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I'm looking for references or a derivation for the classical capacity of the quantum bitflip channel, $$\mathcal{N}(\rho) = (1-p)\rho + pX\rho X.$$

Or if this is not known, are some other quantities like the Holevo information available?

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The classical capacity of the quantum bitflip channel is 1.

The states $|+\rangle\langle+|$ and $|-\rangle\langle-|$ are invariant under bitflip channel.

Let $\mathcal{N}$ be the bitflip map.

$$ \therefore\mathcal{N}(|+\rangle\langle+|) = (1-p)|+\rangle\langle+| + pX|+\rangle\langle+|X =(1-p)|+\rangle\langle+| + p|+\rangle\langle+|=|+\rangle\langle+|\,. $$ Similarly, $$ \mathcal{N}(|-\rangle\langle-|) = |-\rangle\langle-|\,. $$ Therefore, you can encode your 0 and 1 into $|+\rangle\langle+|$ & $|-\rangle\langle-|$ respectively and achieve the classical capacity of 1 since you can always send 1-bit of information per channel usage regardless of the presence of bitflip noise.

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