## 2 The DM Source Preparation via ZPC Operations
### The ZPC-Based DM Source Preparation  
It is known that ZPC, which is a non-Gaussian operation with zero-photon state input/output [21; 22], can enhance the correlation of the two-participant entangled quantum system [25; 28]. In what follows,  
Figure 1: The discrete-modulation (DM)-based sources for four states (**left**) and eight states (**right**). Figure 2: The correlation comparison of \(Z_{4}\), \(Z_{8}\) and \(Z_{G}\). we suggest an improved CVQKD system with ZPC operation for the DM source preparation, as shown in Figure 3. In the auxiliary mode C, the zero-photon Fock state \(\ket{0}\) is injected at one input port of beam splitter (BS) with transmittance \(\tau\), and quantum state \(\ket{\psi}_{in}\) of the mode B is simultaneously injected at another input port. Subsequently, we detect the zero-photon Fock state \(\ket{0}\) at one of the output ports of the BS using a photon number resolving detector (PNRD), resulting in the ZPC-enhanced quantum state \(\ket{\psi}_{out}\) from another output port. Similar to the traditional photon subtraction operation [30; 31; 32; 33], the ZPC operation can be used for performance improvement of CVQKD. In order to describe the ZPC-based DM source preparation, the ZPC operation can be described as [22; 25]  
\[\hat{\mathcal{O}}_{0}=(\sqrt{\tau})^{b\tau b}. \tag{3}\]  
For an input \(\ket{\psi}_{in}\), we have output \(\ket{\psi}_{out}\) given by  
\[\ket{\psi}_{out}=\frac{\hat{\mathcal{O}}_{0}}{\sqrt{P_{0}}}\ket{\psi}_{in}, \tag{4}\]  
where \(P_{0}\) denotes the success probability of the operation given by \(P_{0}=e^{(T-1)|\alpha_{i}|^{2}}\) as function of parameters \(V_{A}\) and \(\tau\), as shown in Figure 4. We find that the large \(V_{A}\) and \(\tau\) result in the high success probability. Note that we have \(\ket{\psi}_{out}=\ket{\psi}_{in}\) for \(\tau=1\), indicating that there is no ZPC effect. Figure 4: Success probabilities of ZPC with transmitances \(\tau\) and modulation variance \(V_{A}\). Figure 3: Schematic diagram of continuous-variable quantum key distribution (CVQKD) with zero-photon-catalysis (ZPC)-based DM source preparation. In the PM version of CVQKD, [PERSON] randomly draws \(n\) quaternary or octal variables, each corresponding to a coherent state of \(\left|\alpha_{k}^{N}\right\rangle\). Subsequently, [PERSON] performs the ZPC operation on these \(n\) coherent states, which is modeled (BS) with transmittance \(\tau\), resulting in the catalyzed state  
\[\left|\tilde{\alpha}_{k}\right\rangle=\frac{e^{(\tau-1)\left|a\right|^{2}}}{ \sqrt{P_{0}}}\left|\sqrt{\tau}\alpha_{k}\right\rangle. \tag{5}\]  
We note that the relationship of the transformed amplitudes can be written as \(\tilde{\alpha}_{k}=\sqrt{\tau}\alpha_{k}\). After that, the catalyzed quantum state is sent to [PERSON] through the Eve-controlled channel. [PERSON] may perform either homodyne (Hom) or heterodyne (Het) detection on the resulting quantum state. Finally, [PERSON] and [PERSON] distill the shared secret key after they conduct the classical post-processing procedure. In the EB version of the CVQKD system, as shown in Figure 3, [PERSON] prepares for \(\left|\Psi\right\rangle_{N}\), and performs heterodyne detection (Het) on mode A. Mode B, which is performed with the ZPC operation, is sent to [PERSON]. The role of the ZPC operation is for source-entanglement enhancement while promoting the mode transition between \(B\) and \(B_{1}\), resulting in the covariance matrix \(\gamma_{AB_{1}}\) of the catalyzed state \(\rho_{AB_{1}}\) given by  
\[\gamma_{AB_{1}}= \begin{pmatrix}(\tau V_{A}+1)\mathbb{I}&Z_{N}^{\prime}\sigma_ {z}\\ Z_{N}^{\prime}\sigma_{z}&(\tau V_{A}+1)\mathbb{I}\end{pmatrix} , \tag{6}\]  
with the parameters  
\[Z_{N}^{\prime}=\tau V_{A}\sum_{k=0}^{N-1}\frac{\tilde{\lambda}_{N-1}^{3/2}}{ \sqrt{\tilde{\lambda}_{k}}}, \tag{7}\]  
where the calculation of \(\tilde{\lambda}_{k}\) is similar to that of \(\lambda_{k}\) in Equation (2), but note that \(\alpha\) is replaced by \(\sqrt{\tau}\alpha\).