A Novel Continuous-Variable Quantum Key Distribution Scheme Based on Multi-Dimensional Multiplexing Technology

Abstract

Dual-polarization division multiplexing (DPDM) is considered to be a potential method to boost the secure key rate (SKR) of the continuous-variable quantum key distribution (CV-QKD) system. In this article, we propose a pilot alternately assisted local local oscillator (LLO) CV-QKD scheme based on multi-dimensional multiplexing, where time division multiplexing and frequency division multiplexing are combined with dual-polarization multiplexing techniques to dramatically isolate the quantum signal from the pilot tone. We establish a general excess noise model for the LLO CV-QKD system to analyze the influence mechanism of various disturbances (e.g., time-domain diffusion, frequency-domain modulation residual, and polarization perturbation) on the key parameters, such as the channel transmittance and excess noise. Specifically, the photon leakage noise from the reference path to the quantum path and that between quantum signals with two different polarization paths are simultaneously analyzed in the dual-polarization LLO CV-QKD scheme for the first time. Furthermore, a series of simulations are established to verify the performance of the proposed scheme. The results show that the maximal isolation degree achieves 84.0 dB~90.4 dB, and the crosstalk between pilot tones and quantum signals can be suppressed to a very small range. By optimizing the system parameters (e.g., modulation variance and repetition frequency), the SKR with 12.801 Mbps@25 km is achieved under the infinite polarization extinction ratio (PER) and 30 dB residual ratio of the frequency modulation in the nanosecond-level pulse width. Moreover, the performance of the proposed DPDM CV-QKD scheme under relatively harsh conditions is simulated; the results show that the SKR with 1.02 Mbps@25 km is achieved under a relatively low PER of 17 dB with the nanosecond-level pulse width and 20 dB residual ratio of the frequency modulation. Our work lays an important theoretical foundation for the practical DPDM LLO CV-QKD system.

1. Introduction

In recent decades, quantum key distribution (QKD) technology has developed rapidly for its information-theoretical security based on the laws of quantum physics, which have potentially wide application in cryptography [1]. The QKD technology can guarantee the information-theoretic secure key exchange between two distant participants, Alice and Bob [2,3]. Due to its high performance in metropolitan areas and compatibility with traditional optical communication systems, the continuous-variable quantum key distribution (CV-QKD) has attracted great interest [4,5,6,7,8,9,10,11].

In the CV-QKD system, the secure key rate (SKR) is severely limited by the excess noise and the channel transmittance. The channel transmittance is influenced by the fiber’s quality and the transmission distance. However, the excess noise is influenced by many factors, among which, time-domain diffusion, frequency-domain modulation residual, and polarization perturbation are the research focuses of this article.

In fiber-based CV-QKD systems, when the quantum signals are transmitted, many kinds of influence, including polarization perturbation, time-domain diffusion, as well as imperfect modulation, will influence the extremely weak quantum signals and introduce excess noise [12,13]. To suppress the photon leakage from the pilot light to the weak quantum light, time division multiplexing (TDM) [12,13,14], polarization division multiplexing (PDM) [12,13,14,15,16], as well as frequency division multiplexing (FDM) techniques [12,17] are used to isolate the quantum signal from the pilot light [12,14,15,16,17,18,19]. In the previous polarization multiplexing LLO CV-QKD scheme, the pilot light and quantum light are modulated on perpendicular polarization modes, respectively [15]. Therefore, the fiber utilization rate is limited. To further improve the performance of the CV-QKD system, the orthogonal dual-polarization scheme has been proposed [16,18]. In this framework, quantum signals are modulated in two polarization directions, and it is expected to double the SKR compared with the single-polarization method. However, the crosstalk between the strong pilot lights and weak signal lights increases dramatically and deteriorates the performance of the CV-QKD system. There is still a lack of research on the quantitative analysis of the crosstalk in the dual-polarization LLO CV-QKD scheme. Therefore, it is of great significance to conduct theoretical research on the orthogonal dual-polarization LLO CV-QKD scheme.

