3080 (2019).
[Crossref]

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T. Sasaki, Y. Yamamoto, and M. Koashi, “Practical quantum key distribution protocol without monitoring signal disturbance,” Nature 509(7501), 475–478 (2014).
[Crossref]

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X. B. Wang, L. Yang, C. Z. Peng, and J. W. Pan, “Decoy-state quantum key distribution with both source errors and statistical fluctuations,” New J. Phys. 11(7), 075006 (2009).
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X. B. Wang, C. Z. Peng, J. Zhang, L. Yang, and J. W. Pan, “General theory of decoy-state quantum cryptography with source errors,” Phys. Rev. A 77(4), 042311 (2008).
[Crossref]

Yin, H. L.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

Yin, Z. Q.

C. Cui, Z. Q. Yin, R. Wang, W. Chen, S. Wang, G. C. Guo, and Z. F. Han, “Twin-Field Quantum Key Distribution without Phase Postselection,” Phys. Rev. Appl. 11(3), 034053 (2019).
[Crossref]

S. Wang, S. L. Zhang, H. W. Li, Z. Q. Yin, Y. B. Zhao, W. Chen, Z. F. Han, and G. C. Guo, “Decoy-state theory for the heralded single-photon source with intensity fluctuations,” Phys. Rev. A 79(6), 062309 (2009).
[Crossref]

You, L. X.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

Yu, Z. W.

Z. W. Yu, X. L. Hu, C. Jiang, H. Xu, and X. B. Wang, “Sending-or-not-sending twin-field quantum key distribution in practice,” Sci. Rep. 9(1), 3080 (2019).
[Crossref]

X. B. Wang, Z. W. Yu, and X. L. Hu, “Twin-field quantum key distribution with large misalignment error,” Phys. Rev. A 98(6), 062323 (2018).
[Crossref]

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
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C. Jiang, Z. W. Yu, and X. B. Wang, “Measurement-device-independent quantum key distribution with source state errors in photon number space,” Phys. Rev. A 94(6), 062323 (2016).
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P. Zeng, W. Wu, and X. Ma, “Symmetry-Protected Privacy: Beating the Rate-Distance Linear Bound Over a Noisy Channel,” Phys. Rev. Appl. 13(6), 064013 (2020).
[Crossref]

X. Ma, P. Zeng, and H. Zhou, “Phase-Matching Quantum Key Distribution,” Phys. Rev. X 8(3), 031043 (2018).
[Crossref]

Zhang, C. M.

K. Liu, J. Li, J. R. Zhu, C. M. Zhang, and Q. Wang, “Decoy-state reference-frame-independent quantum key distribution with both source errors and statistical fluctuations,” Chin. Phys. B 26(12), 120302 (2017).
[Crossref]

Zhang, J.

X. B. Wang, C. Z. Peng, J. Zhang, L. Yang, and J. W. Pan, “General theory of decoy-state quantum cryptography with source errors,” Phys. Rev. A 77(4), 042311 (2008).
[Crossref]

Zhang, Q.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

Zhang, S. L.

S. Wang, S. L. Zhang, H. W. Li, Z. Q. Yin, Y. B. Zhao, W. Chen, Z. F. Han, and G. C. Guo, “Decoy-state theory for the heralded single-photon source with intensity fluctuations,” Phys. Rev. A 79(6), 062309 (2009).
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Zhang, W. J.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

Zhang, Y. Y.

Zhao, S.

W. Li, L. Wang, and S. Zhao, “Phase-Matching Quantum Key Distribution based on Single-Photon Entanglement,” Sci. Rep. 9(1), 15466 (2019).
[Crossref]

L. Wang and S. Zhao, “Round-robin differential-phase-shift quantum key distribution with heralded pair-coherent sources,” Quantum Inf. Process. 16(4), 100 (2017).
[Crossref]

Zhao, S. M.

Q. P. Mao, L. Wang, and S. M. Zhao, “Decoy-state round-robin differential-phase-shift quantum key distribution with source errors,” Quantum Inf. Process. 19(2), 56 (2020).
[Crossref]

Q. P. Mao, L. Wang, and S. M. Zhao, “Plug-and-play round-robin differential phase-shift quantum key distribution,” Sci. Rep. 7(1), 15435 (2017).
[Crossref]

L. Wang, S. M. Zhao, L. Y. Gong, and W. W. Cheng, “Free-space measurement-device-independent quantum-key-distribution protocol using decoy states with orbital angular momentum,” Chin. Phys. B 24(12), 120307 (2015).
[Crossref]

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X. Ma, B. Qi, Y. Zhao, and H. K. Lo, “Practical decoy state for quantum key distribution,” Phys. Rev. A 72(1), 012326 (2005).
[Crossref]

Zhao, Y. B.

S. Wang, S. L. Zhang, H. W. Li, Z. Q. Yin, Y. B. Zhao, W. Chen, Z. F. Han, and G. C. Guo, “Decoy-state theory for the heralded single-photon source with intensity fluctuations,” Phys. Rev. A 79(6), 062309 (2009).
[Crossref]

Zhou, C.

Y. Y. Zhang, W. S. Bao, C. Zhou, H. W. Li, Y. Wang, and M. S. Jiang, “Practical round-robin differential phase-shift quantum key distribution,” Opt. Express 24(18), 20763–20773 (2016).
[Crossref]

C. Zhou, W. S. Bao, and X. Q. Fu, “Decoy-state quantum key distribution for the heralded pair coherent state photon source with intensity fluctuations,” Sci. China Inf. Sci. 53(12), 2485–2494 (2010).
[Crossref]

Zhou, F.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

Zhou, H.

