Product state

A bipartite density operator is a product state when it factors as ${\rho }_A\otimes {\rho }_B$. A pure product state factors as $|{\psi }_A⟩\otimes |{\psi }_B⟩$; its reduced density matrices have rank one and zero entanglement entropy. Mixed product states can have nonzero marginal Von Neumann entropy.

Write a two-qubit pure state as ${\sum }_{i,j=0}^1{c}_{ij}|ij⟩$ and form its coefficient matrix $C=(c_{ij})$. The state is a product state exactly when $C$ has rank one, equivalently when