Two-qubit product-state determinant criterion
= Two-qubit product-state determinant criterion
Write a two-qubit pure state as $\sum_{i,j=0}^1c_{ij}|ij\rangle$ and form its coefficient matrix $C=(c_{ij})$. The state is a <product state> exactly when $C$ has rank one, equivalently when
$$
c_{00}c_{11}=c_{01}c_{10}.
$$