Purity decouples a subsystem from its purification

If subsystem $S$ has pure state $|u⟩⟨u|$ in a normalized joint pure vector $|\Omega {⟩}_{SE}$, then $|\Omega {⟩}_{SE}=|u{⟩}_S\otimes |e{⟩}_E$. Indeed, for an orthonormal basis $\{v_j\}$ of $u^{\perp }$, each squared norm $\Vert (⟨v_j|\otimes I)|\Omega ⟩{\Vert }^2=⟨v_j|{\rho }_S|v_j⟩$ is zero. Therefore the joint vector has no component outside $\mathbb{C}u\otimes {\mathcal{H}}_E$. In particular, if Alice and Bob share a known pure entangled state, an adversary holding a purifying system has no correlations with it. This does not make every pure bipartite state useful for key distribution: a pure product state has no shared random correlations.