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.

In a Hadamard gate, phase gate, Hadamard gate sequence starting from $|0⟩$, let a population-preserving interaction correlate $|j⟩$ with a normalized environment state $|e_j⟩$ between the two Hadamard gates. Use $P_{\phi }=\mathrm{diag}(1,e^{i\phi })$ and $⟨e_0|e_1⟩=v{e}^{i\alpha }$. The environment-valued output-zero coefficient is $(|e_0⟩+e^{i\phi }|e_1⟩)/2$, whose squared norm is the displayed probability. The modulus $v$ controls contrast, while $\alpha$ shifts the quantum interference fringe. For $v=1$ there is no mixing; a deterministic relative quantum phase accounts for the shift.