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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.

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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 $a^2\neq b^2$.

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  • $\begingroup$ I am thinking to have n qubits as ancilla qubits (assuming originally we also have n qubits) so that the phases can be transferred to the ancilla qubits (the ancilla qubits are now 1/√2(eiθ|0⟩+eiϕ|1⟩ and the original state becomes a|0>+b|1>) , so in some sense the absolute values and phases are 'decoupled'. Does this process involve cloning? $\endgroup$ – Zzy1130 Jun 12 '19 at 8:59
  • $\begingroup$ My objective is only to obtain the values of the phases (of course the 'decoupling' I mentioned is better if it is possible). I think it is actually possible through quantum phase estimation it is just it is too tedious. By the way I came across this paper iopscience.iop.org/article/10.1088/1367-2630/aafb8e/pdf which seems to provide a solution but I can't be completely sure since I cannnot understand it completely... $\endgroup$ – Zzy1130 Jun 12 '19 at 9:09
  • $\begingroup$ In my example, any deterministic way that would let you distinguish the phases $\pm 1$ on the $\ket{1}$ component implies cloning. Alternatively you could potentially use optimal cloning-type arguments to put an upper bound on the success probability of any such protocol to determine the $\pm 1$ values. $\endgroup$ – DaftWullie Jun 12 '19 at 9:17

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