Standard Leggett-Garg inequalities (SLGIs) were formulated for testing incompatibility between the classical world view of macrorealism and quantum mechanics. In recent times, various other formulations such as the Wigner form of Leggett-Garg inequalities (WLGIs), entropic Leggett-Garg inequalities (ELGIs), and the no-signaling-in-time (NSIT) condition have also been proposed. It was also recently argued that
no set of SLGIs can provide the necessary and sufficient conditions for macrorealism, but a suitable conjunction of NSIT conditions provides the same. In this paper, we first provide a comparative study of the various formulations of Leggett-Garg inequalities (LGIs) for testing macrorealism pertaining to the two different unsharp measurements. While the violations of WLGIs are more robust than SLGIs and ELGIs for spin positive operator-valued measures (POVMs), here we demonstrate that for the case of biased POVMs the quantum violations of both SLGIs and ELGIs provide the same robustness as WLGIs. Importantly, the violations of all formulations of LGIs can be achieved for any nonzero value of unsharpness parameter. We have also studied the connection between LGIs and NSIT conditions. Further, we investigate the role of the joint measurability of the POVMs in the violation of LGIs and find that there is
no generic connection.
Lüders Bounds of Leggett–Garg Inequalities, PT
PT‐ Symmetric Evolution and Arrow‐of‐Time
Abstract Leggett–Garg inequalities (LGIs) test the incompatibility between the notion of macrorealism and quantum theory. For unitary dynamics, the optimal quantum violation of a Leggett–Garg inequality is constrained by the Lüders bound. However, the LGIs do not provide the necessary and sufficient condition of macrorealism. A suitably formulated set of no‐signaling in time conditions along with the arrow‐of‐time condition provides the same. In this paper, two formulations in the three‐time Leggett–Garg scenario, are studied namely, the standard LGIs and the recently formulated variant of LGIs when the system evolves under PT‐symmetric Hamiltonian. It is first demonstrated that the quantum violations of both forms of LGIs exceed their respective Lüders bounds and can even reach their algebraic maximum. It is further shown that for the case of standard Leggett–Garg inequality, the violation of Lüders bound can be obtained when both no‐signaling in time and arrow‐of‐time conditions are violated. Interestingly, for the case of a variant of LGIs, for suitable choices of relevant parameters, the quantum violation can even be obtained when only the arrow‐of‐time is violated. However, all no‐signaling in time conditions are satisfied. Such a feature is hitherto unexplored.
Leggett-Garg inequalities (LGIs) have been proposed in order to assess how far the predictions of quantum mechanics defy "macroscopic realism." With LGIs, correlations of measurements performed on a single system at different times are described. We report on an experiment that demonstrates the violation of an LGI with neutrons. The final measured value of the Leggett-Garg correlator K=1.120±0.007(stat)±0.019(sys), obtained in a neutron interferometric experiment, is clearly above the limit K=1 predicted by macrorealistic theories. The experimental results are analyzed within the framework of dynamical theory of neutron diffraction, evidently reproducing the obtained values.