Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 10E / a

If the Schmidt rank is one, the decomposition is $|\psi ⟩=|\alpha ⟩|\beta ⟩$ after absorbing its sole coefficient, so the state is a product. Conversely, a product state has coefficient matrix $X=a{b}^T$, which has rank one. Therefore a bipartite pure state is entangled exactly when its Schmidt rank is at least two.