# One-Photon-Interference Quantum Secure Direct Communication
## Appendix B Details of Security Analysis  
In this section, we analyze the security of the OPI-QSDC protocol by analyzing the equivalent entanglement-based protocol. We first introduce the equivalence transformation process of our protocol, as shown in Figure A2. For convenience, we briefly show the process of the OPI-QSDC protocol in Figure A2a. [PERSON] is not credible, so we replaced him with [PERSON]. [PERSON] and [PERSON] use a coherent source, and randomly select one of two modes to send signals to [PERSON]. In coding modes,the encoding of Alice includes all operations in **step 1**. [PERSON] deterministically sends state \(|\alpha\rangle_{a}\) or \(|-\alpha\rangle_{a}\) to [PERSON] depending on the encoding codes, while [PERSON] randomly sends it. In multi-intensity modes, [PERSON] and [PERSON] send different amplitude and phase-randomized coherent states. To randomly transmit bits 0 and 1, [PERSON] can prepares an entangled state \(|\Phi^{+}_{Bb}(\alpha)\rangle=(|+\rangle_{B}|\alpha\rangle_{b}+|-\rangle_{B}|- \alpha\rangle_{b})/\sqrt{2}\). Here, we use the subscript \(B\) to represent the local qubit retained by [PERSON], and use the subscript \(b\) to represent the photon to be sent to [PERSON]. Then [PERSON] uses the X basis to measure the local qubit. Similarly, [PERSON] can prepare an entangled state \(|\Phi^{+}_{Aa}(\alpha)\rangle\) and measures the local qubit in X basis to transmit bits 0 and 1. Without the loss of generality, we assume [PERSON] can make a deterministic measurement in the X basis based on the precoding result. For example, the precoded bit is 0, the measurement result will be \(|+\rangle_{A}\), which makes Figure 1(a) become Figure 1(b). We can also consider an entanglement-based version of this protocol, namely the measurement of [PERSON] and [PERSON] can be delayed after [PERSON]'s announcement of the successful event. Therefore, we obtain Figure 1(c), which is our entanglement-based protocol. This measurement operation commutes with all the operations performed in the other steps. Hence, this entanglement-based protocol is mathematically equivalent to the OPI-QSDC protocol. We will prove its security, thus proving the security of our OPI-QSDC protocol. The detailed steps of the equivalent entanglement-based protocol are as follows. **Step 1: encoding. ** Same as OPI-QSDC protocol. **Step 2: mode preparation. ** [PERSON] and [PERSON] randomly select the coding mode with a probability of \(1-p\) and the multi-intensity mode with a probability of \(p\), where \(0<p<<1\). Coding mode: [PERSON] prepares an entangled state \(|\Phi^{+}_{Aa}(\alpha)\rangle=(|+\rangle_{A}|\alpha\rangle_{a}+|-\rangle_{A}|- \alpha\rangle_{a})/\sqrt{2}\). She sends the photon \(a\) to [PERSON], and retains the local photon \(A\). Similarly, [PERSON] prepares an entangled state \(|\Phi^{+}_{Bb}(\alpha)\rangle\) and sends the photon \(b\) to [PERSON]. Multi-intensity mode: Same as OPI-QSDC protocol. **Step 3: measurement. ** [PERSON] measures the pulses sent by [PERSON] and [PERSON] using single-photon-interferometer and publishes measurement results. [PERSON] and [PERSON] discard the no-click events and two-click events, and retain the one-click events, namely \(m_{C}\oplus m_{D}=1\). Then, if [PERSON] ([PERSON]) sent the coding mode, she (he) measures the retained local photon A (B) using the X basis. Here, we assume [PERSON] can make a deterministic measurement based on the precoding result. **Step 4: mode matching. ** Same as OPI-QSDC protocol. **Step 5: parameter estimation. ** Same as OPI-QSDC protocol. **Step 6: decoding. ** Same as OPI-QSDC protocol.