1. INTRODUCTION

In this highly intelligent age, the privacy of information is vital to personal lives and the management of companies and governments. Recently, researchers turned to physical theory, rather than mathematical complexities, to determine an unconditional security scheme. The significance of quantum key distribution (QKD) [1] has attracted increasing public attention, and extensive theoretical and experimental efforts have been made in this field [2–9].

However, the performance of practical apparatuses should be evaluated in a real QKD system; otherwise, the gap between the theoretical and practical models will have an adverse effect on security [10–18]. Three main approaches have been proposed to close this gap. The first one is the use of a security patch [19,20]. However, this is not suitable for all potential and unnoticed security loopholes. The second one is device-independent (DI) QKD [21–23], which is still not feasible with current technology, as a loophole-free Bell test is required [24]. The third and most promising approach is measurement-device-independent QKD (MDI-QKD) [25,26]. This approach successfully removes all detection-related security loopholes, which indicates that a secure key can be generated even when the measurement unit is fully controlled by the adversary Eve. Furthermore, with current technology, MDI-QKD can provide a solution to build more secure long-distance key distribution links or metropolitan networks [27,28].

The merits of the MDI-QKD protocol have attracted extensive attention in recent decades, and good progress has been made both theoretically [29–33] and experimentally [34–38]. As the relative phase and time bins of pulses can be firmly maintained along the transmission, time-bin encoding is a suitable scheme for a fiber-based QKD system. However, most experiments based on time-bin encoding can only distinguish one Bell state, such as $|{\psi }^{-}⟩$, which will eventually lead to a factor of loss of 3/4 in the final key. In addition, active reference frame alignment is required to ensure a higher secure key rate (SKR). Although additional calibration parts appear feasible, they increase the complexity of the MDI-QKD system, which may lead to extra information leakage through these ancillary processes [16].

As a promising solution to eliminate the requirements for reference frame calibration, a reference-frame-independent (RFI) MDI-QKD protocol is proposed [39]. To the best of our knowledge, only two experimental verifications have been conducted until now [40,41]; these systems were operated at 1 MHz, and the longest distance between Alice and Bob was 20 km. Experiments with a higher clock rate and longer transmission distance have not been reported yet. To demonstrate the robustness of the RFI-MDI-QKD protocol, a comparison with the original MDI-QKD when different misalignments of the reference frame are deployed is also required [42,43].

In this paper, we propose a time-bin encoding scheme based on polarization multiplexing. By also using the detecting scheme proposed in our previous work [44], both Bell states $|{\psi }^{\pm }⟩$ can be effectively distinguished. This indicates that the factor of loss in the final key can be reduced to 1/2. A proof-of-principle experiment based on the RFI-MDI-QKD protocol is conducted to show the feasibility of our scheme. The system clock rate is improved to 50 MHz. In an asymptotic case, we compare the performance of the RFI-MDI-QKD protocol with that of the original MDI-QKD protocol at a transmission distance of 160 km. A key rate of an order of magnitude higher is achieved for the RFI-MDI-QKD protocol when the misalignment of the relative reference frame $\beta$ is controlled at 25°. For real-world applications, we deploy the decoy-state RFI-MDI-QKD protocol with the biased bases proposed in [43]. By employing the elegant statistical fluctuation analysis proposed in [41], positive SKRs are obtained for $\beta =0°$ at a transmission distance of 120 km and for $\beta =25°$ at a transmission distance of 100 km. We believe that this result can further illustrate the feasibility of the RFI-MDI-QKD protocol at a higher clock rate and longer secure transmission distance when a large misalignment of the reference frame occurs. Eliminating the calibration of the primary reference frames of the system will reduce the complexity of the realistic setup and prevent extra information leakage through the ancillary alignment processes.

