Born rule for a product-basis measurement

For a bipartite state $|\psi ⟩$, measurement outcome $(x_j,y_k)$ has probability $|⟨x_j{y}_k|\psi ⟩{|}^2$.

For two labelled orthonormal bases, the event of equal labels is represented by ${\sum }_j|x_j{y}_j⟩⟨x_j{y}_j|$.

For a product state, probabilities of local measurement outcomes multiply.