X. Ma, P. Zeng, and H. Zhou, “Phase-Matching Quantum Key Distribution,” Phys. Rev. X 8(3), 031043 (2018).
[Crossref]

Zhou, Y. H.

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

Zhu, J. R.

K. Liu, J. Li, J. R. Zhu, C. M. Zhang, and Q. Wang, “Decoy-state reference-frame-independent quantum key distribution with both source errors and statistical fluctuations,” Chin. Phys. B 26(12), 120302 (2017).
[Crossref]

Chin. Phys. B (2)

L. Wang, S. M. Zhao, L. Y. Gong, and W. W. Cheng, “Free-space measurement-device-independent quantum-key-distribution protocol using decoy states with orbital angular momentum,” Chin. Phys. B 24(12), 120307 (2015).
[Crossref]

K. Liu, J. Li, J. R. Zhu, C. M. Zhang, and Q. Wang, “Decoy-state reference-frame-independent quantum key distribution with both source errors and statistical fluctuations,” Chin. Phys. B 26(12), 120302 (2017).
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M. Takeoka, S. Guha, and M. M. Wilde, “Fundamental rate-loss tradeoff for optical quantum key distribution,” Nat. Commun. 5(1), 5235 (2014).
[Crossref]

S. Pirandola, R. Laurenza, C. Ottaviani, and L. Banchi, “Fundamental limits of repeaterless quantum communications,” Nat. Commun. 8(1), 15043 (2017).
[Crossref]

Nature (2)

M. Lucamarini, Z. L. Yuan, J. F. Dynes, and A. J. Shields, “Overcoming the rate-distance limit of quantum key distribution without quantum repeaters,” Nature 557(7705), 400–403 (2018).
[Crossref]

T. Sasaki, Y. Yamamoto, and M. Koashi, “Practical quantum key distribution protocol without monitoring signal disturbance,” Nature 509(7501), 475–478 (2014).
[Crossref]

New J. Phys. (1)

X. B. Wang, L. Yang, C. Z. Peng, and J. W. Pan, “Decoy-state quantum key distribution with both source errors and statistical fluctuations,” New J. Phys. 11(7), 075006 (2009).
[Crossref]

npj Quantum Inf. (1)

M. Curty, K. Azuma, and H. K. Lo, “Simple security proof of twin-field type quantum key distribution protocol,” npj Quantum Inf. 5(1), 64 (2019).
[Crossref]

Opt. Express (1)

Phys. Rev. A (7)

X. Ma, B. Qi, Y. Zhao, and H. K. Lo, “Practical decoy state for quantum key distribution,” Phys. Rev. A 72(1), 012326 (2005).
[Crossref]

X. B. Wang, C. Z. Peng, J. Zhang, L. Yang, and J. W. Pan, “General theory of decoy-state quantum cryptography with source errors,” Phys. Rev. A 77(4), 042311 (2008).
[Crossref]

S. Wang, S. L. Zhang, H. W. Li, Z. Q. Yin, Y. B. Zhao, W. Chen, Z. F. Han, and G. C. Guo, “Decoy-state theory for the heralded single-photon source with intensity fluctuations,” Phys. Rev. A 79(6), 062309 (2009).
[Crossref]

C. Jiang, Z. W. Yu, and X. B. Wang, “Measurement-device-independent quantum key distribution with source state errors in photon number space,” Phys. Rev. A 94(6), 062323 (2016).
[Crossref]

C. Jiang, Z. W. Yu, and X. B. Wang, “Measurement-device-independent quantum key distribution with source state errors in photon number space,” Phys. Rev. A 94(6), 062323 (2016).
[Crossref]

X. B. Wang, “Decoy-state protocol for quantum cryptography with four different intensities of coherent light,” Phys. Rev. A 72(1), 012322 (2005).
[Crossref]

X. B. Wang, Z. W. Yu, and X. L. Hu, “Twin-field quantum key distribution with large misalignment error,” Phys. Rev. A 98(6), 062323 (2018).
[Crossref]

Phys. Rev. Appl. (2)

C. Cui, Z. Q. Yin, R. Wang, W. Chen, S. Wang, G. C. Guo, and Z. F. Han, “Twin-Field Quantum Key Distribution without Phase Postselection,” Phys. Rev. Appl. 11(3), 034053 (2019).
[Crossref]

P. Zeng, W. Wu, and X. Ma, “Symmetry-Protected Privacy: Beating the Rate-Distance Linear Bound Over a Noisy Channel,” Phys. Rev. Appl. 13(6), 064013 (2020).
[Crossref]

Phys. Rev. Lett. (4)

H. K. Lo, M. Curty, and B. Qi, “Measurement-Device-Independent Quantum Key Distribution,” Phys. Rev. Lett. 108(13), 130503 (2012).
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W. Y. Hwang, “Quantum Key Distribution with High Loss: Toward Global Secure Communication,” Phys. Rev. Lett. 91(5), 057901 (2003).
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X. B. Wang, “Beating the Photon-Number-Splitting Attack in Practical Quantum Cryptography,” Phys. Rev. Lett. 94(23), 230503 (2005).
[Crossref]

