  We consider a system composed of two qubits and a high-excitation-energy
quantum object used to mediate coupling between the qubits. We treat the entire
system quantum mechanically and analyze the properties of the eigenvalues and
eigenstates of the total Hamiltonian. After reproducing well-known results
concerning the leading term in the mediated coupling, we obtain an expression
for the residual coupling between the qubits in the off state. We also analyze
the entanglement between the three objects, i.e. the two qubits and the
coupler, in the eigenstates of the total Hamiltonian. Although we focus on the
application of our results to the recently realized parametric-coupling scheme
with two qubits, we also discuss extensions of our results to
harmonic-oscillator couplers, couplers that are near resonance with the qubits
and multi-qubit systems. In particular, we find that certain errors that are
absent for a two-qubit system arise when dealing with multi-qubit systems.
