Four-intensity measurement-device-independent quantum key distribution protocol with modified coherent state sources

Cong Jiang; Cong Jiang; Fei Zhou; Xiang-Bin Wang; Xiang-Bin Wang; Xiang-Bin Wang; Xiang-Bin Wang; Xiang-Bin Wang

doi:10.1364/OE.454026

1. Introduction

Quantum key distribution (QKD) can provide information-theoretical secure communication between two remote parties [1–6], traditionally named as Alice and Bob. The first QKD protocol, BB84 protocol was proposed in 1984 [7] and has been widely studied in both theories [8–11] and experiments [12–21]. The mostly used decoy-state method [8–10] can assure the security of a BB84 QKD system with imperfect sources, but in principle, the realistic detectors of the system may cause new loopholes [22,23]. Measurement-device-independent (MDI)-QKD protocol [24,25] can resist any attacks against the detectors and thus attracts many attentions. Combining the decoy state method and MDI-QKD protocol, the security of the system with both the imperfect sources and detectors can be assured.

Many modified schemes were proposed to improve the key rate of the decoy state MDI-QKD protocol, including the global optimization method [26], the joint constraint method [27] and the 4-intensity protocol [28]. Among all those, the 4-intensity MDI-QKD protocol can improve the key rate by several orders of magnitude and has become the mainstream of MDI-QKD protocols [28]. The 4-intensity MDI-QKD protocol has been widely applied in MDI-QKD experiments [29–36]. Recently, we proposed a double-scanning method [37] of the 4-intensity MDI-QKD protocol which can further improve the key rate by up to $280\%$ in typical experiment conditions. The double-scanning 4-intensity MDI-QKD protocol has been applied in recent experiments [38].

Almost all QKD experiments adopted the weak coherent state (WCS) sources due to the maturity. However, the WCS sources are not the best choice for the high rate and long distance QKD due to their low single photon ratios. Here we propose a scheme of double-scanning 4-intensity MDI-QKD protocol with the modified coherent state (MCS) sources to further improve the key rates. The MCS source can be characterized by two positive parameters, $\xi$ and $c$, and the density matrix of a phase randomized MCS source is [39]

(1)$$\rho_{\xi,c}=\sum_{n=0}^{\infty}P_n(\xi,c){|{n}\rangle\langle{n}|},$$

where ${|{n}\rangle \langle {n}|}$ represents the n-photon state and

(2)$$P_n(\xi,c)=\frac{1}{n!\cosh \xi}\left(\frac{\sinh \xi}{2\cosh\xi}\right)^{n}\exp[c\sinh\xi(\sinh\xi-\cosh\xi)]H_n^{2}(\sqrt{c/2}),$$

where $H_n(x)$ is the $n$-order Hermitian Polynomial. In general, $\xi$ is a complex number. while in this paper, we only concern the photon number distribution of the MCS source which is only affected by the modulus of $\xi$. Thus without loss of generality, we assume $\xi$ to be a positive real number. The MCS state can be generated by injecting the coherent state into an optical parametric amplifier with a small gain, and the MCS state introduced here is also known as the squeezed coherent state [39]. The parameter $\xi$ is proportional to the amplitude of the pump field, and the parameter $c=\frac {\mu }{\cosh \xi \sinh \xi }$, where $\mu$ is the intensity of the injected coherent state.

Compared with the WCS source, the MCS source emits less multi-photon incidents and more single-photon incidents. The QKD protocols with MCS sources have been studied [39–43] and shown the advantages in key rates and distances. However, all prior works set $c$ as the same for all decoy and signal sources. Here we show that $c$ doses not need to be the same for different sources, and removing such a constraint can drastically improve the key rates of the protocol with MCS sources.

The paper is arranged as follows. We first introduce the procedure of the 4-intensity MDI-QKD protocol with MCS sources. Then we show the calculation method of the final key rate with double-scanning method. Afterwords, we show some numerical results. The paper is ended with the conclusion.

