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 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 Lüders bound of K Lüders 3 = 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 Lüders 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 Lüders 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 K Lüders 3 is not possible. Hence our findings not only establish a threshold for decoherence which will allow for K3 = K Lüders 3 , but also demonstrate the importance of temporal sequencing of the exposure of a TLS to Markovian baths in obtaining Lüders bound.

Essential role of quantum speed limit in violation of Leggett-Garg inequality across a PT-transition

“…Rest of the set of CP dynamics is comprised of unital and non-unital dynamics. It is known by now, in case of both the unitary and unital dynamics that upper bound on the LG parameter K 3 is Lüders bound 9,19 . However, set of dynamics outside the sub-set of CP dynamics, which is only positive and trace preserving has not been explored much earlier.…”

Investigating Leggett-Garg inequality in neutrino oscillations -- role of decoherence and decay

“…These are known as the Leggett-Garg inequalities (LGI). LGI has been studied in the context of non-hermitian PT symmetric dynamics [65,66], markovian quantum dynamics [67,68]. Tests of LGI have been carried out using photons, electrons, or nuclear spins for which coherence length is limited [62,69].…”