= Purity decouples a subsystem from its purification
If subsystem $S$ has <pure state> $|u\rangle\langle u|$ in a normalized joint pure vector $|\Omega\rangle_{SE}$, then $|\Omega\rangle_{SE}=|u\rangle_S\otimes|e\rangle_E$. Indeed, for an <orthonormal basis> $\{v_j\}$ of $u^\perp$, each squared norm $\|(\langle v_j|\otimes I)|\Omega\rangle\|^2=\langle v_j|\rho_S|v_j\rangle$ is zero. Therefore the joint vector has no component outside $\mathbb Cu\otimes\mathcal H_E$. 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.
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