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×n Hermitian matrices, and [A,B]=C.


I'd like a function μ:Cn×n→[0,∞) such that μ(C)=0 if the two operators commute and maybe obeys some other properties that I haven't quite figured out yet. Otherwise, a function ν(A,B):Cn×n×Cn×n→[0,∞) 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 ν(A,B)=0 in the event that the two operators commute and ν(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.



Reference for measures of commutativity needed
It is better to normalize the matrices -consider $A/||A||,B/||B||$-.


As Robert wote, you can choose $||AB-BA||$.


On the other hand, if $A,B$ are hermitian, then $i(AB-BA)$ is hermitian. Thus, as a measure, you can choose the spectral radius of $AB-BA$.


