# High-Dimensional and Multi-Intensity One-Photon-Interference Quantum Secure Direct Communication
## Appendix B Details of Security Analysis  
First, we define two entangled pairs  
We first consider the case where only detector \(\mathrm{D}_{0}\) clicks, i.e., \(|(\alpha/\sqrt{2})(\mathrm{e}^{\pi\mathrm{i}(b_{A}/M)}-\mathrm{e}^{\pi\mathrm{i}(b _{B}/M)})\rangle_{D_{1}}=|0\rangle_{D_{1}}\), which implies \(b_{B}=b_{A}\). Thus, we obtain  
\[ \begin{split}|\phi\rangle_{ABD_{0}D_{1}}=&\frac{1}{ 2M}\sum_{b_{A}}|b_{A}\rangle_{A}|b_{A}\rangle_{B}|\sqrt{2}\mathrm{e}^{\pi \mathrm{i}\frac{b_{A}}{M}}\rangle_{D_{0}}|0\rangle_{D_{1}}\\ =&\frac{1}{2M}\sum_{b_{A}}|b_{A}\rangle_{A}|b_{A} \rangle_{B}\Bigg{[}\mathrm{e}^{-|a|^{2}}\sum_{l=0}^{\infty}\frac{(\sqrt{2} \mathrm{e}^{\pi\mathrm{i}\frac{b_{A}}{M}})^{l}}{\sqrt{l!}}|l\rangle_{D_{0}} \Bigg{]}|0\rangle_{D_{1}}\\ =&\frac{\mathrm{e}^{-|a|^{2}}}{2M}\Bigg{\{}\sum_{b_{A }}\mathrm{e}^{\pi\mathrm{i}\frac{b_{A}}{M}}|b_{A}\rangle_{A}|b_{A}\rangle_{B} \Bigg{[}\sum_{l=0}^{\infty}\frac{(\sqrt{2}\alpha)^{2 Ml}}{\sqrt{(2 Ml)!}}|2 Ml \rangle_{D_{0}}\Bigg{]}|0\rangle_{D_{1}}\\ &+\sum_{b_{A}}\mathrm{e}^{\pi\mathrm{i}\frac{b_{A}+1}{M}}|b_{A} \rangle_{A}|b_{A}\rangle_{B}\Bigg{[}\sum_{l=0}^{\infty}\frac{(\sqrt{2}\alpha)^ {2 Ml+1}}{\sqrt{(2 Ml+1)!}}|2 Ml+1\rangle_{D_{0}}\Bigg{]}|0\rangle_{D_{1}}+\cdots \\ &+\sum_{b_{A}}\mathrm{e}^{\pi\mathrm{i}\frac{b_{A}+(2M-1)}{M}}|b_{ A}\rangle_{A}|b_{A}\rangle_{B}\Bigg{[}\sum_{l=0}^{\infty}\frac{(\sqrt{2} \alpha)^{2 Ml+2M-1}}{\sqrt{(2 Ml+2M-1)!}}|2 Ml+2M-1\rangle_{D_{0}}\Bigg{]}|0 \rangle_{D_{1}}\Bigg{\}}\\ =&\frac{\mathrm{e}^{-|a|^{2}}}{\sqrt{2M}}\Bigg{[}| \Phi_{00}\rangle_{AB}\Bigg{(}\sum_{l=0}^{\infty}\frac{(\sqrt{2}\alpha)^{2 Ml}} {\sqrt{(2 Ml)!}}|2 Ml\rangle_{D_{0}}\Bigg{)}|0\rangle_{D_{1}}\\ &+|\Phi_{01}\rangle_{AB}\Bigg{(}\sum_{l=0}^{\infty}\frac{(\sqrt{ 2}\alpha)^{2 Ml+1}}{\sqrt{(2 Ml+1)!}}|2 Ml+1\rangle_{D_{0}}\Bigg{)}|0\rangle_{D_{ 1}}+\cdots\\ &+|\Phi_{0(2M-1)}\rangle_{AB}\Bigg{(}\sum_{l=0}^{\infty}\frac{( \sqrt{2}\alpha)^{2 Ml+2M-1}}{\sqrt{(2 Ml+2M-1)!}}|2 Ml+2M-1\rangle_{D_{0}}\Bigg{)} |0\rangle_{D_{1}}\Bigg{]},\end{split}  \tag{10}\]  
where \(|\Phi_{mn}\rangle\) is the \(2M\)-dimensional Bell state  
\[|\Phi_{mn}\rangle=\frac{1}{\sqrt{2M}}\sum_{j=0}^{2M-1}\mathrm{e}^{\pi\mathrm{i }\frac{m}{M}}|j\rangle|j\oplus m\pmod{2M}\rangle. \tag{11}\]  
After [PERSON]'s measurement, we can trace out systems \(D_{0}\) and \(D_{1}\) from \(|\phi\rangle_{ABD_{0}D_{1}}\), and then obtain the joint state of [PERSON] and [PERSON], which is  
\[ \begin{split}\rho_{AB}&=\sum_{n=0}^{2M-1}\rho_{AB} ^{n}\\ &=\frac{1}{Q}\sum_{n=0}^{2M-1}\left(\sum_{l=0}^{\infty}\mathrm{e}^ {-|a|^{2}}\frac{\left(\sqrt{2}\alpha\right)^{2 Ml+n}}{\sqrt{(2 Ml+n)!}}\right)^{2 }|\Phi_{0n}\rangle\langle\Phi_{0n}|\\ &\equiv\frac{1}{Q}\sum_{n=0}^{2M-1}\left(\sum_{l=0}^{\infty}C_{2 Ml+n}\right)^{2}|\Phi_{0n}\rangle\langle\Phi_{0n}|,\end{split}  \tag{12}\]  
where \(Q\) is a normalization coefficient. We assume that [PERSON] performs a coherent attack using an auxiliary system \(|E\rangle\) and further consider it as a collective attack, applying the quantum de [PERSON] theorem [69]. Therefore, the state that [PERSON], [PERSON], and [PERSON] jointly form is  
\[|\phi\rangle_{ABE}=\sum_{m,n}\sqrt{\lambda_{nm}}|\Phi_{mn}\rangle|E_{mn}\rangle, \tag{13}\]  
where \(|E_{mn}\rangle\) is the orthogonal basis of the auxiliary system \(|E\rangle\).