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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?

pcalc
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relativeentropy
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    what did you try? – glS Aug 30 '23 at 22:07
  • Sadly, I don't know how to handle this. Do you have a suggestion or better, a solution? – relativeentropy Sep 06 '23 at 10:54
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    if you want to ask how to find the stinespring representation of a given channel, there's already many examples on the site, see eg https://quantumcomputing.stackexchange.com/q/24511/55 and links therein. If you have trouble applying the general procedure to the particular problem you have, you should clarify what exactly is the issue. Also, you wrote "I trailed, but I fail[ed]", so I was assuming you did indeed try something? If so, you should edit the post to add that information – glS Sep 06 '23 at 13:42
  • The examples are all for channels that map from a space with dimension n to an equal dimension space. Mine is changing the dimension, therefore I do not know how to handle it properly and sadly, the list cannot help me. Do you have an idea? – relativeentropy Nov 25 '23 at 08:45

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