The non-negative eigenvalues then imply a corresponding non-negativity condition on the determinant, det [ 1 ⟨ ⟨ x ⟩ ⟩ ⟨ ⟨ p ⟩ ⟩ ⟨ ⟨ x ⟩ ⟩ ⟨ ⟨ x ⋆ ⋆ x ⟩ ⟩ ⟨ ⟨ x ⋆ ⋆ p ⟩ ⟩ ⟨ ⟨ p ⟩ ⟩ ⟨ ⟨ p ⋆ ⋆ x ⟩ ⟩ ⟨ ⟨ p ⋆ ⋆ p ⟩ ⟩ ] = det [ 1 ⟨ ⟨ x ⟩ ⟩ ⟨ ⟨ p ⟩ ⟩ ⟨ ⟨ x ⟩ ⟩ ⟨ ⟨ x 2 ⟩ ⟩ ⟨ x p + i ℏ ℏ 2 ⟩ ⟨ ⟨ p ⟩ ⟩ ⟨ x p − − i ℏ ℏ 2 ⟩ ⟨ ⟨ p 2 ⟩ ⟩ ] ≥ ≥ 0 , or, explicitly, after algebraic manipulation, σ σ x 2 σ σ p 2 = ( ⟨ ⟨ x 2 ⟩ ⟩ − − ⟨ ⟨ x ⟩ ⟩ 2 ) ( ⟨ ⟨ p 2 ⟩ ⟩ − − ⟨ ⟨ p ⟩ ⟩ 2 ) ≥ ≥ ( ⟨ ⟨ x p ⟩ ⟩ − − ⟨ ⟨ x ⟩ ⟩ ⟨ ⟨ p ⟩ ⟩ ) 2 + ℏ ℏ 2 4 .