1. Introduction

Quantum key distribution (QKD) can provide secure symmetric key distribution for distant legitimate parties with information-theoretic security, which relies on fundamental laws of quantum physics [1,2]. QKD using weak coherent states and coherent detection, known as continuous-variable (CV) QKD [3–5], is a promising candidate for practical implementation thanks to its compatibility with the current telecommunication infrastructure and the room-temperature operation. In the past two decades, remarkable progress have been achieved both in theory and experiments of CVQKD [6–16], paving the way for building high secret key rate and classical compatible CVQKD.

In the Gaussian-modulated coherent-state (GMCS) CVQKD protocol [3,4], Alice encodes her secret information on the position or momentum quadrature of coherent states with Gaussian random numbers, and sends them to Bob for coherent detection over a lossy channel. Since the variables of each quadrature are independent of each other, even a small misalignment can lead to a significant reduction of the correlation between the strings of Alice and Bob, consequently the mutual information between them, making it impossible for them to distill identical secret key strings. Therefore, accurate synchronization is of the essence for a CVQKD system to operate efficiently. Frame synchronization is usually one of the synchronization steps in the CVQKD process, which is supposed to find the head of each string to align the frames of Alice and Bob correctly. In practice, there are challenges for frame synchronization that need to be addressed. First, the signal received by the receiver is very weak after long-distance transmission. At the same time, quantum signals are affected by various noise during transmission and detection, which results in a low signal-to-noise ratio (SNR). Second, the quadrature components of the optical field of quantum states will suffer phase shift during signal transmissions. In the transmitted-local-oscillator (TLO) scenario [17–19], the phase difference between quantum signal and local-oscillator (LO) will drift inevitably due to fiber vibration, temperature changes, and asymmetry of interferometers. In addition to this slow phase drift, there is also a fast phase drift that originates from the random phase difference and the frequency offset between two lasers in the local-local-oscillator (LLO) scenario [20–22]. Therefore, it is desirable to design a reliable frame synchronization scheme to cope with low SNR and phase drift.

Primitively, frame synchronization schemes based on modulation of special synchronization frames [23,24] are proposed by referring to the synchronization approach in classical communication. However, these schemes show poor performances under low SNR and random phase drift conditions. Moreover, the synchronization frames of these schemes can be easily distinguished by the eavesdropper from Gaussian distributed quantum signals due to their specific modulations, compromising the security of the systems. To overcome the disadvantages, different approaches with synchronization frames that have the same power with quantum signals have been put forward [25,26]. In these approaches, synchronization signals with excellent auto-correlation property are adopted to adapt the low SNR environment, such as Gaussian distributed signals with phase dissembling and matching [25], or Baker codes encoded pseudo-random sequences [26]. Nevertheless, synchronization frames in these schemes are usually inserted before data frames with designed modulation, furthermore, preset phase shift in some synchronization signals with additional modulations is also indispensable to resist the phase drift. As a result, this modulation-based frame synchronization scheme will increase the complexity of the QKD systems and reduce the duty cycle of quantum signals.

Fortunately, a frame synchronization scheme with quantum signals based on the incremental label and Hamming distance is presented [27]. This scheme is modulation-free that without specified synchronization frames, and can endure high environment noise and phase drift. However, in the case of homodyne detection, only half of the quantum pulses that chosen randomly as synchronization frames can be used in this scheme. In addition, the synchronization frames are discarded after synchronization, which cannot be reused for parameter estimation. More importantly, these schemes are proposed aiming at the TLO scenario, the adaptability for that of the LLO case has not been studied. Recently, a frame synchronization scheme based on Zadoff-Chu sequence encoded signals has been proposed [28], which can resist fast phase drift and can be used for both the TLO and LLO systems. While this scheme still need modulation to produce synchronization frames.

In order to circumvent the aforementioned defects, we propose here a robust modulation-free frame synchronization scheme for CVQKD systems with only quantum states transmitted from Alice to Bob, which is not only capable of resisting high level noise and arbitrary phase drift, but also can be reused for parameter estimation. Moreover, this scheme can be applied in both the LLO and TLO CVQKD systems.

