Hadamard interference with environment overlap ($P_0=\frac{1}{2}[1+v\cos (\phi +\alpha )]$)

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.