Past exam of the computer science course of the University of Oxford / 2019 / Quantum information / 2 / a

A normalized bipartite pure state is entangled when it cannot be written as a product state $|a{⟩}_A\otimes |b{⟩}_B$. Equivalently, its Schmidt rank exceeds one. This definition concerns the specified partition into systems A and B; the mixed-state definition of separability requires a different convex-mixture condition.

A bipartite unitary operator is an entangling gate if it maps at least one product state to an entangled state. It need not entangle every product input. For example, the controlled-NOT gate entangles $|+⟩|0⟩$ but leaves $|00⟩$ as a product. A gate that preserves every product input is not entangling, even if it exchanges the two subsystems.