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I am trying to replicate the solution I have to this problem provided by the instructor in the class where I am trying to use the Choi-Jamialkoski theorem to prove that Phase damping channel is completely positive.

The phase damping channel is given by the following map: $\Lambda(\rho)=\begin{pmatrix} 1 & 1-P\\ 1-P & 1 \end{pmatrix}$

Now the Choi-Jamialkoski theorem states that $I_n\otimes(|\psi><\psi|)\geq0$

Where $|\psi><\psi|=\frac{1}{2}\begin{pmatrix} 1 & 0 & 0 & 1\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 1 \end{pmatrix}$

and $I_n$ represents a 2x2 identity matrix in case of a qubit system.

What I don't understand is why I don't get a 8x8 matrix if I am doing a tensor product with Identity and also shouldn't the terms at off-diagonal extremities be zero. Please Help.

The problem I am facing is marked in red

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    $\begingroup$ Can you please write the equations out instead of use a photo graph? $\endgroup$ Sep 12, 2022 at 9:58

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