= Solution
A compression-decompression scheme consists of <quantum channels> $\mathcal E_n:\mathcal D(\mathcal H^{\otimes n})\to\mathcal D(\mathcal K_n)$ and $\mathcal D_n:\mathcal D(\mathcal K_n)\to\mathcal D(\mathcal H^{\otimes n})$, where $\limsup_n n^{-1}\log_2\dim\mathcal K_n\leq R$. For the emitted pure-state ensemble, reliability means that the mean squared <quantum fidelity> tends to one:
$$
\boxed{\sum_kp_k\langle\Psi_k^{(n)}|
(\mathcal D_n\circ\mathcal E_n)(|\Psi_k^{(n)}\rangle\langle\Psi_k^{(n)}|)
|\Psi_k^{(n)}\rangle\longrightarrow1}.
$$
The average is taken over the actual source distribution, rather than the worst possible input vector.
A standard stronger formulation of <reliable quantum source compression> requires
$$
F_e(\pi^{\otimes n},\mathcal D_n\circ\mathcal E_n)\longrightarrow1,
$$
where <entanglement fidelity> tests preservation of a purification and its reference-system correlations. It implies the mean-fidelity condition for every pure-state ensemble of $\pi^{\otimes n}$. <Schumacher compression> achieves this at any rate $R>S(\pi)$: choose $0<\varepsilon<R-S(\pi)$, encode the <quantum typical subspace>, and map atypical outcomes to a fallback state. The composed channel has a Kraus term $P_\varepsilon^{(n)}$, so its <entanglement fidelity> is at least $[\operatorname{Tr}(\pi^{\otimes n}P_\varepsilon^{(n)})]^2\to1$.
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