In this paper, we propose a pilot alternately assisted scheme of orthogonal dual-polarization joint time and frequency division multiplexing for the LLO CV-QKD system. Compared with the earlier works (shown in Figure 1) [20], we further optimize the pulse distribution by applying the pilot alternately assisted scheme (shown in Figure 2) to suppress the crosstalk between signal pulses. By the application of polarization multiplexing and time and frequency division multiplexing techniques, the simulation results show that the overall isolation degree can achieve higher than 85.8 dB~90.4 dB, and the crosstalk between pilot tones and quantum signals can be suppressed to a very small range. By optimizing the system parameters (e.g., modulation variance and repetition frequency), the SKR with 12.801 Mbps@25 km is achieved under the infinite polarization extinction ratio (PER) and 30 dB residual ratio of the frequency modulation in the nanosecond-level pulse width. Further, even when the polarization extinction ratio (PER) deteriorates to 17 dB approximately under the nanosecond-level pulse width condition, the SKR achieves higher than 1.02 Mbps@25 km. In addition, we also studied the LLO CV-QKD system for 50 km transmission distance. The results show that the requirement for PER should be above 23 dB, and greater than 0.331 Mbps SKR can still be achieved even if the PER deteriorates to 24 dB. Our work lays an important theoretical foundation for the practical dual-polarization division multiplexing (DPDM) LLO CV-QKD system.

The rest of the paper is organized as follows. In Section 2, the pilot alternately assisted orthogonal dual-polarization CV-QKD scheme based on multi-dimensional multiplexing technology is described. In Section 3, we established a theoretical model of the dual-polarization LLO CV-QKD system, which is based on the influence mechanism of polarization perturbation, time-domain diffusion, and the residual ratio of the frequency modulation. Section 4 describes the simulation results to verify the effectiveness of the proposed dual-polarization scheme, and a brief conclusion is given in Section 5. Finally, in Appendix A, the security framework of the GMCS protocol is introduced.

2. The Pilot Alternately Assisted Orthogonal Dual-Polarization CV-QKD Scheme Based on Multi-Dimensional Multiplexing Technology

According to different arrangements of local oscillator (LO), there are two essential CV-QKD schemes, namely the transmitting LO (TLO) and local LO (LLO) CV-QKD [21,22,23,24,25]. Compared with the TLO scheme, in the LLO scheme, instead of transmitting local oscillator light in the optical fiber channel, the reference light named the pilot tone is transmitted for phase recovery and polarization compensation. As shown in Figure 1, in the previous polarization multiplexing scheme [15,20], due to the fact that the pilot pulse has the same optical frequency as the quantum signal pulse, the photon leakage from the pilot pulse to the quantum signal is mainly isolated by the time division multiplexing technique. As a result, the degree of pulse isolation is only limited by the interval between the pilot pulse and quantum signal, which is related to the repetition frequency. To control the photon leakage, the repetition frequency and SKR are limited. For higher repetition frequency and SKR, a larger isolation degree between the pilot pulse and quantum signal is required.

To further isolate the pilot pulse and the quantum light, the frequency multiplexing method is used in our scheme. As is shown in Figure 2b, the pilot tone is modulated by single-sideband modulation, which can increase the optical bandwidth efficiency and increase the isolation degree. Comparing with modulating the quantum signal, in the actual experiment, it is difficult to modulate the quantum signal with single-sideband modulation, and the quantum signal will become unstable. By the application of single-sideband modulation, the isolation degree between the pilot tone and the quantum signal is increased.

Furthermore, by optimizing the pulse distribution of the quantum signal and pilot tone, the crosstalk can be further reduced. As shown in Figure 2a, the quantum signal and pilot tone are loaded in the two polarization directions. Without loss of generality, the proportion of 4:1 for signal and pilot pulse is set in this paper, as shown in Figure 2a. Here, the quantum signal is modulated as a Gaussian modulated coherent state (GMCS). Specifically, the quantum signals 1 (QS1) and 3 (QS3) together with PT1 are modulated at the vertical polarization. The GMCS quantum signals 2 (QS2) and 4 (QS4) together with the PT2 are modulated at the horizontal polarization. In previous works, we found that the crosstalk is mainly from PT1 and PT2, which constrained the repetition frequency and limited the SKR of the dual-polarization LLO CV-QKD system. To further optimize the isolation between quantum and pilot tone signals, frequency division multiplexing (FDM) technology is also used. The frequency of pilot tones, including PT1 and PT2, is shifted in the frequency domain by the single-sideband modulation. As shown in Figure 2b, the center frequency of PT1 (i.e., $f_{PT1}$) and PT2 (i.e., $f_{PT2}$) is $f_A+f_m$, while the center frequency of the quantum signal (i.e., $f_q$) is $f_A$. Due to the symmetrical distribution of quantum signal lights and the alternate distribution of reference lights, when we analyze the influence mechanism of polarization disturbance, time-domain photon diffusion, and frequency-domain modulation residual, QS1 and QS4 are equivalent on average. Accordingly, QS2 and QS3 are also equivalent. Therefore, we only analyze QS1 and QS2 in the following contents. Figure 3 is a schematic diagram of the multi-dimensional multiplexing LLO CV-QKD scheme.

