  The resource framework of quantum coherence was introduced by Baumgratz,
Cramer and Plenio [PRL 113, 140401 (2014)] and further developed by Winter and
Yang [PRL 116, 120404 (2016)]. We consider the one-shot problem of distilling
pure coherence from a single instance of a given resource state. Specifically,
we determine the distillable coherence with a given fidelity under incoherent
operations (IO) through a generalisation of the Winter-Yang protocol. This is
compared to the distillable coherence under maximal incoherent operations (MIO)
and dephasing-covariant incoherent operations (DIO), which can be cast as a
semidefinite programme, that has been presented previously by Regula et al.
[PRL 121, 010401 (2018)]. Our results are given in terms of a smoothed
min-relative entropy distance from the incoherent set of states, and a variant
of the hypothesis-testing relative entropy distance, respectively. The one-shot
distillable coherence is also related to one-shot randomness extraction.
  Moreover, from the one-shot formulas under IO, MIO, DIO, we can recover the
optimal distillable rate in the many-copy asymptotics, yielding the relative
entropy of coherence. These results can be compared with previous work by some
of the present authors [Zhao et al., PRL 120, 070403 (2018)] on one-shot
coherence formation under IO, MIO, DIO and also SIO. This shows that the amount
of distillable coherence is essentially the same for IO, DIO, and MIO, despite
the fact that the three classes of operations are very different. We also
relate the distillable coherence under strictly incoherent operations (SIO) to
a constrained hypothesis testing problem and explicitly show the existence of
bound coherence under SIO in the asymptotic regime.
