How can I fit an unknown quantum channel?

Question

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!

Answer 1

The 2-norm difference typically isn't particularly physical. So no, this is most likely not the right distance. What you want from a physical point of view is a distance measure which measures the maximal distance between the two channels on any possible input (possibly on a larger system), such as the diamond norm.

Note that the fact that "norms are equivalent" just means that they are bounded with respect to each other, but this does by no means imply that the minimum distance is obtained at the same point, or takes the same value! (Just consider the 1- and $\infty$-norm in $\mathbb R^2$, where one is a square, where the lines of constant distance are a square and a diamond, respectively.)

One reason why people might be using the 2-norm distance, though, is that this is a quadratic problem, so it is easy to minimize by solving an eigenvalue problem. But this is a purely pragmatic reason.