Absorption-free discrimination between semitransparent objects

Absorption-free (also known as “interaction-free”) measurement aims to detect the presence of an opaque object using a test particle without that particle being absorbed by the object. Here we consider semitransparent objects that have an amplitude $\alpha$ of transmitting a particle while leaving the state of the object unchanged and an amplitude $\beta$ of absorbing the particle. The task is to devise a protocol that can decide which of two known transmission amplitudes is present while ensuring that no particle is absorbed by the object. We show that the probabilities of being able to achieve this are limited by an inequality. This inequality implies that absorption-free distinction between complete transparency and any partial transparency is always possible with probabilities approaching 1, but that two partial transparencies can only be distinguished with probabilities less than 1.