With similar arguments, one can derive a relation with a convex roof on the right-hand side σ σ A 2 F Q [ ϱ ϱ , B ] ≥ ≥ 4 [ min p k , Ψ Ψ k ∑ ∑ k p k L ( | Ψ Ψ k ⟩ ⟩ ⟨ ⟨ Ψ Ψ k | ) ] 2 where F Q [ ϱ ϱ , B ] denotes the quantum Fisher information and the density matrix is decomposed to pure states as ϱ ϱ = ∑ ∑ k p k | Ψ Ψ k ⟩ ⟩ ⟨ ⟨ Ψ Ψ k | . The derivation takes advantage of the fact that the quantum Fisher information is the convex roof of the variance times four.