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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.

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  • $\begingroup$ Trace of the commutator is always $0$, so that's not useful... If you want something that's nonzero whenever the commutator is nonzero and zero whenever the commutator is zero, a norm would seem to be the obvious choice. $\endgroup$
    – Robert Israel
    Commented May 20, 2015 at 18:49
  • $\begingroup$ Also posted to m.se, without notification to either site of the other posting: math.stackexchange.com/questions/1291238/… $\endgroup$
    – Gerry Myerson
    Commented May 21, 2015 at 22:44
  • $\begingroup$ You might be interested in Geher and Nagy, Maps on classes of Hilbert space operators preserving measure of commutativity, Lin. Alg. and its Appl. 463 (2014) 205-227, which discusses the norm of the commutator, and refers to earlier work. $\endgroup$
    – Gerry Myerson
    Commented May 21, 2015 at 22:49

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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 $AB-BA$.

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  • $\begingroup$ The spectral radius actually sounds like it's probably the right way to go, with some modification. I can't normalize the matrices for reasons inherent to the problem. Let mr think about this... $\endgroup$
    – Michael Jarret
    Commented May 21, 2015 at 20:40

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