2. PROTOCOL

In both the RFI-MDI-QKD and original MDI-QKD protocols, Alice and Bob first prepare their states via a random selection in several mutually orthogonal bases, which are $Z$-basis states ($|0⟩$, $|1⟩$), $X$-basis states ($|+⟩=(|0⟩+|1⟩)/\sqrt{2}$, $|-⟩=(|0⟩-|1⟩)/\sqrt{2}$) for the original MDI-QKD protocol, and additional $Y$-basis states ($|+i⟩=(|0⟩+i|1⟩)/\sqrt{2}$, $|-i⟩=(|0⟩-i|1⟩)/\sqrt{2}$) for the RFI-MDI-QKD protocol. They subsequently send them to an untrusted relay Charlie, who performs a Bell state measurement (BSM) and announces the corresponding measurement results. Charlie’s measurement projects the incoming states into one of two Bell states, $|{\psi }^{+}⟩=(|01⟩+|10⟩)/\sqrt{2}$ or $|{\psi }^{-}⟩=(|01⟩-|10⟩)/\sqrt{2}$. Alice and Bob retain the data that conform to these instances and discard the rest. After basis sifting and error estimation, they can obtain the total counting rate $Q_{i_A{i}_B}^{{\lambda }_A{\lambda }_B}$ and quantum bit error rate (QBER) $E_{i_A{i}_B}^{{\lambda }_A{\lambda }_B}$, where ${\lambda }_{A(B)}\in \{{\mu }_i,{\nu }_i,o\}$ denotes the signal states ${\mu }_i$, decoy states ${\nu }_i$ for basis $i_{A(B)}\in \{Z,X,Y\}$, or vacuum states $o$ randomly prepared by Alice (Bob). Notably, Alice and Bob do not choose any bases for vacuum states.

If the deviation of the practical reference from the ideal one ${\beta }_{A(B)}$ is considered, the $Z$ basis is assumed to be well aligned ($Z_A=Z_B=Z$), and the $X$ and $Y$ bases can be written as follows [39,40]:

The secure key is extracted from the data when both Alice and Bob encode their bits using signal states ($\mu$) in the $Z$ basis. The rest of the data are applied to estimate the parameters used in the calculation of the SKR. The SKR is given by [25,41]

When state preparations are assumed perfectly, Eve’s information $I_E$ in Eq. (2) can be estimated using $I_E=H(e_{XX}^{11,U})$ for the original MDI-QKD protocol, where $e_{XX}^{11,U}$ is an upper bound of the quantum error rate of single-photon states in the $X$ basis. As for the RFI-MDI-QKD protocol, $I_E$ can be bounded by [39]

3. EXPERIMENTAL SETUP

The time-bin encoding method is used in our system, and the experimental setup is shown in Fig. 1. For both Alice and Bob, we employ a distributed feedback (DFB) laser combined with a home-built drive board. By operating the laser below and above the threshold, we first generate phase-randomized laser pulses with a temporal width of 2 ns and repetition rate of 50 MHz, which eliminates the possibility of an unambiguous-state-discrimination attack [45]. The generation, control, and acquisition of electrical pulses are realized by a digital waveform generator (NI-PXIe-6556) and board-level controller (NI-sbRIO-9606) (not pictured in Fig. 1). To calibrate the wavelength, the laser pulses are injected into an optical spectrum analyzer (OSA, YOKOGAWA AQ6370D) through the beam splitters (BSs) after the two lasers. The OSA, which has a resolution of 10–20 pm, is used to monitor the wavelength difference of the two independent lasers, which can be minimized by precisely adjusting the operating temperature of the lasers using the temperature controllers on the laser drive boards.

 

Fig. 1. Experimental setup of our scheme. Laser, a distributed feedback (DFB) laser combined with a home-built drive board; EPC, electronic polarization controller; PC, polarization controller; IM, intensity modulator; PM, phase modulator; PS, phase shifter; Cir, circulator, whose ports and directions are labeled above; ATT, attenuator; SPD, single photon detector; QC, an SMF-28 fiber spool, which has a channel attenuation of $\alpha =0.195\mathrm{dB}/\mathrm{km}$; BS, beam splitter; PBS, polarizing beam splitter; FR, 90° Faraday rotator; AMZI, asymmetric Mach–Zehnder interferometer; SI, Sagnac interferometer.

