  Quantum coherence can be used to infer the presence of a detector without
triggering it. Here we point out that, according to quantum mechanics, such
interaction-free measurements cannot be perfect, i.e., in a single-shot
experiment one has strictly positive probability to activate the detector. We
formalize the extent to which such measurements are forbidden by deriving a
trade-off relation between the probability of activation and the probability of
an inconclusive interaction-free measurement. Our description of
interaction-free measurements is theory independent and allows derivations of
similar relations in models generalizing quantum mechanics. We provide the
trade-off for the density cube formalism, which extends the quantum model by
permitting coherence between more than two paths. The trade-off obtained hints
at the possibility of perfect interaction-free measurements and indeed we
construct their explicit examples. Such measurements open up a paradoxical
possibility where we can learn by means of interference about the presence of
an object in a given location without ever detecting a probing particle in that
location. We therefore propose that absence of perfect interaction-free
measurement is a natural postulate expected to hold in all physical theories.
As shown, it holds in quantum mechanics and excludes the models with multipath
coherence.
