Finding the HSW capacity is an optimization problem which I believe is moderately tractable. There is an iterative numerical method outlined in this paper of mine ("Capacities of quantum channels and how to find them."). A different, although somewhat similar, method was detailed in the paper "Qubit channels which require four inputs to achieve capacity: implications for additivity conjectures" by Masahito Hayashi, Hiroshi Imai, Keiji Matsumoto, Mary Beth Ruskai and Toshiyuki Shimono. If the number of qubits $n$ is not quite small, however, the high dimensionality of the space is going to keep these techniques from working.

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