  Semi-device-independent certification of an unsharp instrument has recently
been demonstrated [New J. Phys. 21, 083034 (2019)] based on the sequential
sharing of quantum advantages in a prepare-measure communication game by
assuming the system to be qubit. In this work, we provide device-independent
(DI) self-testing of the unsharp instrument through the quantum violation of
two Bell inequalities where the devices are uncharacterized and the dimension
of the system remains unspecified. We introduce an elegant sum-of-squares
approach to derive the dimension-independent optimal quantum violation of Bell
inequalities which plays a crucial role. Note that the standard Bell test
cannot self-test the post-measurement states and consequently cannot self-test
unsharp instrument. The sequential Bell test possess the potential to self-test
an unsharp instrument. We demonstrate that there exists a trade-off between the
maximum sequential quantum violations of the Clauser-Horne-Shimony-Holt
inequality, and they form an optimal pair that enables the DI self-testing of
the entangled state, the observables, and the unsharpness parameter. Further,
we extend our study to the case of elegant Bell inequality and we argue that it
has two classical bounds - the local bound and the non-trivial preparation
non-contextual bound, lower than the local bound. Based on the sharing of
preparation contextuality by three independent sequential observers, we
demonstrate the DI self-testing of two unsharpness parameters. Since an actual
experimental scenario involves losses and imperfection, we demonstrate
robustness of our certification to noise.
