TeX source:

\rho' = \sum_{i=1}^n p_i|\mathbf{U}\psi_i\rangle\langle\mathbf{U}\psi_i| = \sum_{i=1}^n p_i\mathbf{U}|\psi_i\rangle\langle\psi_i|\mathbf{U}^\dagger = \mathbf{U}\rho\mathbf{U}^\dagger.