Assignment:
Consider a Mach-Zehnder interferometer where in one arm we place a filter with a transmission coefficient f << 1. The system is set up such that at one detector site we have constructive interference and at the other side we have destructive interference. Demonstrate that the expected number of detected photons N needed before a deviation of k sigma's from the case f = 0 is obtained, is given by N = k^2/(2 f). Therefore, had we just measured the transmission coefficient by shining light through the filter, we could expected to detect N f = k^2/2 photons, so the interferometer doesn't seem to offer any advantage.
Consider how the presence of noise that leads to a spurious background photon detection rate, can change this conclusion.