Is there a biijection between all the markov chains represented by the stochastic matrices, and all the positive unital maps from $\Phi : \mathbb{C}^N \rightarrow \mathbb{C}^N$?
I mean, given any unital positive map of this kind, will this be a stochastic matrix and therefore, a representation of a certain markov chain? And why is this true?
Thanks
"Unital" in your setup is precisely "stochastic". Given one of your $\Phi$, you have $\Phi e=e$, where $e$ is the identity of $\mathbb C^N$. This equality, row by row, is $$ 1=(\Phi e)_k=\sum_j\Phi_{kj}e_j=\sum_j\Phi_{kj}. $$
Similarly, in this context $\Phi\geq0$, means that $\Phi x\geq0$ whenever $x\geq0$. It follows that $\Phi\geq0$ if and only if $\Phi_{kj}\geq0$ for all $k,j$. Indeed, suppose that $\Phi\geq0$. Thus $$ 0\leq \sum_k \Phi_{kj}a_j, \ \ \text{ each $k$ and for all } a_1,\ldots,a_n\geq0. $$ Taking a single $a_j$ to be nonzero, you get $\Phi_{kj}\geq0$ for all $k,j$. The converse is trivial.