This paper is arranged as follows. In Sec. 2, we describe the proposed frame synchronization scheme. In Sec. 3, the performance of the proposed scheme is analyzed through simulations under different parameter settings, the influence on the secret key rate is compared with the previous schemes are given, and the complexity and security are analyzed as well. Finally, we make a conclusion in Sec. 4.

2. Frame synchronization scheme

We first review the signal model of the GMCS LLO CVQKD protocol, where we take the scheme proposed in Ref. [29] for example, the schematic setup of which is depicted in Fig. 1. At the sender, Alice first uses a continuous-wave (CW) laser to generate coherent light and then uses an amplitude modulator (AM) to convert CW light to pulsed light. She then splits one pulse into two pulses, one of which is modulated by another AM and a phase modulator (PM) as the GMCS quantum signal $\left | {{\alpha _S}} \right \rangle = \left | {{X_A} + i{P_A}} \right \rangle$ with two independent Gaussian random variables $X_A$ and $P_A$ which obey the same zero-centered Gaussian distribution $\mathcal {N}(0,V_A)$, where $V_A$ is the modulation variance. Then Alice sends the quantum signal to Bob through the insecure quantum channel. The other pulse, acting as the phase reference, is also sent to Bob along with the quantum signal via time-multiplexing and polarization-multiplexing techniques by using of a delay line and a polarization beam combiner (PBC), for the sake of recovering arbitrary phase rotation of the quantum signal in phase space. After repeating this process many times, an interleaved quantum signal pulse and phase reference pulse are simultaneously transmitted from Alice to Bob. At the receiver, a polarization controller (PC) is exploited to compensate for the polarization disturbance of signals after fiber channel transmission. Then the quantum pulses and phase reference pulses are separated by a polarization beam splitter (PBS). Bob adopts an independent CW laser to locally generate a high-power LO to measure the quantum signals with a homodyne detector and the phase reference pulses with a heterodyne detector, respectively.

 

Fig. 1. Schematic setup of the GMCS LLO CVQKD. CW, continuous wave laser; BS, beam splitter; AM, amplitude modulator; PM, phase modulator; VOA, variable optical attenuator; PBC, polarization beam combiner; PC, polarization controller; PBS, polarization beam splitter.

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The measurement outcomes of quantum signals can be written as [30]

According to the measurement outcomes $X_R$ and $P_R$ of the phase reference, $\theta _f$ can be estimated as [21]

The difference between the actual phase rotation value of the phase reference pulse and its estimation value, referred to as the phase estimation error $\varphi =\theta _f-{\hat \theta _f}$, is a Gaussian variable such that $\varphi \sim \mathcal {N}(0,V_\varphi )$, where $V_\varphi$ is its variance [29,30].

Considering that Bob has only a single quadrature component due to the homodyne detection, this phase correction has to be performed on Alice’s data as a rotation of her data:

However, since the data of Alice and Bob are misaligned before frame synchronization, this phase correction may be applied to the irrelevant data of Alice, thus leads to no correlation between the strings of Alice and Bob (take the $X$ quadrature as an example):

Once the data of Alice and Bob happen to be aligned, their correlation can be established:

Equation (5) also indicates that the cross-correlation is affected by the phase estimation error $\varphi$ and the slow phase drift $\theta _s$. In the practical scenario, the fast phase drift $\theta _f$ can be precisely estimated by using a relatively intense pilot pulse [30], hence $\varphi$ has a negligible effect on the cross-correlation. While the slow phase drift $\theta _s$ cannot be compensated for prior to frame synchronization, it can have a significant impact on cross-correlation. For instance, the cross-correlation reaches a peak value when $\theta _s=0$, while drops to 0 when $\theta _s =\pi /2$.

Similarly, for the $P$ quadrature we have

To tackle this problem, we can take advantage of the correlations of the two conjugate quadratures as complements, as can be easily obtained as

It is because of the presence of the phase drift, the cross-correlation between the two conjugate quadratures of the states from Alice and Bob could be nonzero. Apparently, these two complementary covariances reach peak values when the slow phase drift is ${\pi \mathord {\left / {\vphantom {\pi 2}} \right.} 2}$. That is, we can always have a nonzero cross-correlation by picking the one with a larger absolute value from the four covariances.

