  The resource theory of coherence studies the operational value of
superpositions in quantum technologies. A key question in this theory concerns
the efficiency of manipulation and interconversion of this resource. Here we
solve this question completely for mixed states of qubits by determining the
optimal probabilities for mixed state conversions via stochastic incoherent
operations. This implies new lower bounds on the asymptotic state conversion
rate between mixed single-qubit states which in some cases is proven to be
tight. Furthermore, we obtain the minimal distillable coherence for given
coherence cost among all single-qubit states, which sheds new light on the
irreversibility of coherence theory.