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As Alice and Bob’s parts are symmetrical, here we use Alice’s part as an example to illustrate our experimental setup. To realize the preparation of decoy states, an intensity modulator (IM, Photline, MXER-LN-10) is used to modulate the laser pulses into different intensities, and the vacuum states are prepared by stopping the trigger on the lasers. The circulator (Cir) is used to transmit the incident pulses from port 1 to port 2. Each of the pulses is thereafter divided into two adjacent pulses with a separation of 5 ns by the first modified asymmetric Mach–Zehnder interferometer (AMZI1), which is composed of a BS and a polarizing beam splitter (PBS). The relative phase of these two successive pulses is modulated by the phase modulator (PM1, Photline, MPZ-LN-10) in the Sagnac interferometer (SI). When the phase of 0 or $\pi$ is modulated, the $Z$-basis state can be prepared. We define the light passing through the upper path of the second AMZI (AMZI2) as the time-bin state $|0⟩$ and that passing through the lower path of AMZI2 as the time-bin state $|1⟩$. These two time bins are separated by a time delay of 4.2 ns. When the phase of $\pi /2$ is modulated by PM1, the phases modulated by PM2 in AMZI2 are 0, $\pi$ for the $X$ basis and $\pi /2$, $3\pi /2$ for the $Y$ basis. We present our time-bin encoding scheme in Table 1.

Table 1. Details of the Time-Bin Encoding Scheme

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By adjusting the driven voltage of PM1, the average photon number of pulses in the two time bins can be normalized. However, an intensity modulator (IM) or variable optical attenuators (VOA) were used for this purpose in previous works [27,28,41]. Thus, our encoding scheme also reduces the complexity of the system to some extent. Furthermore, orthogonal polarization states ($H$, $V$) are multiplexed to the time bins because of the PBS at the output of AMZI2. For a comparison of the performance at different misalignments, phase shifters (PS) in AMZI2 are applied to control the reference frame. The quantum error rate in the $X$ basis $E_{X_A{X}_B}^{{\lambda }_A{\lambda }_B}$ can be regarded as a guide to set the deviation of the relative reference frame. Note that the whole time-bin encoding units are strictly thermally and mechanically isolated to enhance their stability.

At the measurement site, as the time bins are multiplexed with the orthogonal polarization states ($H$ and $V$), we can use the PBS to demultiplex them easily. Two electric polarization controllers (EPC, General Photonics, PCD-M02-4X) are used to control the polarization fluctuations. By changing the polarization of input light until the single photon detector (SPD) count rate is maximized, all the polarization changes during photon transmission can be compensated for. Two BSs are used to realize the interference. Four commercial InGaAs SPDs (ID210) with an efficiency of ${\eta }_d=12.5\%$, a dark count rate of $P_d=1.2\times {10}^{-6}$ and dead time of 5 $\mu s$ are placed at each output of the BSs. Therefore, all the results of BSM are effectively detected. We define the Bell state $|{\psi }^{+}⟩$ as the clicks of D1 and D4 or D2 and D3 in Fig. 1 simultaneously, and the clicks of D1 and D3 or D2 and D4 are represented by $|{\psi }^{-}⟩$. The parameters for the experiment and numerical simulations are listed in Table 2.

Table 2. Parameters for the Experiment and Numerical Simulations

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4. RESULTS AND DISCUSSION

We first test the indistinguishability of the photons from Alice and Bob by measuring the visibility of Hong–Ou–Mandel (HOM) interference. We obtain a visibility of 42.7%, which is less than the maximally possible value of 50% for a weak coherence source. The low visibility of the HOM interference is mainly caused by the detector-side imperfections owing to after-pulses. It has been determined that the after-pulses effect of SPDs has a greater impact on the measurement of HOM visibility [46]. Furthermore, two PBSs are used before the interference in our scheme. The change in polarization of the incident pulses after a long transmission distance will lead to a fluctuating intensity, and the finite extinction ratio (approximately 20 dB) of the PBS will also lower the visibility. Moreover, the beam-splitting ratio and detection-efficiency mismatch of the detectors can partly influence the visibility of the HOM interference as discussed in [46]. The central wavelength of the laser pulses is 1558.18 nm after calibration. Subsequently, we will discuss our experiment results for an asymptotic case and finite-size pulses case separately.

