  Any quantum resource theory is based on free states and free operations,
i.e., states and operations which can be created and performed at no cost. In
the resource theory of coherence free states are diagonal in some fixed basis,
and free operations are those which cannot create coherence for some particular
experimental realization. Recently, some problems of this approach have been
discussed, and new sets of operations have been proposed to resolve these
problems. We propose here the framework of genuine quantum coherence. This
approach is based on a simple principle: we demand that a genuinely incoherent
operation preserves all incoherent states. This framework captures coherence
under additional constrains such as energy preservation and all genuinely
incoherent operations are incoherent regardless of their particular
experimental realization. We also introduce the full class of operations with
this property, which we call fully incoherent. We analyze in detail the
mathematical structure of these classes and also study possible state
transformations. We show that deterministic manipulation is severely limited,
even in the asymptotic settings. In particular, this framework does not have a
unique golden unit, i.e., there is no single state from which all other states
can be created deterministically with the free operations. This suggests that
any reasonably powerful resource theory of coherence must contain free
operations which can potentially create coherence in some experimental
realization.
