The von Neumann entropy S of a mixture can be expressed in terms of the eigenvalues of ρ ρ or in terms of the trace and logarithm of the density operator ρ ρ . Since ρ ρ is a positive semi-definite operator, it has a spectral decomposition such that ρ ρ = ∑ ∑ i λ λ i | φ φ i ⟩ ⟩ ⟨ ⟨ φ φ i | , where | φ φ i ⟩ ⟩ are orthonormal vectors, λ λ i ≥ ≥ 0 , and ∑ ∑ λ λ i = 1 . Then the entropy of a quantum system with density matrix ρ ρ is