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Abstract
Free-space continuous-variable quantum key distribution (CV-QKD) is an important technology that enables all-day quantum key distribution. Precise frame synchronization is a prerequisite for establishing a correlation between legitimate users of CV-QKD. In free-space CV-QKD, channel transmittance fluctuation caused by atmospheric turbulence increases the difficulty of synchronization. Also, as the channel transmittance is monitored in many reported experiments, the transmittance data also needs to be synchronized. We propose a novel method to solve the above problems by inserting two kinds of synchronization frames, i.e., data synchronization frames and transmittance synchronization frames. The performance of the proposed method is analyzed and Monte Carlo simulation is conducted to test its performance. The results demonstrate the feasibility and efficiency of this method. The proposed method paves the way for the realization of free-space CV-QKD.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quantum key distribution (QKD) has become a hot issue, for its ability to distribute secret keys between two legitimate parties routinely called Alice and Bob with theoretical unconditional security. Continuous-variable (CV) quantum key distribution, one of the two branches of QKD, has experienced rapid development in recent years, due to its high secret key rate, low implementation cost and compatibility with existing optical communication networks [1–4]. A binary phase modulated CV-QKD protocol is constructed and its security in the finite-key-size regime against general coherent is proved [5]. So far, CV-QKD experiment over the recordbreaking distance of about 200 km of fiber channel has been realized [6]. To solve the last mile problem of QKD and achieve satellite-to-ground, satellite-to-satellite QKD, many scholars have been concerned of free-space QKD. Some free-space QKD experiments has been performed [7–11]. Due to inherent characteristics of CV-QKD, that is, the resistance to background noise [12–14], the feasibility of free-space CV-QKD through atmospheric channel, has been theoretically and experimentally demonstrated [15–17]. Some technologies, such as phase compensation [18], dynamic polarization control [19] and excess noise suppression [20], have paved the way for realization of free-space CV-QKD.
It should be noted that accurate synchronization is a prerequisite for a CV-QKD system to operate efficiently. A worse performing synchronization scheme will result in no correlation and extremely high excess noise. The synchronization process is usually combined of the clock synchronization and the frame synchronization. In brief, the purpose of the clock synchronization is to precisely find the correct sampling time that collects the peak value of the pulse with the help of homodyne or heterodyne detection. The purpose of frame synchronization is to find the head of each string. According to the Gaussian-modulated coherent-state (GMCS) CV-QKD protocol, ${X_A}$ and ${P_A}$ are true random variables with a Gaussian distribution, which means that even a small misalignment can lead to a significant reduction of the correlation of the data between Alice and Bob. Hence a failed frame synchronization method may cause a misalignment between string $X$ and string $Y'$ and reduce the mutual information $I\left ( {X:Y'} \right )$ to $0$, which makes it impossible for the two legitimate communications parties to negotiate an string of identical secret keys after secret reconciliation process. In general, a good frame synchronization method should have high success rates at low signal noise ratio (SNR), ability to resist channel phase drift, and low computational complexity.
So far some frame synchronization methods for fiber-based CV-QKD protocol [21–23] have been proposed. Inspired by the synchronization approach of classical communication, a frame synchronization method based on modulation of special synchronization frames is put forward [21]. However, since this particular synchronization frame does not have an elaborate structure, more specifically, its auto-correlation function does not have a sharp peak value and low values at points adjacent to the peak point, so it is difficult to synchronize successfully under low SNR conditions. To meet the need of long-distance CV-QKD, a frame synchronization scheme including a phase decomposition step and a phase matching step is proposed [22], which is able to function under low SNR and suffer arbitrary phase drift. In some passive-state-preparation (PSP) CV-QKD schemes [24], the frame synchronizations that require special modulation process are technically impossible to implement. To overcome this disadvantage, a frame synchronization scheme based on finding a robust feature [23], i.e., the Hamming distance between different characters, is presented.
Unfortunately, the channel characteristics of free-space are completely different from that of optical fiber. It means that the above synchronization method will have a compromised efficiency or be completely useless in the free-space CV-QKD protocol. So far, no synchronization method for the result of Bob’s measurements and the monitored channel transmittance data has been reported. Fluctuation of losses [25,26] is a major obstacle to the synchronization of free-space QKD. Besides, as the channel transmittance is monitored by measuring the light intensity of the local oscillator in many reported experiments, the transmittance data also needs to be synchronized. Note that to ensure the protocol security, the channel transmittance parameter could be given only by the parameter estimation process. The way to obtain transmittance parameters by monitoring classical light may cause security loopholes, but in many free-space CV-QKD experiments classical light is monitored only for experimental purposes [13,17,27]. As discussed in subsection 3.2, the correlation coefficient between the Gaussian Bob’s data and the monitored channel transmittance data is $0$ on the basis of probability theory. Therefore, it is difficult to synchronize the monitored transmittance data through the previous method.
