  We establish an operational theory of coherence (or of superposition) in
quantum systems, by focusing on the optimal rate of performance of certain
tasks. Namely, we introduce the two basic concepts - "coherence distillation"
and "coherence cost" in the processing quantum states under so-called
incoherent operations [Baumgratz/Cramer/Plenio, Phys. Rev. Lett. 113:140401
(2014)]. We then show that in the asymptotic limit of many copies of a state,
both are given by simple single-letter formulas: the distillable coherence is
given by the relative entropy of coherence (in other words, we give the
relative entropy of coherence its operational interpretation), and the
coherence cost by the coherence of formation, which is an optimization over
convex decompositions of the state.
  An immediate corollary is that there exists no bound coherent state in the
sense that one would need to consume coherence to create the state but no
coherence could be distilled from it. Further we demonstrate that the coherence
theory is generically an irreversible theory by a simple criterion that
completely characterizes all reversible states.
