The linear entropy then is obtained by expanding ln ρ = ln (1−(1− ρ)), around a pure state, ρ = ρ ; that is, expanding in terms of the non-negative matrix 1− ρ in the formal Mercator series for the logarithm, − − ⟨ ⟨ ln ⁡ ⁡ ρ ρ ⟩ ⟩ = ⟨ ⟨ 1 − − ρ ρ ⟩ ⟩ + 1 2 ⟨ ⟨ ( 1 − − ρ ρ ) 2 ⟩ ⟩ + 1 3 ⟨ ⟨ ( 1 − − ρ ρ ) 3 ⟩ ⟩ + ⋯ ⋯ , and retaining just the leading term. Both the linear and the von Neumann entropy measure the degree of mixing of a state, although the linear entropy is easier to calculate, as it does not require diagonalization of the density matrix. Some authors define linear entropy with a different normalization S L = ˙ ˙ d d − − 1 ( 1 − − tr ⁡ ⁡ ( ρ ρ 2 ) ) , which ensures that the quantity ranges from zero to unity.