Distillability for non-full-rank coherent states in the probabilistic framework

We adequately characterize the distillability of quantum coherence under the maximally incoherent operations (MIO) in the probabilistic distillation's framework. In particular, we prove that every non-full-rank coherent state exhibits a nonzero probability in the task of probabilistic deterministic distillation. Moreover, we find that the maximal coherence, a computable coherence monotone under strictly incoherent operations (SIO), is a coherence monotone under incoherent operations (IO) and add the proof that the maximal coherence fulfills strong monotonicity under SIO. It is suggested that the maximal success probability of distillation from all coherent states whose density matrix does not contain any rank-one submatrix is less than 1 under IO and equals 0 under SIO. Finally, we present an explicit example for probabilistic distillation under IO and show that a class of non-full rank 3-dimensional states possesses the probabilistic distillability.