  In this paper we consider quantum resources required to maximize the mean
values of any nontrivial quantum observable. We show that the task of
maximizing the mean value of an observable is equivalent to maximizing some
form of coherence, up to the application of an incoherent operation. As such,
for any nontrivial observable, there exists a set of preferred basis states
where the superposition between such states is always useful for optimizing a
quantum observable. The usefulness of such states is expressed in terms of an
infinitely large family of valid coherence measures which is then shown to be
efficiently computable via a semidefinite program. We also show that these
coherence measures respect a hierarchy that gives the robustness of coherence
and the $l_1$ norm of coherence additional operational significance in terms of
such optimization tasks.
