"Interaction-free" discrimination between semi-transparent objects

Standard ``interaction-free'' measurement aims to detect the presence of an opaque object without the photon interacting with the object. Here we consider semi-transparent objects which have an amplitude $\alpha$ of transmitting a photon while leaving the state of the object unchanged and an amplitude $\beta$ of interacting with a photon. The task is to devise a protocol that can decide which of two known transmission amplitudes is present while ensuring that no photon interacts with the object. We show that the probabilities of being able to achieve this are limited by an inequality. This inequality implies that 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.