A. Asymptotic Case

In the asymptotic case, we adopt a symmetrical three-intensity decoy-state protocol for the sake of simplicity, i.e., ${\mu }_i={\mu }_{i^{\prime }}=\mu$ for the signal states and ${\nu }_i={\nu }_{i^{\prime }}=\nu$ for the decoy states. Through simulation of our system (see Appendices A and B for details), we determine that the optimal value of the average photon number for the original MDI-QKD protocol (O-MDI) and the RFI-MDI-QKD protocol (R-MDI) is almost the same at $\beta =0°$, as depicted by the blue and purple dashed lines in Fig. 3. This indicates that the SKR for both protocols can be obtained from a single experiment. The simulation and experimental results are presented in Table 3 and Fig. 2, which shows that the two curves are almost overlapped (red line for R-MDI and blue dashed line for O-MDI). We set the average photon number of the vacuum state to be 0, as there are no pulses emitted when the trigger on the lasers is paused. The value of $C$ for R-MDI is estimated as 1.668. A QBER of 0.6% is obtained for the $Z$ basis, according to the successful BSM declared by Charlie when Alice and Bob prepared the same states in the $Z$ basis. In the ideal case, the QBER of the $Z$ basis should be 0. The dark counts of the detector and the finite extinction ratio of the first AMZI in Fig. 1 will lead to incorrect coincidence counts and thus increase the QBER of the $Z$ basis. Moreover, the vacuum and multiphoton components of weak coherent states cause accidental coincidences, which introduce an error rate of 50%. Thus, the error rate of the $X$ basis has an expected value of 25%, as does that of the $Y$ basis. However, when the visibility of the HOM interference is lower than 50%, the QBER of the $X$ basis will be higher than 25%, as the error counts originate from the situation when the Bell state $|{\psi }^{+}⟩$ was announced as Alice and Bob prepared the same states in the $X$ basis or $|{\psi }^{-}⟩$ was declared as the orthogonal states were prepared. In our system, the QBER of the $X$ basis is measured to be 27.9%.

 

Fig. 2. Lower secure key rate bound of the RFI-MDI-QKD protocol (R-MDI) and the original MDI-QKD protocol (O-MDI) when the deviations of the reference frame are controlled at $\beta =0°$ and $\beta =25°$. Except for simulating the green curve and purple dashed curve, all the average photon number settings of the signal state and decoy state are optimized for each transmission distance. The inserted figure is a partial magnification of the experimental results. The horizontal axis represents the distance between Alice (Bob) and Charlie, and the quantum channel is symmetric.

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Fig. 3. Lower secure key rate bound and optimal average photon number of the RFI-MDI-QKD protocol (R-MDI) and the original MDI-QKD protocol (O-MDI) versus the different misalignments of the reference frame at the distance of 160 km. As the simulation results are symmetrical about $\beta =45°$, this figure only shows the variation in $\beta$ from 0° to 45°.

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Table 3. Experimental Results when Misalignments of the Reference Frame are $\beta =0°$ and $\beta =25°$

