# High-Dimensional and Multi-Intensity One-Photon-Interference Quantum Secure Direct Communication
## 3 Security Analysis  
According to quantum wiretap channel theory, when the capacity of the main channel is higher than that of the wiretap channel, a feasible coding scheme can be found that achieves secure and reliable information transmission. We introduce an equivalent entanglement-based protocol, the details of which are given in Appendix B, and analyze its security so as to determine the achievable secrecy capacity \(R\) of our HDOPI-QSDC protocol. Generally, we know that [51]  
\[R=\max\{I(A:B)-I(A:E),0\}, \tag{1}\]  
where \(I(A:B)\) is the mutual information of [PERSON] and [PERSON] and \(I(A:E)\) is the mutual information of [PERSON] and [PERSON]. Firstly, we consider the achievable secrecy capacity \(R^{D_{0}}\) when only detector \(\mathrm{D}_{0}\) clicks. We assume that [PERSON] and [PERSON] use the \(Z\) basis to transmit information and the \(X\) basis to estimate the amount of information leakage. The channels are treated as cascaded channels of a binary erasure channel (BEC) and binary symmetric channel (BSC). The QBER \(E_{\mu}^{Z,D_{0}}\) and \(E_{\mu}^{X,D_{0}}\), the gain \(Q_{\mu}^{D_{0}}\), and the inefficiency function for FEC \(f\) can be determined through experiments, where \(\mu=|\alpha|^{2}\) represents the light intensity of [PERSON] and [PERSON]. Thus, the mutual information \(I(A:B)\) satisfies  
\[I(A:B)\leq Q_{\mu}^{\mathrm{D_{0}}}\cdot\Big{[}1-fH_{2}\Big{(}E_{\mu}^{Z,D_{0}} \Big{)}\Big{]}, \tag{2}\]  
where \(H_{2}(x)\) is the binary entropy function \(H_{2}(x)=-x\log(x)-(1-x)\log(1-x)\). The upper bound of \(I(A:E)\) is given by  
\[ \begin{split} I(A:E)&\leq Q_{\mu}^{\mathrm{D_{0}} }\cdot H_{2M}\Big{(}E_{\mu}^{X,D_{0}}\Big{)}\\ &=-Q_{\mu}^{\mathrm{D_{0}}}\sum_{n=0}^{2M-1}\lambda_{0n}\log( \lambda_{0n}),\end{split}  \tag{3}\]  
where \(\lambda_{0n}\) is defined as  
\[ \begin{split}\lambda_{0n}&=\frac{1}{Q_{\mu}^{ \mathrm{D_{0}}}}\left(\sum_{l=0}^{\infty}C_{2 Ml+n}\sqrt{Y_{2 Ml+n}^{D_{0}}} \right)^{2}\\ &=\frac{1}{Q_{\mu}^{\mathrm{D_{0}}}}\left(\sum_{l=0}^{\infty} \mathrm{e}^{-|\alpha|^{2}}\frac{(\sqrt{2}\kappa)^{2 Ml+n}}{\sqrt{(2 Ml+n)!}} \sqrt{Y_{2 Ml+n}^{D_{0}}}\right)^{2},\end{split}  \tag{4}\]  
and \(C_{2 Ml+n}\) is the probability amplitude when the number of photons in the channel is \(2 Ml+n\) in the event that only detector D\({}_{0}\) clicks. \(Y_{2 Ml+n}^{D_{0}}\) represents the yield of the \(|2 Ml+n\rangle\) photon state when only detector D\({}_{0}\) clicks, and the details are explained in Appendix B. Therefore, we know that  
\[R^{D_{0}}=\frac{q}{M}\cdot Q_{\mu}^{\mathrm{D_{0}}}\cdot\Big{[}1-fH_{2}\Big{(} E_{\mu}^{Z,D_{0}}\Big{)}-H_{2M}\Big{(}E_{\mu}^{X,D_{0}}\Big{)}\Big{]}, \tag{5}\]  
where \(q=1-2p(1-p)\) is the successful rate of mode matching. The result is similar for the achievable secrecy capacity \(R^{\mathrm{D_{1}}}\) when only detector D\({}_{1}\) responds; that is,  
\[R^{\mathrm{D_{1}}}=\frac{q}{M}\cdot Q_{\mu}^{\mathrm{D_{1}}}\cdot\Big{[}1-fH_ {2}\Big{(}E_{\mu}^{Z,D_{1}}\Big{)}-H_{2M}\Big{(}E_{\mu}^{X,D_{1}}\Big{)}\Big{]}. \tag{6}\]  
Finally, the total achievable secrecy capacity \(R\) of the HDOPI-QSDC protocol is given by  
\[R=\max\Big{\{}R^{\mathrm{D_{0}}},0\Big{\}}+\max\Big{\{}R^{\mathrm{D_{1}}},0 \Big{\}}. \tag{7}\]  
Although the above information-theoretic framework guarantees the feasibility of secure information transmission within our protocol, its resistance to certain well-known attacks should be discussed further. One such attack is the intercept-resend attack, where an adversary [PERSON] attempts to extract information by intercepting, measuring, and then resending quantum states to the intended recipient. However, our protocol is inherently resistant to this attack for the following reasons: First, single-photon interference eliminates the vulnerabilities in direct transmission. Unlike traditional QSDC, [PERSON] and [PERSON] do not exchange qubits directly.