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.
Abstract In two-party, two-input and two-output measurement scenario only relevant Bell’s inequality is the Clauser–Horne–Shimony–Holt (CHSH) form. They also provide the necessary and sufficient conditions (NSCs) for local realism. Any other form, such as, Clauser–Horne and Wigner forms reduce to the CHSH one. The standard Leggett–Garg inequalities, proposed for testing incompatibility between macrorealism and quantum theory, are often considered to be the temporal analog of CHSH inequalities. However, they do not provide the NSCs for macrorealism. There is thus scope of formulating new macrorealist inequalities inequivalent and stronger than the standard Leggett–Garg inequalities. In this paper, we propose two different classes of macrorealistic inequalities. A class of inequalities which are equivalent to the standard ones in macrorealist model but inequivalent and stronger in quantum theory, and the other class of inequalities are inequivalent to the all the other formulations of Leggett–Garg inequalities both in macrorealist model and in quantum theory. The latter class of macrorealist inequalities reveals the incompatibility between macrorealism and quantum theory for specific cases even when any other formulation of Leggett–Garg inequalities fails to do so. We extend the formulations of inequivalent Leggett–Garg inequalities to the four-time and two-time measurement scenarios. Further, we provide a brief discussion about the alternative formulation of macrorealism known as the no-signaling in time (NSIT) conditions.