## 2 Theoretical Framework and Simulation Experiment Setup
Similarly, the FPC\({}_{4}\) couplesthe modulated \(S_{Ex}(n)\) and \(S_{Ey}(n)\) into a single beam, which is subsequently injected into the excited state of the reservoir QD spin-VCSEL. Within the reservoir, the GS and ES of the QD spin-VCSEL are both influenced by dual feedback mechanisms. The feedback loops for the GS are denoted by subscripts 1 and 2, while those for the ES are indicated by subscripts 3 and 4. Each loop employs a NDF and a PCL to adjust the feedback intensity and polarization direction of the feedback beam, respectively. The delay time established by the delay lines (DL\({}_{1}\)-DL\({}_{4}\)) is defined as \(\tau\). In the output layers (OLs), the GS-PC\({}_{x}^{{}^{\prime}}\) and GS-PC\({}_{y}^{{}^{\prime}}\) emissions from the QD spin-VCSEL are bifurcated using FPBS\({}_{5}\). Similarly, the ES-PC\({}_{x}^{{}^{\prime}}\) and ES-PC\({}_{y}^{{}^{\prime}}\) emissions are split through FPBS\({}_{6}\). The intensity values of GS-PC\({}_{x}^{{}^{\prime}}\), GS-PC\({}_{y}^{{}^{\prime}}\), ES-PC\({}_{x}^{{}^{\prime}}\) and ES-PC\({}_{y}^{{}^{\prime}}\) are sampled at intervals of \(\theta\) and are considered as virtual nodes. Accordingly, the total number \(N\) of virtual nodes along each delay line is determined by the ratio \(N=\tau/\theta\). The states of the \(N\) virtual nodes along the DL\({}_{1}\)-DL\({}_{4}\) are weighted and linearly summed up. The combined weighted states from the DL\({}_{1}\) and DL\({}_{2}\) are represented as \(y_{Gx}^{{}^{\prime}}(n)\) and \(y_{Gy}^{{}^{\prime}}(n)\), respectively, while those from the DL\({}_{3}\) and DL\({}_{4}\) are signified as \(y_{Ex}^{{}^{\prime}}(n)\) and \(y_{Ey}^{{}^{\prime}}(n)\). In this setup, by calibrating the output weights, \(y_{Gx}^{{}^{\prime}}(n)\) and \(y_{Gy}^{{}^{\prime}}(n)\) can achieve synchronization with \(C_{Gx}(n-L_{Gx})\) and \(C_{Gy}(n-L_{Gy})\), respectively. Likewise, \(y_{Ex}^{{}^{\prime}}(n)\) and \(y_{Ey}^{{}^{\prime}}(n)\) can be attuned to synchronize with \(C_{Ex}(n-L_{Ex})\) and \(C_{Ey}(n-L_{Ey})\). Under these synchronization conditions, the concealed messages QPSK and 4 QAM are decoded by the synchronous subtraction of \(y_{Gx}^{{}^{\prime}}(n)\) from \(C_{Gx}(n-L_{Gx})\) and \(y_{Gy}^{{}^{\prime}}(n)\) from \(C_{Gy}(n-L_{Gy})\), with the retrieved messages designated as \(S_{1}^{{}^{\prime}}(t)\) and \(S_{2}^{{}^{\prime}}(t)\), respectively. In a similar fashion, the messages 8 QAM and 16 QAM are decoded by the synchronous subtraction of \(y_{Ex}^{{}^{\prime}}(n)\) from \(C_{Ex}(n-L_{Ex})\) and \(y_{Ey}^{{}^{\prime}}(n)\) from \(C_{Ey}(n-L_{Ey})\), with their decoded equivalents presented as \(S_{3}^{{}^{\prime}}(t)\) and \(S_{4}^{{}^{\prime}}(t)\), correspondingly. As illustrated in Figure 2c, the decoded messages, \(S_{1}^{{}^{\prime}}(t)\), \(S_{2}^{{}^{\prime}}(t)\), \(S_{3}^{{}^{\prime}}(n)\) and \(S_{4}^{{}^{\prime}}(n)\), are initially modulated with the optical field phases of the CW\({}_{5}\)-CW\({}_{8}\) using intensity modulators (IM\({}_{1}\)-IM\({}_{4}\)) and then each injected into its corresponding coherent demodulation unit (CDU). Each CDU comprises a polarization-diversity digital coherent receiver (PDDCR), a submatrix finder (SF), a set of five digital signal processors (DSPs), and a bit error rate estimation module (BEREM). The PDDCR, depicted in VPI [23], models an optical coherent quadrature receiver that encompasses a local oscillator, optical hybrids, post-detection electrical filters, and analog-to-digital converters. The SF is used to extract specified elements of the input matrix. The DSPs with subscripts 1, 6, 11 and 16 address the compensation of group velocity dispersion and nonlinear effects within the optical fiber, whereas the DSPs labeled with subscripts 2, 7, 12 and 17 are designated to down-sample the in-phase and quadrature signals to match the baud rate. The DSPs marked with subscripts 3, 8, 13 and 18 are dedicated to estimating and correcting frequency discrepancies between the received optical signal and the local oscillator.