# High-Dimensional and Multi-Intensity One-Photon-Interference Quantum Secure Direct Communication
## 4 Performance  
In this section, we analyze the performance of the HDOPI-QSDC protocol. We denote the channel transmittance from [PERSON] and [PERSON] to [PERSON] as \(\eta=\eta_{d}\sqrt{\eta_{c}}\), where \(\eta_{d}\) is the detection efficiency and \(\eta_{c}\) is the channel loss function. The gain is expressed as  
\[Q_{\mu}^{D_{0}}=Q_{\mu}^{D_{1}}=1-\mathrm{e}^{-2\eta\mu}+2p_{d}\mathrm{e}^{-2 \eta\mu}, \tag{8}\]  
where \(p_{d}\) is the rate of dark counts. The QBER is given by  
\[E_{\mu}^{X,D_{0}}=E_{\mu}^{X,D_{1}}=\frac{\mathrm{e}^{-2\eta\mu}}{Q_{\mu}^{D_{ 0}}}\left(p_{d}+2\eta\mu\delta\right), \tag{9}\]  
where \(\delta\) is the misalignment error. We assume that [PERSON] and [PERSON] use light pulses with infinite numbers of intensities in the multi-intensity mode, thus the yield of the \(\left|2 MI+n\right\rangle\) photon state is  
\[\gamma_{2 MI+n}^{D_{0}}=\gamma_{2 MI+n}^{D_{1}}=1-(1-2p_{d})(1-\eta)^{2M\mathrm{ }+n}. \tag{10}\]  
The simulation parameters are as described in Table 1. Following Equation (7), we illustrate the performance of the HDOPI-QSDC protocol and compare it to the PLOB bound [42], the performance of OPI-QSDC (with an optimized intensity \(u=0.046\) as stated in [49]), DL04 ([14], INCUM-enhanced and with ideal sources), and MDI-QSDC ([19], INCUM-enhanced and with ideal sources) in Figure 2. OPI-QSDC can be regarded as an HDOPI-QSDC protocol with \(M=1\), since it uses \(0\) and \(\pi\) phases in its coding mode. In the case where \(M=2\), our high-dimensional protocol has a higher secrecy capacity and a roughly \(50\,\mathrm{km}\) longer transmission distance than the original. As \(M\) increases, its secrecy capacity starts to reduce but its maximum transmission distance grows slightly. When \(M\geq 5\), its secrecy capacity lags behind that of the original, though within a relatively short range, while its maximum distance outperforms the original by nearly \(60\,\mathrm{km}\). The benefit of further increasing \(M\) diminishes, since the maximum transmission distance hardly lengthens any further and the secrecy channel capacity continues to drop. The determination of a proper \(M\) should be guided by the specific needs of the system in practical applications. That is to say, \(M=1\), i.e., the original OPI-QSDC, provides a balance between practicality, secrecy capacity, and transmission distance, while a more complicated experimental setup leads to a considerable extension in communication distance and secrecy channel capacity when \(M=2,3\), or \(4\). Table 2 provides a concise comparison between our work and previous studies. It highlights key differences in terms of quantum resources, encoding methodologies, security guarantees, reliance on quantum memory, and performance metrics, thereby clarifying the advantages of our approach. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Description** \\ \hline \(\zeta\) & \(0.2\,\mathrm{dB}/\mathrm{km}\) & attenuation coefficient \\ \(\eta_{d}\) & \(14.5\%\) & detector efficiency \\ \(p_{d}\) & \(8\times 10^{-8}\) & dark count rate \\ \(\delta\) & \(1.5\%\) & misalignment error \\ \(f\) & \(1.2\) & FEC efficiency \\ \(u\) & \(0.15\) & light intensity \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Key parameter settings of the simulation.