2. Protocol

In the 4-intensity MDI-QKD protocol, each side (Alice or Bob) has four different sources which are the vacuum source $o_A$ ($o_B$), the decoy sources $x_A$ and $y_A$ ($x_B$ and $y_B$) in the $X$ basis and the signal source $z_A$ ($z_B$) in the $Z$ basis of Alice’s side (Bob’s side). We denote the parameters of source $l_\alpha$ for $l=o,x,y,z$ and $\alpha =A,B$ by $(\xi _{l_\alpha },c_{l_\alpha })$. Specially, we have $\xi _{o_A}=c_{o_A}=\xi _{o_B}=c_{o_B}=0$. The density matrix of source $l_\alpha$ can be expressed as

(3)$$\rho_{l_A}=\sum_{n=0}^{\infty}a_{n}^{l}{|{n}\rangle\langle{n}|}, \quad \rho_{l_B}=\sum_{n=0}^{\infty} b_{n}^{l}{|{n}\rangle\langle{n}|},$$

where $a_n^{l}=P_n(\xi _{l_A},c_{l_A})$ and $b_n^{l}=P_n(\xi _{l_B},c_{l_B})$, and $P_n(\xi,c)$ is defined in Eq. (2).

For each time window, Alice (Bob) randomly chooses one source $l_A, l=o,x,y,z$ ($r_B, r=o,x,y,z$) from all sources with probability $p_{l_A}$ ($p_{r_B}$) and prepares the pulse in the corresponding basis. Then Alice and Bob send the prepared pulses to Charlie, who is assumed to perform Bell measurement to the received pulse pairs. If two specific detectors click, Charlie would announce the detection result to Alice and Bob, and they take it as an effective event. Only the effective events in the protocol are used to estimate the final key rate and extract the final keys.

3. Calculation method

In the whole protocol, Alice and Bob send $N$ pulse pairs to Charlie and acquire a series of data. For clarity, we denote the pulse pairs source as $lr$ for $l,r=o,x,y,z$ if Alice chooses the source $l_A$ and Bob chooses the source $r_B$. We denote the observed values of the numbers of effective events and wrong effective events of source $lr$ by $n_{lr}$ and $m_{lr}$ respectively, and their corresponding expected values by ${\langle {n_{lr}}\rangle }$ and ${\langle {m_{lr}}\rangle }$ respectively. Likewise, we denote the counting rate and error counting rate of source $lr$ by $S_{lr}$ and $T_{lr}$ respectively, and their corresponding expected values by ${\langle {S_{lr}}\rangle }$ and ${\langle {T_{lr}}\rangle }$ respectively. We have

(4)$$S_{lr}=\frac{n_{lr}}{Np_{l_A}p_{r_B}}, \quad {\langle {S_{lr}}\rangle}=\frac{{\langle {n_{lr}}\rangle}}{Np_{l_A}p_{r_B}},\quad T_{lr}=\frac{m_{lr}}{Np_{l_A}p_{r_B}}, \quad {\langle {T_{lr}}\rangle}=\frac{{\langle {m_{lr}}\rangle}}{Np_{l_A}p_{r_B}}.$$

By applying the Chernoff bound [44], we can estimate the expected value according to its observed values and vice versa. We denote the estimated lower and upper bounds of the expected value by the subscripts $L$ and $U$ respectively.

In the following content of this paper, we always assume

(5)$$c_{x_A}=c_{y_A}=c_{a}, \quad c_{x_B}=c_{y_B}=c_{b},$$

but $c_{z_A}$ does not need to be the same as $c_{a}$, and $c_{z_B}$ does not need to be the same as $c_{b}$. The reason we assume Eq. (5) holds is that in this condition, we can easily prove that if $\xi _{x_A}\le \xi _{y_A}$ and $\xi _{x_B}\le \xi _{y_B}$, we have [40]

(6)$$\frac{a_j^{y}}{a_j^{x}}\ge \frac{a_2^{y}}{a_2^{x}}\ge \frac{a_1^{y}}{a_1^{x}}, \quad \frac{b_k^{y}}{b_k^{x}}\ge \frac{b_2^{y}}{b_2^{x}}\ge \frac{b_1^{y}}{b_1^{x}},$$

holding for any $j>2$ and $k>2$. If $\frac {a_1^{x}b_2^{x}}{a_2^{x}b_1^{x}}\ge \frac {a_1^{y}b_2^{y}}{a_2^{y}b_1^{y}}$, we have the lower bound of the expected value of the counting rate of single-photon pairs, ${\langle {s_{11}}\rangle }^{L}$, that satisfies [37]

(7)$${\langle {s_{11}}\rangle}^{L}=\frac{{\langle {S_+}\rangle}^{L}+{a_1^{y}b_2^{y}}\mathcal{M}-{\langle {S_-}\rangle}^{U}-a_1^{y}b_2^{y}\mathcal{H}}{a_1^{x}a_1^{y}(b_1^{x}b_2^{y}-b_2^{x}b_1^{y})},$$

where

(8)$$ {\langle {S_+}\rangle}={a_1^{y}b_2^{y}}{\langle {\bar{T}_{xx}}\rangle}+{a_1^{x}b_2^{x}a_0^{y}}{\langle {S_{oy}}\rangle}+{a_1^{x}b_2^{x}b_0^{y}}{\langle {S_{yo}}\rangle}, $$

