To conclude that the inequality is saturated by some POVM, we need only consider the projective measurement with elements corresponding to the eigenvectors of ρ ρ − − σ σ . With this choice, δ δ ( P Π Π , ρ ρ , P Π Π , σ σ ) = 1 2 ∑ ∑ i | Tr ⁡ ⁡ ( Π Π i ( ρ ρ − − σ σ ) ) | = 1 2 ∑ ∑ i | λ λ i | = T ( ρ ρ , σ σ ) , where λ λ i are the eigenvalues of ρ ρ − − σ σ .