## 3 Results and Discussions
However, for the ES-PC\({}_{x}\) and ES-PC\({}_{y}\), the prediction errors slightly increase with an increase in \(\theta\), suggesting a potential sensitivity to the chosen \(\theta\). The results obtained from Figures 4 and 5 demonstrate that the four parallel reservoirs, based on the reservoir QD spin-VCSEL, are capable of reproducing the chaotic dynamics of the GS-PC\({}_{x}\), GS-PC\({}_{y}\), ES-PC\({}_{x}\) and ES-PC\({}_{y}\) emitted by the driving QD spin-VCSEL. This indicates that the delayed GS-PC\({}_{x}\), GS-PC\({}_{y}\), ES-PC\({}_{x}\) and ES-PC\({}_{y}\) can successfully synchronize with the GS-PC\({}_{x}^{\prime}\), GS-PC\({}_{y}^{\prime}\), ES-PC\({}_{x}^{\prime}\) and ES-PC\({}_{y}^{\prime}\) outputs by the reservoir QD spin-VCSEL, respectively. To further analyze the qualities of their chaos synchronizations, the correlation coefficients are introduced and defined as follows. Figure 4: Samples of four delayed polarization components emitted by the driving QD spin-VCSEL (blue solid line) and the outputs of four parallel reservoir based on the reservoir QD Spin-VCSEL (red dashed line). Here, (**a**) \(C_{Gx}(n-L_{Gx})\) and \(y^{{}^{\prime}}_{Gx}(n)\); (**b**) \(C_{Gy}(n-L_{Gy})\) and \(y^{{}^{\prime}}_{Gy}(n)\); (**c**) \(C_{Ex}(n-L_{Ex})\) and \(y^{{}^{\prime}}_{Ex}(n)\); and (**d**) \(C_{Eg}(n-L_{Eg})\) and \(y^{{}^{\prime}}_{Ey}(n)\). \[\rho_{jx} =\frac{\left\langle\left[C_{jx}(n-L_{jx})-\left\langle C_{jx}(n-L_{jx} )\right\rangle\right]\left[y^{{}^{\prime}}_{jx}(n)-\left\langle y^{{}^{\prime}}_ {jx}(n)\right\rangle\right]\right\rangle}{\left\langle\left[C_{jx}(n-L_{jx})- \left\langle C_{jx}(n-L_{jx})\right\rangle\right]^{2}\right\rangle^{1/2} \right\rangle^{1/2}\right\rangle\left\langle\left[y^{{}^{\prime}}_{jx}(n)- \left\langle y^{{}^{\prime}}_{jx}(n)\right\rangle\right]^{2}\right\rangle^{1 /2}}, \tag{24}\] \[\rho_{jy} =\frac{\left\langle\left[C_{jy}(n-L_{jy})-\left\langle C_{jy}(n-L_ {jy})\right\rangle\right]\left[y^{{}^{\prime}}_{jy}(n)-\left\langle y^{{}^{ \prime}}_{jy}(n)\right\rangle\right]\right\rangle}{\left\langle\left[C_{jy}(n -L_{jy})-\left\langle C_{jy}(n-L_{jy})\right\rangle\right]^{2}\right\rangle^{1 /2}\right\rangle\left\langle\left[y^{{}^{\prime}}_{jy}(n)-\left\langle y^{{}^{ \prime}}_{jy}(n)\right\rangle\right]^{2}\right\rangle^{1/2}},\]  
where \(j=G\), \(E\) (the same below). The symbol \(\left\langle\right\rangle\) represents the time average. \(\rho\) ranges from \(-1\) to \(1\). With the bigger absolute value of \(\rho\), the higher quality of synchronization can be obtained. When \(\rho\) equals to \(\pm 1\), the in-phase and anti-phase synchronous solutions in this system exist. In Figure 6, the correlations \(\rho_{Gx}\), \(\rho_{Gy}\), \(\rho_{Ex}\) and \(\rho_{Ey}\) are shown as a function of \(k_{inj}\) and \(k_{f}\). It can be observed that \(\rho_{Gx}\), \(\rho_{Gy}\), \(\rho_{Ex}\) and \(\rho_{Ey}\) exhibit minimal changes as \(k_{inj}\) and \(k_{f}\) increase in the range of \(0.1\) ns\({}^{-1}\) to \(50\) ns\({}^{-1}\). Within these two parameter spaces, \(\rho_{Gx}\) and \(\rho_{Gy}\) both range between \(0.9849\) and \(0.9857\), while \(\rho_{Ex}\) and \(\rho_{Ey}\) fluctuate between \(0.949\) and \(0.96\). This indicates that \(\rho_{Gx}\), \(\rho_{Gy}\), \(\rho_{Ex}\) and \(\rho_{Ey}\) possess strong robustness to variations in \(k_{inj}\) and \(k_{f}\), which are key parameters of the reservoir QD spin-VCSEL. Furthermore, as all \(\rho_{Gx}\), \(\rho_{Gy}\), \(\rho_{Ex}\) and \(\rho_{Ey}\) are greater than \(0.949\), it can be concluded that the GS-PC\({}^{\prime}_{x}\), GS-PC\({}^{\prime}_{y}\), ES-PC\({}^{\prime}_{x}\) and ES-PC\({}^{\prime}_{y}\) can effectively synchronize with the delayed GS-PC\({}_{x}\), GS-PC\({}_{y}\), ES-PC\({}_{x}\) and ES-PC\({}_{y}\), respectively. Notably, \(\rho_{Gx}\) and \(\rho_{Gy}\) are higher than \(\rho_{Ex}\) and \(\rho_{Ey}\), respectively.