= Solution
{c}
Suppose the conditional atomic <pure state> were a <product state>
$$
(\alpha|e\rangle+\beta|g\rangle)\otimes(\gamma|e\rangle+\delta|g\rangle).
$$
Comparing its four <probability amplitudes> with $(|eg\rangle+|ge\rangle)/\sqrt2$ gives
$$
\alpha\gamma=0,\quad \beta\delta=0,\quad
\alpha\delta=\frac1{\sqrt2},\quad\beta\gamma=\frac1{\sqrt2}.
$$
The last two equations make all four factors nonzero, contradicting either of the first two. Equivalently, the coefficient <matrix> has <determinant> $-1/2$, while a <product state> has a rank-one coefficient <matrix> and zero <determinant>. Thus the atoms are entangled; indeed each <reduced density matrix> is $I/2$, making the conditional state maximally entangled.
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