 Solution

The quantum bit-flip channel has Kraus operatorsThusThe orthonormal diagonal linear polarizations $|D⟩=(|H⟩+|V⟩)/\sqrt{2}$ and $|A⟩=(|H⟩-|V⟩)/\sqrt{2}$ are the Pauli X eigenstates. Each is transmitted perfectly as a density operator for every $p$; the possible minus sign on $|A⟩$ is global. Encode a classical zero and one as $D$ and $A$, then measure in that basis at the receiver. This sends one classical bit per transmitted qubit without error. General superpositions of $D$ and $A$ need not be preserved, because their relative phase is randomized.