We propose a novel method here to solve the above problems by inserting two kinds of synchronization frames, i.e., data synchronization frames and transmittance synchronization frames. The data synchronization frames are constructed by choosing the synchronization codes which have good auto-correlation characteristics and appropriate length, and modulating signal pulses by a special modulation method. The transmittance synchronization frames is constructed by modulating constantly at a frequency similar to the transmittance fluctuation. Moreover, excess rotated synchronization frames for both kinds of frames could help resist random phase drift.
This paper is organized as follow. In section 2, we introduce the channel model of free-space CV-QKD. In section 3, we introduce the proposed synchronization method for free-space CV-QKD; and analyze the proposed method theoretically. In section 4, simulations of the proposed method are performed to test the performance under different conditions. Finally, the conclusion is given in section 5.
2. Model of signal transmission
So far free-space GMCS CV-QKD is still in the stage of experimental verification. In the free-space CV-QKD channel model, the channel transmittance $T$ is a variable resulted from the effect of beam wandering, broadening, deformation and scintillation [18,27], unlike the fiber-based CV-QKD, where $T$ can be regarded as a constant. In the case of a lossy and noisy channel (see Fig. 1), the transformation of the quadrature variable with encoded information in phase space can be described by the following equation (we take $X-$quadrature as an example, which is basically the same for $P-$quadrature)
It is worth noting that, as in the fiber channel, the quantum state is also affected by the phase drift in the free-space channel. Considering the influence of phase drift, the correlation is given by
Fig. 1. The signal transmission model of free-space CV-QKD. $X_A$ is the Gaussian distributed signal prepared by Alice with a variance of $V_A$, $X_B$ is the measured value normalized to SNU of the received state, ${{\eta _D}}$ is the detection efficiency, $T$ is channel transmittance, $\varepsilon$ is excess noise, ${X_\delta }$ and ${X_{vb}}$ are vacuum states with unity variance.
3. Synchronization for free-space CV-QKD
Here we proposed a high-performance frame synchronization method for CV-QKD systems with variable channel transmittance by inserting two kinds of synchronization frames. We can modulate a special data synchronization frame at the head of each string of quantum signals and channel transmittance synchronization frames at intervals of each string (see Fig. 2). This scheme is able to run at low SNR and suffer random phase drift in the signal transmission process. Moreover, the proposed scheme can ignore the effect of random selection of measurement bases in CV-QKD protocol with homodyne detection, i.e., half of the synchronization frames are saved. Because in the previous methods, the $X-$quadratures or $P-$quadratures of the synchronized frame are used for synchronization and the rest is discarded. In addition, the proposed method can synchronize the monitored channel transmittance data for some experimental scheme that need to monitor channel transmittance.
Fig. 2. The structure of data frame in the free-space CV-QKD system. From top to bottom, the three series are the coherent states sequence sent by Alice, the states sequence received by Bob, and the monitored channel transmittance data.
3.1 Data synchronization
The process of data frame synchronization can be divided into four steps (see Fig. 3). Firstly, an appropriate synchronization code is chosen. Secondly, the selected synchronization code is modulated on the optical pulses, according to a special modulation method. Thirdly, in order to resist arbitrary phase drift, additional rotated synchronization frames are modulated on the optical pulses. Fourthly, legitimate two users find the synchronization point by calculating the correlation.
Fig. 3. The proposed data synchronization scheme. The diagrams of step 2 and step 3 represent the modulation position in phase space.
Step 1: choosing the appropriate synchronization code
In classical communications, frame synchronization is also known as group synchronization. There are two major ways to insert group synchronization codes, one is concentrated insertion, the other is scattered insertion. The proposed method, like most of the previous CV-QKD synchronization method, belongs to concentrated insertion. Synchronization codes, such as Barker codes [28], Willard codes [29], Linde codes [30], etc., are widely used on account of excellent performance. A high-performance synchronization code must satisfy two conditions: first, its auto-correlation function shows a sharp single peak, and second, its sidelobe level is sufficiently low.
