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 \(\rho \to U\rho U^{\dagger} = \mathcal{E}(\rho)\).