(9)$$ {\langle {S_-}\rangle}={a_1^{x}b_2^{x}}{\langle {S_{yy}}\rangle}+{a_1^{x}b_2^{x}a_0^{y}b_0^{y}}{\langle {S_{oo}}\rangle}, $$

(10)$$ \mathcal{H}={a_0^{x}}{\langle {S_{ox}}\rangle}+{b_0^{x}}{\langle {S_{xo}}\rangle}-{a_0^{x}b_0^{x}}{\langle {S_{oo}}\rangle}, $$

${\langle {\bar {T}_{xx}}\rangle }={\langle {S_{xx}}\rangle }-{\langle {T_{xx}}\rangle }$ and $\mathcal {M}={\langle {T_{xx}}\rangle }$. With the joint constraint method and Chernoff bound [27,37], we can get ${\langle {S_+}\rangle }^{L}$, ${\langle {S_-}\rangle }^{U}$, $\mathcal {H}^{L}$ and $\mathcal {H}^{U}$. With Chernoff bound, we can get $\mathcal {M}^{L}$ and $\mathcal {M}^{U}$. For the case $\frac {a_1^{x}b_2^{x}}{a_2^{x}b_1^{x}}<\frac {a_1^{y}b_2^{y}}{a_2^{y}b_1^{y}}$, we can get similar formulas. As shown in Ref. [28], the density matrix of the single-photon pairs in the $X$ basis is the same as that of the single-photon pairs in the $Z$ basis, thus Eq. (7) represents the lower bound of the expected value of the counting rate of single-photon pairs in both the $X$ basis and $Z$ basis.

Then we can use the bit-flip error rate of the single-photon pairs in the $X$ basis to estimate the phase-flip error rate of the single-photon pairs in the $Z$ basis, and we can get the upper bound of the expected value of the phase-flip error rate of the single-photon pairs in $Z$ basis as

(11)$${\langle {e_{11}}\rangle}^{U}=\frac{\mathcal{M}-\mathcal{H}/2}{a_1^{x}b_1^{x}{\langle {s_{11}}\rangle}^{L}}.$$

With Chernoff bound, we can get the real value of the lower bound of the yield of the single photon pulse pairs from source $zz$, $s_{11,Z}^{L}$, and its corresponding upper bound of the phase flip error rate, $e_{11}^{ph,U}$,

(12)$$ s_{11,Z}^{L}=\frac{O^{L}(N_{zz}a_1^{z}b_1^{z}{\langle {s_{11}}\rangle}^{L})}{N_{zz}a_1^{z}b_1^{z}}, $$

(13)$$ {e_{11}^{ph,U}=\frac{O^{U}(N_{zz}a_1^{z}b_1^{z} s_{11,Z}^{L}{\langle {e_{11}^{ph}}\rangle}^{U})}{N_{zz}a_1^{z}b_1^{z}s_{11,Z}^{L}},} $$

where $O^{U}(Y)$ and $O^{L}(Y)$ are defined in Eqs. (21) and (22). Since only those $N_{zz}a_1^{z}b_1^{z} s_{11,Z}^{L}$ effective events of the single photon pulse pairs from source $zz$ are valid for the extraction of the final keys, and we have known the upper bound of the expected value of the phase-flip error rate in those events through decoy-state analysis, we only need to consider the difference between the expected value and real value of the phase-flip error rate in those events, thus we get Eq. (13).

Then for each group of $(\mathcal {H},\mathcal {M})$, we have

(14)$$\begin{aligned} R(\mathcal{H},\mathcal{M})=&p_{z_A}p_{z_B}\{a_1^{z}b_1^{z}s_{11,Z}^{L}[1-h(e_{11}^{ph,U})]-fS_{zz}h(E_{zz})\} \\ & -\frac{1}{N}\left(\log_2\frac{8}{\varepsilon_{cor}}+2\log_2\frac{2}{\varepsilon^{\prime}\hat{\varepsilon}}+2\log_2\frac{1}{2\varepsilon_{PA}}\right), \end{aligned}$$

where $E_{zz}=T_{zz}/S_{zz}$ is the bit-flip error rate of the effective events in source $zz$; $h(x)=-x\log _2(x)-(1-x)\log _2(1-x)$ is the Shannon entropy; $\varepsilon _{cor}$ is the failure probability of error correction; $\varepsilon _{PA}$ is the failure probability of privacy amplification; and $\varepsilon ^{\prime }$ and $\hat {\varepsilon }$ are the coefficients while using the chain rules of smooth min- and max-entropy.

