= Solution
Combine the hypothetical protocol with <superdense coding>. Before a message is chosen, Alice and Bob share a further <Bell pair> $|\phi_+\rangle_{QR}$, with $Q$ at Alice and $R$ at Bob, in addition to any resources needed for the hypothetical <quantum teleportation> protocol. To encode two classical <bits> $u,v$, Alice applies the <Pauli gates> $Z_Q^uX_Q^v$, preparing
$$
|\beta_{uv}\rangle_{QR}=(Z_Q^uX_Q^v\otimes I_R)|\phi_+\rangle_{QR}.
$$
The four choices are the four mutually <orthogonal> <Bell states>. Alice now uses the hypothetical protocol to teleport $Q$ to Bob using only one classical <bit>. Its assumed preservation of the joint <quantum state> gives Bob both $B$ and $R$ in
$$
|\beta_{uv}\rangle_{BR}=(Z_B^uX_B^v\otimes I_R)|\phi_+\rangle_{BR}.
$$
Bob performs a <Bell-basis measurement>. Explicitly, a <controlled-NOT gate> $B\to R$, followed by a <Hadamard gate> on $B$, gives
$$
\boxed{(H_B\otimes I_R)\operatorname{CNOT}_{B\to R}|\beta_{uv}\rangle_{BR}=|u\rangle_B|v\rangle_R.}
$$
A <computational-basis measurement> therefore recovers \b[both message bits with certainty after only one transmitted classical bit]. The initial <Bell pairs> are shared before the message is chosen and carry no message themselves. This proves the <classical communication lower bound for exact qubit teleportation>: a perfect protocol preserving an external reference needs at least two transmitted classical <bits>.
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