H. L. Yin, T. Y. Chen, Z. W. Yu, H. Liu, L. X. You, Y. H. Zhou, S. J. Chen, Y. Mao, M. Q. Huang, W. J. Zhang, H. Chen, M. J. Li, D. Nolan, F. Zhou, X. Jiang, Z. Wang, Q. Zhang, X. B. Wang, and J. W. Pan, “Measurement-Device-Independent Quantum Key Distribution Over a 404 km Optical Fiber,” Phys. Rev. Lett. 117(19), 190501 (2016).
[Crossref]

Phys. Rev. X (1)

X. Ma, P. Zeng, and H. Zhou, “Phase-Matching Quantum Key Distribution,” Phys. Rev. X 8(3), 031043 (2018).
[Crossref]

Quantum Inf. Process. (2)

L. Wang and S. Zhao, “Round-robin differential-phase-shift quantum key distribution with heralded pair-coherent sources,” Quantum Inf. Process. 16(4), 100 (2017).
[Crossref]

Q. P. Mao, L. Wang, and S. M. Zhao, “Decoy-state round-robin differential-phase-shift quantum key distribution with source errors,” Quantum Inf. Process. 19(2), 56 (2020).
[Crossref]

Sci. China Inf. Sci. (1)

C. Zhou, W. S. Bao, and X. Q. Fu, “Decoy-state quantum key distribution for the heralded pair coherent state photon source with intensity fluctuations,” Sci. China Inf. Sci. 53(12), 2485–2494 (2010).
[Crossref]

Sci. Rep. (4)

Q. Wang and X. B. Wang, “Simulating of the measurement-device independent quantum key distribution with phase randomized general sources,” Sci. Rep. 4(1), 4612 (2015).
[Crossref]

Q. P. Mao, L. Wang, and S. M. Zhao, “Plug-and-play round-robin differential phase-shift quantum key distribution,” Sci. Rep. 7(1), 15435 (2017).
[Crossref]

W. Li, L. Wang, and S. Zhao, “Phase-Matching Quantum Key Distribution based on Single-Photon Entanglement,” Sci. Rep. 9(1), 15466 (2019).
[Crossref]

Z. W. Yu, X. L. Hu, C. Jiang, H. Xu, and X. B. Wang, “Sending-or-not-sending twin-field quantum key distribution in practice,” Sci. Rep. 9(1), 3080 (2019).
[Crossref]

Science (1)

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[Crossref]

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Figures (5)

Fig. 1.
Fig. 1. Secure key rates for the four-intensity decoy-state PM-QKD, original PM-QKD, new PM-QKD in [32] with the same parameter in Table 1, and new PM-QKD with the original parameters in [32].
Fig. 2.
Fig. 2. Key rates for the four-intensity decoy-state PM-QKD under the data size $N=1\times 10^{15}$, $1\times 10^{16}$, $1\times 10^{17}$ and infinitely large
Fig. 3.
Fig. 3. Key rate comparison between MDI-QKD protocol, the proposed protocol with and without source error.
Fig. 4.
Fig. 4. Key rate ratio against transmission distance for different $\delta$.
Fig. 5.
Fig. 5. Key rate ratio against $\delta$ for different transmission distance $z$.

Tables (1)

Tables Icon

Table 1. Parameters of the simulation

Equations (50)

Equations on this page are rendered with MathJax. Learn more.

