I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if $A$ and $B$ are two $n \times n$ Hermitian matrices, and $[A,B]=C$.
I'd like a function $\mu : \mathbb{C}^{n \times n} \rightarrow [0,\infty)$ such that $\mu(C)=0$ if the two operators commute and maybe obeys some other properties that I haven't quite figured out yet. Otherwise, a function $\nu(A,B): \mathbb{C}^{n \times n} \times \mathbb{C}^{n \times n} \rightarrow [0,\infty)$ could be good. Clearly the absolute value of the trace of the commutator, determinant, norms, etc, are candidates, but I'm wondering if anyone knows of a review of different measures and their relative strengths/weaknesses.
I'm ultimately looking to construct a measure that behaves something like the Kullbeck-Leibler divergence (quantum relative entropy), but where we wind up with a value $\nu(A,B)=0$ in the event that the two operators commute and $\nu(A,B) > 0$ if they do not.
A link to some lecture notes, a paper/book reference would be appreciated, or just being pointed in the right direction would be appreciated.