= Solution
Use <deterministic two-qubit entanglement dilution>, retaining the specified complex <probability amplitudes>. Alice implements a <quantum instrument> on $A$ with <Kraus operators>
$$
M_0=\begin{pmatrix}\alpha&0\\0&\beta\end{pmatrix},
\qquad
M_1=\begin{pmatrix}\beta&0\\0&\alpha\end{pmatrix}.
$$
The <Kraus representation> completeness condition holds even for complex $\alpha,\beta$:
$$
M_0^\dagger M_0+M_1^\dagger M_1=(|\alpha|^2+|\beta|^2)I=I.
$$
To realize this using the permitted <quantum gates> and a <basic measurement>, Alice initializes an <ancilla qubit> $C$ to $|0\rangle$. Conditional on her <computational-basis state> being $j$, she applies to $C$ the <unitary matrix>
$$
V_0=\begin{pmatrix}\alpha&-\overline\beta\\\beta&\overline\alpha\end{pmatrix},
\qquad
V_1=\begin{pmatrix}\beta&-\overline\alpha\\\alpha&\overline\beta\end{pmatrix}.
$$
Each has <orthonormal> columns. Therefore $W=|0\rangle\langle0|_A\otimes V_0+|1\rangle\langle1|_A\otimes V_1$ is a <unitary gate>, with
$$
W\bigl(|u\rangle_A|0\rangle_C\bigr)
=M_0|u\rangle_A|0\rangle_C+M_1|u\rangle_A|1\rangle_C.
$$
A <computational-basis measurement> of $C$ implements the desired <quantum instrument>. Its two unnormalized branches on $AB$ are
$$
(M_0\otimes I)|\phi_+\rangle
=\frac{\alpha|00\rangle+\beta|11\rangle}{\sqrt2},
\qquad
(M_1\otimes I)|\phi_+\rangle
=\frac{\beta|00\rangle+\alpha|11\rangle}{\sqrt2}.
$$
Each has <probability> $1/2$. Alice sends her outcome $m$ as one classical <bit>. For $m=0$, both parties keep their <qubits> unchanged. For $m=1$, both apply a <Pauli X gate>, giving
$$
(X\otimes X)(\beta|00\rangle+\alpha|11\rangle)
=\alpha|00\rangle+\beta|11\rangle.
$$
Thus \b[both outcomes produce the exact target using one classical bit and local operations]. The construction includes $\alpha=0$ or $\beta=0$ and arbitrary <relative quantum phases>; it requires no <quantum channel> between the parties.
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