= Solution
The <quantum bit-flip channel> has <Kraus operators>
$$
K_0=\sqrt p\,I,\qquad K_1=\sqrt{1-p}\,X,\qquad
\sum_jK_j^\dagger K_j=I.
$$
Thus
$$
\boxed{\rho_{\rm out}=\mathcal C(\rho_{\rm in})=p\rho_{\rm in}+(1-p)X\rho_{\rm in}X.}
$$
The orthonormal diagonal linear <polarizations> $|D\rangle=(|H\rangle+|V\rangle)/\sqrt2$ and $|A\rangle=(|H\rangle-|V\rangle)/\sqrt2$ are the <Pauli X eigenstates>. Each is transmitted perfectly as a <density operator> for every $p$; the possible minus sign on $|A\rangle$ 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.
Back to article page
