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.

A nonzero bipartite pure state is a product state if and only if all its partial inner products on the first subsystem lie in a common one-dimensional vector subspace of the second. For a product $|a⟩\otimes |b⟩$, every contraction equals $⟨u|a⟩|b⟩$. Conversely, contract against each vector of an orthonormal basis to write $|\Psi ⟩={\sum }_i|i⟩\otimes |{\chi }_i⟩$. If $|{\chi }_i⟩=c_i|b⟩$ for a common nonzero $|b⟩$, then $|\Psi ⟩=({\sum }_i{c}_i|i⟩)\otimes |b⟩$. Zero conditional vectors are allowed. A nontrivial linear dependence relation requires multipliers not both zero; allowing both multipliers to vanish would give no test at all.

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