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To investigate the performance of the RFI-MDI-QKD protocol and the original MDI-QKD protocol at a nonzero $\beta$, we can control the voltage of the PS in Fig. 1 to simulate the deviation of the reference frame according to the simulation result of $E_{XX}^{\mu \mu }$. Figure 3 presents the SKR and the optimal average photon number versus different deviations of the reference frame $\beta$ when the transmission distance between Alice and Bob is 160 km. It is apparent that O-MDI is particularly dependent on the change in $\beta$. However, the SKR and the optimal average photon number of R-MDI are almost identical at different deviations of the reference frame. Thus, for R-MDI at $\beta =25°$, we maintain the values of the average photon number to be consistent with the setting at $\beta =0°$. In this case, the simulation results in Fig. 2 show that the red curve for $\beta =0°$ is almost overlapped with the green curve marked with crosses for $\beta =25°$. For comparison, the optimal values of $\mu$ and $\nu$ for O-MDI at $\beta =25°$ are used to conduct the experimental test. The related experimental results are presented in Table 3 and Fig. 2. The value of $C$ for R-MDI is estimated as 1.595. At $\beta =25°$, the SKR of R-MDI is close to that at $\beta =0°$ and is an order of magnitude higher than that of the O-MDI at a transmission distance of 160 km. Thus, unlike the O-MDI, the changes in the reference frame can barely influence the SKR of R-MDI and the optimal average photon number. These results demonstrate the robustness of the RFI-MDI-QKD protocol against the deviations of the relative reference frame.

B. Finite-Size Pulses Case

In real-world applications, the key size is always finite; thus, we must consider the effect of statistical fluctuation caused by a finite size of pulses. Such an analysis is crucial to ensure the security of RFI-MDI-QKD. The three-intensity decoy-state RFI-MDI-QKD protocol with biased bases proposed in [43] has demonstrated that the achievable SKR and transmission distance can be evidently improved compared with those of the original protocol, as this protocol avoids the futility in the $Z$ basis for decoy states; thus, it can simplify the operation of the system. Recently, a universal analysis appropriate for fluctuating systems with an arbitrary number of observables has been developed in [41]. It is shown that, by adopting both the collective constraints and joint parameter estimation techniques, the SKR and transmission distance can be evidently improved for the four-intensity decoy-state RFI-MDI-QKD protocol.

Here, by using this elegant fluctuation analysis method, we employ the four-intensity decoy-state RFI-MDI-QKD protocol with biased bases in our experiment. In this scheme, except for the vacuum states, Alice and Bob must prepare the signal states ${\mu }_z$ for the $Z$ basis and ${\mu }_x$ for both the $X$ basis and $Y$ basis owing to the symmetry of the $X$ basis and $Y$ basis in Eq. (4), whereas the decoy states ${\nu }_x$ are prepared only for the $X$ basis and $Y$ basis. All the related parameters, including ${\mu }_z$, ${\mu }_x$, ${\nu }_x$, $P_z$, $P_x$, and $P_x^{{\mu }_x}$, should be optimized to achieve the highest SKR. It is observed that the achievable SKR and transmission distance in this scheme can also be notably improved as shown in Fig. 4.

 

Fig. 4. Lower secure key rate bound of the RFI-MDI-QKD protocol (R-MDI) with biased bases when statistical fluctuations are considered. The black dashed line shows the results at $\beta =25°$ using the original method proposed in [43]. The total number of pulse pairs sent from Alice and Bob is $N=3\times {10}^{12}$, the failure probability is $ϵ={10}^{-10}$, and the full parameter optimization method is applied. The horizontal axis represents the distance between Alice (Bob) and Charlie, and the quantum channel is symmetric.

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We apply the Chernoff bound for the fluctuation estimation in our experiment, with a fixed failure probability of $ϵ={10}^{-10}$ and a total number of pulse pairs $N=3\times {10}^{12}$. After the simulation with full parameter optimization shown in Fig. 4, we observe that the results are slightly different as compared with the asymptotic case. It is evident that RFI-MDI-QKD deteriorates with the increase in $\beta$ when statistical fluctuations are considered, which can be explained by the fact that the correlations of $e_{X_A{X}_B}^{11}$, $e_{Y_A{Y}_B}^{11}$, $e_{X_A{Y}_B}^{11}$, and $e_{Y_A{X}_B}^{11}$ are smeared with the increase in $\beta$; thus, it leads to poor estimation of the value of $C$ in Eq. (4). Furthermore, the setup of optimal values for the experiment will change with the increase in $\beta$, whereas it almost remains the same in the asymptotic case as shown in Fig. 3. For instance, when the transmission distance is 100 km, the optimal signal intensity setting for the $Z$ basis ${\mu }_z$ at $\beta =0°$ is 0.4407, whereas it is 0.2648 at $\beta =25°$.

