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arXiv:2408.10363 · Physical Review A

Self-testing of multiple unsharpness parameters through sequential violations of a noncontextual inequality

Rajdeep Paul, Souradeep Sasmal, A. K. Pan

Published Jul 17, 2024• 6 citations
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Abstract

Self-testing protocols refer to novel device-independent certification schemes wherein the devices are uncharacterized and the dimension of the system remains unspecified. The optimal quantum violation of 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 noncontextual 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 suboptimal sequential quantum violations of the noncontextual inequality form an optimal set, eventually enabling the self-testing of a 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.

DOI
10.1103/physreva.110.012444

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Self-testing protocols refer to novel device-independent certification schemes wherein the devices are uncharacterized and the dimension of the system remains unspecified. The optimal quantum violation of 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 noncontextual 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 suboptimal sequential quantum violations of the noncontextual inequality form an optimal set, eventually enabling the self-testing of a 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.
Abstract Semi-device-independent certification of an unsharp instrument has recently been demonstrated (2019 New J. Phys. 21 083034) 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.
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Robust self-testing of the m−m-m−partite maximally entangled state and observables
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