# Unitary evolution from a mixed state to a pure state

Why is it not possible to have an unitary evolution from a mixed state to a pure state ?

Because if $A\sim B$ then $$A^2\sim B^2$$
Recall that a state is pure if and only if $\rho^2=\rho$, and that time-evolution is a similarity transformation.
Because the Von Neumann entropy $$S(\rho)=-tr(\rho\log{\rho})$$ is conserved under unitary transformations, and $$S(\rho)=0$$ if and only if $$\rho$$ is a pure state.