In the practical experiment, we can not know the exact value of $(\mathcal {H},\mathcal {M})$, but just the upper and lower bounds of $\mathcal {H}$ and $\mathcal {M}$. To ensure the security, we need to scan $R(\mathcal {H},\mathcal {M})$ in the ranges of $[\mathcal {H}^{L},\mathcal {H}^{U}]$ and $[\mathcal {M}^{L},\mathcal {M}^{U}]$, and take the worst case of $R(\mathcal {H},\mathcal {M})$ as the final key rate. Finally, we have the final key rate

(15)$$ R=\min_{\substack{\mathcal{H}\in[\mathcal{H}^{L},\mathcal{H}^{U}],\\ \mathcal{M}\in [\mathcal{M}^{L},\mathcal{M}^{U}]}} R(\mathcal{H},\mathcal{M}), $$

4. Numerical simulation

In this part, we consider the symmetry case that Alice and Bob take the same source parameters, which are $p_{l_A}=p_{l_B}=p_l$ and $\xi _{l_A}=\xi _{l_B}=\xi _l$ for $l=o,x,y,z$. Assume $c_a=c_b=c_x$ and $c_{z_A}=c_{z_B}=c_z$. The experiment parameters are listed in Table 1. In order to obtain the optimized results with high accuracy, we use the random direction method to optimize all parameters [37]. The details are shown in Appendix C. Unlike the WCS sources in which we can use the Bessel function to directly simulate the observed values in the experiment, we have to truncate the number state of the photon at a specific position, saying $M_{max}$, which means we approximate the real density matrix in Eq. (1) by

(16)$$\rho_{\xi,c}=\sum_{n=0}^{M_{max}}P_n(\xi,c){|{n}\rangle\langle{n}|}.$$

A larger $M_{max}$ results in a more accurate simulate observed value and more time-consuming. Thus we set $M_{max}=14$ to balance accuracy and computational complexity. Such a truncation method has been introduced in Refs. [45,46] where the WCS sources have been considered, but the situation for the MCS sources is different.

Table 1. List of experimental parameters used in numerical simulations. Here $p_d$ is the dark counting rate per pulse of Charlie’s detectors; $e_d$ is the misalignment-error probability; $\eta _d$ is the detection efficiency of Charlie’s detectors; $f$ is the error correction inefficiency; $\alpha _f$ is the fiber loss coefficient ($dB/km$); $\varepsilon$ is the failure probability while using Chernoff bound; $N$ is the number of total pulse pairs sent out in the protocol.

View Table

Figure 1 is the photon-number probability distributions of the MCS sources and WCS sources. Here we set two groups of typical parameters of the MCS source, $\xi =0.5$ and $c=1$ which corresponds to the $0.4877$ average photon number, $\xi =0.5$ and $c=3$ which corresponds to the $0.92$ average photon number. Fig. 1 shows that the probability that the WCS source emits a multi-photon with a photon number larger than $14$ is less than $10^{-18}$, which is $11$ orders of magnitude lower than the dark counting rate, but the corresponding probability for the MCS source is about $10^{-6}$ which is even larger than the dark counting rate. This means that the accuracy by setting $M_{max}=14$ is enough for the calculation of WCS sources, but not for the calculation of MCS sources. And we need some special tricks to properly simulate the observed values for the MDI-QKD protocol with MCS sources. The details of our simulation method are shown in Appendix B. Fig. 1 also shows that for MCS sources, although the overall trend is that the larger the photon number, the lower the emission probability, the emission probability of the state with a larger photon number is not necessarily smaller than that with a smaller photon number. A typical example is while $c=1$, the two-photon state is eliminated and thus all the emission probabilities of the states with more than three photon number is larger than the emission probability of two-photon state.

Fig. 1. The photon-number probability distributions of the MCS sources and WCS sources.

