Product States

Canonical treatment: Product States maintains the complete tests, dynamics, examples, exercises, and references.

This bridge is intentionally limited to the factorization criterion and notation needed by the composition postulate.

Factorization criterion

A bipartite pure state is product across $A|B$ when

$$\lvert\Psi\rangle_{AB}=\lvert\psi\rangle_A\otimes\lvert\phi\rangle_B.$$

In a product basis, $\lvert\Psi\rangle=\sum_{ij}C_{ij}\lvert i\rangle\lvert j\rangle$ is product exactly when the coefficient matrix $C$ has rank one. For two qubits this is equivalent to $\det C=0$ for a nonzero state.

What factorization predicts

For local observables,

$$\langle A\otimes B\rangle = \langle A\rangle_{\psi}\langle B\rangle_{\phi}.$$

Local superposition does not imply entanglement: $\lvert+\rangle_A\otimes\lvert-\rangle_B$ remains product. Conversely, a sum of product-basis vectors need not factor.

Always state the subsystem split before calling a vector product. Continue to the canonical page for mixed product states, separability, local dynamics, and multipartite factorization.