# How to quantify trace distance between two matrices representing two quantum optical networks?

This is my first post here, so I'm sorry if this question could be ill-formulated. I have performed measurements on a 12x12 optical quantum network, so that I have a stochastic matrix $$P^{exp}$$ where each element $$P^{exp}_{i,j}$$ correspond to the probability of measuring an output photon in the $$i$$ channel if I have injected an input photon in the $$j$$ channel.

I have a $$P^{theo}$$ matrix of the expected ideal behaviour, and I want to quantify how much the experimental $$P^{exp}$$ moves away from the $$P^{theo}$$.

Referring to p. 400-403 of Nielsen-Chuang, I read about the trace distance between two $$\{p_x\}, \{q_x\}$$ 1D probability distributions over the same index set $$x$$, defined as

$$D(p_x, q_x) = \frac{1}{2}\sum_x |p_x-q_x|_1$$

Now, I want to use this metric to treat my stochastic matrices. I thought initially that this can be accomplished by using the $$L_1$$ norm on matrices:

$$D(P^{theo},P^{exp}) = ||P^{theo}-P^{exp}||_1 = max_{1

which performs the sum over all the elements of all the columns, and takes the max among all the sums. However, I am unsure this has some physical insightful meaning, as I would rather do the average between all the $$\sum_{i=1}^m |P^{theo}_{i,j}-P^{exp}_{i,j}|$$,

$$D(P^{theo},P^{exp}) = \frac{\sum_{j=1}^{12}\sum_{i=1}^{12} |P^{theo}_{i,j}-P^{exp}_{i,j}|}{12}$$

Is this reasoning correct? Do you have other suggestions about how to quantify the "diversity" between an experimental and a theoretical stochastic matrix?