  In the task of assisted coherence distillation via the set of operations X,
where X is either local incoherent operations and classical communication
(LICC), local quantum-incoherent operations and classical communication
(LQICC), separable incoherent operations (SI), or separable quantum incoherent
operations (SQI), two parties, namely Alice and Bob, share many copies of a
bipartite joint state. The aim of the process is to generate the maximal
possible coherence on the subsystem of Bob. In this paper, we investigate the
assisted coherence distillation of some special mixed states, the states with
vanished basis-dependent discord and Werner states. We show that all the four
sets of operations are equivalent for assisted coherence distillation, whenever
Alice and Bob share one of those mixed quantum states. Moreover, we prove that
the assisted coherence distillation of the former can reach the upper bound,
namely QI relative entropy, while that of the latter can not. Meanwhile, we
also present a sufficient condition such that the assistance of Alice via the
set of operations X can not help Bob improve his distillable coherence, and
this condition is that the state shared by Alice and Bob has vanished
basis-dependent discord.