It is worth noting that homodyne detection is employed by Bob on the quantum signals with random basis selection. When using quantum signals for frame synchronization, approximately half of the signals in the frame can be considered as noise due to the independence of the two conjugate quadratures of a quantum state. Fortunately, this problem can be solved by simply reconstructing the sequences of the rotated Alice’s quadratures $X'_A$ and $P'_A$, according to Bob’s basis selection $B$:

Correspondingly, the measurement results of Bob can be written in a general form

When the measurement basis $B=0$ ($B=1$), the quadrature $X$ ($P$) of the received signal is measured, resulting in a measurement result of $M=X_B$ ($M=P_B$).

We can calculate the peak value of the cross-correlation of $S_\nu$ ($\nu =1,2$) and $M$ as (see Appendix A).

By comparing the absolute values of them and always picking the larger one that denoted by ${C_{\max }} =\sqrt {\eta }V_A {e^{ - \frac {{{V_\varphi }}}{2}}} \mathop {\max }_{\nu = 1,2} \left \{ {\left | {{\mu _\nu }} \right |} \right \}$, the normalized peak value of the cross-correlation remains no less than $1/\sqrt 2$ regardless of the slow phase drift, as is clearly shown in Fig. 2.

 

Fig. 2. The normalized $C_\text {max}$ under different values of slow phase drift $\theta _s$.

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As a result, the two covariances in Eq. 10) can be used to conduct frame synchronization by selecting the one with a larger absolute value. Based on this, we proposed a frame synchronization scheme that only use quantum states after the quantum signals transmission and detection phase, as depicted in Fig. 3. The detailed procedures are concluded as follows:

 

Fig. 3. The diagram of the frame-synchronization scheme based on quantum states. Alice multiplexes the quantum signal sequence with the pilot sequence, and sends them to Bob through quantum channel. Bob demultiplexes the received sequence, measuring them using homodyne detection for the quantum signal sequence and heterodyne detection for the pilot sequence. Then Alice randomly selects a segment of the quantum signals, and sends the quadratures of those signals to Bob. Bob follows the proposed frame synchronization scheme to conduct the necessary procedures.

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It is noteworthy that the proposed scheme is applicable to the TLO CVQKD system, eliminating the need for pilot pulses and the corresponding phase rotation. Additionally, the quantum states published to conduct the frame synchronization can be reused in phase compensation and parameter estimation. For phase compensation, from the peak value of cross correlation $C_{S_\nu M}$ one can obtain the slow phase drift parameter $\theta _s$ as

Combining the sign of the correlation value in Eq. (10), the value of phase drift can be ascertained. Subsequently, the synchronization frame associate with more publicly disclosed data are utilized to perform parameter estimation, since there is no difference between them.

3. Performance analysis

In theory, the correlation of two independent Gaussian variables is 0, that is, the transmitted quantum states and the measurement results are totally irrelevant variables when they are not aligned. While in a practical implementation, finite-size effect should be taken into account due to the finite length of $L$, in which case the outcome of the cross-correlation will fluctuate with finite samples.

3.1 Synchronization success rate

For the $i$th bit in the correlation operation, we denote the realizations of $S_{\nu,i}$ and $M_i$ with $s^j_{\nu,i}$ and $m^j_i$ ($\nu =1,2$, and $j \in \{1,2, \ldots,L \}$), respectively. We shall omit all the subscript $i$ for simplicity in the following without causing any confusion. The covariance of $S_\nu$ and $M$ can be estimated by using the maximum-likelihood estimator

When the transmitted and measured sequences are misaligned, they become totally independent of each other and their covariance acts as the noise floor of the cross-correlation profile with a centered normal distribution as follows (see Appendix B):

While when they are aligned, the absolute value of the cross-correlation reaches a peak, whose estimator obeys the following distribution (see Appendix C):

After the correlation operation, the synchronization success rate can be characterized by the probability that the peak value of the cross-correlation exceeds a threshold $C_{th}$,

The error-synchronization probability, also referred to as the false alarm rate $P_\text {FA}$, is defined as the probability that a correlation value in the asynchronous case exceeds the threshold value, which is given by

One effective way to evaluate the synchronization performance is to determine $P_\text {S}$ for a given constant FAR (CFAR). In the following simulations, we set the CFAR to $10^{-3}$, indicating that the error-synchronization probability is less than $0.1{\% }$. This low probability implies that the occurrence of synchronization errors is highly unlikely, providing a strong level of confidence in the synchronization process.

