The purity of a normalized quantum state satisfies 1 d ≤ ≤ γ γ ≤ ≤ 1 , where d is the dimension of the Hilbert space upon which the state is defined. The upper bound is obtained by tr ⁡ ⁡ ( ρ ρ ) = 1 and tr ⁡ ⁡ ( ρ ρ 2 ) ≤ ≤ tr ⁡ ⁡ ( ρ ρ ) (see trace).