 Solution

A bipartite pure state is entangled when it cannot be written $|\alpha {⟩}_a\otimes |\beta {⟩}_b$. 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):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.