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:

$$D(P^{theo},P^{exp}) = ||P^{theo}-P^{exp}||_1 = max_{1<j<n}\sum_{i=1}^{12} |P^{theo}_{i,j}-P^{exp}_{i,j}|$$

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?

Thank you in advance.



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