  The manipulation of quantum coherence is one of the principal issues in the
resource theory of coherence, with two critical topics being the purification
and enhancement of coherence. Here, we present two no-go theorems for the
deterministic purification of coherence and the probabilistic enhancement of
coherence, respectively. Specifically, we prove that a quantum state cannot be
deterministically purified if it can be expressed as a convex combination of an
incoherent state and a coherent state. Besides, we give an easy-to-verified
sufficient and necessary condition to determine whether a state can be
probabilistically enhanced via a stochastic strictly incoherent operation
(sSIO). Our findings provide two feasibility criteria for the deterministic
purification and the probabilistic enhancement of coherence, respectively.
These results have repercussions on the understanding of quantum coherence in
real quantum systems.