If we set the threshold $C_{th}=\kappa \sigma _f$ with $\sigma _f=\sqrt {V_f}$, the coefficient $\kappa$ is obtained to be approximately 3.09 based on Eq. (19). Then the synchronization success rate can be written as

3.2 Performance evaluation

From Eq. (20) we can figure out that the synchronization success rate depends on the SNR, the length of synchronization frame $L$, the phase noise $V_\varphi$ (variance of the fast phase drift estimation error $\varphi$) and the slow phase drift $\theta _s$.

In Fig. 4 we present contour maps showing the relationship between the slow phase drift and the other three parameters. The range of the slow phase drift is set to $[0,2\pi )$ in all three sub-graphs of Fig. 4, while the other two parameters remain fixed in each sub-graph. It is obvious that the synchronization success rate reaches a minimum value at slow phase drifts $\theta _s=\pi /4,3\pi /4,5\pi /4,7\pi /4$, irrespective of the SNR, $L$, and $V_\varphi$. Additionally, when $\theta _s$ is fixed, the success rate increases as the SNR and $L$ increase, while it decreases with higher values of $V_\varphi$. In order to ensure robust performance of the frame synchronization under arbitrary phase drift during the transmission process, we will set the $\theta _s=\pi /4$ in the subsequent simulations to further studies of the other three parameters. This choice allows us to evaluate the frame synchronization algorithm under worst-case conditions and ensure its effectiveness in challenging scenarios.

 

Fig. 4. Contour maps of the slow phase drift and the other three parameters. (a) Contour map of the slow phase drift and SNR with $L=2^{11},V_\varphi =0.05$ $rad^2$. (b) Contour map of the slow phase drift and $L$ with SNR=$-18$ dB, $V_\varphi =0.05$ $rad^2$. (c) Contour map of the slow phase drift and $V_\varphi$ with $L=2^{11}$,SNR=$-18$ dB.

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Figure 5 demonstrates the impact of SNR on the synchronization success rate, with the frame lengths $L=2^{10},2^{11},2^{12}$. As can be seen, the success rate reduces as the SNR decrease. Conversely, increasing the frame length $L$ leads to improved performance under the same phase drift condition. This is attributed to the reduction in statistical fluctuation of the correlation value resulting from a longer frame length. The performances with different phase noise, i.e., $V_\varphi =0.001,0.005,0.01,0.05,0.1$ (in $rad^2$), are also illustrated in Fig. 5 for comparison. In general, the synchronization success rate with different $V_\varphi$ show analogical trends with different frame lengths. When the frame length is fixed, the synchronization success rate reduces with the growth of $V_\varphi$. However, this reduction can be neglected when the SNR is high or $V_\varphi \leqslant 0.01$. Specifically, the synchronization success rate will be higher than $90{\% }$ when SNR is larger than −20 dB in the case of $L=2^{12}$ and $V_\varphi \leqslant 0.05$, regardless of the slow phase drift. Since the phase noise can be reduced to values as low as 0.01 $rad^2$ or even 0.001 $rad^2$ [32], it has a negligible influence on the performance of frame synchronization. Consequently, the simulation results strongly support the robustness of our proposed scheme when faced with phase noise and arbitrary slow phase drift, even under low SNR conditions.

 

Fig. 5. The success rate versus SNR under different parameter sets. The slow phase drift is set to $\pi /4$ for all curves. The red curves, orange curves and light blue curves are the success rate with $L=2^{10}$, $L=2^{11}$, $L=2^{12}$, respectively. Different symbols with the same color represent the results with different variances of the phase estimation error, i.e., $V_\varphi =0.001,0.005,0.01,0.05,0.1$ (in $rad^2$). The inset is the results for the case of $L=2^{12}$ with the SNR from −18 dB to −20 dB.

