Lüders Bounds of Leggett–Garg Inequalities, PT$\mathcal {PT}$‐ Symmetric Evolution and Arrow‐of‐Time

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. The Lüders bounds of the standard and variant of Leggett‐Garg inequalities in unitary quantum mechanics requires the violation of no‐signaling in time conditions. In PT‐symmetric quantum mechanics the quantum violation of both form of Leggett‐Garg inequalities exceeds their Lüders bound and reach their algebraic maxima. PT‐symmetric evolution results in violation of arrow‐of‐time along with no‐signaling in time.