Suppose we have a single qubit mixed state described by a density matrix $\rho$, and we want to find a unitary transformation that brings $\rho$ to the pure state $|0\rangle\langle 0|$. Is there a matrix $U$ such that $U^{\dagger}U=I$ and that $U \rho U^\dagger=|0\rangle\langle 0|$, independent of the matrix elements of $\rho$?. That is, if for an arbitrary single qubit $\rho$, we can always find such projection with a fixed unitary $U$, with constant matrix elements, always the same $U$ regardless of the entries of $\rho$.
EDIT: Probably this is not possible since $\rho$ lies inside the unit sphere and a unitary transformation necessarily preserves the norm, so one needs a non-unitary operation in order to do that. Is there any good reference where I could look into this issue?