 Solution

For normalized vector representatives in the same Hilbert space, two pure states are $\epsilon$-close in pure-state norm distance whenHere $\Vert \cdot {\Vert }_2$ is the Euclidean norm of the coefficient vector in any orthonormal basis. Equivalently, $2-2\mathrm{Re}⟨\psi |\varphi ⟩\le {\epsilon }^2$. If the convention identifies vector representatives differing by a global phase, first minimize over that phase, giving ${\min }_{\gamma }\Vert |\psi ⟩-e^{i\gamma }|\varphi ⟩{\Vert }_2=\sqrt{2-2|⟨\psi |\varphi ⟩|}$; the vector-norm convention above specifies the representatives.