Towards minimal self-testing of qubit states and measurements in prepare-and-measure scenarios

Self-testing is a promising approach to certifying quantum states or measurements. Originally, it relied solely on the output statistics of the measurements involved in a device-independent setup. Extra physical assumptions about the system make the setup semi-device-independent. In the latter setup, 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 constructions in the prepare-and-measure setup on IBM and IonQ quantum processors and certify the existence of a complex qubit Hilbert space based on the data obtained from these experiments.