= Solution
{c}
A bipartite <pure state> is entangled when it cannot be written $|\alpha\rangle_a\otimes|\beta\rangle_b$. For <mixed states>, the corresponding definition excludes every convex mixture of <product states>, not just a single <product state>.
Apply a <Hadamard gate> to ion $a$, followed by the <CNOT gate> of part (d):
$$
|gg\rangle\xrightarrow{H_a}\frac{|gg\rangle+|eg\rangle}{\sqrt2}
\xrightarrow{C_X}\boxed{\frac{|gg\rangle+|ee\rangle}{\sqrt2}.}
$$
The lower network in the figure shows this <Bell pair> preparation. Either ion alone has <reduced density matrix> $I/2$, whereas the joint <quantum state> is pure; hence the joint <quantum state> cannot be a <product state>. The entangling operation can be expanded into $H_b,G,Z_a,H_b$ exactly as above.
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