Optimal probabilistic distillation of quantum coherence

We complete the task of optimal probabilistic coherence distillation protocol, whose aim is to transform a general state into a set of n-level maximally coherent states via strictly incoherent operations (SIO). Specifically, we present the necessary and sufficient condition for the transformation from a mixed state into a pure-state ensemble via SIO. With the aid of this condition and the simplex algorithm, we can accomplish the probabilistic distillation protocol by presenting an analytical expression of the maximal average distillable coherence of a general state and constructing the corresponding operation achieving this bound. Our protocol is a universal protocol since it can be applied to any coherence measure.