Let $\Phi$ be a CPTP map on density operators for a fixed $n-$dimensional state space and fix a basis $\{ | j\rangle \}$. I'm trying to understand the relationship between the Choi matrix $$M_\Phi:= \sum_{i,j} \Phi(|i\rangle \langle j|) \otimes |i\rangle \langle j|$$ and a set of Kraus operators $\{V_j \}$ where $$\Phi(\rho)=\sum_j V_j\rho V_j^\dagger.$$

The wikipedia page states without proof or reference that:

The Kraus operators correspond to the square roots of $M_\Phi$: For any square root B of $M_\Phi$, one can obtain a family of Kraus operators $V_i$ by undoing the Vec operation to each column $b_i$ of $B$.

I have a few questions about this statement.

  1. Does anybody have a proof of this statement? It seems strange that a simple rearrangement of the square root of $M_\Phi$ gives rise to the Kraus operators.

  2. Given a set of Kraus operators $\{V_i \}$, can one construct the Choi Matrix by reversing the operation described above? That is, from stacking the columns of each $V_i$ to form the columns $b_i= \text{Vec}(V_i)$ of a matrix $B$, then would $M_\Phi=B^*B$. This seems strange as the Kraus operators $V_j$ can be randomly relabelled and still represent the same channel (i.e. map $V_1 \to V_2$ and $V_2 \to V_1$). Ultimately this will change the matrix $B$ by some permutation $\pi$ which result in a different Choi matrix $M_\Phi= B^*\pi^* \pi B$.

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    $\begingroup$ It would be good if you could show your thoughts on the problem. $\endgroup$ – Norbert Schuch Aug 2 '17 at 13:39

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