= Solution
\b[No.] Before any operation, Bob's <reduced density matrix> is $I/2$, independent of $|\alpha\rangle$. Any trace-preserving operation performed only by Alice leaves Bob's reduced state unchanged when her uncommunicated outcomes are averaged. This is the <no-communication theorem>, or <quantum no-signalling>.
If Bob received the unknown state, his reduced state would instead be the pure state $|\alpha\rangle\langle\alpha|$, which depends on Alice's input. That would let Alice signal by choosing different inputs, contradicting no-signalling. The two classical outcome bits are therefore essential.
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