Repeated weak-rotation bomb-test efficiency ($P_N=\frac{q}{1+q}(1-q^{2N}),q=1-p$)

For an Elitzur-Vaidman bomb tester using a rotation and its inverse, let $p$ be the probability of entering the absorbing path and $q=1-p$. Conditional on a live absorber, one trial has explosion probability $p$, successful safe identification probability $pq$, and inconclusive probability $q^2$. The inconclusive projective measurement resets the input, so summing a geometric series gives the displayed success probability after at most $N$ trials. Its limit is $q/(1+q)$, approaching one half as $p$ tends to zero through positive values. Exactly zero coupling produces no information.