  Quantum coherence is one of the fundamental aspects distinguishing classical
and quantum theories. Coherence between different energy eigenstates is
particularly important, as it serves as a valuable resource under the law of
energy conservation. A fundamental question in this setting is how well one can
prepare good coherent states from low coherent states and whether a given
coherent state is convertible to another one. Here, we show that any low
coherent state is convertible to any high coherent state arbitrarily well in
two operational settings: asymptotic and catalytic transformations. For a
variant of asymptotic coherence manipulation where one aims to prepare desired
states in local subsystems, the rate of transformation becomes unbounded
regardless of how weak the initial coherence is. In a non-asymptotic
transformation with a catalyst, a helper state that locally remains in the
original form after the transformation, we show that an arbitrary state can be
obtained from any low coherent state. Applying this to the standard asymptotic
setting, we find that a catalyst can increase the coherence distillation rate
significantly -- from zero to infinite rate. We also prove that such anomalous
transformation requires small but non-zero coherence in relevant modes,
establishing the condition under which a sharp transition of the operational
capability occurs. Our results provide a general characterization of the
coherence transformability in these operational settings and showcase their
peculiar properties compared to other common resource theories such as
entanglement and quantum thermodynamics.