In the alternately positioned pulses distribution of the quantum signal and pilot tone (pilots alternately assisted in the two polarization directions), together with the application of frequency division multiplexing, time division multiplexing, and dual-polarization division multiplexing, namely the multi-dimensional multiplexing scheme we proposed, smaller crosstalk from the strong reference light to the weak quantum signal is expected to be achieved.

3. Theoretical Analysis of the Crosstalk between Pilot and Quantum Signal in the Multi-Dimensional Multiplexing LLO CV-QKD System

In this section, we analyze how the leakage, including the time- and frequency-domain diffusion and polarization disturbance, influences the key parameters (i.e., channel transmittance and excess noise) of the LLO CV-QKD system. As a result, we establish a theoretical model for the dual-polarization scheme, which provides a systematic theoretical framework for analyzing the SKR of the dual-polarization CV-QKD system.

Firstly, we consider the ideal situation. The quadrature component of the signal light can be written as:

$${\hat{X}}_{\theta }={\hat{a}}_s{e}^{-i\theta }+{\hat{a}}_s^{†}{e}^{i\theta }$$

(1)

where ${\hat{a}}_s$ is the annihilation operator of the signal light, and $\theta$ denotes the relative phase difference between the signal light and LO.

Then, we consider the case that the situation is not perfect. Polarization disturbance, time-domain diffusion, and frequency modulation residual all lead to photon leakage from the reference path to the weak quantum signal mode [13]. Compared with the previous single-polarization scheme, the leakage from the quantum signal at the other polarization should also be considered.

Moreover, due to the use of LLO, the local oscillator is not affected by polarization disturbance. Therefore, the output of the balanced homodyne detector (BHD) can be written as:

$${{\hat{V}}^{\prime }}_{peak}=\left|{\alpha }_l\right|{{\hat{X}}^{\prime }}_{\theta }$$

(2)

where $\left|{\alpha }_l\right|$ is the average amplitude of LO. In this case, the normalized quadrature components of the signal light $x_B^{\prime }$ and its variance ${\hat{V}}_B^{\prime }$ are written, respectively, as

$${x^{\prime }}_B=\frac{{{\hat{V}}^{\prime }}_{peak}}{\sqrt{N_0}}={{\hat{X}}^{\prime }}_{\theta }$$

(3)

$${\widehat{V^{\prime }}}_B=\frac{V({{\hat{V}}^{\prime }}_{peak})}{N_0}=V({{\hat{X}}^{\prime }}_{\theta })$$

(4)

where $N_0$ is the shot noise. Based on the previous analysis in Section 2, we take QS1 and QS2 as examples for the following analysis. The optical signal is attenuated by a variable optical attenuator (VOA) to be quantum signals with GMCS [17], $\left|{\mathrm{x}}_A+j{p}_A\right.〉=\left|A_{sig}{e}^{j{\varphi }_A}\right.〉$. Due to the fact that the result of frequency modulation is not ideal, there will exist a modulation residual in the frequency domain. After the single-sideband modulation, the pilot tone is given by:

$$E_{\mathrm{ref}}=\sqrt{1-h}{A}_{ref}{J}_1(m_f)\cos (2\pi f_{A}t\pm 2\pi f_{m}t+{\phi }_A)$$

(5)

Then, the quantum signal can be written as:

$$\begin{array}{l}{E}_{sig1}^{\text{'}}=\sqrt{1-h}{A}_{sig}\cos (2\pi f_{A}t+{\varphi }_A+{\phi }_A)+\sqrt{h}{A}_{sig}\cos (2\pi f_{A}t+{\varphi }_A+{\phi }_A)\\ +\sqrt{(1-h)q_{1}s}{A}_{ref}{J}_1(m_f)\cos (2\pi f_{A}t\pm 2\pi f_{m}t+{\phi }_A)\\ +\sqrt{h{q}_{2}s}{A}_{ref}{J}_1(m_f)\cos (2\pi f_{A}t\pm 2\pi f_{m}t+{\phi }_A)\end{array}$$

