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

EDIT: As another comment pointed out I was wrong about this, please ignore.

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.

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