= Solution
\b[Reverse communication is also possible.] Prepare Alice in $|+\rangle_A$. Bob encodes $m=0$ as $|+\rangle_B$ and $m=1$ as $|-\rangle_B$. Since $X|\pm\rangle=\pm|\pm\rangle$, <phase kickback> gives
$$
\operatorname{CNOT}_{A\to B}\bigl(|+\rangle_A|(-)^m\rangle_B\bigr)
=\frac{|0\rangle_A+(-1)^m|1\rangle_A}{\sqrt2}\otimes|(-)^m\rangle_B,
$$
where $|(-)^0\rangle=|+\rangle$ and $|(-)^1\rangle=|-\rangle$. Alice measures in the <Hadamard basis>: $+$ means zero and $-$ means one, deterministically.
The target's <Pauli X gate> <eigenvalue> becomes a <relative phase> on the control. This is the operational content of <CNOT control reversal in the Hadamard basis>; the <computational basis> control arrow alone does not determine every possible direction of communication.
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