(6)

$$\begin{array}{l}{E}_{sig2}^{\text{'}}=\sqrt{1-h}{A}_{sig}\cos (2\pi f_{A}t+{\varphi }_A+{\phi }_A)+\sqrt{h}{A}_{sig}\cos (2\pi f_{A}t+{\varphi }_A+{\phi }_A)\\ +\sqrt{(1-h)q_{2}s}{A}_{ref}{J}_1(m_f)\cos (2\pi f_{A}t\pm 2\pi f_{m}t+{\phi }_A)\\ +\sqrt{h{q}_{1}s}{A}_{ref}{J}_1(m_f)\cos (2\pi f_{A}t\pm 2\pi f_{m}t+{\phi }_A)\end{array}$$

(7)

where $q_1$ and $q_2$ are the time-domain attenuation factor due to the finite time-domain extinction ratio of PT1 and PT2, respectively. $\overline{h}={10}^{-R/10}$ is the average attenuation coefficient of the signal light due to the finite PER. R is the recovered PER of Bob’s side. $\mathrm{s}=J_0\left({\mathrm{m}}_f\right)/J_1\left({\mathrm{m}}_f\right)$ denotes the residual ratio of the frequency modulation, among which $J_0\left({\mathrm{m}}_f\right)$ denotes the residual intensity of the frequency modulation in the original frequency position of $f_A$, and $J_1\left({\mathrm{m}}_f\right)$ denotes the first-order Bessel function of the first kind and the actual intensity of the frequency modulation in the subsequent frequency position of $f_A+f_m$. $A_{ref}$ and $A_{sig}$ denote the modulated amplitude of the pilot tone and the quantum signal. $f_A$ denotes the center frequency of the CW laser. $f_m$ and $m_f$ correspond to modulation frequency and modulation index of the frequency modulation. ${\phi }_A$ denotes the phase of Alice’s laser.

After the channel disturbance, the resulting annihilation operator of the pilot ${\hat{\alpha }}_{ref}^{\prime }$ and the quantum signal lights and signal field ${\hat{a}}_{s1}^{\prime }$, ${\hat{a}}_{s2}^{\prime }$ are given by:

$${\widehat{{\alpha }^{\prime }}}_{ref}\approx \sqrt{(1-h)}\left|{\alpha }_{ref}\right|e^{i{\beta }_0}$$

(8)

$${{\hat{a}}^{\prime }}_{s1}=\sqrt{1-h}{\hat{a}}_{s1}+\sqrt{h}({\hat{a}}_{s2}+\delta {\hat{a}}_V)+\sqrt{(1-h)q_{1}s}\left|{\alpha }_{ref}\right|e^{i{\phi }_1}+\sqrt{h{q}_{2}s}\left|{\alpha }_{ref}\right|e^{i{\phi }_2}$$

(9)

$${{\hat{a}}^{\prime }}_{s2}=\sqrt{1-h}{\hat{a}}_{s2}+\sqrt{h}({\hat{a}}_{s1}+\delta {\hat{a}}_V)+\sqrt{h{q}_{1}s}\left|{\alpha }_{ref}\right|e^{i\phi 1}+\sqrt{(1-h)q_{2}s}\left|{\alpha }_{ref}\right|e^{i{\phi }_2}$$

(10)

where ${\beta }_0$ is the relative phase difference between the signal light and pilot light. ${\phi }_1$ is the relative phase difference between the leaked photons of PT1 and quantum signal (QS1 and QS2), while ${\phi }_2$ is the relative phase difference between the leaked photons of PT2 and quantum signal (QS1 and QS2).

