= Controlled-unitary entangling criterion
{title2=$C(V)\text{ entangles some product input}\iff V\not\propto I$}
For a <qubit> control and finite-dimensional target with <unitary operator> V, the <controlled unitary gate> $C(V)=|0\rangle\langle0|\otimes I+|1\rangle\langle1|\otimes V$ 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 $|b\rangle=(|v_0\rangle+|v_1\rangle)/\sqrt2$ and use product input $|+\rangle|b\rangle$. The two conditional target <vectors> $|b\rangle$ and $V|b\rangle$ 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.
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