Interferometric information gain versus interaction-free measurement

Interaction-free measurement schemes with ideal Mach-Zehnder interferometers promised to distinguish absorptive samples with lower average absorption than simple transmission schemes. We show that this is only true for an ensemble of two kinds of samples, where one kind is highly absorptive and the other is highly transmissive. As soon as a third kind of sample with intermediate transmission is introduced, but no phase shift is permitted, the cost of information gain in terms of absorbed particles in the samples is higher in the interferometric scheme. We also investigate the general case of samples with a continuous range of transmission and phase shift values, such that an interferometer’s ability to measure both sample characteristics can be exploited. With an interferometer the number of principally distinguishable samples increases linearly with the number of probe particles, but with a simple transmission setup it increases as the square root. When wishing to distinguish twice as many samples from a continuous sample distribution with an interferometric scheme, the number of absorbed particles per sample only doubles, but it quadruples with a simple transmission scheme.