# Relation between Kraus operators and the Choi matrix

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 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 $$B$$.
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 $$M_\Phi= B^*\pi^* \pi B$$.