= Global phase of a controlled quantum gate
{title2=$C(e^{i\phi}V)=(\operatorname{diag}(1,e^{i\phi})\otimes I)C(V)$}
Multiplying a target <unitary operator> by $e^{i\phi}$ changes the controlled operation by a <relative quantum phase> on its control-one branch. Substitution in $C(V)=|0\rangle\langle0|\otimes I+|1\rangle\langle1|\otimes V$ proves the displayed identity. In particular, $C(-I)=Z\otimes I$ rather than the <identity operator>. Preparing the control at $|+\rangle$ 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.
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