= Solution
Let $Q$ be Alice's unknown <qubit>, with <pure state> $|\phi\rangle_Q=\alpha|0\rangle+\beta|1\rangle$. Alice and Bob initially share the <Bell state> $|\Phi^+\rangle_{AB}=(|00\rangle+|11\rangle)/\sqrt2$. Alice holds $Q,A$ and Bob holds $B$. Define the <Bell basis>
$$
|\Phi^\pm\rangle=\frac{|00\rangle\pm|11\rangle}{\sqrt2},\qquad
|\Psi^\pm\rangle=\frac{|01\rangle\pm|10\rangle}{\sqrt2}.
$$
Expanding in this <orthonormal basis> gives the <Bell-basis teleportation identity>
$$
\begin{aligned}
|\phi\rangle_Q|\Phi^+\rangle_{AB}=\frac12\bigl(&|\Phi^+\rangle_{QA}|\phi\rangle_B
+|\Phi^-\rangle_{QA}Z|\phi\rangle_B\\
&+|\Psi^+\rangle_{QA}X|\phi\rangle_B
+|\Psi^-\rangle_{QA}XZ|\phi\rangle_B\bigr).
\end{aligned}
$$
For example, the last branch on $B$ is $\alpha|1\rangle-\beta|0\rangle$, which is $XZ|\phi\rangle$; the ordering of the <Pauli operators> therefore matters.
Alice performs a <projective measurement> in the <Bell basis> on her two local <qubits>. Each outcome has <probability> $1/4$, independent of the unknown amplitudes. She encodes the four possible outcomes as two classical bits and sends them to Bob. For outcomes $\Phi^+,\Phi^-,\Psi^+,\Psi^-$, respectively, Bob applies the <local unitary operations>
$$
I,\qquad Z,\qquad X,\qquad ZX.
$$
These are inverses of the corresponding branch <Pauli operators>, so \b[Bob obtains exactly the unknown input qubit in every branch]. Using $XZ$ instead of $ZX$ as the last correction also succeeds up to an irrelevant overall phase. Alice need not know $\alpha$ or $\beta$. Every quantum operation is local to one of the parties, and the only transmitted information is the two-bit classical outcome. The shared <Bell pair> is consumed.
Before the message arrives, Bob's <density operator> is
$$
\frac14\left(\rho+Z\rho Z+X\rho X+XZ\rho ZX\right)=\frac I2,
$$
so no unknown-state information is available without the <classical communication>. Alice's original <qubit> has been included in the destructive <Bell-basis measurement>; the protocol creates no extra copy. Since the corrected transformation is the identity on every input vector and is linear, it also preserves <entanglement> with an external reference and transfers mixed inputs. Thus this is deterministic <quantum teleportation>, using precisely one shared maximally entangled <Bell pair>, two classical bits, and local operations.
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