| φ A φ B k π | 2 π M ( k = 0 , 1 ) ,
m = 0 n p μ A ( m ) p μ B ( n m ) = m = 0 n e μ / 2 ( μ 2 ) m m ! e μ / 2 ( μ 2 ) n m ( n m ) ! = e μ μ n m = 0 n 1 2 n 1 m ! ( n m ) ! = e μ μ n n ! = p μ ( n ) .
R = 2 M Q μ [ 1 f H ( E μ Z ) H ( E μ X ) ] ,
E μ X = k = 0 q 2 k + 1 e 2 k + 1 Z + k = 0 q 2 k ( 1 e 2 k Z ) = k = 0 q 2 k + 1 e 2 k + 1 Z + q 0 e 0 Z + k = 1 q 2 k k = 1 q 2 k e 2 k = k = 0 q 2 k + 1 e 2 k + 1 Z + q 0 e 0 Z + ( 1 q 0 q o d d ) k = 1 q 2 k e 2 k k = 0 q 2 k + 1 e 2 k + 1 Z + q 0 e 0 Z + ( 1 q 0 q o d d ) .
E μ X = k = 0 q 2 k + 1 e 2 k + 1 Z + k = 0 q 2 k ( 1 e 2 k Z ) q 0 e 0 Z + ( q 1 e 1 Z + q 3 e 3 Z + q 5 e 5 Z ) + ( 1 q 0 q 1 q 3 q 5 ) ,
E μ X = k = 0 q 2 k + 1 e 2 k + 1 Z + k = 0 q 2 k ( 1 e 2 k Z ) = k = 0 q 2 k + 1 e 2 k + 1 Z + q 0 e 0 Z + q 2 ( 1 e 2 Z ) + k = 2 q 2 k k = 2 q 2 k e 2 k k = 0 q 2 k + 1 e 2 k + 1 Z + q 0 e 0 Z + q 2 ( 1 e 2 Z ) + ( 1 q 0 q 2 q o d d ) = k = 0 q 2 k + 1 e 2 k + 1 Z + q 0 e 0 Z q 2 e 2 Z + ( 1 q 0 q o d d ) .
E μ X = q 0 e 0 Z + q 1 e 1 Z q 2 e 2 Z + ( 1 q 0 q 1 ) .
Q x e x = k = 0 Y k ( x ) k k ! , E x Q x e x = k = 0 e k Z Y k ( x ) k k ! ( x = μ , v 1 , v 2 , v 3 ) .
e 2 Z Y 2 ( v 1 v 2 ) ( v 2 v 3 ) ( v 1 v 3 ) ( μ v 1 v 2 v 3 ) 2 μ ( v 2 v 3 ) ( E v 1 Q v 1 e v 1 E v 2 Q v 2 e v 2 ) ( v 1 v 2 ) ( E v 2 Q v 2 e v 2 E v 3 Q v 3 e v 3 ) ( v 1 v 2 ) ( v 2 v 3 ) ( v 1 v 3 ) ( v 1 + v 2 + v 3 ) μ 3 ( E μ Q μ e μ e 0 Z Y 0 e 1 Z Y 1 μ ) .
e 2 Z 2 μ Y 2 U ( v 1 v 2 ) ( v 1 v 3 ) ( v 2 v 3 ) ( μ v 1 v 2 v 3 ) [ ( v 2 v 3 ) ( E v 1 Q v 1 e v 1 E v 2 Q v 2 e v 2 ) ( v 1 v 2 ) ( E v 2 Q v 2 e v 2 E v 3 Q v 3 e v 3 ) ] 2 ( v 1 + v 2 + v 3 ) Y 2 U μ 2 ( μ v 1 v 2 v 3 ) [ E μ Q μ e μ e 0 Z Y 0 L ( e 1 Z ) L Y 1 L μ ] = ( e 2 Z ) L .
Y 2 2 [ ( v 2 v 3 ) Q v 1 e v 1 ( v 1 v 3 ) Q v 2 e v 2 + ( v 1 v 2 ) Q v 3 e v 3 ] ( v 1 v 2 ) ( v 1 v 3 ) ( v 2 v 3 ) = Y 2 U .
e 1 Z μ ( E v 1 Q v 1 e v 1 E v 2 Q v 2 e v 2 ) ( v 1 v 2 ) ( μ v 1 v 2 ) Y 1 U v 1 + v 2 μ ( μ v 1 v 2 ) Y 1 U ( E μ Q μ e μ e 0 Z Y 0 L ) = ( e 1 Z ) L ,
( Q x ) L = Q x ( 1 n α N x Q x ) k = 0 Y k e x ( x ) k k ! Q x ( 1 + n α N x Q x ) = ( Q x ) U , ( E x Q x ) L = E x Q x ( 1 n α N x E x Q x ) k = 0 e k Z Y k e x ( x ) k k ! E x Q x ( 1 + n α N x E x Q x ) = ( E x Q x ) U .
1 e r f ( n α 2 ) = ε
Q x = 1 e η x + 2 p d e η x , E x = e η x ( p d + η x e δ ) Q μ ,
P x i | k = P x p x i ( k ) P μ p μ i ( k ) + P v 1 p v 1 i ( k ) + P v 2 p v 2 i ( k ) + P v 3 p v 3 i ( k ) .
N x = k = 0 n k , x = k = 0 i s k P x i | k = i s 0 P x i | 0 + i s 1 P x i | 1 + k = 2 i s k P x i | k .
N x = k = 0 i s k P x p x i ( k ) d k i = P x k = 0 i s k p x i ( k ) d k i .
N x P x = Q x N = N ( Q 0 , x + Q 1 , x + k = 2 Q k , x ) = i s 0 p x i ( 0 ) d 0 i + i s 1 p x i ( 1 ) d 1 i + k = 2 i s k p x i ( k ) d k i ,
