= Solution
\b[The normalized ray produces a secure key; the literal printed vector is not normalized.] The squared norm of the coefficients in the PDF is $8/9+1/81=73/81$. Thus the physical <pure state> on that ray has normalized coefficients $6\sqrt2/\sqrt{73}$ and $1/\sqrt{73}$. This is a genuine <entangled state> with two nonzero <Schmidt coefficients>, and Eve is independent of it by <purity decouples a subsystem from its purification>.
For an explicit way to obtain uniform key bits, Alice applies a two-outcome local <quantum measurement> with <Kraus operators>
$$
K_s=\begin{pmatrix}1/(6\sqrt2)&0\\0&1\end{pmatrix},\qquad K_f=\begin{pmatrix}\sqrt{71/72}&0\\0&0\end{pmatrix}.
$$
Their squared products sum to the identity. The successful unnormalized state is $(|00\rangle+|11\rangle)/\sqrt{73}$, so
$$
\boxed{p_s=2/73,\qquad p_f=71/73.}
$$
Alice announces only success or failure. Success leaves an exact <Bell pair>, yielding one uniform secret bit; failure leaves $|00\rangle$ and is discarded. With many copies the successful fraction is positive. Alternatively, direct <computational basis> measurements give a shared biased bit with probabilities $72/73$ and $1/73$, from which <privacy amplification> can extract uniform secret bits. The coefficient $1/9$ has not been silently replaced by $1/3$.
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