  The ability to distill quantum coherence is pivotal for optimizing the
performance of quantum technologies; however, such a task cannot always be
accomplished with certainty. Here we develop a general framework of
probabilistic distillation of quantum coherence in a one-shot setting,
establishing fundamental limitations for different classes of free operations.
We first provide a geometric interpretation for the maximal success
probability, showing that under maximally incoherent operations (MIO) and
dephasing-covariant incoherent operations (DIO) the problem can be simplified
into efficiently computable semidefinite programs. Exploiting these results, we
find that DIO and its subset of strictly incoherent operations (SIO) have equal
power in probabilistic distillation of coherence from pure input states, while
MIO are strictly stronger. We then prove a fundamental no-go result: distilling
coherence from any full-rank state is impossible even probabilistically. We
further find that in some conditions the maximal success probability can vanish
suddenly beyond a certain threshold in the distillation fidelity. Finally, we
consider probabilistic coherence distillation assisted by a catalyst and
demonstrate, with specific examples, its superiority to the unassisted and
deterministic cases.
