Suppose that I have one noisy channel $\mathcal{E}$ and I want to fit it with another one $\mathcal{E}_0(p)$ that depends on some fitting parameter $p$.
As both of this processes for me are represented by matrices, I would simply minimize the distance that I define as $$d(p) = ||\mathcal{E} - \mathcal{E}_0(p)||_2$$ where for the norm $||\cdot||_2 $ I would just take the square difference between the elements of the two matrices that represent the processes.
Would it be correct using this distance? I read in Gilchrist et al. PRA 71, 062310 (2005)(https://arxiv.org/pdf/quant-ph/0408063.pdf) that there are several metrics for processes, but I think that they should be all equivalent, as I have a finite Hilbert space. Am I missing something?
Thanks!