Conditions for coherence transformations under incoherent operations

We build in this paper the counterpart of the celebrated Nielsen theorem for coherence manipulation. This offers an affirmative answer to the open question: whether, given two states $\ensuremath{\rho}$ and $\ensuremath{\sigma}$, either $\ensuremath{\rho}$ can be transformed into $\ensuremath{\sigma}$ or vice versa under incoherent operations [Baumgratz et al., Phys. Rev. Lett. 113, 140401 (2014)]. As a consequence, we find that there exist essentially different types of coherence. Moreover, incoherent operations can be enhanced in the presence of certain coherent states. These extra states are coherent catalysts: they allow uncertain incoherent operations to be realized without being consumed in any way. Our main result also sheds light on the construction of coherence measures.