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 [3–5], measurement-device-independent quantum key distribution(MDI-QKD) protocols [6–8], and round-robin differential-phase-shift quantum key distribution(RRDPS-QKD) protocols [9–11]. 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 [26–28] 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

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

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

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]

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

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

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

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

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

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

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

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,

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,

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

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.

Table 1. Parameters of the simulation

View Table

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]

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

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

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

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)

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

3.1.2 Bounds of $e_k^Z$

The overall quantum bit error rate is given by

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

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

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.