N μ P μ = i s 0 p μ i ( 0 ) d 0 i + i s 1 p μ i ( 1 ) d 1 i + k = 2 i s k p μ i ( k ) d k i ,
N v 1 P v 1 = i s 0 p v 1 i ( 0 ) d 0 i + i s 1 p v 1 i ( 1 ) d 1 i + k = 2 i s k p v 1 i ( k ) d k i ,
N v 2 P v 2 = i s 0 p v 2 i ( 0 ) d 0 i + i s 1 p v 2 i ( 1 ) d 1 i + k = 2 i s k p v 2 i ( k ) d k i ,
N v 3 P v 3 = i s 0 p v 3 i ( 0 ) d 0 i + i s 1 p v 3 i ( 1 ) d 1 i + k = 2 i s k p v 3 i ( k ) d k i .
D 1 U = p v 2 U ( 0 ) N v 1 P v 1 p v 1 L ( 0 ) N v 2 P v 2 p v 2 U ( 0 ) p v 1 L ( 1 ) p v 1 L ( 0 ) p v 2 U ( 1 ) ,
D 2 U = b 1 [ p v 2 U ( 0 ) N v 1 P v 1 p v 1 L ( 0 ) N v 2 P v 2 ] b 2 [ p v 3 L ( 0 ) N v 2 P v 2 p v 2 U ( 0 ) N v 3 P v 3 ] b 1 [ p v 2 U ( 0 ) p v 1 L ( 2 ) p v 1 L ( 0 ) p v 2 U ( 2 ) ] b 2 [ p v 3 L ( 0 ) p v 2 U ( 2 ) p v 2 U ( 0 ) p v 3 L ( 2 ) ] ,
Q 1 , μ U = p μ U ( 1 ) [ p v 2 U ( 0 ) Q v 1 p v 1 L ( 0 ) Q v 2 ] p v 2 U ( 0 ) p v 1 L ( 1 ) p v 1 L ( 0 ) p v 2 U ( 1 ) ,
Q 2 , μ U = p μ U ( 2 ) { b 1 [ p v 2 U ( 0 ) Q v 1 p v 1 L ( 0 ) Q v 2 ] b 2 [ p v 3 L ( 0 ) Q v 2 p v 2 U ( 0 ) Q v 1 ] } b 1 [ p v 2 U ( 0 ) p v 1 L ( 2 ) p v 1 L ( 0 ) p v 2 U ( 2 ) ] b 2 [ p v 3 L ( 0 ) p v 2 U ( 2 ) p v 2 U ( 0 ) p v 3 L ( 2 ) ] .
N x E x = k = 0 n k , x e k Z = k = 0 i s k P x p x i ( k ) d k i e k Z ,
N μ E μ = P μ [ i s 0 p μ i ( 0 ) d 0 i e 0 Z + i s 1 p μ i ( 1 ) d 1 i e 1 Z + i s 2 p μ i ( 2 ) d 2 i e 2 Z + k = 3 i s k p μ i ( k ) d k i e k Z ] ,
N v 1 E v 1 = P v 1 [ i s 0 p v 1 i ( 0 ) d 0 i e 0 Z + i s 1 p v 1 i ( 1 ) d 1 i e 1 Z + i s 2 p v 1 i ( 2 ) d 2 i e 2 Z + k = 3 i s k p v 1 i ( k ) d k i e k Z ] ,
N v 2 E v 2 = P v 2 [ i s 0 p v 2 i ( 0 ) d 0 i e 0 Z + i s 1 p v 2 i ( 1 ) d 1 i e 1 Z + i s 2 p v 2 i ( 2 ) d 2 i e 2 Z + k = 3 i s k p v 2 i ( k ) d k i e k Z ] ,
N v 3 E v 3 = P v 3 [ i s 0 p v 3 i ( 0 ) d 0 i e 0 Z + i s 1 p v 3 i ( 1 ) d 1 i e 1 Z + i s 2 p v 3 i ( 2 ) d 2 i e 2 Z + k = 3 i s k p v 3 i ( k ) d k i e k Z ] .
( e 1 Z ) U = [ p v 2 U ( 0 ) E v 1 Q v 1 p v 1 L ( 0 ) E v 2 Q v 2 ] p μ L ( 1 ) [ p v 2 U ( 0 ) p v 1 L ( 1 ) p v 1 L ( 0 ) p v 2 U ( 1 ) ] Q 1 , μ L ,
( e 1 Z ) L = { [ p v 2 L ( 0 ) E v 1 Q v 1 p v 1 U ( 0 ) E v 2 Q v 2 ] p μ L ( 2 ) a 1 [ E μ Q μ Q 0 , μ L e 0 Z ] } p μ L ( 1 ) [ a 3 p μ L ( 2 ) a 1 p μ L ( 1 ) ] Q 1 , μ U ,
( e 2 Z ) L = { { { a 2 [ p v 2 L ( 0 ) E v 1 Q v 1 p v 1 U ( 0 ) E v 2 Q v 2 ] a 3 [ p v 3 U ( 0 ) E v 2 Q v 2 p v 2 L ( 0 ) E v 3 Q v 3 ] } p μ L ( 3 ) ( a 2 a 4 a 3 a 5 ) ( E μ Q μ Q 0 , μ L e 0 Z ) } p μ U ( 2 ) } / { { { a 2 a 1 a 3 [ p v 3 U ( 0 ) p v 2 L ( 2 ) p v 2 L ( 0 ) p v 3 U ( 2 ) ] } p μ L ( 3 ) ( a 2 a 4 a 3 a 5 ) p μ L ( 2 ) } Q 2 , μ U } ,
p x U ( 0 ) = p x ( 1 δ ) ( 0 ) = e x ( 1 δ ) , p x L ( 0 ) = p x ( 1 + δ ) ( 0 ) = e x ( 1 + δ ) ,
