Leggett-Garg inequality, Wigner form of Leggett-Garg inequality and no-signaling-in-time condition under coarsening measurement

In this paper, we investigate the Leggett-Garg inequality (LGI), the Wigner form of Leggett-Garg inequality (WLGI) and the no-signaling-in-time (NSIT) condition under projective and coarsening measurements. And we focus on studying effects of the coarsening measurement in final resolution, in reference and in temporal reference on the LGI, the WLGI and the NSIT condition. For the projective measurement, we find that the NSIT condition can be violated for a wider parameter regime than the WLGI, and the WLGI can be violated for a wider parameter regime than the LGI. Then, for the coarsening measurement in final resolution, in reference and in temporal reference, we find that the robustness of the LGI and the WLGI are equally vulnerable, and more vulnerable than the NSIT condition, i.e., the NSIT condition is the most robust. In addition, for the LGI, the WLGI and the NSIT condition, we find that the robustness of the coarsening measurement in temporal reference is the most robust, and that of the coarsening in final measurement resolution is the most vulnerable.

Automated summary

In this paper, the authors investigate the effects of projective and coarsening measurements on the Leggett-Garg inequality (LGI), the Wigner form of Leggett-Garg inequality (WLGI), and the no-signaling-in-time (NSIT) condition. It is found that the NSIT condition can be violated over a wider parameter range compared to the WLGI, and the WLGI can be violated over a wider parameter range than the LGI under projective measurement. Furthermore, the study shows that the robustness of the LGI and the WLGI are equally vulnerable under coarsening measurements, while the NSIT condition is the most robust. Additionally, the robustness of the coarsening measurement in temporal reference is found to be the most robust, while that of the coarsening in final measurement resolution is the most vulnerable.