For simplicity, Barker codes and their derived synchronization codes are used as synchronization codes in subsequent simulations, because of their well auto-correlation qualities. It is worth noting that not only the Barker code is applicable to our proposed synchronization method. Any synchronization code with excellent auto-correlation characteristics can be used in the synchronization method proposed in this article.
According to our previous experimental experience, even the longest 13-bit Barker code (see Fig. 4) as the frame synchronization code is far from the synchronization requirements of the CV-QKD system. Fortunately, it is not difficult to find a long enough synchronization code. For example, by performing two Kroneker production operations on three sets of 13-bit Barker codes, we can generate a 2197-bit frame synchronization code. A 2197-bit code is more likely to be correctly recognized and is more suitable for frame synchronization code in long-term CV-QKD.
Step 2: modulating field quadratures by special modulations
In the CV-QKD protocol with homodyne detection, randomly selecting the measurement bases is an indispensable step to ensure its theoretical security. This makes the application of traditional communication synchronization codes in CV-QKD seem out of reach. Fortunately, with the help of a special modulation method, the synchronization code in the classical communication can be applied to CV-QKD.
According to CV-QKD protocols, the secret key information is encoded on the quadrature position and the quadrature momentum of quantum signals. However, the encoding process is normally accomplished by using an amplitude modulator and a phase modulator in a practical CV-QKD system.
Synchronization codes can be applied to CV-QKD systems by the modulation methods described below. One can modulate the amplitude value of the optical field to a constant value $\sqrt {2{V_A}}$, and modulate the phase of the optical field chosen from a binary alphabet ${\{\pi /4, 5\pi /4\}}$. More specifically, if the synchronization code value is $+ 1$, one can modulate the phase to $\pi /4$. Similarly, if the synchronization code value is $- 1$, one can modulate the phase to $5\pi /4$. As long as the numbers of $+ 1$ and $- 1$ in the synchronization code are close, the average value is close to 0. Therefore, the power of the optical pulse of the data synchronization frame is almost the same as the optical power of the key data pulse and some security loopholes caused by excessive optical intensity can be avoided.
After modulating the amplitude and the phase of the light field through this special modulation method, the $X-$quadrature and the $P-$quadrature are the same without considering the phase drift. This means that when performing homodyne detection on the optical pulses of data synchronization frame, the random selection of the measurement base will not cause different measurement results. It does not matter which measurement base is chosen, and the results can be used directly for correlation calculations. Therefore, half of the synchronization frames are saved. Alice can negotiate the synchronization code with Bob in advance or announce the synchronization code after Bob measures the quantum state. Then Bob calculates the correlation between the synchronization code and the homodyne detection result, and synchronizes at the position where the function reaches its peak. The peak value of the correlation can be expressed as
From the synchronization codes proposed in step 1 and the modulation method proposed in step 2, we now find a robust synchronization method. In free-space CV-QKD system, the phase drift is a tough problem we have to tackle with, just like the situation in fiber-based CV-QKD system. Here we adopt a commonly used assumption that phase drift $\Delta \varphi$ varies so small during the time of a stable frame, so the phase drift in this frame can be reasonably assumed as a constant. By taking into account the influence of the phase drift of quantum channel, the measurement results of received quantum coherent state $\left | {{X} + i{P}} \right \rangle$ with a homodyne or a heterodyne detector is given by
From the above formulas, we know that phase drift will inevitably cause a decrease in the peak value of the correlation. Compared with the peak value $Cor{r_{peak}} = {V_A} \cdot \sqrt { {\eta _D}} \cdot \langle {\sqrt T } \rangle$ when the phase drift is 0, the peak value of the correlation is approximately $Cor{r_{peak}}/\sqrt 2$ when the phase drift is $\pi /4$ and the peak value decrease to 0 when the phase drift is close to $\pi /2$. If $\Delta \varphi \in ( \pi /2,3\pi /2 )$, the correlation is negative. When the phase drift is $3\pi /4$, the peak value of the correlation is $- Cor{r_{peak}}/\sqrt 2$ , and when the phase drift is $\pi$, the peak value is $- Cor{r_{peak}}$. If $\Delta \varphi$ is close to $\pi$, we can calculate the absolute value of the correlation, and then take the peak value point as the frame synchronization point. Therefore, the main challenge is to successfully synchronize as $\Delta \varphi$ approaches $\pi /2$ or $3\pi /2$, where the correlation is close to 0.
