# High-Dimensional and Multi-Intensity One-Photon-Interference Quantum Secure Direct Communication
## 2 Our Protocol  
We assume that [PERSON] and [PERSON] use weak laser pulses with phase locking and have agreed upon a reasonable number of total base slices \(M\) before completing the following steps. [PERSON], an untrusted third party, is in between them, as illustrated in Figure 1. The steps of the HDOPI-QSDC protocol are as follows. **Step 1: Encoding. ** [PERSON] encodes the message to be transmitted into ciphertext using local random numbers shared with [PERSON]. Note that the shared key can be obtained by running the rest of the procedures in this protocol, in which case random numbers are sent instead of the ciphertext. The encoding process includes forward error correction (FEC) coding, secure coding [37], and INCUM (increase capacity using masking) [50]. These processes eliminate the protocol's reliance on quantum memory, and their details are provided in Appendix A. **Step 2: State preparation. ** [PERSON] and [PERSON] independently select a mode to operate in: coding mode, with a probability of \(1-p\), or multi-intensity mode, with a probability of \(p\), where \(p\ll 1\). The specifics of the coding mode and multi-intensity mode are detailed below. Figure 1: Schematic diagram of the HDOPI-QSDC protocol. BS, 50:50 beam splitter; D\({}_{0}\) and D\({}_{1}\), single-photon detectors; VOA, variable optical attenuator; PM, phase modulator; IM, intensity modulator. Red dashed boxes represent the coding modes, and blue dashed boxes represent the multi-intensity modes. In coding modes, [PERSON]’s pulses contain \(M\) same bases, while [PERSON]’s contain \(M\) different bases, and the order of these bases is random as well. In multi-intensity modes, [PERSON] and [PERSON]’s \(M\) pulses contain random intensities and phases, chosen from the sets \(\{\beta_{0},\beta_{1},\ldots\}\) and \([0,2\pi)\), respectively. \(j,k\in\{0,1,\ldots,M-1\}\) are the bases’ indices and \(b,b^{\prime}\in\{0,1\}\) are the information bits. [PERSON] conducts one-photon interferences and publishes the untrustworthy measurement results, which are utilized by [PERSON] and [PERSON] to estimate the channel parameters and extract the original secret message. Coding mode: [PERSON] sends \(M\) weak coherent states (WCSs) with the same base \(|\mathsf{ae}^{\mathrm{i}\pi(b_{A}/M)}\rangle=|\mathsf{ae}^{\mathrm{i}\pi([bM+A )/M]}\rangle\), where \(b=0\) or \(1\), which is the information bit value and is encoded in the phases as \(b_{A}=bM+A\in\{0,1,\ldots,2M-1\}\), with \(A\in\{0,1,\ldots,M-1\}\) being the base index and \(M\) the total number of bases. [PERSON] sends \(M\) WCSs with different bases \(|\mathsf{ae}^{\mathrm{i}\pi(b_{B}/M)}\rangle\in\left\{|\mathsf{ae}^{\mathrm{i} \pi(b_{0}/M)}\rangle,|\mathsf{ae}^{\mathrm{i}\pi(b_{1}/M)}\rangle,\ldots,| \mathsf{ae}^{\mathrm{i}\pi(b_{M-1}/M)}\rangle\right\}\), where \(b_{B}=b^{\prime}M+B\in\{0,1,\ldots,2M-1\}\) and \(b^{\prime}\) is a random bit in \(0\) or \(1\). The order of [PERSON]'s WCSs is random. Multi-intensity mode: [PERSON] and [PERSON] send \(M\) WCSs with random intensities and random phases \(|\beta_{a}\mathrm{e}^{\mathrm{i}\phi_{a}}\rangle,|\beta_{b}\mathrm{e}^{\mathrm{ i}\phi_{b}}\rangle\). \(\beta_{a}\) and \(\beta_{b}\) are randomly selected light intensities in \(\{\beta_{0},\beta_{1},\ldots\}\), and \(\phi_{a},\phi_{b}\in[0,2\pi)\) are random phases. Step 3: [PERSON]'s measurement.One-photon interferences between pulses from [PERSON] and [PERSON] are conducted by [PERSON], and the measurement results are announced on the public channel. Let \(D_{0}\) and \(D_{1}\) denote the measurement outcomes of the detectors \(\mathrm{D}_{0}\) and \(\mathrm{D}_{1}\), respectively; and their values can be set to \"0\", indicating a no-click event, or \"1\", indicating a click event. [PERSON] and [PERSON] discard the events where no detector clicks or both of them click, retaining only the one-click events, namely \(D_{0}\oplus D_{1}=1\).