We experimentally demonstrate the feasibility of the four-intensity biased decoy-state scheme when statistical fluctuations are considered. The SKRs for the transmission distances of 120 km and 100 km are obtained, as presented in Table 4 and Fig. 4. The respective deviations of the reference frame are controlled at $\beta =0°$ and $\beta =25°$. At a transmission distance of 120 km, the result of $Q_{ZZ}^{\mu \mu }$ is $7.463\times {10}^{-7}$. The values of the other optimized parameters are set as ${\mu }_x={\mu }_y=0.34$, ${\nu }_x={\nu }_y=0.066$, $P_z=0.219$, $P_x=P_y=0.339$, $P_z^{{\mu }_z}=1$, and $P_x^{{\mu }_x}=P_y^{{\mu }_y}=0.236$. For the experiment at 100 km, these values are $Q_{ZZ}^{\mu \mu }=1.351\times {10}^{-6}$, ${\mu }_x={\mu }_y=0.3$, ${\nu }_x={\nu }_y=0.047$, $P_z=0.156$, $P_x=P_y=0.377$, $P_z^{{\mu }_z}=1$, and $P_x^{{\mu }_x}=P_y^{{\mu }_y}=0.216$. At extreme distances, we noted that the statistical fluctuation of the finite key has a striking impact on the experimental outcomes, as a small change in the count rate may not generate a positive key rate. In this experiment, we have performed repeated trials to obtain reasonable results.

Table 4. Experimental Results when Statistical Fluctuations are Considered

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5. CONCLUSION

In conclusion, with a high-speed clock rate of 50 MHz and a transmission distance of more than 100 km, RFI-MDI-QKD is demonstrated based on time-bin encoding and polarization multiplexing. Two of the four Bell states $|{\psi }^{\pm }⟩$ can be distinguished without a loss. Moreover, the states in different bases can be prepared by only using phase modulators without intensity modulators. The value of the quantum error rate of the $Z$ basis $E_{ZZ}^{\mu \mu }$ demonstrates the feasibility of this scheme. In the asymptotic case, we experimentally compare the performances of the RFI-MDI-QKD protocol and original MDI-QKD protocol for different deviations of the reference frame at a transmission distance of 160 km. It is shown that the SKR of the RFI-MDI-QKD protocol is an order of magnitude higher than that of the original MDI-QKD when the misalignment of the reference frame is $\beta =25°$. Moreover, a simulation model for the RFI-MDI-QKD protocol is provided and combined with the experimental results; the robustness of the RFI-MDI-QKD protocol against a change in the reference frame has been demonstrated through the invariance of SKR and optimal average photon numbers for different deviations of the reference frame. The four-intensity decoy-state RFI-MDI-QKD protocol with biased bases is employed to consider statistical fluctuation of the finite key in our experiment. By adopting both the collective constraints and joint parameter estimation techniques, the achievable SKR and transmission distance are evidently improved compared with those of the original biased decoy-state RFI-MDI-QKD protocol. We also experimentally implemented this protocol at a transmission distance of 120 km when the deviation of the reference frame is controlled at $\beta =0°$ and at a distance of 100 km when $\beta =25°$.

APPENDIX A: SIMULATION MODEL

To simulate the protocol performance and obtain the optimal value of the average photon number for the experimental system, we must first derive the model for the total counting rate $Q_{i_A{i}_B}^{{\lambda }_A{\lambda }_B}$ and the error counting rate $E{Q}_{i_A{i}_B}^{{\lambda }_A{\lambda }_B}$. According to the method in [47], we obtain

