Complex environment overlap in qubit dephasing ($c=⟨R_0|R_1⟩$)

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 $c=r{e}^{i\phi }$, the channel is a phase-flip channel of probability $(1-r)/2$ followed by the diagonal unitary operator $\mathrm{diag}(1,e^{i\phi })$. The coherent phase cannot be discarded just because the populations are unchanged.