Self-testing positive operator-valued measures and certifying randomness

In the device-independent scenario, positive operator-valued measures (POVMs) can certify more randomness than projective measurements. This paper self-tests a three-outcome extremal qubit POVM in the X-Z plane of the Bloch sphere by achieving the maximal quantum violation of a newly constructed Bell expression \(\mathcal {C}_3^{'} \), adapted from the chained inequality \(\mathcal {C}_3\). Using this POVM, approximately 1.58 bits of local randomness can be certified, which is the maximum amount of local randomness achievable by an extremal qubit POVM in this plane. Further modifications of \(\mathcal {C}_3^{'} \) produce \(\mathcal {C}_3^{''} \), enabling the self-testing of another three-outcome extremal qubit POVM. Together, these POVMs are used to certify about 2.27 bits of global randomness. Both local and global randomness surpass the limitations certified from projective measurements. Additionally, the Navascués-Pironio-Acín hierarchy is employed to compare the lower bounds on global randomness certified by \(\mathcal {C}_3\) and several other inequalities. As the extent of violation increases, \(\mathcal {C}_3\) demonstrates superior performance in randomness certification.