= Balanced Hadamard interaction-free measurement
{title2=$P_{\rm success}=1/4,\quad P_{\rm absorb}=1/2,\quad P_{\rm inconclusive}=1/4$}
With two <Hadamard gates> as mixers, an unobstructed zero-phase arm gives only port 0 for upper-port incidence. An ideal absorber in arm 1 acts on the intermediate superposition $(|0\rangle+|1\rangle)/\sqrt2$. It absorbs with probability $1/2$; its no-absorption branch is the unnormalized state $|0\rangle/\sqrt2$. Recombination gives amplitude $1/2$ at each surviving output, so the dark-port success probability is $1/4$. The result assumes a calibrated transparent alternative and an ideal absorber; a dark-port event alone does not establish those apparatus assumptions.
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