Abstract
We study Leggett-Garg inequalities (LGIs) for a two level system (TLS) undergoing Markovian dynamics described by unital maps. We find analytic expression of LG parameter K₃ (simplest variant of LGIs) in terms of the parameters of two distinct unital maps representing time evolution for intervals: t₁ to t₂ and t₂ to t₃. We show that the maximum violation of LGI for these maps can never exceed well known L\"{u}ders bound of K₃^Luders=3/2 over the full parameter space. We further show that if the map for the time interval t₁ to t₂ is non-unitary unital then irrespective of the choice of the map for interval t₂ to t₃ we can never reach L\"{u}ders bound. On the other hand, if the measurement operator eigenstates remain pure upon evolution from t₁ to t₂, then depending on the degree of decoherence induced by the unital map for the interval t₂ to t₃ we may or may not obtain L\"{u}ders bound. Specifically, we find that if the unital map for interval t₂ to t₃ leads to the shrinking of the Bloch vector beyond half of its unit length, then achieving the bound K₃^Luders is not possible. Hence our findings not only establish a threshold for decoherence which will allow for K₃ = K₃^Luders, but also demonstrate the importance of temporal sequencing of the exposure of a TLS to Markovian baths in obtaining L\"{u}ders bound.