= Solution
In <superdense coding>, Alice and Bob initially share $|\Psi^+\rangle_{AB}$. Alice selects the same one-qubit <Pauli gate> $I,Z,X,XZ$ according to her message and sends her single encoded <qubit> to Bob. He now has both halves and measures in the <Bell basis>. For the chosen assignment,
$$
\boxed{\Pr(Y=j\mid X=i)=\delta_{ij},\qquad I(X:Y)=H(X),}
$$
which reaches two bits for the uniform distribution. A circuit implementation first converts the resource convention with $X_B$, then applies $\operatorname{CNOT}_{A\to B}$ and $H_A$, and measures both in the <computational basis>. The four outcomes are distinct; relabeling them restores the selected message order.
Before Alice's transmission, Bob's reduced <density operator> is $I/2$ for every choice of her local operation. Thus distributing the message-independent entangled resource has transmitted no information about the subsequently chosen message. It is nevertheless a real physical resource: the two-bit rate counts the one additional transmitted <qubit> after entanglement has been shared, not a free initial distribution. The deterministic conditional probabilities above establish perfect correlation rather than merely matching output and input entropies.
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