## 1 Introduction
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., \(f_{\text{PT1}}\)) and PT2 (i.e., \(f_{\text{PT2}}\)) 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. 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:  
\[\hat{X}_{\theta}=\hat{a}_{s}e^{-i\theta}+\hat{a}_{s}^{\dagger}e^{i\theta} \tag{1}\]  
Figure 3: Schematic diagram of the multi-dimensional multiplexing LLO CV-QKD scheme. AM, MZ amplitude modulator; PM, phase modulator; BS, beam splitter; PBS, polarizing beam splitter; VOA, variable optical attenuator; PBC, polarizing beam combiner; SMF, single mode fiber; BHD, balanced homodyne detector; PMOC, polarization maintaining optical couplers; DSO, digital storage oscilloscope. 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:  
\[\hat{V}^{\prime}_{peak}=|\alpha_{l}|\hat{X}^{\prime}_{\theta} \tag{2}\]  
where \(|\alpha_{l}|\) is the average amplitude of LO. In this case, the normalized quadrature components of the signal light \(x^{\prime}_{B}\) and its variance \(\hat{V}^{\prime}_{B}\) are written, respectively, as  
\[x^{\prime}_{B}=\frac{\hat{V}^{\prime}_{peak}}{\sqrt{N_{0}}}=\hat{X}^{\prime}_ {\theta} \tag{3}\]  
\[\hat{V^{\prime}}_{B}=\frac{V(\hat{V}^{\prime}_{peak})}{N_{0}}=V(\hat{X}^{ \prime}_{\theta}) \tag{4}\]  
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], \(|\mathbf{x}_{A}+j\rho_{A}\rangle=|A_{sig}e^{i\phi_{A}}\rangle\). Due to the fact that the result of frequency modulation is not ideal, there will exist a modulation residual in the frequency domain.