 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 $\pi ={\sum }_{u}p(u)|u⟩⟨u|$. The projector onto eigenvectors labelled by the classical typical set has dimension at most $2^{n(S(\pi )+\epsilon )}$ and expectation in ${\pi }^{\otimes n}$ tending to one. For $R>S(\pi )$ choose $\epsilon$ 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.