Trace distance for qubit dephasing
= Trace distance for qubit dephasing
{title2=$d(\rho,\mathcal D_c(\rho))=|\alpha\beta|\,|1-c|$}
For the pure <qubit> $\alpha|0\rangle+\beta|1\rangle$, subtracting the output of <dephasing channel> leaves a Hermitian matrix with zero diagonal and off-diagonal modulus $|\alpha\beta|\,|1-c|$. Its two <eigenvalues> are that number and its negative, giving the displayed <trace distance>. For real $c=1-2p$, this is $2p|\alpha\beta|$.