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

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.

• You can't measure unknown state without altering it. The "proof" is through no cloning theorem. If you could clone an unknown state - then you could teleport it arbitrary far, make a tomography of it and know $|\Psi\rangle$ upto arbitrary precision, i.e - superluminal signaling. – Alexander Jul 13 '15 at 2:25