  Self-testing is a promising approach to certifying quantum states or
measurements. Originally, it relied solely on the outcome statistics of the
measurements involved in a device-independent (DI) setup. Extra physical
assumptions about the system make the setup semi-DI. In the latter approach, we
consider a prepare-and-measure scenario in which the dimension of the mediating
particle is assumed to be two. In a setup involving four (three) preparations
and three (two) projective measurements in addition to the target, we exemplify
how to self-test any four- (three-) outcome extremal positive operator-valued
measure using a linear witness. One of our constructions also achieves
self-testing of any number of states with the help of as many projective
measurements as the dimensionality of the space spanned by the corresponding
Bloch vectors. These constructions are conjectured to be minimal in terms of
the number of preparations and measurements required. In addition, we implement
one of our prepare-and-measure constructions on IBM and IonQ quantum processors
and certify the existence of a complex qubit Hilbert space based on the data
obtained from these experiments.