After the channel disturbance, the normalized quadrature components of QS1 and QS2 are given by:

$${\mathrm{x}\prime }_{\mathrm{B}1}=\sqrt{1-\mathrm{h}}{\hat{\mathrm{X}}}_{\theta 1}+\sqrt{\mathrm{h}}{\hat{\mathrm{X}}}_{\theta 2}+\sqrt{\mathrm{h}}{\hat{\mathrm{X}}}_{\mathrm{V}}+2\sqrt{{(1-\mathrm{h})\mathrm{q}}_{1}s}\left|{\mathsf{\alpha }}_{\mathrm{ref}}\right|{\cos (\mathsf{\theta }-\mathsf{\phi }}_1)+2\sqrt{{\mathrm{hq}}_{2}s}\left|{\mathsf{\alpha }}_{\mathrm{ref}}\right|{\cos (\mathsf{\theta }-\mathsf{\phi }}_2)$$

(11)

$${\mathrm{x}\prime }_{\mathrm{B}2}=\sqrt{1-\mathrm{h}}{\hat{\mathrm{X}}}_{\theta 2}+\sqrt{\mathrm{h}}{\hat{\mathrm{X}}}_{\theta 1}+\sqrt{\mathrm{h}}{\hat{\mathrm{X}}}_{\mathrm{V}}+2\sqrt{{\mathrm{hq}}_{1}s}\left|{\mathsf{\alpha }}_{\mathrm{ref}}\right|{\cos (\mathsf{\theta }-\mathsf{\phi }}_1)+2\sqrt{{(1-\mathrm{h})\mathrm{q}}_{2}s}\left|{\mathsf{\alpha }}_{\mathrm{ref}}\right|{\cos (\mathsf{\theta }-\mathsf{\phi }}_2)$$

(12)

yielding the variances:

$$V({x^{\prime }}_{B1})=V({\hat{\mathrm{X}}}_{\theta })+\langle n_{e1}\rangle +4(\langle n_{e2}\rangle +\langle n_{e3}\rangle )\langle \cos {(\theta -\phi )}^2\rangle +\overline{h}$$

(13)

$$V({x^{\prime }}_{B2})=V({\hat{\mathrm{X}}}_{\theta })+\langle n_{e4}\rangle +4(\langle n_{e5}\rangle +\langle n_{e6}\rangle )\langle \cos {(\theta -\phi )}^2\rangle +\overline{h}$$

(14)

where $\langle n_{e1}\rangle$ is the leaked photons from QS2 to QS1, and $\langle n_{e4}\rangle$ is the leaked photons from QS1 to QS2. $\langle n_{e2}\rangle$ and $\langle n_{e3}\rangle$ denote the photon leakage from the PT1 and PT2 to the QS1 mode, respectively. Correspondingly. $\langle n_{e5}\rangle$ and $\langle n_{e6}\rangle$ denote the photon leakage from the PT1 and PT2 to the QS2 mode, respectively.

Specifically,

$$\langle n_{e1}\rangle =\langle n_{quantum}\rangle h$$

(15)

$$\langle n_{e2}\rangle =\langle n_{pilot}\rangle q_1(1-h)s$$

(16)

$$\langle n_{e3}\rangle =\langle n_{pilot}\rangle q_{2}hs$$

(17)

$$\langle n_{e4}\rangle =\langle n_{quantum}\rangle h=\langle n_{e1}\rangle$$

(18)

$$\langle n_{e5}\rangle =\langle n_{pilot}\rangle q_{1}hs$$

(19)

$$\langle n_{e6}\rangle =\langle n_{pilot}\rangle q_2(1-h)\mathrm{s}$$

(20)

Correspondingly, the resulting channel transmittance and excess noise are given by:

$${T^{\prime }}_0=\frac{{\langle x_A{x^{\prime }}_B\rangle }^2}{\eta {\langle x_A^2\rangle }^2}=T_0(1-\overline{h})$$

(21)

$${{\epsilon }^{\prime }}_{01}={\epsilon }_0+\frac{\langle n_{e1}\rangle +4(\langle n_{e2}\rangle +\langle n_{e3}\rangle )\langle \cos {(\theta -\phi )}^2\rangle +\overline{h}}{\eta T_0(1-\overline{h})}+\frac{(1+v_{ele})\left(1-\frac{1}{1-\overline{h}}\right)}{\eta T_0}$$

(22)

$${{\epsilon }^{\prime }}_{02}={\epsilon }_0+\frac{\langle n_{e4}\rangle +4(\langle n_{e5}\rangle +\langle n_{e6}\rangle )\langle \cos {(\theta -\phi )}^2\rangle +\overline{h}}{\eta T_0(1-\overline{h})}+\frac{(1+v_{ele})\left(1-\frac{1}{1-\overline{h}}\right)}{\eta T_0}$$

(23)

where $T_0$ and ${\epsilon }_0$ denote the initial channel transmittance and excess noise when the recovered situation is perfect. ${\epsilon }_{01}^{\prime }$ and ${\epsilon }_{02}^{\prime }$ denote the excess noise of QS1 and QS2 modes after channel disturbance, respectively.

