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

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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.

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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.

Physically if we were able to transforme mixed states to pure states via unitary transformations we would decrease the entropy of the universe, and that would be funny.

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