(A1)$$\begin{gathered} Q_{Z_A{Z}_B}^{{\lambda }_A{\lambda }_B}=Q_C+Q_E, \\ {Q}_{Z_A{Z}_B}^{{\lambda }_A{\lambda }_B}{E}_{Z_A{Z}_B}^{{\lambda }_A{\lambda }_B}=e_d{Q}_C+(1-e_d)Q_E, \\ {Q}_{X_A{X}_B}^{{\lambda }_A{\lambda }_B}=2y^2[2y^2-4y{I}_0(x)+I_0(\mathrm{B})+I_0(\mathrm{E})], \\ {Q}_{X_A{X}_B}^{{\lambda }_A{\lambda }_B}{E}_{X_A{X}_B}^{{\lambda }_A{\lambda }_B}=2y^2[y^2-2y{I}_0(x)+e_d{I}_0(\mathrm{B})+(1-e_d)I_0(\mathrm{E})], \\ {Q}_{X_A{Y}_B}^{{\lambda }_A{\lambda }_B}=2y^2\{2y^2-4y{I}_0(x)+I_0[\mathrm{\Theta }]+I_0[\mathrm{\Xi }]\}, \\ {Q}_{X_A{Y}_B}^{{\lambda }_A{\lambda }_B}{E}_{X_A{Y}_B}^{{\lambda }_A{\lambda }_B}=2y^2\{y^2-2y{I}_0(x)+e_d{I}_0[\mathrm{\Xi }]+(1-e_d)I_0[\mathrm{\Theta }]\}, \\ {Q}_{Y_A{X}_B}^{{\lambda }_A{\lambda }_B}{E}_{Y_A{X}_B}^{{\lambda }_A{\lambda }_B}=2y^2\{y^2-2y{I}_0(x)+e_d{I}_0[\mathrm{\Theta }]+(1-e_d)I_0[\mathrm{\Xi }]\}, \end{gathered}$$

where

(A2)$$\begin{gathered} Q_C=2{(1-P_d)}^2{e}^{-{\mu }^{\prime }/2}[1-(1-P_d)e^{-{\eta }_A{\lambda }_A/2}] \\ \times [1-(1-P_d)e^{-{\eta }_B{\lambda }_B/2}], \\ {Q}_E=2P_d{(1-P_d)}^2{e}^{-{\mu }^{\prime }/2}[I_0(2x)-(1-P_d)e^{-{\mu }^{\prime }/2}], \\ \mathrm{B}=2x\cos \beta , \\ \mathrm{E}=2x\sin \beta , \\ \mathrm{\Theta }=\sqrt{2}x(\cos \beta +\sin \beta ), \\ \mathrm{\Xi }=\sqrt{2}x(\cos \beta -\sin \beta ), \end{gathered}$$

where $I_0(·)$ is the modified Bessel function of the first kind, $e_d=0.005$ is the misalignment-error probability, $P_d$ is the dark count of a single-photon detector, ${\eta }_A({\eta }_B)$ is the transmission of Alice (Bob), and ${\mu }^{\prime }={\eta }_A{\lambda }_A+{\eta }_B{\lambda }_A$, $x=\sqrt{{\eta }_A{\lambda }_A{\eta }_B{\lambda }_B}/2$, and $y=(1-P_d)e^{-{\mu }^{\prime }/4}$. Owing to the symmetry of the quantum channel and the $X$ basis and $Y$ basis in Eq. (4), we treat the parameters of the $X$ basis and $Y$ basis and the average photon number setting for Alice and Bob equivalently for the sake of simplicity. Accordingly, ${\mu }_A={\mu }_B$, $Q_{X_A{X}_B}^{{\lambda }_A{\lambda }_B}=Q_{Y_A{Y}_B}^{{\lambda }_A{\lambda }_B}$, $Q_{X_A{Y}_B}^{{\lambda }_A{\lambda }_B}=Q_{Y_A{X}_B}^{{\lambda }_A{\lambda }_B}$, and $E{Q}_{X_A{X}_B}^{{\lambda }_A{\lambda }_B}=E{Q}_{Y_A{Y}_B}^{{\lambda }_A{\lambda }_B}$. The quantum error rate can be calculated as $E_{i_A{i}_B}^{{\lambda }_A{\lambda }_b}=E{Q}_{i_A{i}_B}^{{\lambda }_A{\lambda }_b}/Q_{i_A{i}_B}^{{\lambda }_A{\lambda }_b}$, and it is apparent that $E_{X_A{Y}_B}^{{\lambda }_A{\lambda }_b}$ and $E_{Y_A{X}_B}^{{\lambda }_A{\lambda }_b}$ are symmetrical about 0.5. As mentioned above, we assume that the quantum error rate is less than 0.5 for the sake of simplicity; thus, $e_{X_A{Y}_B}^{{\lambda }_A{\lambda }_b}=1-E_{X_A{Y}_B}^{{\lambda }_A{\lambda }_b}$ if $E_{X_A{Y}_B}^{{\lambda }_A{\lambda }_b}>0.5$.

