Assume that we have an unknown quantum state and we need to learn that unknown state with arbitrary precision.
Under what conditions can we learn the unknown state with arbitrary precision?
One condition that I understand is if I can get an arbitrary number of copies of the unknown states, I can estimate the probabilities and consequently conclude about the amplitudes.
Another condition is the ability to measure in the standard basis. But do we need the ability to measure in all bases other than the standard basis? Put another way, if I restrict myself to measurements in the standard basis only, would I fail to learn the unknown quantum state with arbitrary precision?