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Figure 2 is the comparison of the key rates of the double-scanning 4-intensity MDI-QKD with different sources. As shown in Ref. [40], if we set $c=1$, the two-photon state is eliminated in the MCS source, and such a source is also called as MCS-2; if we set $c=3$, the three-photon state is eliminated in the MCS source, and such a source is also called as MCS-3. MCS-2 and MCS-3 sources are often studied in the QKD protocol. However, in the prior works [39–43], MCS-2 or MCS-3 sources were used for all the decoy and signal sources. The key rates of such cases are shown by the dashed lines ‘MCS, $c_x=c_z=1$’ and ‘MCS, $c_x=c_z=3$’ in Fig. 2, which correspond to the MCS-2 sources and MCS-3 sources respectively. In fact, we only need to set $c_x$ as a constant and the $c_z$ can be taken as an optimization parameter to improve the key rates. The key rates of such cases are shown by the dashed lines ‘MCS, $c_x=1$’ and ‘MCS, $c_x=3$’ in Fig. 2. The optimized values of $c_z$ while $c_x=1$ and $c_x=3$ are shown in Fig. 3. Figure 3 shows that in both cases that $c_x=1$ and $c_x=3$, the optimized values of $c_z$ gradually decrease from $2.5$ to $1.6$ as the distance increases. Thus in Fig. 2, we can see that for the MCS-2 sources, optimizing $c_z$ can significantly improve the key rates in the short distances, and for the MCS-3 sources, optimizing $c_z$ can significantly improve the key rates in the long distances especially in the tail of the curve. The key rates of $c_x=1$ are little higher than those of $c_x=3$ in the short distances, but lower than those of $c_x=3$ in the long distances. However, if we use MCS-2 or MCS-3 sources for all the decoy and signal sources, the key rates of MCS-2 sources are lower than those of the MCS-3 sources in all distances. Comparing with the key rates of WCS sources, the key rates of MCS sources can be improved by several orders of magnitude, and the secure distance is improved by about 40 km.

Fig. 2. The comparison of the key rates of the double-scanning 4-intensity MDI-QKD with different sources.

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Fig. 3. The optimized values of $c_z$ while $c_x=1$ and $c_x=3$.

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5. Conclusion

We propose a scheme of double-scanning 4-intensity MDI-QKD protocol with the MCS sources and introduce the calculation method of the final key rates and the simulation method of the observed values. We show that the source parameter $c$ can be different for the sources in the $X$ basis and the sources in the $Z$ basis. Numerical results show that removing such a constraint can drastically improve the key rates of the protocol with MCS sources. In the typical experiment conditions, comparing with the key rates of WCS sources, the key rates of MCS sources can be improved by several orders of magnitude, and the secure distance is improved by about 40 km.

Although in theory, the key rates of QKD with the MCS sources are higher than those with the WCS sources. It is still a challenge to realize the MDI-QKD protocol with the MCS sources in practice. Unlike the WCS source which can be easily obtained by modulating the laser, there are many difficulties in realizing the stable MCS source, such as the precise controlling of various cavity lengths and optical field phases, the suppression of additional losses in optical components and non-linear crystals during squeezing, the influence of the classical noise of the injected coherent state and the quantum fluctuations of the pump light, etc [47]. To rapidly switch the MCS source between different $\xi$ and $c$, we need to rapidly modulate the intensities of the injected coherent state and the pump light, which makes the realization of the MDI-QKD protocol with the MCS sources more difficult.

Appendix A: Chernoff bound

The Chernoff bound can help us estimate the expected value from their observed values [44,48]. Let $X_1,X_2,\dots,X_n$ be $n$ random samples, detected with the value 1 or 0, and let $X$ denote their sum satisfying $X=\sum _{i=1}^{n}X_i$. $E$ is the expected value of $X$. We have

(17)$$ E^{L}(X)=\frac{X}{1+\delta_1(X)}, $$

(18)$$ E^{U}(X)=\frac{X}{1-\delta_2(X)}, $$

where we can obtain the values of $\delta _1(X)$ and $\delta _2(X)$ by solving the following equations

(19)$$ \left(\frac{e^{\delta_1}}{(1+\delta_1)^{1+\delta_1}}\right)^{\frac{X}{1+\delta_1}}=\varepsilon, $$

(20)$$ \left(\frac{e^{-\delta_2}}{(1-\delta_2)^{1-\delta_2}}\right)^{\frac{X}{1-\delta_2}}=\varepsilon, $$

where $\varepsilon$ is the failure probability.

