= Solution
{c}
Without interaction, the two-<qubit> <Hamiltonian operator> is $H_a\otimes I+I\otimes H_b$, so its evolution factors as $U_a\otimes U_b$. It maps each <product state> to another <product state>, and cannot implement an entangling <two-qubit gate> such as a <CNOT gate>.
In an <ion trap>, the <Coulomb interaction> couples the ions' displacements into collective vibrational <normal modes>. The <qubit> states are internal atomic states; lasers coupling internal excitation to creation or annihilation of a shared vibrational quantum allow that motion to carry conditional information between ions. Thus the physical mediator is a collective motional <normal mode> of the <Coulomb>-coupled chain, rather than a direct resonant exchange between isolated internal <qubit> transitions. Ground-state cooling and resolved motional sidebands support the ideal pulse model used here.
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