As shown in Equations (21)–(23), due to the channel disturbance, the polarization rotation angle shifted, which leads to both channel transmittance and excess noise deteriorated. Therefore, the SKR of the CV-QKD system decreases accordingly, namely deteriorating the performance of the CV-QKD system.

In the following part of this paper, several simulations are conducted based on the dual-polarization model established in this section. Furthermore, the system parameters are optimized to reduce the requirement for higher PER.

4. Simulations and Performance Estimation

In this section, a CV-QKD system is demonstrated under the optimal parameters (e.g., modulation variance and repetition frequency) for the orthogonal dual-polarization joint time and frequency division multiplexing model established in Section 2 by numerical simulations.

4.1. Influence Mechanism of the PER on SKR under Different Transmission Distances

In this section, a simulation design is conducted to compare the accuracy requirements of polarization extinction ratio (PER) under different transmission distances, including 15 km, 25 km, 50 km, and 100 km. Here, we set the residual ratio of the frequency modulation s = 0.01.

As shown in Figure 4, with increasing transmission distance, to maintain a relatively high SKR, the requirement for PER increases correspondingly. Specifically, the SKRs of 15 km, 25 km, 50 km, and 100 km transmission distances reduce to zero when the PER is below 21.93 dB, 22.23 dB, 25.95 dB, and 30.9 dB, respectively. Take the 25 km transmission distance as an example; the SKR descends sharply during the PER range from 22.23 dB to 27 dB (approximately 60% of the SKR under the perfect PER case). This work provides the accuracy requirement of PER for subsequent polarization compensation algorithm design in digital signal processing methods.

4.2. Research on Joint Optimization of Multiple Parameters

In this section, the parameters are optimized for the dual-polarization scheme we proposed in Section 2. For the purpose of researching the dual-polarization LLO CV-QKD under relatively low PERs, as an example, the PER is set to 20 dB. Figure 5 shows the SKR versus different modulation variance (V_A) and transmission distance. The red curve denotes that when the transmission distance is 25 km, to maximize the SKR, the optimal V_A is set to 1.6. In this case, the maximum value of SKR appears at the point indicated by the small red circle, and the maximum SKR is 5.858 Mbps. Compared with previous research works [20], the SKR has been greatly improved. Furthermore, our work also laid a solid foundation for further solving the high requirements problem of PER for the dual-polarization CV-QKD system in the future. We will analyze the details in the following contents.

4.3. The Influence of the Pulse Width and Research on Repetition Frequency

To match the accuracy requirement of polarization extinction ratio (PER) in the practical systems, we designed a simulation analysis for a relatively low polarization extinction ratio (PER) below 20 dB and conducted a multi-parameter joint optimization study under the condition of PER ≤ 20 dB.

According to the earlier works [12], the photon leakage between the pilot tone and quantum signal can be estimated as:

$$〈n_{pilot}^e〉=q〈n_{pilot}〉$$

(24)

where $<n_{pilot}>$ is the average photon number of the pilot tone, $<n_{pilot}^e>$ denotes the leaked photon number to the quantum signal mode, and q is the overlapping factor, which is related to the pulse extinction ratio. Assuming a Gaussian pulse shape, the overlapping factor q is given by:

$$q=e^{-\frac{{\Delta }_t^2}{2{\sigma }_t^2}}$$

(25)

where ${∆}_t$ is the time delay between the pilot and the quantum signal. In this scheme, it is related to the repetition frequency $f_{rep}$, which can be written as:

$${\Delta }_t=\frac{1}{f_{rep}}$$

(26)

${\sigma }_t$ is related to the full width at half maximum of the pulse ${\sigma }_{FW}$,

$${\sigma }_t=\frac{{\sigma }_{FW}}{2\sqrt{\ln 2}}$$

(27)

In the dual-polarization joint time and frequency division multiplexing CV-QKD system we proposed, the pulse width and repetition frequency will affect the leaked photons < n_e1 >, < n_e2 >, < n_e3 >, < n_e4 >, < n_e5 >, and < n_e6 > in Equations (18)–(23). In the following analysis, we take the 1 ns pulse width as an example to conduct research on repetition frequency under the optimal modulation variance that we concluded in Section 4.2. Here, we set the transmission distance as 25 km and the modulation variance VA as 1.6.

