Title #1:  How distinguishable are two quantum processes?

Title #2:  What is the error rate of a quantum gate?

Abstract:  I will try to convince you that the two titles of this talk are, in fact, synonymous -- that the "error rate" and "distinguishability of quantum processes" are the same thing.  Whether or not I succeed, I will go on to discuss (1) the various ways that this has been quantified, (2) the state of the art in doing so, and (3) why I'm not (and you shouldn't be!) satisfied.  Having spent 45 minutes just establishing what the "right" problem is, I will then propose to solve it by sandwiching "distinguishability" between "distillable distinguishability" and "distinguishability of formation".  To demonstrate the utility of this approach, I'll prove that the diamond norm is not always the right measure of distinguishability (or even close to it!).  I will then do a 180-degree turn and argue that for most of the cases that we care about, the diamond norm is a good measure of distinguishability, and

finally conclude with another 180-degree turn in which I argue that maybe it's not.