= Complex environment overlap in qubit dephasing
{title2=$c=\langle R_0|R_1\rangle$}
An environment interaction preserving the computational populations multiplies a qubit's upper off-diagonal density entry by $\overline c$. A real <phase-flip channel> representation exists for every input precisely when $c$ is real, with probability $(1-c)/2$. For $c=re^{i\varphi}$, the channel is a <phase-flip channel> of probability $(1-r)/2$ followed by the diagonal <unitary operator> $\operatorname{diag}(1,e^{i\varphi})$. The coherent phase cannot be discarded just because the populations are unchanged.
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