Pure-state norm distance ($d_2(\psi ,\varphi )=\Vert |\psi ⟩-|\varphi ⟩{\Vert }_2$)

The Euclidean norm distance between normalized pure state vector representatives in a common Hilbert space. Its square is $2-2\mathrm{Re}⟨\psi |\varphi ⟩$. If vectors differing by a global phase are identified, minimize over that phase to obtain $\sqrt{2-2|⟨\psi |\varphi ⟩|}$. This is a different distance convention from trace distance or one minus squared quantum fidelity; stating the convention is necessary when defining $\epsilon$-closeness.