As shown in Figure 6, the SKR increases with the repetition frequency for different PERs when the repetition frequency ≤302.5 MHz. In this case, the maximum SKR can be achieved during the repetition frequency of 292.5–302.5 MHz. Then, with increasing repetition frequency, the SKR goes into decline. This is mainly due to the fact that the pulse extinction ratio, or the overlapping factor q of pilot tones in the QS1 time-domain position, increased when the pulse repetition frequency increases to a relatively high level.

From Figure 6 and Figure 7, it can also be concluded that different PERs have different optimal repetition frequencies. As the dashed line shows, when the PER increases, the system has more redundancy for time-domain diffusion. At the bottom right of the dashed line, the repetition frequency becomes the main limiting factor for improving the SKRs. However, in the upper-left corner of the dashed line, PER is the main factor restricting the improvement of SKRs. First, we consider the ideal case in Figure 6, namely the PER is infinite. As the red solid line shows, the optimal repetition frequency can be higher than 302.5 MHz, and the maximum SKR is 11.74 Mbit/s. In this case, the time-domain attenuation factor $q_1 \text{is} 2.6337\times {10}^{-7}$, equivalent to −65.8 dB, and $q_2 \text{is} \text{equal} \text{to} 4.8110\times {10}^{-27}$ or −263.1 dB. When the PER deteriorates to 20 dB, the maximum SKR is about 5.863 Mbps under the 297.5 MHz optimal repetition frequency. In this situation, $q_1=1.5760\times {10}^{-7}$, or −68.0 dB; $q_2=6.1684\times {10}^{-28}$ or −272.1 dB. Then, when the PER deteriorates to 18 dB, an SKR of 2.933 Mbps@25 km can still be achieved based on the dual-polarization LLO CV-QKD system under the theoretical model we established in Section 3. The corresponding time-domain attenuation factor $q_1=9.1832\times {10}^{-8}$, or −70.4 dB, and $q_2=7.1116\times {10}^{-29}$, or −281.5 dB. Furthermore, as Figure 7 shows, when the residual ratio of the frequency modulation s is set to 0.01, the maximum SKR can achieve 12.801 Mbps@25 km, and the optimal repetition frequency increases to 334 MHz correspondingly.

By comparing Figure 6 and Figure 7, we can conclude that the optimal repetition frequencies and SKRs increase due to the application of the frequency division multiplexing technique. By optimizing the frequency-domain residual ratio from s = 0.01 to s = 0.001, the scale of optimal repetition frequency increases to 319.0–334.0 MHz, and the SKRs increase correspondingly. The reason for this is that the use of frequency division multiplexing reduces the high requirement for time-domain isolation degree.

As is shown in

Table 1. The total degree of isolation between the pilot tones and quantum signals.

s	PER	Optimal Repetition Frequency	q1	q2	The Total Degree of Isolation	SKR
−20 dB	inf	302.5 MHz	−65.8 dB	−263.1 dB	≈−85.8 dB	11.740 Mbit/s
−20 dB	20 dB	297.5 MHz	−68.0 dB	−272.1 dB	≈−88.0 dB	5.863 Mbit/s
−20 dB	18 dB	292.5 MHz	−70.4 dB	−281.5 dB	≈−90.4 dB	2.933 Mbit/s
−30 dB	inf	334.0 MHz	−54.0 dB	−215.9 dB	≈−84.0 dB	12.801 Mbit/s
−30 dB	20 dB	325.0 MHz	−57.0 dB	−228.0 dB	≈−87.0 dB	6.373 Mbit/s
−30 dB	18 dB	319.0 MHz	−59.2 dB	−236.7 dB	≈−89.2 dB	3.175 Mbit/s

The factor s denotes the residual ratio of the frequency modulation, inf means positive infinity. PER denotes the polarization perturbation in the quantum channel. q₁ and q₂ are the time-domain attenuation factor due to the finite time-domain extinction ratio of PT1 and PT2, respectively.