Besides, we can use the Chernoff bound to help us estimate their real values from their expected values. Similar to Eqs. (17)–(20), the observed value, $O$, and its expected value, $Y$, satisfy

(21)$$ O^{U}(Y)=[1+\delta_1^{\prime}(Y)]Y, $$

(22)$$ O^{L}(Y)=[1-\delta_2^{\prime}(Y)]Y, $$

where we can obtain the values of $\delta _1^{\prime }(Y,\varepsilon )$ and $\delta _2^{\prime }(Y,\varepsilon )$ by solving the following equations

(23)$$ \left(\frac{e^{\delta_1^{\prime}}}{(1+\delta_1^{\prime})^{1+\delta_1^{\prime}}}\right)^{Y}=\varepsilon, $$

(24)$$ \left(\frac{e^{-\delta_2^{\prime}}}{(1-\delta_2^{\prime})^{1-\delta_2^{\prime}}}\right)^{Y}=\varepsilon. $$

Appendix B: Simulation method of observed values

By truncating the photon number in $M_{max}$, we can get the simulate values of $S_{lr}$ and $T_{lr}$ by applying the method proposed in Ref. [46]. In order to truly reflect the advantages of the MCS sources, and to avoid parameter optimization errors caused by truncation errors, we need to replace some simulate values by the following equations:

(25)$$ T_{xx}:=T_{xx}+0.5\times(1-P_{x_A}^{trunc}*P_{x_B}^{trunc}), $$

(26)$$ S_{yy}:=S_{yy}+(1-P_{y_A}^{trunc}*P_{y_B}^{trunc}), $$

(27)$$ T_{zz}:=T_{zz}+0.5\times(1-P_{z_A}^{trunc}*P_{z_B}^{trunc}), $$

(28)$$ S_{zz}:=S_{zz}+(1-P_{z_A}^{trunc}*P_{z_B}^{trunc}), $$

where the values in the right of the equations are the simulate values calculated by the method proposed in Ref. [46], the values in the left of the equations are used in this work, and $P_{l_A}^{trunc}=\sum _{n=0}^{M_{max}}a_n^{l},P_{r_B}^{trunc}=\sum _{n=0}^{M_{max}}b_n^{r}$ for $l,r=x,y,z$.

Appendix C: Random direction method

We define $Para$ as the list of all parameters to be optimized, which means

(29)$$Para=[p_x,p_y,p_z,\xi_x,\xi_y,\xi_z,c_z].$$

When the properties of the detection set-ups and the channel between Alice and Bob are given, $R$ is the function of $Para$ in the view of simulation. We can apply the random direction method in optimizing $Para$ and get the optimized $R$ [37]. The procedure of the random direction method is as follows:

Initialization Find an original point $Para$ that makes $R(Para)> 0$, which can be done by setting a group of typical parameters or constantly trying the combination of random numbers until $R(Para)>0$. Set the initial step length $d_{step}$ and the minimum step length $d_{min}$. Set the maximum number of cycles $C_{max}$. Denote $V$ as the number of elements in $Para$.

(i). If $d_{step}<d_{min}$, stop the optimization program and output the value of $R_{opt}=R(Para)$ as the optimal key rate, where $Para$ is the list of optimal parameters; if $d_{step}>d_{min}$, set the cycle count $C=1$. Then go to step (ii).

(ii). If $C>C_{max}$, let $d_{step}:=d_{step}/5$, then go to step (i); if $C\le C_{max}$, go to step (iii).

(iii). Use a Gaussian random number generator to generate $V$ random numbers, then normalize these random numbers and put them into the list $D_{dir}$. Then calculate $R_{temp} = R_f (Para + d_{step} \times D_{dir})$. If $R_{temp}> R_{opt}$, then let $Para: = Para + d_{step} \times D_{dir}$, $R_{opt} = R_{temp}$, $C = 1$; if $R_{temp} \le R_{opt}$, then let $C: = C + 1$. Finally go to step (ii).

Funding

National Key Research and Development Program of China (2020YFA0309701, 2017YFA0303901); National Natural Science Foundation of China (12174215, 12104184, 11974204 ); Natural Science Foundation of Shandong Province (ZR2021LLZ007, ZR2019LLZ004); Open Research Fund Program of the State Key Laboratory of Low-Dimensional Quantum Physics (KF202110); ); Leading Talents of Quancheng Industry (Jinan, Shandong Province).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Four-intensity measurement-device-independent quantum key distribution protocol with modified coherent state sources

Abstract

1. Introduction

2. Protocol

3. Calculation method

4. Numerical simulation

5. Conclusion

Appendix A: Chernoff bound

Appendix B: Simulation method of observed values

Appendix C: Random direction method

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (3)

Tables (1)

Equations (29)

Optics Express