The Robertson–Schrödinger uncertainty relation can be trivial if the state of the system is chosen to be eigenstate of one of the observable. The stronger uncertainty relations proved by Lorenzo Maccone and Arun K. Pati give non-trivial bounds on the sum of the variances for two incompatible observables. (Earlier works on uncertainty relations formulated as the sum of variances include, e.g., Ref. due to Yichen Huang.) For two non-commuting observables A and B the first stronger uncertainty relation is given by σ σ A 2 + σ σ B 2 ≥ ≥ ± ± i ⟨ ⟨ Ψ Ψ ∣ ∣ [ A , B ] | Ψ Ψ ⟩ ⟩ + ∣ ∣ ⟨ ⟨ Ψ Ψ ∣ ∣ ( A ± ± i B ) ∣ ∣ Ψ Ψ ¯ ¯ ⟩ ⟩ | 2 , where σ σ A 2 = ⟨ ⟨ Ψ Ψ | A 2 | Ψ Ψ ⟩ ⟩ − − ⟨ ⟨ Ψ Ψ ∣ ∣ A ∣ ∣ Ψ Ψ ⟩ ⟩ 2 , σ σ B 2 = ⟨ ⟨ Ψ Ψ | B 2 | Ψ Ψ ⟩ ⟩ − − ⟨ ⟨ Ψ Ψ ∣ ∣ B ∣ ∣ Ψ Ψ ⟩ ⟩ 2 , | Ψ Ψ ¯ ¯ ⟩ ⟩ is a normalized vector that is orthogonal to the state of the system | Ψ Ψ ⟩ ⟩ and one should choose the sign of ± ± i ⟨ ⟨ Ψ Ψ ∣ ∣ [ A , B ] ∣ ∣ Ψ Ψ ⟩ ⟩ to make this real quantity a positive number.