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A controlled unitary gate applies a unitary operator $U$ to a target system when a control qubit is $|1⟩$, and leaves the target unchanged when the control is $|0⟩$:For a coherent control, linearity givesThe control has not been measured; the output may be entangled. The controlled-unitary entangling criterion states exactly when some product input becomes entangled.

The global phase of a controlled quantum gate matters: if $U=e^{i\theta }I$, then $C(U)$ acts as a phase gate on the control. A phase invisible on the isolated target is a relative phase between control branches. ^[ref]

By the spectral theorem, the nonscalar unitary V has orthonormal eigenvectors $|v_0⟩,|v_1⟩$ with distinct eigenvalues ${\lambda }_0\ne {\lambda }_1$. Choose $|b⟩=(|v_0⟩+|v_1⟩)/\sqrt{2}$. Then $V|b⟩=({\lambda }_0|v_0⟩+{\lambda }_1|v_1⟩)/\sqrt{2}$ is not proportional to $|b⟩$.

On the product input $|+⟩\otimes |b⟩$, the controlled unitary gate producesIts two conditional target vectors are linearly independent, so the conditional-vector product-state criterion proves that the output is entangled. Equivalently, in the control computational basis and target eigenbasis its coefficient matrix is $\frac{1}{2}\left(\begin{array}{cc}1& 1\\ {\lambda }_0& {\lambda }_1\end{array}\right)$, whose determinant is $({\lambda }_1-{\lambda }_0)/4\ne 0$. Thus U is an entangling gate. The excluded scalar case would only apply a phase gate to the control and could not entangle the factors; this is the controlled-unitary entangling criterion.