= Solution
A quantum rate-$R$ code uses a compression channel into a space of dimension at most $2^{nR}$ followed by a decompression channel; reliability means that its entanglement fidelity, and hence its average source-state fidelity, tends to one on $\pi^{\otimes n}$. Diagonalize $\pi=\sum_up(u)|u\rangle\langle u|$. The projector onto eigenvectors labelled by the classical <typical set> has dimension at most $2^{n(S(\pi)+\varepsilon)}$ and expectation in $\pi^{\otimes n}$ tending to one. For $R>S(\pi)$ choose $\varepsilon$ so this typical subspace fits in $2^{nR}$ dimensions, encode it isometrically, and send the orthogonal complement to a fixed state. The typical-subspace probability and the gentle-measurement estimate make the fidelity tend to one. This is <Schumacher compression>.
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