A pure quantum state can be represented as a single vector | ψ ψ ⟩ ⟩ in the Hilbert space. In the density matrix formulation, a pure state is represented by the matrix ρ ρ pure = | ψ ψ ⟩ ⟩ ⟨ ⟨ ψ ψ | . However, a mixed state cannot be represented this way, and instead is represented by a convex combination of pure states ρ ρ mixed = ∑ ∑ i p i | ψ ψ i ⟩ ⟩ ⟨ ⟨ ψ ψ i | , while ∑ ∑ i p i = 1 for normalization. The purity parameter is related to the coefficients: If only one coefficient is equal to 1, the state is pure. Indeed, the purity is 1/ d when the state is completely mixed, i.e. ρ ρ completely mixed = 1 d ∑ ∑ i = 1 d | ψ ψ i ⟩ ⟩ ⟨ ⟨ ψ ψ i | = 1 d I d , where | ψ ψ i ⟩ ⟩ are d orthonormal vectors that constitute a basis of the Hilbert space.