For non-specifically modulated frame synchronization methods, an effective solution of the phase drift is called phase matching [22]. To resist the correlation degradation caused by phase drift, phase matching is to rotate the date published by Alice in phase space and then calculate the correlation between the rotated data and the synchronization code. Unfortunately, this method does not work with our scheme, because $X-$quadrature and $P-$quadrature of our synchronization frames are not Gaussian. Here we can modulate $N$ real rotated synchronization frames following the frame proposed previously. More specifically, we can modulate the phase to $\pi /4 + \Delta {\theta _i}$ if the synchronization code value is $+1$, and modulate the phase to $5\pi /4 + \Delta {\theta _i}$ if the synchronization code value is $-1$, where $\Delta {\theta _i} = \frac {{\pi \cdot i}}{{2 \cdot N}},i = 1, \ldots ,N$. As shown later, while a larger $N$ ensures a larger correlation, but $N = 1$ provides adequate performance for most cases with fewer frame overheads. The correlation coefficient between the measured result without rotated synchronization frames is shown in Fig. 5. The correlation coefficient of the unrotated synchronization frames shows a significant peak at phase drift of $0$ and $\pi$. The correlation coefficient between the measured result with $\pi /2$ rotated synchronization frames is shown in Fig. 6. The correlation coefficient of the rotated synchronization frames shows a significant peak at phase drift of $\pi /2$ and $3\pi /2$. So by modulating additional rotated frames, synchronization at arbitrary phase drift can be accomplished.
The fourth step is calculating the correlation [22,23] between the measured values of received states and the synchronization code. There is no need to describe it specifically here.
3.2 Channel transmittance synchronization
The data synchronization frame insures that Alice’s data is aligned with Bob’s data, and the channel transmittance synchronization frame guarantees that Bob knows the transmittance for each received optical pulse. The proposed channel transmittance synchronization method here belongs to the scattered insertion method, another method of group synchronization.
According to Eq. (1) and Eq. (2), the correlation of Bob’s measurement results of Gaussian distributed $X-$quadrature and $\sqrt T$ is given by (the same for $P-$quadrature)
Simply speaking, by calculating the correlation between $\sqrt {T}$ and the measured data when the transmitted data is Gaussian distributed, we cannot find the correct frame synchronization point. The effect of additive noise can be ignored in analyzation of the peak value of the correlation on account of that it is independent of $\sqrt {T}$ and the quadrature. Then $\sqrt {T}$ can be regarded as a random variable, and the measured data can be considered as the product of $\sqrt {T}$ and another random variable, i.e., the quadrature. According to the probability theory, we know that the correlation between $\sqrt {T}$ and the measured data is close to 0.
By constantly modulating the channel transmittance synchronization frame, the corresponding measured date is related to $\sqrt {T}$. In other words, multiplying a constant modulation value by a variable $\sqrt T$, we can obtain a variable related to $\sqrt T$. Assuming that we modulate amplitude to $A$ and phase to $\varphi$, then we find that ${X_A} = A \cdot \cos \varphi$, ${X_B} = \sqrt {{\eta _D}} \cdot \sqrt T \cdot A \cdot \cos \varphi + {X_N}$, and
In theory, the data synchronization frame can be regarded as a constant modulation. The transmittance of the free-space channel, usually referred to as the atmospheric channel, typically fluctuates at a frequency of several kHz, whereas the operating frequency of most CV-QKD systems is on the order of hundreds of MHz. Therefore, an ideal solution is to insert channel transmittance synchronization frames at a frequency similar to the channel jitters as shown in Fig. 2, which ensures that the features of the channel fluctuation are fully captured. It is worth noting that in the data synchronization process we use one frame to achieve synchronization. However, for channel transmittance synchronization, we need to use as many synchronization frames, or synchronization points as possible to calculate the correlation to improve the success rate.
