In an asymmetric multislit interference experiment, a quanton is more likely to pass through certain slits than some others. In such a situation one may be able to predict which slit a quanton is more likely to go through, even without using any path-detecting device. This allows one to talk of path predictability. It has been shown earlier that for a two-slit interference, the predictability and fringe visibility are constrained by the inequality P 2 + V 2 ≤ 1. Generalizing this relation to the case of more than two slits is still an unsolved problem. A new definition for predictability for multi-slit interference is introduced. It is shown that this predictability and quantum coherence follow a duality relation P 2 + C 2 ≤ 1, which saturates for all pure states. For the case of two slits, this relation reduces to the previously known one.

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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 instruments. The sequential Bell test possesses 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-Halt 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 experimental scenario involves losses and imperfection, we demonstrate the robustness of our certification to noise.

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