  A semi-device-independent framework for prepare-and-measure experiments is
introduced in which an experimenter can tune the degree of distrust in the
performance of the quantum devices. In this framework, a receiver operates an
uncharacterised measurement device and a sender operates a preparation device
that emits states with a bounded fidelity with respect to a set of target
states. No assumption on Hilbert space dimension is required. The set of
quantum correlations is investigated and bounded from both the interior and the
exterior. Furthermore, the optimal performance of quantum state discrimination
with bounded distrust is derived and applied to certification of detection
efficiency. Quantum-over-classical advantages are demonstrated and the
magnitude of distrust compatible with such advantages is explored. Finally,
efficient schemes for semi-device-independent random number generation are
developed.
