The purity of a quantum state is conserved under unitary transformations acting on the density matrix in the form ρ ρ ↦ ↦ U ρ ρ U † † , where U is a unitary matrix. Specifically, it is conserved under the time evolution operator U ( t , t 0 ) = e − − i ℏ ℏ H ( t − − t 0 ) , where H is the Hamiltonian operator.