Global phase of a controlled quantum gate ($C(e^{i\varphi }V)=(\mathrm{diag}(1,e^{i\varphi })\otimes I)C(V)$)

Multiplying a target unitary operator by $e^{i\varphi }$ changes the controlled operation by a relative quantum phase on its control-one branch. Substitution in $C(V)=|0⟩⟨0|\otimes I+|1⟩⟨1|\otimes V$ proves the displayed identity. In particular, $C(-I)=Z\otimes I$ rather than the identity operator. Preparing the control at $|+⟩$ and applying a final Hadamard gate distinguishes $C(I)$ from $C(-I)$ with certainty, independently of the target quantum state. The global phase of an uncontrolled quantum gate is unobservable, but a specified controlled unitary gate provides a branch relative to which that phase can be measured.