Probing various formulations of macrorealism for unsharp quantum measurements

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.

Figure 1

Plot showing the violation of the ELGI given by inequality (14) for the case of spin POVMs ($x=0$). The values of the relevant parameters are taken to be $\theta =1.7$ and $\varphi =\pi /2$.

Figure 2

Plot showing the violation of the ELGI given by inequality (14) for any nonzero value of $\eta$ for the case of biased POVMs ($x=\eta -1$). The values of the relevant parameters are taken to be $\theta =1.7$ and $\varphi =\pi /2$.

Figure 3

All 24 WLGIs are plotted against $\tau$ for the state $\rho \left(t_0\right)=|+\rangle \langle +|$ or $\rho \left(t_0\right)=I/2.$ It is seen that at least one of the WLGIs is violated for all values of $\tau$ except at $\tau =\pi /4$.

Figure 4

All 24 WLGIs are plotted against $\tau$ for the state $\rho \left(t_0\right)=|0\rangle \langle 0|$. It is seen that no WLGI is violated for a considerable range around $\tau =\pi /3$.