The n-shot classical capacity of the quantum erasure channel

We compute the n-shot classical capacity of the quantum erasure channel, providing upper bounds and almost-matching lower bounds for it, the latter achievable via large-minimum-distance classical linear codes for any n. The protocols are in full product form, i.e. no entanglement is needed either at the encoder or decoder to attain the capacity, and they explicitly adapt to the transition between different error regimes as the erasure probability increases. Finally, we show that our upper and lower bounds on the capacity are tighter than those obtainable from the general theory of finite-size capacity via generalized divergences.