= Product-input classical-capacity converse
For a code with <product state> inputs to $n$ uses of a memoryless <quantum channel>, <Subadditivity of Von Neumann entropy> bounds its output <Holevo quantity> by $n\chi^*(\Lambda)$. The <Holevo bound> and <Fano's inequality> then give, for asymptotic rate $R$, average and hence maximum error at least $1-\chi^*(\Lambda)/R$ in the limit inferior. In particular, no code above that rate can have vanishing maximum error, even with a collective decoder measurement.
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