Main Question or Discussion Point
I am a little bit versed in the formalism of Hilbert space, the state vector and the matricies and their eigenvector basis of observables in QM. I understand that a pure state in one observable basis may be (is) a superposition in another basis (spin xUP) is superposition of spinyUP, spin yDOWN eigenvectors. And that a superposition may also be a pure state if you pick the right observable. My question: is EVERY allowable superposition in quantum mechanics, every state the system could end up in, a pure state for some observable, even if this observable is very complex to realize? I'm speaking in principle rather than practice. If so, would it be correct to say that the state of the system is always in principle an observable pure state but we just have to keep changing bases? I apologize if I'm wording things awkwardly.