Density matrix Physically, the density matrix is a way to represent pure states and mixed states. The density matrix of pure state whose ket is | α ⟩ {\displaystyle |\alpha \rangle } is | α ⟩ ⟨ α | {\displaystyle |\alpha \rangle \langle \alpha |} . Mathematically, a density matrix has to satisfy the following conditions: Tr ⁡ ( ρ ) = 1 {\displaystyle \operatorname {Tr} (\rho )=1} ρ † = ρ {\displaystyle \rho ^{\dagger }=\rho }