  The resource theory of quantum coherence studies the off-diagonal elements of
a density matrix in a distinguished basis, whereas the resource theory of
purity studies all deviations from the maximally mixed state. We establish a
direct connection between the two resource theories, by identifying purity as
the maximal coherence which is achievable by unitary operations. The states
that saturate this maximum identify a universal family of maximally coherent
mixed states. These states are optimal resources under maximally incoherent
operations, and thus independent of the way coherence is quantified. For all
distance-based coherence quantifiers the maximal coherence can be evaluated
exactly, and is shown to coincide with the corresponding distance-based purity
quantifier. We further show that purity bounds the maximal amount of
entanglement and discord that can be generated by unitary operations, thus
demonstrating that purity is the most elementary resource for quantum
information processing.
