The Robertson–Schrödinger uncertainty can be improved noting that it must hold for all components ϱ ϱ k in any decomposition of the density matrix given as ϱ ϱ = ∑ ∑ k p k ϱ ϱ k . Here, for the probabilities p k ≥ ≥ 0 and ∑ ∑ k p k = 1 hold. Then, using the relation ∑ ∑ k a k ∑ ∑ k b k ≥ ≥ ( ∑ ∑ k a k b k ) 2 for a k , b k ≥ ≥ 0 , it follows that σ σ A 2 σ σ B 2 ≥ ≥ [ ∑ ∑ k p k L ( ϱ ϱ k ) ] 2 , where the function in the bound is defined L ( ϱ ϱ ) = | 1 2 tr ⁡ ⁡ ( ρ ρ { A , B } ) − − tr ⁡ ⁡ ( ρ ρ A ) tr ⁡ ⁡ ( ρ ρ B ) | 2 + | 1 2 i tr ⁡ ⁡ ( ρ ρ [ A , B ] ) | 2 . The above relation very often has a bound larger than that of the original Robertson–Schrödinger uncertainty relation. Thus, we need to calculate the bound of the Robertson–Schrödinger uncertainty for the mixed components of the quantum state rather than for the quantum state, and compute an average of their square roots. The following expression is stronger than the Robertson–Schrödinger uncertainty relation σ σ A 2 σ σ B 2 ≥ ≥ [ max p k , ϱ ϱ k ∑ ∑ k p k L ( ϱ ϱ k ) ] 2 , where on the right-hand side there is a concave roof over the decompositions of the density matrix. The improved relation above is saturated by all single-qubit quantum states.