# Finding a unitary transformation for a qubit mixed state that projects onto a pure state

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?

If you want to work out things yourself, here's a (probably incomplete) reasoning. Suppose that $$\rho$$ is not pure. As such $$\DeclareMathOperator{tr}{tr}\tr\left(\rho^2\right)<1$$. Denote $$\sigma=U\rho U^{\dagger}$$. Then $$\tr\left(\sigma^2\right)=\tr\left(U\rho^2U^\dagger\right)=\tr\left(\rho^2\right)<1$$. Thus, $$\sigma$$ cannot be pure.
For the task of mapping an arbitrary state $$\rho$$ to $$|0\rangle\!\langle0|$$, this is forbidden by the fact that the transformation you're asking for is unitary, and as such invertible.