Suppose I have many copies of a qubit in the pure state $\vert\psi\rangle$. How many distinct observables do I need to choose to determine that I indeed hold $\vert\psi\rangle$?
For example, if $\vert\psi\rangle=\vert 0\rangle$, then checking $\langle 0\vert Z\vert 0\rangle = 1$ by doing many $Z$ measurements seems enough. This suggests if one has some idea of what the state might be, then one observable will do to confirm or deny this.
Secondly, if the state is unknown, is knowing $\langle \psi\vert X\vert \psi\rangle, \langle \psi\vert Y\vert \psi\rangle$ and $\langle \psi\vert Z\vert \psi\rangle$ enough? Is there a higher-dimensional analogue too?
