Self-testing of multiple unsharpness parameters through sequential violations of non-contextual inequality

Abstract: The self-testing protocols refer to novel device-independent certification schemes wherein the devices are uncharacterised, and the dimension of the system remains unspecified. The optimal quantum violation of a Bell's inequality facilitates such self-testing. In this work, we put forth a protocol for self-testing of noisy quantum instruments, specifically, the unsharpness parameter of smeared projective measurements in any arbitrary dimension. Our protocol hinges on the sequential quantum violations of a bipartite Bell-type preparation non-contextual inequality, involving three measurement settings per party. First, we demonstrate that at most three sequential independent Bobs manifest simultaneous preparation contextuality with a single Alice through the violation of this inequality. Subsequently, we show that the sub-optimal sequential quantum violations of the non-contextual inequality form an optimal set, eventually enabling the self-testing of shared state, local measurements and unsharpness parameters of one party. Notably, we derive the optimal set of quantum violations without specifying the dimension of the quantum system, thereby circumventing the constraint that may arise due to Naimark's theorem. Furthermore, we extend our investigation to quantify the degree of incompatible measurements pertaining to the sequential observers, exploring how variations in the degree of incompatibility impact the values of unsharp parameters necessary for sequential quantum violation.