Comparing Leggett–Garg inequality for work moments with Leggett–Garg inequality and NSIT

Abstract

In this paper, we investigate non-violations of the work Leggett–Garg (WLG) inequality, the Leggett–Garg (LG) inequality and the no-signaling-in-time (NSIT) condition in a two-level system. We consider two kinds of initial states: the thermal state and the state with coherence. We find that the non-violation condition of WLG inequality for the first work moment is similar to the NSIT conditions. The WLG inequality of the first work moment cannot be violated in the high temperature limit or for a specific driving intensity for the initial thermal state, while for the initial state with coherence, the non-violation of it strongly depends on the relative phase in the quantum coherence. For the initial thermal state and state with coherence, the NSIT conditions cannot be violated for a specific driving intensity when the projective measurement operators at different measurement times are the same, while when the projective measurement operators at different measurement times are different, we cannot find any circumstance to simultaneously make all the NSIT conditions non-violated. We find that Gaussian measurements have no effect on the non-violation conditions of the WLG inequality for the first work moment and the NSIT conditions. We also find that the non-violation condition of WLG inequality for the second work moment is similar to the LG inequality. The non-violation condition of the WLG inequality for the second work moment is the same as the LG inequality under projective measurements, while it cannot be violated for a wider parameter regime than the LG inequality under Gaussian measurements.