Classical table decoding is not quantum recovery

The decoder $\mathrm{Dec}$ in (eq:decoder) is a mathematical interpretation of exact classical codes. It is not a channel recovering an unknown quantum state from a classical record. For precision, the two relevant finite-dimensional comparisons are proved here.

For channels with the same input and output quantum spaces, use the half-diamond convention

The diamond norm permits an untouched reference; in the statements below it suffices to test a maximally entangled input for a lower bound.