Leggett-Garg inequality in Markovian quantum dynamics: role of temporal sequencing of coupling to bath

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 $K_{3}$ (simplest variant of LGIs) in terms of the parameters of two distinct unital maps representing time evolution for intervals: $t_{1}$ to $t_{2}$ and $t_{2}$ to $t_{3}$. We show that the maximum violation of LGI for these maps can never exceed well known L\"{u}ders bound of $K_{3}^{L\ddot{u}ders}=3/2$ over the full parameter space. We further show that if the map for the time interval $t_{1}$ to $t_{2}$ is non-unitary unital then irrespective of the choice of the map for interval $t_{2}$ to $t_{3}$ we can never reach L\"{u}ders bound. On the other hand, if the measurement operator eigenstates remain pure upon evolution from $t_{1}$ to $t_{2}$, then depending on the degree of decoherence induced by the unital map for the interval $t_{2}$ to $t_{3}$ we may or may not obtain L\"{u}ders bound. Specifically, we find that if the unital map for interval $t_{2}$ to $t_{3}$ leads to the shrinking of the Bloch vector beyond half of its unit length, then achieving the bound $K_{3}^{L\ddot{u}ders}$ is not possible. Hence our findings not only establish a threshold for decoherence which will allow for $K_{3} = K_{3}^{L\ddot{u}ders}$, but also demonstrate the importance of temporal sequencing of the exposure of a TLS to Markovian baths in obtaining L\"{u}ders bound.