p x U ( k ) = p x ( 1 + δ ) ( k ) = e x ( 1 + δ ) ( x ( 1 + δ ) ) k k ! ( k 1 ) , p x L ( k ) = p x ( 1 δ ) ( k ) = e x ( 1 δ ) ( x ( 1 δ ) ) k k ! ( k 1 ) .
p v 2 U ( 0 ) N v 1 P v 1 p v 1 L ( 0 ) N v 2 P v 2 i s 0 [ p v 2 U ( 0 ) p v 1 L ( 0 ) p v 1 L ( 0 ) p v 2 U ( 0 ) ] d 0 i + i s 1 [ p v 2 U ( 0 ) p v 1 L ( 1 ) p v 1 L ( 0 ) p v 2 U ( 1 ) ] d 1 i + k = 2 i s k [ p v 2 U ( 0 ) p v 1 L ( k ) p v 1 L ( 0 ) p v 2 U ( k ) ] d k i = [ p v 2 U ( 0 ) p v 1 L ( 1 ) p v 1 L ( 0 ) p v 2 U ( 1 ) ] D 1 + k = 2 [ p v 2 U ( 0 ) p v 1 L ( k ) p v 1 L ( 0 ) p v 2 U ( k ) ] D k .
p v 2 U ( 0 ) N v 1 P v 1 p v 1 l ( 0 ) N v 2 P v 2 [ p v 2 U ( 0 ) p v 1 L ( 1 ) p v 1 L ( 0 ) p v 2 U ( 1 ) ] D 1 ,
D 1 p v 2 U ( 0 ) N v 1 P v 1 p v 1 l ( 0 ) N v 2 P v 2 p v 2 U ( 0 ) p v 1 L ( 1 ) p v 1 L ( 0 ) p v 2 U ( 1 ) = D 1 U .
p v 2 U ( 0 ) N v 1 P v 1 p v 1 L ( 0 ) N v 2 P v 2 i s 0 [ p v 2 U ( 0 ) p v 1 L ( 0 ) p v 1 L ( 0 ) p v 2 U ( 0 ) ] d 0 i + i s 1 [ p v 2 U ( 0 ) p v 1 L ( 1 ) p v 1 L ( 0 ) p v 2 U ( 1 ) ] d 1 i + i s 2 [ p v 2 U ( 0 ) p v 1 i ( 2 ) p v 1 L ( 0 ) p v 2 i ( 2 ) ] d 2 i + k = 3 i s k [ p v 2 U ( 0 ) p v 1 L ( k ) p v 1 L ( 0 ) p v 2 U ( k ) ] d k i = [ p v 2 U ( 0 ) p v 1 L ( 1 ) p v 1 L ( 0 ) p v 2 U ( 1 ) ] D 1 + [ p v 2 U ( 0 ) p v 1 L ( 2 ) p v 1 L ( 0 ) p v 2 U ( 2 ) ] D 2 + k = 3 i s k [ p v 2 U ( 0 ) p v 1 L ( k ) p v 1 L ( 0 ) p v 2 U ( k ) ] d k i .
p v 3 L ( 0 ) × E q . ( 22 ) p v 2 U ( 0 ) × E q . ( 23 ) = p v 3 L ( 0 ) N v 2 P v 2 p v 2 U ( 0 ) N v 3 P v 3 i s 0 [ p v 3 L ( 0 ) p v 2 U ( 0 ) p v 2 U ( 0 ) p v 3 L ( 0 ) ] d 0 i + i s 1 [ p v 3 L ( 0 ) p v 2 U ( 1 ) p v 2 U ( 0 ) p v 3 L ( 1 ) ] d 1 i + i s 2 [ p v 3 L ( 0 ) p v 2 U ( 2 ) p v 2 U ( 0 ) p v 3 L ( 2 ) ] d 2 i + k = 3 i s k [ p v 3 L ( 0 ) p v 2 i ( k ) p v 2 U ( 0 ) p v 3 i ( k ) ] d k i = [ p v 3 L ( 0 ) p v 2 U ( 1 ) p v 2 U ( 0 ) p v 3 L ( 1 ) ] D 1 + [ p v 3 L ( 0 ) p v 2 U ( 2 ) p v 2 U ( 0 ) p v 3 L ( 2 ) ] D 2 + k = 3 i s k [ p v 3 L ( 0 ) p v 2 i ( k ) p v 2 U ( 0 ) p v 3 i ( k ) ] d k i .
[ p v 3 L ( 0 ) p v 2 U ( 1 ) p v 2 U ( 0 ) p v 3 L ( 1 ) ] [ p v 2 U ( 0 ) N v 1 P v 1 p v 1 L ( 0 ) N v 2 P v 2 ] [ p v 2 U ( 0 ) p v 1 L ( 1 ) p v 1 L ( 0 ) p v 2 U ( 1 ) ] [ p v 3 L ( 0 ) N v 2 P v 2 p v 2 U ( 0 ) N v 3 P v 3 ] { [ p v 3 L ( 0 ) p v 2 U ( 1 ) p v 2 U ( 0 ) p v 3 L ( 1 ) ] [ p v 2 U ( 0 ) p v 1 L ( 2 ) p v 1 L ( 0 ) p v 2 U ( 2 ) ] [ p v 2 U ( 0 ) p v 1 L ( 1 ) p v 1 L ( 0 ) p v 2 U ( 1 ) ] [ p v 3 L ( 0 ) p v 2 U ( 2 ) p v 2 U ( 0 ) p v 3 L ( 2 ) ] } D 2 + k = 3 { [ p v 3 L ( 0 ) p v 2 U ( 1 ) p v 2 U ( 0 ) p v 3 L ( 1 ) ] [ p v 2 U ( 0 ) p v 1 L ( 2 ) p v 1 L ( 0 ) p v 2 U ( 2 ) ] [ p v 2 U ( 0 ) p v 1 L ( 1 ) p v 1 L ( 0 ) p v 2 U ( 1 ) ] [ p v 3 L ( 0 ) p v 2 U ( 2 ) p v 2 U ( 0 ) p v 3 L ( 2 ) ] } / p μ U ( k ) i s k p μ i ( k ) ,
