Can you explain? That's not how I interpret either. Every quantum state can be realized as a classical mixture of pure quantum states (i.e. as a density matrix).
EDIT: As another comment pointed out I was wrong about this, please ignore.
It seems true at least if we accept that a pure state is a mixture of itself with nothing else.
Any quantum state CAN be represented by a density matrix. That's what density matrices are for. That quantum state may be pure or may be a mixture. (A mixture density matrix could also be a partial trace representing a subsystem of a larger system instead of a true mixture of pure states.)