As will be mentioned later, the variance of channel transmittance $Var( {\sqrt T } )$ is much smaller than both the variance of signal and noise. According to Eq. (10), a small $Var( {\sqrt T } )$ will result in a small peak value of correlation, which makes it difficult to successfully synchronize. There are two ways to improve the performance. Firstly, by increasing the number of frames, we can make the correlation value calculated by the above formula for the incorrect synchronization point closer to the ideal value 0. Inserting more synchronization frames will increase computational cost and decrease the secret key rate. However, the increase in key rate due to an increase in the synchronization success rate significantly outweighs the decrease in key rate due to an increase in the length of synchronized frames. For example, for a frame with a length of $10^5$ pulses, if the synchronization frame is increased by 1000 pulses, which only accounts for $1\%$ of the total frame length, but the increase in synchronization success rate may reach $50\%$. Secondly, by increasing the amplitude value $A$ of the constant modulation, the peak value of the correlation can be significantly increased. It is worth noting that in this case, whether the optical power will exceed the linear response range of the detector and whether it will introduce additional practical security loopholes are questions we need to consider in depth. If $Var( {\sqrt T } )$ is close to 0, the situation is similar to fiber-based CV-QKD, where monitoring and synchronizing channel transmittance data is not necessary.
The effect of phase drift here is similar to date synchronization, so an effective method to deal with it is to insert additional rotated frames. As we will discuss later, inserting a frame that is rotated $\pi /2$ in phase space is sufficient to achieve transmittance frame synchronization in most case.
4. Performance test
In order to explore the effect of channel fluctuation, SNR and phase drift on performance of the presented frame synchronization method, channel transmittance data with different variance are generated by simulation to perform Monte Carlo simulation. Here the process of random selection of the measurement base is simulated, and the result shows that our synchronization method can synchronize modulation data, receive data, and channel transmittance data under the influence of very low SNR and random phase drift.
4.1 Performance test of data synchronization frame
The length of both channel transmittance data and Gaussian signals generated for simulation is 100,000 and each Monte Carlo simulation under the same condition is repeated for 1000 times. The difference in channel transmittance between day and night is mainly reflected in the variance and the mean. Atmospheric turbulence is relatively strong during the day than at night, which results in a wider transmittance distribution in the day. Therefore, the variance of channel transmittance is larger and the mean value is smaller [13,31]. A uniformly distributed model of transmittance for fast-fading channels is explored, which leads to a very high variance of the fluctuations of the beam [31]. The elliptic beam approximation model, which takes the influence of beam wandering, beam shape deformation, and beam-broadening into consideration [26], fits for their experimental data perfectly.
It is shown in Fig. 7 that the distribution of three strings of transmittance which show different levels of fluctuation and a similar mean value. It is obviously that three sets of transmittance have a similar mean value and the larger the variance, the flatter the distribution. The results in Fig. 8 show the values of the variance does not significantly affect the success rate when the mean value is similar, but a larger variance resulted from stronger atmospheric turbulence means a lower mean value in general. On the conditions that the SNR is greater than $-17dB$, the success rate with $2197-$bit synchronization frame can reach almost $100 \%$. As discussed previously, the performance is related to the peak value of correlation $Cor{r_{peak}} = {V_A} \cdot \sqrt {{\eta _D}} \cdot \langle {\sqrt T } \rangle$, where the fluctuating characteristic of $\sqrt T$ can be ignored. Because the main difference between day and night is the fluctuation level of the channel, the above analysis demonstrates that the proposed synchronization method is applicable for both day and night.
Fig. 7. Distribution of channel transmittance data with different variances. The bin width of the distribution is $0.025$. (a) $\langle {\sqrt T } \rangle = 0.62,Var( {\sqrt T } ) = 0.005$. (b) $\langle {\sqrt T } \rangle = 0.61,Var( {\sqrt T } ) = 0.02$. (c) $\langle {\sqrt T } \rangle = 0.60,Var( {\sqrt T } ) = 0.04$.
Fig. 8. Synchronization success rate under different variance of channel transmittance. The three curves correspond to the three sets of channel transmittance data shown in Fig. 7. The synchronization code used is the 2197-bit code obtained by performing two Kronecker productions of the 13-bit Barker code.
The success rate of the presented synchronization method under different SNR conditions is depicted in Fig. 9 with the frame lengths $N = 605,1331,2197$, respectively. Here the 605-bit synchronization code is obtained by performing two Kronecker products of two 11-bit Barker codes and one 5-bit Barker code, the 1331-bit synchronization code is obtained by performing two Kronecker products of three 11-bit Barker codes and the 2197-bit synchronization code is obtained by performing two Kronecker products of three 13-bit Barker codes, respectively. On the conditions that the SNR is greater than $-18dB$, the success rate with $2197-$bit synchronization frame can reach almost $100 \%$. It is obviously that synchronization can be achieved at a lower SNR conditions by applying a longer frame.
