Obtaining phases of all basis states

Question

I’m wondering that is there a way to separate phases of basis states from the magnitudes (namely putting phases on some new basis states with the same probability magnitude) if the exact values of phases and magnitudes are unknown? One way I can think of is the quantum phase estimation routine. But to me it seems this method can only estimate one single phase at a time, so if there are a lot of basis states it will take a lot of time to estimate all the phases.

Answer 1

So, are you asking whether it's possible (in the qubit case) to perform an arbitrary map of the form $$ ae^{i\theta}|0\rangle+be^{i\phi}|1\rangle\rightarrow \frac{1}{\sqrt{2}}(e^{i\theta}|0\rangle+e^{i\phi}|1\rangle), $$ if $a$ and $b$ are unknown real numbers satisfying $a^2+b^2=1$? If this is the case, this is impossible.

Consider a simple case where instead of completely arbitrary values, you're either given $a|0\rangle+b|1\rangle$ or $a|0\rangle-b|1\rangle$. Your map would require these to be converted into the states $(|0\rangle\pm|1\rangle)/\sqrt{2}$, which are orthogonal. This means that your map would facilitate cloning of those two original states (because I can measure the outcome and know which state I had, so I can prepare many copies), which must be impossible because the two states are not orthogonal if $a^2\neq b^2$.