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 Lüders 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 Lüders 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 Lüders 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 Lüders bound.

9 pages, 2 figures

References in corpus (3)

• The Fidelity and Trace Norm Distances for Quantifying Coherence
• The conditions for quantum violation of macroscopic realism
• Decoherence and maximal violations of the Leggett-Garg inequality

Cited by in corpus (2)

• Essential role of quantum speed limit in violation of Leggett-Garg inequality across a PT-transition
• Fractional Conformal Map, Qubit Dynamics and the Leggett-Garg Inequality