APPENDIX B: SECURE KEY RATE ESTIMATION

The SKR of Eq. (2) is calculated using an analytical method with two decoy states according to [41,48]. The lower bound and upper bound of the single-photon yield and the error yield are given by

(B1)$$\begin{gathered} m^{11L}\ge \frac{T_1-T_2-a_1^{\prime }{b}_2^{\prime }{T}_3}{a_1{a}_1^{\prime }(b_1{b}_2^{\prime }-b_1^{\prime }{b}_2)}, \\ {m}^{11U}\le \frac{M^{v_i{v}_i}-T_3}{a_1{b}_1}, \end{gathered}$$

where

(B2)$$\begin{gathered} T_1=a_1^{\prime }{b}_2^{\prime }{M}^{v_i{v}_i}+a_1{b}_2{a}_0^{\prime }{M}^{o{\mu }_i}+a_1{b}_2{a}_0^{\prime }{M}^{{\mu }_{i}o}, \\ {T}_2=a_1{b}_2{M}^{{\mu }_i{\mu }_i}+a_1{b}_2{a}_0^{\prime }{b}_0^{\prime }{M}^{oo}, \\ {T}_3=a_0{M}^{o{v}_i}+b_0{M}^{v_{i}o}-a_0{b}_0{M}^{oo}, \\ {a}^{\prime }{(b^{\prime })}_k={\mu }_i^k{e}^{-{\mu }_i}/k!, \\ a{(b)}_k=v_i^k{e}^{-v_i}/k!. \end{gathered}$$

In the above formula, $M^{{\lambda }_A{\lambda }_B}\in \{Q^{{\lambda }_A{\lambda }_B},E{Q}^{{\lambda }_A{\lambda }_B}\}$, $m^{11}\in \{S^{11},e{S}^{11}\}$, $i\in \{Z,X,Y\}$, and $e^{11L(U)}=e{S}^{11L(U)}/S^{11U(L)}$.

Notably, the expression in Eq. (B1) is independent of $\omega$; thus, we can use the above equations to estimate the parameters in Eq. (2) for the asymptotic case, which are listed in Table 5, and thereafter to calculate the SKR. However, as there are only signal states for the $Z$ basis in the biased decoy-state protocol, we emphasize that $e_{ZZ}^{11U}$ and $e_{XX}^{11U}$ may be different and should be estimated individually. By using the following formula:

(B3)$$m^{11U}\le \frac{M^{u_z{u}_z}-T_3^{\prime }}{a_1^{\prime }{b}_1^{\prime }},$$

the upper bound of the error yield for the $Z$ basis can be estimated, where

(B4)$$T_3^{\prime }=a_0^{\prime }{M}^{o{u}_z}+b_0^{\prime }{M}^{u_{z}o}-a_0^{\prime }{b}_0^{\prime }{M}^{oo}.$$

By using the fluctuation analysis method proposed in [41], the parameters used for SKR estimation are listed in Table 5.

Table 5. Parameters Estimated in the Process of Secure Key Rate Calculation

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Funding

National Natural Science Foundation of China (NSFC) (11674397); State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications (BUPT)) (IPOC2017ZT04).

Acknowledgment

The authors thank Chao Wang for helpful discussion on statistical fluctuations analysis.

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