Let | ψ ψ ⟩ ⟩ be an arbitrary quantum state in some Hilbert space and let there be a physical process that transforms | ψ ψ ⟩ ⟩ ⟨ ⟨ ψ ψ | → → ρ ρ with ρ ρ = ∑ ∑ k p k | k ⟩ ⟩ ⟨ ⟨ k | . If ρ ρ is independent of the input state | ψ ψ ⟩ ⟩ , then in the enlarged Hilbert space the mapping is of the form | ψ ψ ⟩ ⟩ ⊗ ⊗ | A ⟩ ⟩ → → ∑ ∑ k p k | k ⟩ ⟩ ⊗ ⊗ | A k ( ψ ψ ) ⟩ ⟩ = ∑ ∑ k p k | k ⟩ ⟩ ⊗ ⊗ ( | q k ⟩ ⟩ ⊗ ⊗ | ψ ψ ⟩ ⟩ ⊕ ⊕ 0 ) , where | A ⟩ ⟩ is the initial state of the environment, | A k ( ψ ψ ) ⟩ ⟩ 's are the orthonormal basis of the environment Hilbert space and ⊕ ⊕ 0 denotes the fact that one may augment the unused dimension of the environment Hilbert space by zero vectors.