= Solution
The relevant security model gives Eve the purifying system and access to the public transcript. A useful preliminary fact is that <purity decouples a subsystem from its purification>. If the Alice–Bob marginal is $|u\rangle\langle u|$ and $|\Omega\rangle_{ABE}$ is a <purification of a density operator>, then for every $v\perp u$,
$$
\|(\langle v|_{AB}\otimes I_E)|\Omega\rangle\|^2=\langle v|\rho_{AB}|v\rangle=0.
$$
Hence $|\Omega\rangle_{ABE}=|u\rangle_{AB}|e\rangle_E$: Eve is uncorrelated with their joint <pure state>. A <Bell pair> then supplies a uniform shared secret bit when both parties measure in the <computational basis>. Pure <product states> have no shared random correlations, while a <mixed state> must be assessed with its purifying information included. The classifications below use the usual <quantum key distribution> setting with an authenticated public classical channel; no full security proof is required.
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