  We consider the problem of certification of arbitrary ensembles of pure
states and projective measurements solely from the experimental statistics in
the prepare-and-measure scenario assuming the upper bound on the dimension of
the Hilbert space. To this aim, we propose a universal and intuitive scheme
based on establishing perfect correlations between target states and
suitably-chosen projective measurements. The method works in all finite
dimensions and allows for robust certification of the overlaps between
arbitrary preparation states and between the corresponding measurement
operators. Finally, we prove that for qubits, our technique can be used to
robustly self-test arbitrary configurations of pure quantum states and
projective measurements. These results pave the way towards the practical
application of the prepare-and-measure paradigm to certification of quantum
devices.

Quantum Physics