No, the fact that what you said is not true is basically the core concept of quantum mechanics.

In what sense is that not true?

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.)

Yes, I you're right. I misinterpreted "pure quantum states" as "basis states" (I'm not much in the habit of thinking about QM these days). I believe the "(i.e. as a density matrix)", which makes my mistake more obvious, was added after I commented.