# Exercise 8.1

A pure state $|\psi\rangle$ evolves under unitary transform $U$, which is described as $|\psi\rangle \to U|\psi\rangle$. Here $|\phi\rangle = U|\psi\rangle$ is still a pure state. 

The density matrix of a pure state $|\psi\rangle$ is given by $\rho = |\psi\rangle\langle \psi|$. Similarly, the density matrix of $|\phi\rangle$ is given by

$$
\rho' = |\phi\rangle\langle \phi| = U|\psi\rangle\langle \psi|U^{\dagger} = U\rho U^{\dagger}.
$$

Therefore, the process of unitary transform can be written as $\rho \to U\rho U^{\dagger} = \mathcal{E}(\rho)$.