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Let $x\in \mathcal{X}$ be a symbol from an input alphabet, let $p(y|x)$ be a conditional probability distribution corresponding to a discrete memoryless channel and let $y\in\mathcal{Y}$ be an output alphabet. Given that the channel has capacity $C$, is it true that one can compress any $n-$bit output to $nC$ bits with error in the compression vanishing as $n\rightarrow\infty$?

The motivation for this comes from the reverse Shannon theorem established in Section IV of https://arxiv.org/pdf/quant-ph/0106052.pdf. To simulate a channel, Alice applies the channel locally and then transfers the output to Bob. She does this in a clever way such that only $2^{nC}$ bits of communication is actually required.

I was wondering if her protocol is equivalent to compressing the channel output and sending it over to Bob.

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