Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 10D / a / Solution

A pure bipartite state is entangled when it is not a product state $|u{⟩}_A\otimes |v{⟩}_B$.

Sincewe haveIts coefficient matrix is proportional towhose determinant is $-4\ne 0$. By the two-qubit product-state determinant criterion, a product state would have determinant zero. Therefore $|\alpha ⟩$ is entangled.