 Solution

For normalized vector representatives in the same Hilbert space, the vector-norm convention is $\Vert |\psi ⟩-|\varphi ⟩{\Vert }_2\le \epsilon$. Since physical pure states identify global phases, a phase-insensitive version isThe 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.