# One-Photon-Interference Quantum Secure Direct Communication
## 2 Details of Protocol  
In this section, we propose our OPI-QSDC protocol. As illustrated in Figure 1a, we suppose that [PERSON] and [PERSON] utilize phase locking techniques to lock the frequency and global phase of their lasers. They then simultaneously send light pulses to [PERSON], an untrusted third party right in the middle between them. We use \(M\in\{0,1\}^{m}\) to represent the message that [PERSON] wants to transmit to [PERSON], and \(C\in\{0,1\}^{c}\) represents the ciphertext. In particular, The detailed steps of our protocol are as follows. **Step 1: encoding. ** [PERSON] encodes the message \(M\) to be transmitted to form the codeword \(C\). Encoding operations use forward error correction codes, secure codes [36] and INCUM [26]. Details of the encoding process can be found in the Appendix A. **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 \(0<p<<1\). The quantum states sent in each mode are as follows. Coding mode: a weak coherent state \(\ket{\alpha}\) for logical 0 or a weak coherent state \(\ket{-\alpha}\) for logical 1, where \(\ket{\alpha}^{2}=u\) is the intensity of the coherent state. In this mode, [PERSON] selects the coherent state to be sent based on the encoding result. For example, if the encoding result is 0, [PERSON] sends a \(\ket{\alpha}\) state; otherwise, she sends a \(\ket{-\alpha}\) state. [PERSON] randomly chooses to send these two states. Multi-intensity mode: three different intensity and phase-randomized weak coherent states \(\hat{\rho}_{\beta_{a}}=\ket{\beta_{a}e^{i\phi_{a}}}\) and \(\hat{\rho}_{\beta_{b}}=\ket{\beta_{b}e^{i\phi_{b}}}\), where \(\beta_{a}\) and \(\beta_{b}\) are randomly chosen from the set of \(\{\sqrt{v_{1}},\sqrt{v_{2}},0\}\), while \(u>>v_{1}>v_{2}>0\), and \(\phi_{a}\), \(\phi_{b}\) are randomly chosen from \([0,2\pi)\). **Step 3: measurement. ** [PERSON] measures the pulses sent by [PERSON] and [PERSON] using single-photon interferometer, and publishes measurement results. Let \(m_{C}\) and \(m_{D}\) denote the measurement outcome of \(\mathrm{D}_{0}\) and \(\mathrm{D}_{1}\), where value \"0\" indicates a no-click event and value \"1\" indicates a click event. As shown in Figure 1c, [PERSON] and [PERSON] discard the no-click events and two-click events, and retain the one-click events, namely \(m_{C}\oplus m_{D}=1\). **Step 4: mode matching. ** After all measurements are completed, [PERSON] and [PERSON] publish the modes information. They retain the measurement results of the same modes, and discard the measurement results of different modes, as shown in Figure 1b. Note that there is a probability of \(2p(1-p)\) for a mode mismatch, resulting in the loss of information transmitted by [PERSON]. However, [PERSON] and [PERSON] can utilize error correcting codes in **Step 1** to recover the lost information. If they both send the multi-intensity mode, they publish the intensity and phase of the weak coherent state. Then they retain pulses with \(\beta_{a}=\beta_{b}\) and \(\ket{\phi_{a}-\phi_{b}}=0\) or \(\pi\). **Step 5: parameter estimation. ** [PERSON] and [PERSON] randomly publish the bit values in some coding modes to estimate the quantum bit error rate (QBER), and then use different  
Figure 1: **Schematic diagram of OPI-QSDC protocol and the process of mode matching and measurement. ** BS, 50:50 beam splitter; \(\mathrm{D}_{0}\), \(\mathrm{D}_{1}\), single-photon detector. (**a**) In the OPI-QSDC protocol, we assume that the laser source and coding process of [PERSON] and [PERSON] will not be attacked by [PERSON], which is marked with a solid box. The untrusted [PERSON] uses a single-photon interferometer to measure the signals sent by [PERSON] and [PERSON], but the measurement is completely controlled by [PERSON], which is marked with a dashed box. The three communication parties also need to use the open classic channel marked with the green box to exchange information.