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Suppose we have a depolarizing channel operation $$E(\rho)=\frac{p}{2}\textbf{1}+(1-p)\rho$$ acting on a Spin$\frac{1}{2}$ density matrix of the form $\rho=\frac{1}{2}(\textbf{1}+\textbf{s}\cdot\textbf{$\sigma$})$. I have found the Kraus operators to be: $$E_1=\sqrt{\left(1-\frac{3}{4}p\right)}\textbf{1}, E_2=\frac{\sqrt{p}}{2}\sigma_x,E_3=\frac{\sqrt{p}}{2}\sigma_y \text{ and } E_4=\frac{\sqrt{p}}{2}\sigma_z$$ I am now supposed to find the unitary matrix U such that the Operation can be expressed in a bigger system i.e. after adding a System S. As far as I understand it, the new operation can be written as: $$E(\rho)=\sum_kE_k\rho E_k^\dagger=\text{Tr}_S(U\rho\otimes\rho_EU^\dagger)$$ Supposing the new system S is prepared on a state $|e_0\rangle$, How do I find the correct unitary matrix?

I appreciate your cooperation.

Crossposted to physics: https://physics.stackexchange.com/questions/576952/how-to-find-the-unitary-operation-of-a-depolarizing-channel

1If you have access to a copy of Nielsen and Chuang this is explained in Box 8.1 at the end of section 8.2.3. – Condo – 2020-09-01T17:39:39.030

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See https://physics.stackexchange.com/questions/27657/explicit-construction-for-unitary-extensions-of-completely-positive-and-trace-pr

– Norbert Schuch – 2020-09-01T19:28:42.483