= Hadamard interference with environment overlap
{c}
{title2=$P_0=\tfrac12[1+v\cos(\varphi+\alpha)]$}
In a <Hadamard gate>, <phase gate>, <Hadamard gate> sequence starting from $|0\rangle$, let a population-preserving interaction correlate $|j\rangle$ with a normalized environment state $|e_j\rangle$ between the two <Hadamard gates>. Use $P_\varphi=\operatorname{diag}(1,e^{i\varphi})$ and $\langle e_0|e_1\rangle=ve^{i\alpha}$. The environment-valued output-zero coefficient is $(|e_0\rangle+e^{i\varphi}|e_1\rangle)/2$, 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.
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