D 2 { [ p v 3 L ( 0 ) p v 2 U ( 1 ) p v 2 U ( 0 ) p v 3 L ( 1 ) ] [ p v 2 U ( 0 ) N v 1 P v 1 p v 1 L ( 0 ) N v 2 P v 2 ] [ p v 2 U ( 0 ) p v 1 L ( 1 ) p v 1 L ( 0 ) p v 2 U ( 1 ) ] [ p v 3 L ( 0 ) N v 2 P v 2 p v 2 U ( 0 ) N v 3 P v 3 ] } / { [ p v 3 L ( 0 ) p v 2 U ( 1 ) p v 2 U ( 0 ) p v 3 L ( 1 ) ] [ p v 2 U ( 0 ) p v 1 L ( 2 ) p v 1 L ( 0 ) p v 2 U ( 2 ) ] [ p v 2 U ( 0 ) p v 1 L ( 1 ) p v 1 L ( 0 ) p v 2 U ( 1 ) ] [ p v 3 L ( 0 ) p v 2 U ( 2 ) p v 2 U ( 0 ) p v 3 L ( 2 ) ] } = D 2 U .
p v 2 L ( 0 ) N v 1 E v 1 P v 1 p v 1 U ( 0 ) N v 2 E v 2 P v 2 [ p v 2 L ( 0 ) p v 1 U ( 0 ) p v 1 U ( 0 ) p v 2 L ( 0 ) ] D 0 e 0 Z + [ p v 2 L ( 0 ) p v 1 U ( 1 ) p v 1 U ( 0 ) p v 2 L ( 1 ) ] D 1 e 1 Z + k = 2 p v 2 L ( 0 ) p v 1 U ( k ) p v 1 U ( 0 ) p v 2 L ( k ) p μ L ( k ) i s k p μ i ( k ) d k i e k Z [ p v 2 L ( 0 ) p v 1 U ( 1 ) p v 1 U ( 0 ) p v 2 L ( 1 ) ] D 1 e 1 Z + p v 2 L ( 0 ) p v 1 U ( 2 ) p v 1 U ( 0 ) p v 2 L ( 2 ) p μ L ( 2 ) [ N μ E μ P μ p μ ( 0 ) D 0 e 0 Z p μ ( 1 ) D 1 e 1 Z ] .
e 1 Z p v 2 L ( 0 ) N v 1 E v 1 P v 1 p v 1 U ( 0 ) N v 2 E v 2 P v 2 p v 2 L ( 0 ) p v 1 U ( 2 ) p v 1 U ( 0 ) p v 2 L ( 2 ) p μ L ( 2 ) [ N μ E μ P μ p μ L ( 0 ) D 0 L e 0 Z ] p v 2 L ( 0 ) p v 1 U ( 1 ) p v 1 U ( 0 ) p v 2 L ( 1 ) [ p v 2 L ( 0 ) p v 1 U ( 2 ) p v 1 U ( 0 ) p v 2 L ( 2 ) ] D 1 U p μ L ( 2 ) = { { [ p v 2 L ( 0 ) N v 1 E v 1 P v 1 p v 1 U ( 0 ) N v 2 E v 2 P v 2 ] p μ L ( 2 ) [ p v 2 L ( 0 ) p v 1 U ( 2 ) p v 1 U ( 0 ) p v 2 L ( 2 ) ] [ N μ E μ P μ p μ L ( 0 ) D 0 L e 0 Z ] } p μ U ( 1 ) } / { { [ p v 2 L ( 0 ) p v 1 U ( 1 ) p v 1 U ( 0 ) p v 2 L ( 1 ) ] p μ L ( 2 ) [ p v 2 L ( 0 ) p v 1 U ( 2 ) p v 1 U ( 0 ) p v 2 L ( 2 ) ] p μ L ( 1 ) } D 1 U p μ U ( 1 ) } = { { [ p v 2 L ( 0 ) E v 1 Q v 1 p v 1 U ( 0 ) E v 2 Q v 2 ] p μ L ( 2 ) [ p v 2 L ( 0 ) p v 1 U ( 2 ) p v 1 U ( 0 ) p v 2 L ( 2 ) ] [ E μ Q μ Q 0 , μ L e 0 Z ] } p μ U ( 1 ) } / { { [ p v 2 L ( 0 ) p v 1 U ( 1 ) p v 1 U ( 0 ) p v 2 L ( 1 ) ] p μ L ( 2 ) [ p v 2 L ( 0 ) p v 1 U ( 2 ) p v 1 U ( 0 ) p v 2 L ( 2 ) ] p μ L ( 1 ) } Q 1 , μ U } = ( e 1 Z ) L .
p v 2 L ( 0 ) N v 1 E v 1 P v 1 p v 1 U ( 0 ) N v 2 E v 2 P v 2 [ p v 2 L ( 0 ) p v 1 U ( 0 ) p v 1 U ( 0 ) p v 2 L ( 0 ) ] D 0 e 0 Z + [ p v 2 L ( 0 ) p v 1 U ( 1 ) p v 1 U ( 0 ) p v 2 L ( 1 ) ] D 1 e 1 Z + [ p v 2 L ( 0 ) p v 1 U ( 2 ) p v 1 U ( 0 ) p v 2 L ( 2 ) ] D 2 e 2 Z + k = 3 i s k [ p v 2 L ( 0 ) p v 1 i ( k ) p v 1 U ( 0 ) p v 2 i ( k ) ] d k i e k Z = [ p v 2 L ( 0 ) p v 1 U ( 1 ) p v 1 U ( 0 ) p v 2 L ( 1 ) ] D 1 e 1 Z + [ p v 2 L ( 0 ) p v 1 U ( 2 ) p v 1 U ( 0 ) p v 2 L ( 2 ) ] D 2 e 2 Z + k = 3 i s k [ p v 2 L ( 0 ) p v 1 i ( k ) p v 1 U ( 0 ) p v 2 i ( k ) ] d k i e k Z .
