= Solution
For normalized <vector> representatives in the same <Hilbert space>, the vector-norm convention is \b[$\||\psi\rangle-|\phi\rangle\|_2\le\varepsilon$]. Since physical <pure states> identify <global phases>, a phase-insensitive version is
$$
\boxed{\min_{\theta\in\mathbb R}\||\psi\rangle-e^{i\theta}|\phi\rangle\|_2=\sqrt{2-2|\langle\psi|\phi\rangle|}\le\varepsilon.}
$$
The equality follows by choosing the phase to make the overlap real and nonnegative. This states an explicit <pure-state norm distance> convention; fidelity-based or trace-distance conventions use different numerical error parameters. We use the phase-insensitive convention for the universality claim below.
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