, we set the residual ratio of the frequency modulation as −20 dB and −30 dB to evaluate the overall isolation degree. By comparing the cases of s = −20 dB and s = −30 dB, we can conclude that the isolation degrees of frequency and time division multiplexing are correlated with each other. When the frequency-domain residual ratio s reduces, the time-domain attenuation factors q₁ and q₂ increase accordingly, and the result is that the repetition frequencies increase to the optimal values, and the SKRs increase to the maximum. Furthermore, from Equations (15)–(20), we can conclude that < n_e2 > is the main influencing factor that increases the total excess noise. Thus, the total degree of isolation is mainly determined by s and q₁. Based on the dual-polarization LLO CV-QKD model we proposed in Section 3, we achieved an overall isolation degree higher than 84.0 dB~90.4 dB, and the crosstalk between pilot tones and quantum signals can be limited to a very small range. Our work lays an important theoretical foundation for the practical LLO CV-QKD system.

By the above research and analysis, the orthogonal dual-polarization multiplexing LLO CV-QKD system under relatively low PER conditions has been achieved, which is of great significance for the practicality of the dual-polarization CV-QKD system.

4.4. SKR Analysis under Relatively low PER

Based on the optimal parameters from the above sections, we investigate the SKR of different transmission distances, especially 25 km and 50 km under relatively low PER conditions. The repetition frequency is set to 292.5 MHz, and the modulation variance V_A is set to 1.6. The results are shown in Figure 8. As can be seen from Figure 8, different PERs have different transmission distance limitations. When the transmission distance is 25 km, the required PER should be above 17 dB, which is a relatively low scale for the polarization division multiplexing (PDM) system in the actual experiment. The results show that under the proposed orthogonal dual-polarization scheme in Section 2, higher than 1.02 Mbps SKR can still be achieved even if the PER deteriorates to 17 dB. When the transmission distance is 50 km, as the yellow and green lines show, the required PER should be above 23 dB, which is a relatively low scale for the middle-length transmission distance PDM CV-QKD system. The results show that higher than 0.331 Mbps SKR can still be achieved even if the PER deteriorates to 24 dB. Thus, the proposed scheme has an excellent PER tolerance performance for the dual-polarization LLO CV-QKD system.

5. Conclusions

In this paper, we theoretically proposed the pilot alternately assisted scheme of orthogonal dual-polarization joint time and frequency division multiplexing for the LLO CV-QKD system. By analyzing the influence mechanism of polarization perturbation, time-domain diffusion, and frequency-domain modulation residual, the orthogonal dual-polarization theoretical model was established to analyze the influence of the polarization extinction ratio (PER) on the key parameters of the CV-QKD system quantitatively. To investigate the performance of the proposed orthogonal dual-polarization scheme, a series of simulations were conducted to verify this theoretical model. Simulation results indicate that as the transmission distance increases, the required PER increases correspondingly to guarantee the performance of quantum key distribution. The overall isolation degree between pilot tones and quantum signals achieves 85.8 dB~90.4 dB, which means the noise due to crosstalk between pilot tones and quantum signals was suppressed efficiently. Then, by optimizing the modulation variance, the required PER reduces accordingly. In the 25 km transmission distance, when the pulse width is fixed to 1 nanosecond, the scale of optimal pulse repetition frequency rate is 292.5–302.5 MHz during the PER≥18 dB scale. Compared with the previous literature, the SKR with 0.0197 bit/pulse@25 km is achieved in the proposed dual-polarization scheme based on multi-dimensional multiplexing, which is significantly higher than 0.0078 bit/pulse@25 km in ref. [26]. Furthermore, we analyzed the SKR under a relatively low PER. The results show that more than 1.02 Mbps SKR can still be achieved even if the PER deteriorates to 17 dB. Our work greatly extends the application scenarios of the orthogonal dual-polarization multiplexing CV-QKD system. Consequently, the proposed pilot alternately assisted scheme of dual-polarization joint time and frequency division multiplexing has high potential for improving the performance of LLO CV-QKD. With the research deepening (i.e., larger block size, narrower pulse width, higher pulse isolation degree, and higher repetition frequency), we believe the proposed dual-polarization LLO CV-QKD scheme can be widely applied in practical confidential communication scenarios in the future.