= Global phase of an uncontrolled quantum gate is unobservable
{title2=$\mathcal U_{e^{i\gamma}V}=\mathcal U_V$}
A physical <unitary gate> used without coherent control gives the same <quantum channel> for $V$ and $e^{i\gamma}V$:
$$
(e^{i\gamma}V)\rho(e^{i\gamma}V)^\dagger=V\rho V^\dagger.
$$
Interleaving any preparations, known operations and measurements preserves this equality. Equivalently, each fixed measurement branch with $k$ uses gains only the overall factor $e^{ik\gamma}$, which cancels in its <probability>. Thus unrestricted repeated uncontrolled uses do not determine the common phase of the <eigenvalues>. A supplied <controlled unitary gate> is a different resource: $|0\rangle\langle0|\otimes I+|1\rangle\langle1|\otimes V$ changes by a relative control phase when $V$ is replaced by $e^{i\gamma}V$. Such controlled access permits <quantum phase estimation> to measure that phase relative to the identity branch.
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