Is there any restriction on the ability to measure the full quantum state of a system without inducing backaction?

Question

Suppose an arbitrary quantum system is in the state $ \mid \Psi \rangle $, which may or may not be a function of time.

An initially ignorant obsevrer would like to figure out what $ \mid \Psi \rangle $ is at some time, without destroying or altering it or its evolution (let's say up to some arbitrarily small tolerance).

My question is: do there exist states for which this is even in principle impossible? Or should all states allow for such "complete and nondestructive information retrieval" (leaving it then only to the experimentalist to figure out the question of "how")?

Just to be clear I am not asking here whether one may measure all possible observables of a system simultaneously, as that is obviously forbidden by things like $ [q,p] = i \hbar$ and the uncertainty principle. I am asking about knowlege of the quantum state itself - i.e. can one always completely determine the position of a system in Hilbert space.

Edit: Up to phase.

Answer 1

Short answer: This is impossible in principle for all states.

A bit more elaborate: Given any state, by the postulate of quantum mechanics, it will be projected into the eigenspace of the eigenvalue measured when measuring an observable (this can be extended to quantum instruments). This means that the state will not be altered if and only if it is in the eigenspace of the observable you measure. Since the state is unknown to the observer, this can only happen by chance and if you choose the observable to measure at random from the set of all possible observables the probability is zero in any usual measure.

Only if you have more information - e.g. the knowledge that the state is in one of a certain set of orthogonal vectors with certainty - can you perform such a "nondemolition" measurement and actually learn something.