Fig. 9. The success rate of the proposed synchronization method under different SNR conditions. The frame lengths from bottom to top are $N = 605,1331,2197$. The mean and variance of the transmittance used for the simulation are $\langle {\sqrt T } \rangle = 0.61,Var( {\sqrt T } ) = 0.02$.
In Fig. 10, we show the influence of different phase drifts $\Delta \varphi$ on the proposed synchronization method. It is obviously that the success rate reaches its lowest value when the phase drifts $\Delta \varphi = 45^\circ ,135^\circ ,225^\circ ,315^\circ$, regardless the length of the synchronization frame. If the aforementioned $\pi /2$ rotated synchronization frames are not inserted, the success rate will close to 0 at phase drifts $\Delta \varphi = 90^\circ ,270^\circ$ and reach to $- Cor{r_{peak}}$ at phase drifts $\Delta \varphi = 180^\circ$. This is because when the phase drift equals $90^\circ$ or $270^\circ$, the received state is given by $S = A \cdot {e^{i\left ( {\varphi + \Delta \varphi } \right )}}$, where $A = \sqrt {2{V_A}}$, $\varphi = \pi /4$ or $5\pi /4$, $\Delta \varphi = \pi /2$ or $3\pi /2$. Then the values of the $X-$quadrature and $P-$quadrature are reversed, and the correlation obtained after random selection of the measurement bases is 0. By inserting additional rotated synchronization frames and taking absolute values for the correlations, the proposed synchronization method works successfully with arbitrary phase drift. When the frame length is $605$ bits, the success rate of this method in the worst case of phase drift is only about $10\%$. When the frame length is 1331 bits, the success rate of the proposed synchronization method is higher than $80\%$, even in the worst case of phase drift. With a synchronization frame length of $2197$ bits, the success rate with arbitrary phase drift is almost always $100\%$.
Fig. 10. The success rate of the proposed synchronization under different phase drift conditions. The frame lengths from bottom to top are $N = 605,1331,2197$. The SNR is $-11dB$. The channel transmittance data and synchronizations frames with different length are the same as in Fig. 7.
4.2 Performance test of transmittance synchronization frame
A set of transmittance data of length 1,000,000 is simulated for performance test of transmittance synchronization frames.
The transmittance distribution used for transmittance synchronization test is shown in Fig. 11(a). The variance of transmittance is $0.05$, which corresponds to the aforementioned high fluctuation condition. The spectrum of these transmittance data is derived by computing their fast Fourier transform (FFT) as shown in Fig. 11(b). It is apparently that the channel transmittance fluctuation is on the order of kHz.
Fig. 11. The channel transmittance data used for Monte Carlo simulation. $\langle {\sqrt T } \rangle = 0.61,Var( {\sqrt T } ) = 0.05$. (a) Distribution of channel transmittance data. The bin width of the distribution is $0.025$. (b) The amplitude spectrum of channel transmittance data.
The success rate under different SNR conditions with different amplitude of the constant modulated signals, i.e., transmittance synchronization frames, is shown in Fig. 12. It increases as the SNR decreases and vice versa. Simulations on the same condition are repeated for 1,000 times. Since the dominate frequency spectrum is below 1kHz, one transmittance synchronization frame is inserted every 500 points, which means the transmittance frames are inserted at a frequency of 2 kHz. So the characteristics of the channel fluctuation are fully captured. The total number of frames used to calculate the correlation is 1000. Note in the simulation each frame contains only one point for convenience, but each frame should contains more points in practical system in order to improve the robustness of the system. In addition, since the transmittance fluctuation does not exceed 1 kHz, the peak point of the correlation can be considered a correct synchronization point as long as it is not more than 500 points away from the preset synchronization point.
Fig. 12. The success rate of the transmittance synchronization method under different SNR conditions. The amplitudes of constant modulation are $A = \sqrt {2{V_A}} ,2\sqrt {2{V_A}}$, where $V_A$ is the modulated variance of the Gaussian signals.
If the SNR is less that $-10dB$, modulating the amplitude to $A = \sqrt {2{V_A}}$, which means the power of the transmittance synchronization frames is equal to the Gaussian signals, is enough to synchronize successfully with a probability of over $80\%$. And if modulating the amplitude to $A = 2\sqrt {2{V_A}}$, the success rate will be higher than $90\%$ when SNR is less than -$15dB$. Synchronization at a lower SNR can be achieved by increasing the constant modulation amplitude, but care must be taken not to saturate the detector. Besides, by increasing the total number of frames used for synchronization, and keeping the frequency of inserting frames, the synchronization success rate can also be improved.
