  Quantum coherence is a key element in topical research on quantum resource
theories and a primary facilitator for design and implementation of quantum
technologies. However, the resourcefulness of quantum coherence is severely
restricted by environmental noise, which is indicated by the loss of
information in a quantum system, measured in terms of its purity. In this work,
we derive the limits imposed by the mixedness of a quantum system on the amount
of quantum coherence that it can possess. We obtain an analytical trade-off
between the two quantities that upperbound the maximum quantum coherence for
fixed mixedness in a system. This gives rise to a class of quantum states,
"maximally coherent mixed states," whose coherence cannot be increased further
under any purity-preserving operation. For the above class of states, quantum
coherence and mixedness satisfy a complementarity relation, which is crucial to
understand the interplay between a resource and noise in open quantum systems.
