A graphical intuition of purity may be gained by looking at the relation between the density matrix and the Bloch sphere, ρ ρ = 1 2 ( I + a ⋅ ⋅ σ σ ) , where a is the vector representing the quantum state (on or inside the sphere), and σ σ = ( σ σ x , σ σ y , σ σ z ) is the vector of the Pauli matrices.