In Fig. 13, we show the influence of different phase drift on the proposed transmittance synchronization method. As with methods that resist lower SNR, the success rate at worse phase drift can be improved by increasing the constant modulation amplitude or increasing the synchronization frame. The effect of different phase drifts on transmittance synchronization is similar to that of data synchronization, so it will not be discussed here.
Fig. 13. The success rate for phase drift from $0^\circ$ to $360^\circ$ with the amplitude of the constant modulation is $A = \sqrt {2{V_A}} ,2\sqrt {2{V_A}}$. The SNR is $-13dB$, the total frame number used for transmittance synchronization is 1,000 and each frame contains one point.
4.3 Algorithm complexity and secret key rate
Computation complexity is also an important metric for synchronization schemes. In a practical CV-QKD system, the synchronization method needs to be implemented in real time, so the computational complexity should not be too high. The time complexity and spatial complexity of the previously proposed synchronization method based on computational correlation are both $O\left ( N \right )$, where N is the length of the synchronization frame. Here the time complexity of our proposed method, which includes two sets of correlation calculations, is given by $O\left ( {{N_{data}} + {N_{trans}} \cdot L} \right )$, where ${N_{data}}$ is the length of one data synchronization frame, $N_{trans}$ is the number of the transmittance frames and $L$ is the length of each transmittance frame, so the time complexity of our method is of the same order as that of the previous algorithms in Refs. [22,23].
In our proposed synchronization method, a portion of the data is used for synchronization. In addition, the actual key rate is also affected by the synchronization success rate, which depends on the length of the synchronized frame and the SNR. The key rate can be derived from the following modified key rate formula [18]
It should be mentioned that we only consider the effect of beam extinction on the transmittance. The influence of temporal pulse broadening on transmittance is relatively small compared to beam extinction, so it can be neglected here. As for the influence of beam wandering, broadening, deformation and scintillation is mainly demonstrated in $Var(\sqrt T )$. It is obviously that the method sacrifices only a small portion of the data for synchronization in free-space CV-QKD. In Fig. 14, we show that the theoretical secret key rate without considering the synchronization frame overhead and the practical secret key with considering the synchronization frame overhead at the same $Var( \sqrt T)$ are very close to each other. So the effects on the secret key rate can be negligible.
Fig. 14. Solid lines indicate the theoretical secret key rate without considering the synchronization frame overhead, and dotted lines indicate the practical secret key with considering the synchronization frame overhead and the impact of synchronization success rate. The block length is $M = {10^5}$, the length of data used for synchronization is $M - N = {N_{data}} + {N_{tran}}$, where $N_{data}=2197$ and $N_{tran}=1000$. The components of the total extinction coefficient are $\alpha _{sca}^{aer}( \lambda ) = 1.64 \times {10^{ - 4}}$, $\alpha _{abs}^{aer}( \lambda ) = 3.35 \times {10^{ - 3}}$, $\alpha _{sca}^{mol}( \lambda ) = 2.52 \times {10^{ - 2}}$, $\alpha _{abs}^{mol}( \lambda ) = 5.49 \times {10^{ - 3}} (k{m^{ - 1}})$, which are the typical parameters for summer. Other Parameters: $V_{A}=1$, $\beta =0.95$, $\eta =0.35$, $\varepsilon =0.01$, $V_{ele}=0.1$.
5. Conclusion
In this paper, we propose a method to synchronize the data for the fluctuation channel by inserting two kinds of synchronization frames, i.e., data synchronization frames and transmittance synchronization frames. We can realize data synchronization by four steps. Besides, for CV-QKD experiments that require monitoring the channel transmittance, the proposed method can be also used to synchronize the channel transmittance with the corresponding received data. Then we analyze the performance of the proposed method under different SNR and phase drift by Monte Carlo simulation. It is indicated that the performance can be significantly improved by increasing the synchronization frame length or the amplitude of constant modulation. Besides, the random selection of the measurement base does not affect the performance of the proposed method. We verify the proposed synchronization scheme with simulation data. The results show the feasibility and high-efficiency of the proposed scheme for practical implementation of atmospheric CV-QKD.
Funding
Science and Technology Commission of Shanghai Municipality (2019SHZDZX01); National Natural Science Foundation of China (61631014, 61671287, 61971276); National Key Research and Development Program of China (2016YFA0302600).
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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