Indexing an "unknown" quantum state

Assuming I have a state
$$|xrangle = frac{1}{sqrt{n}}sum_n |x_nrangle$$
where $|x_nrangle$ are quantum state vectors
$$|x_nrangle = frac{1}{|x_n|}sum_i x_{in}|irangle$$
and that I have a unitary $U:|x_nrangle mapsto e^{2pi itheta_n}|x_nrangle$ such that I can use the phase estimation procedure to get the state
$$|xrangle = frac{1}{sqrt{n}}sum_n |x_nrangle|theta_nrangle$$

Question:
I am wondering whether there is a way to compute the state
$$|xrangle = frac{1}{sqrt{n}}sum_n |x_nrangle|nrangle$$
I was thinking to modify the Phase Estimation Algorithm, but I still find it difficult to understand if I can prepare a unitary $U = sum_n e^{2pi in}|x_nranglelangle x_n|$ for instance.
I am not insterested in ordering the vectors $|x_nrangle$ in any way, I just wonder if there is a way to index them easily.
I don’t know if this problem has been raised before in literature and I don’t know where to look at. I’d be glad if someone had some insights.