= Solution
\b[Alice cannot send a <bit> to Charlie either.] An argument based only on the apparent chronological chain would be insufficient: Alice interacts with Bob before Bob interacts with Charlie. The extra fact is that these particular gates commute.
In Bob's <computational basis> the complete gate is
$$
U=I_A\otimes|0\rangle\langle0|_B\otimes I_C
+X_A\otimes|1\rangle\langle1|_B\otimes X_C.
$$
Both <controlled-NOT gates> preserve Bob's control label, and their target operations act on different systems. Thus $WV=VW$. We may consequently regard Bob–Charlie as the first interaction and Bob–Alice as the last one. The final operation on $AB$ cannot change Charlie's <reduced density matrix>, and Charlie's earlier state cannot depend on Alice's initially separate preparation.
The <target no-signalling in a common-control unitary> formula makes the independence explicit:
$$
\boxed{\rho_C'=\langle0|\rho_B|0\rangle\rho_C
+\langle1|\rho_B|1\rangle X\rho_CX.}
$$
Alice's <density matrix> does not appear. The formula includes arbitrary complex initial amplitudes and every final measurement on Charlie. This conclusion uses the specified pair of gates; a general intervening operation on Bob could change the communication possibilities.
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