p v 3 U ( 0 ) N v 2 E v 2 P v 2 p v 2 L ( 0 ) N v 3 E v 3 P v 3 [ p v 3 U ( 0 ) p v 2 L ( 0 ) p v 1 L ( 0 ) p v 3 U ( 0 ) ] D 0 e 0 Z + [ p v 3 U ( 0 ) p v 2 L ( 1 ) p v 2 L ( 0 ) p v 3 U ( 1 ) ] D 1 e 1 Z + [ p v 3 U ( 0 ) p v 2 L ( 2 ) p v 2 L ( 0 ) p v 3 U ( 2 ) ] D 2 e 2 Z + k = 3 i s k [ p v 3 U ( 0 ) p v 2 i ( k ) p v 2 L ( 0 ) p v 3 i ( k ) ] d k i e k Z = [ p v 3 U ( 0 ) p v 2 L ( 1 ) p v 2 L ( 0 ) p v 3 U ( 1 ) ] D 1 e 1 Z + [ p v 3 U ( 0 ) p v 2 L ( 2 ) p v 2 L ( 0 ) p v 3 U ( 2 ) ] D 2 e 2 Z + k = 3 i s k [ p v 3 U ( 0 ) p v 2 i ( k ) p v 2 L ( 0 ) p v 3 i ( k ) ] d k i e k Z .
[ p v 3 U ( 0 ) p v 2 L ( 1 ) p v 2 L ( 0 ) p v 3 U ( 1 ) ] [ p v 2 L ( 0 ) N v 1 E v 1 P v 1 p v 1 U ( 0 ) N v 2 E v 2 P v 2 ] [ p v 2 L ( 0 ) p v 1 U ( 1 ) p v 1 U ( 0 ) p v 2 L ( 1 ) ] [ p v 3 U ( 0 ) N v 2 E v 2 P v 2 p v 2 L ( 0 ) N v 3 E v 3 P v 3 ] { [ p v 3 U ( 0 ) p v 2 L ( 1 ) p v 2 L ( 0 ) p v 3 U ( 1 ) ] [ p v 2 L ( 0 ) p v 1 U ( 2 ) p v 1 U ( 0 ) p v 2 L ( 2 ) ] [ p v 2 L ( 0 ) p v 1 U ( 1 ) p v 1 U ( 0 ) p v 2 L ( 1 ) ] [ p v 3 U ( 0 ) p v 2 L ( 2 ) p v 2 L ( 0 ) p v 3 U ( 2 ) ] } D 2 e 2 Z + { [ p v 3 U ( 0 ) p v 2 L ( 1 ) p v 2 L ( 0 ) p v 3 U ( 1 ) ] [ p v 2 L ( 0 ) p v 1 U ( 3 ) p v 1 U ( 0 ) p v 2 L ( 3 ) ] [ p v 2 L ( 0 ) p v 1 U ( 1 ) p v 1 U ( 0 ) p v 2 L ( 1 ) ] [ p v 3 U ( 0 ) p v 2 L ( 3 ) p v 2 L ( 0 ) p v 3 U ( 3 ) ] } / p μ L ( 3 ) [ N μ E μ P μ p μ L ( 0 ) D 0 L e 0 Z p μ L ( 1 ) D 1 L ( e 1 Z ) L p μ L ( 2 ) D 2 e 2 Z ] ,
e 2 Z { { a 2 [ p v 2 L ( 0 ) N v 1 E v 1 P v 1 p v 1 U ( 0 ) N v 2 E v 2 P v 2 ] a 3 [ p v 3 U ( 0 ) N v 2 E v 2 P v 2 p v 2 L ( 0 ) N v 3 E v 3 P v 3 ] } p μ L ( 3 ) ( a 2 a 4 a 3 a 5 ) [ N μ E μ P μ p μ L ( 0 ) D 0 L e 0 Z p μ L ( 1 ) D 1 L ( e 1 Z ) L ] } / { D 2 U { { a 2 a 1 a 3 [ p v 3 U ( 0 ) p v 2 L ( 2 ) p v 2 L ( 0 ) p v 3 U ( 2 ) ] } p μ L ( 3 ) ( a 2 a 4 a 3 a 5 ) p μ L ( 3 ) } } = { p μ U ( 2 ) { { a 2 [ p v 2 L ( 0 ) N v 1 E v 1 P v 1 p v 1 U ( 0 ) N v 2 E v 2 P v 2 ] a 3 [ p v 3 U ( 0 ) N v 2 E v 2 P v 2 p v 2 L ( 0 ) N v 3 E v 3 P v 3 ] } p μ L ( 3 ) ( a 2 a 4 a 3 a 5 ) [ N μ E μ P μ p μ L ( 0 ) D 0 L e 0 Z p μ L ( 1 ) D 1 L ( e 1 Z ) L ] } } / { p μ U ( 2 ) D 2 U { { a 2 a 1 a 3 [ p v 3 U ( 0 ) p v 2 L ( 2 ) p v 2 L ( 0 ) p v 3 U ( 2 ) ] } p μ L ( 3 ) ( a 2 a 4 a 3 a 5 ) p μ L ( 3 ) } } = { { { a 2 [ p v 2 L ( 0 ) E v 1 Q v 1 p v 1 U ( 0 ) E v 2 Q v 2 ] a 3 [ p v 3 U ( 0 ) E v 2 Q v 2 p v 2 L ( 0 ) E v 3 Q v 3 ] } p μ L ( 3 ) ( a 2 a 4 a 3 a 5 ) ( E μ Q μ Q 0 , μ L e 0 Z ) } p μ U ( 2 ) } / { { { a 2 a 1 a 3 [ p v 3 U ( 0 ) p v 2 L ( 2 ) p v 2 L ( 0 ) p v 3 U ( 2 ) ] } p μ L ( 3 ) ( a 2 a 4 a 3 a 5 ) p μ L ( 2 ) } Q 2 , μ U } = ( e 2 Z ) L ,