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3.3 Secret key rate

In previous literatures, frame synchronization are commonly implemented by using special modulated frames that inserted at the head of quantum signal strings [23–27], or the incremental labels of the randomly selected parts of quantum signals [28], which make no contributions to the parameter estimation. In our proposed synchronization scheme, however, quantum states regarded as synchronization frames can be exploited in the parameter estimation, which will not sacrifice the samples used for final key generation, thereby slightly improving the secret key rate compared with the previous schemes. We can compare their performances by calculating the secret key rate per pulse when considering the finite-size effects [7,33]

Figure 6 shows the secret key rate (SKR) curves with our proposed scheme or with the previous schemes. In Fig. 6(a), these two types of curves almost overlap, which indicates that the lengths of synchronization frames have no significant influence on the secret key rate when the overhead ratio is $a=1.024{\% }$. While the differences between the two types of curves increase with the overhead ratio for all block sizes, as demonstrated in Fig. 6(b)–6(d), in which the overhead ratios are $a=2.048{\% },4.096{\% },8.192{\% }$, respectively. That is, the SKR can be slightly enhanced with our scheme than that with previous schemes when a longer synchronization frame is used.

 

Fig. 6. The the secret key rate curves with our proposed scheme (dashed lines with circle symbols) or with the previous schemes (solid lines with triangle symbols). The overhead ratio for frame synchronization is set to (a) $a=1.024{\% }$; (b) $a=2.048{\% }$; (c) $a=4.096{\% }$; (d) $a=8.192{\% }$. The curves with different colors from left to right respectively correspond to block lengths of $10^8,10^9,10^{10},10^{11}$. Other parameters: $V_A=6$, $\beta =0.95$, $b=10{\% }$, $\eta _d=0.6$, $\nu _{el}=0.1$, $\xi =0.05$, where the last two parameters are the electronic noise of the detector and the excess noise (including phase noise), respectively. The variances are all in shot noise unit.

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3.4 Complexity and security analysis

In a practical CVQKD system, an efficient frame synchronization scheme with a low computational complexity is essential to guarantee the synchronization function well with instantaneity. Since the proposed synchronization scheme is based on the cross-correlation operation, it needs lots of multipliers to be implemented, which is quite costly.

For the sake of simplifying the multiplications, we can apply the method proposed in Ref. [25]. By quantizing the amplitude of the Gaussian distributed quantum signals used as frame synchronization, one can realize the multiplications by the left-shift operation in binary field, which has less complexity. Such quantization may sacrifice the synchronization precision because of the induced quantization noise though, it is worthy for reducing the complexity to ensure the instantaneity of frame synchronization. Consequently, the complexity of multiplier can be reduced to the level same to the adder, which is acceptable in the case of hardware implementation. Besides, the cross-correlation will decrease by half due to the measurement with random basis selection with the scheme presented in Ref. [25], while in our scheme the cross-correlation will not decrease thanks to the construction of sequences $S_{\nu }$, thus reducing requirement of the synchronization frame length.

It is worth mentioning that our scheme is proposed for the homodyne detection scenario where Bob measuring the quantum states with random basis selection, it is also adaptable for heterodyne detection case.

As the synchronization frames are segments of quantum states that randomly selected from the whole data frames in our scheme, so there are no differences between the signals and synchronization frames, i.e., they have the identical features, making it impossible to distinguish for an eavesdropper who intends to attack the CVQKD system through the potential loopholes in the frame synchronization scheme without being founded by the legitimate parties. More importantly, the revealing of these synchronization frames does not leave any useful information about the secret key, and they can be reused in the parameter estimation stage.

4. Conclusion

In conclusion, we have presented a robust frame synchronization scheme for CVQKD systems utilizing quantum signals. The proposed scheme eliminates the need for additional modulation by randomly selecting synchronization frames from the quantum signals. We have provided both a theoretical description and a practical model of the frame synchronization scheme for the LLO CVQKD scenario that consider the finite-size effect. Through thorough simulations, we have demonstrated the excellent performance of our scheme under low SNR conditions, significant phase noise, and slow phase drift. The results highlight its strong tolerance to noise and phase shifts. Furthermore, our scheme slightly improves the secret key rate compared to the use of inserted synchronization frames, without introducing potential loopholes. Since the key idea of the proposed scheme is to use the correlations of the two conjugate quadratures as complements, it is suitable for the discrete modulation protocols that use four or more states [35–37] as well. These results support the feasibility of implementing the proposed scheme for practical long-distance CVQKD applications.