= Solution
\b[No secure key can be produced from this state alone.] Factoring the printed amplitudes gives the normalized <product state>
$$
\left(\frac{\sqrt2}{3}|0\rangle_A+\frac{\sqrt7}{3}|1\rangle_A\right)\otimes|+\rangle_B.
$$
Eve is uncorrelated with the joint <pure state>, but Alice and Bob also have no correlations with each other: local outcome probabilities factor. They may generate independent private random bits locally, but <local operations and classical communication> cannot make those bits shared and secret using only public messages. A <product state> supplies no shared secret resource.
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