  In 2018, Renes [IEEE Trans. Inf. Theory, vol. 64, no. 1, pp. 577-592 (2018)]
(arXiv:1701.05583) developed a general theory of channel duality for
classical-input quantum-output (CQ) channels. That result showed that a number
of well-known duality results for linear codes on the binary erasure channel
could be extended to general classical channels at the expense of using dual
problems which are intrinsically quantum mechanical. One special case of this
duality is a connection between coding for error correction (resp. wire-tap
secrecy) on the quantum pure-state channel (PSC) and coding for wire-tap
secrecy (resp. error correction) on the classical binary symmetric channel
(BSC). While this result has important implications for classical coding, the
machinery behind the general duality result is rather challenging for
researchers without a strong background in quantum information theory. In this
work, we leverage prior results for linear codes on PSCs to give an alternate
derivation of the aforementioned special case by computing closed-form
expressions for the performance metrics. The noted prior results include
optimality of the square-root measurement (SRM) for linear codes on the PSC and
the Fourier duality of linear codes. We also show that the SRM forms a
suboptimal measurement for channel coding on the BSC (when interpreted as a CQ
problem) and secret communications on the PSC. Our proofs only require linear
algebra and basic group theory, though we use the quantum Dirac notation for
convenience.
