= Solution
Assume no previously shared <entanglement>. Alice's encoded ensemble $\{p_x,\rho_x\}$ lives in a <Hilbert space> of dimension $2^n$. Its <Holevo quantity> satisfies
$$
\chi_{\rm in}=S\left(\sum_xp_x\rho_x\right)-\sum_xp_xS(\rho_x)
\leq\log_2(2^n)=n,
$$
using nonnegativity of <Von Neumann entropy> and the <maximum entropy of a quantum state>. The <data-processing inequality for quantum relative entropy> implies $\chi_{\rm out}\leq\chi_{\rm in}$ for the transmission <quantum channel>. Finally, the <Holevo bound> applied to Bob's <quantum measurement> gives
$$
\boxed{I(X:Y)\leq\chi_{\rm out}\leq n\text{ bits}}.
$$
This argument also works when the output system has larger dimension than the input. Prior shared <entanglement> changes the communication resource: <superdense coding> can transmit two classical bits per sent qubit, so the absence of that resource is part of this bound's setting.
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