  We characterize the operational task of environment-assisted distillation of
quantum coherence under different sets of free operations when only a finite
supply of copies of a given state is available. We first evaluate the one-shot
assisted distillable coherence exactly, and introduce a semidefinite
programming bound on it in terms of a smooth entropic quantity. We prove the
bound to be tight for all systems in dimensions 2 and 3, which allows us to
obtain computable expressions for the one-shot rate of distillation, establish
an analytical expression for the best achievable fidelity of assisted
distillation for any finite number of copies, and fully solve the problem of
asymptotic zero-error assisted distillation for qubit and qutrit systems. Our
characterization shows that all relevant sets of free operations in the
resource theory of coherence have exactly the same power in the task of
one-shot assisted coherence distillation, and furthermore resolves a conjecture
regarding the additivity of coherence of assistance in dimension 3.
