Controlled-unitary entangling criterion ($C(V)\text{ entangles some product input}⟺V/\propto I$)

For a qubit control and finite-dimensional target with unitary operator V, the controlled unitary gate $C(V)=|0⟩⟨0|\otimes I+|1⟩⟨1|\otimes V$ is entangling if and only if V is not scalar. If $V=e^{i\theta }I$, the controlled operation is a phase gate on the control tensored with the target identity and preserves every product state.

Otherwise the spectral theorem supplies two orthonormal eigenvectors with distinct eigenvalues ${\lambda }_0,{\lambda }_1$. Set $|b⟩=(|v_0⟩+|v_1⟩)/\sqrt{2}$ and use product input $|+⟩|b⟩$. The two conditional target vectors $|b⟩$ and $V|b⟩$ are linearly independent because their eigenbasis coefficients would otherwise require ${\lambda }_0={\lambda }_1$. The conditional-vector product-state criterion shows that their coherent controlled superposition is entangled. Thus even a target gate arbitrarily close to scalar can be entangling, although the resulting entanglement need not be large.