I need to conduct a numerical analysis of a quantum channel using SDP and therefore, I need the Stinespring representation of this quantum channel:
$$\Phi=\mathrm{Tr}_{\mathrm{DE}}\Big((\mathrm{Id}^{\mathrm{AC}}\otimes\tilde{M}_{0}^\mathrm{D}(0)\otimes\mathrm{Id}^{\mathrm{E}})\tilde{\rho}^{(6)}{(\mathrm{Id}^{\mathrm{AC}}\otimes\tilde{M}_{0}^\mathrm{D}(0)\otimes\mathrm{Id}^{\mathrm{E}})}^{\dagger}+(\mathrm{Id}^{\mathrm{A}}\otimes O^\mathrm{C}\otimes\tilde{M}_{1}^\mathrm{D}(0)\otimes\mathrm{Id}^{\mathrm{E}})\tilde{\rho}^{(6)}{(\mathrm{Id}^{\mathrm{A}}\otimes O^\mathrm{C}\otimes\tilde{M}_{1}^\mathrm{D}(0)\otimes\mathrm{Id}^{\mathrm{E}})}^{\dagger}\Big)- \mathrm{Tr}_{\mathrm{DE}}\Big((\mathrm{Id}^{\mathrm{AC}}\otimes\tilde{M}_{0}^\mathrm{D}(\theta^\mathrm{D})\otimes\mathrm{Id}^{\mathrm{E}})\tilde{\rho}^{(6)}{(\mathrm{Id}^{\mathrm{AC}}\otimes\tilde{M}_{0}^\mathrm{D}(\theta^\mathrm{D})\otimes\mathrm{Id}^{\mathrm{E}})}^{\dagger}+(\mathrm{Id}^{\mathrm{A}}\otimes O^\mathrm{C}\otimes\tilde{M}_{1}^\mathrm{D}(\theta^\mathrm{D})\otimes\mathrm{Id}^{\mathrm{E}})\tilde{\rho}^{(6)}{(\mathrm{Id}^{\mathrm{A}}\otimes O^\mathrm{C}\otimes\tilde{M}_{1}^\mathrm{D}(\theta^\mathrm{D})\otimes\mathrm{Id}^{\mathrm{E}})}^{\dagger}\Big)$$
Here $$\tilde{M}_{0}^\mathrm{D}(\theta_{j_\mathrm{D}}^\mathrm{D})=\frac{\mathrm{Id}^\mathrm{D}+\cos\left(\frac{\pi}{4}+\theta\right)X-\sin\left(\frac{\pi}{4}+\theta\right)Z}{2},$$ $$\tilde{M}_{1}^\mathrm{D}+(\theta^\mathrm{D})=\frac{\mathrm{Id}^\mathrm{D}-\cos\left(\frac{\pi}{4}+\theta\right)X+\sin\left(\frac{\pi}{4}+\theta\right)Z}{2}$$ and $X$ and $Z$ are the Pauli-X and -Z matrices. $O^\mathrm{C}=\tilde{M}_{0}^\mathrm{D}(0)-\tilde{M}_{1}^\mathrm{D}(0)$.
I tried, but I fail to find the representation. Can you please help me?
