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

We study Leggett-Garg inequalities (LGIs) for a two level system (TLS) undergoing Markovian dynamics described by unital maps. We first find the 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 then show that the maximum violation of LGI for all possible unital maps can never exceed well known Luders bound of K-3(Luders ) = 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 Luders 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 Luders 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-Luders (3) is not possible. Hence our findings not only establish a threshold for decoherence which will allow for K-3 = K-3(Luders ) , but also demonstrate the importance of temporal sequencing of the exposure of a TLS to Markovian baths in obtaining Luders bound.

Automated summary

We study Leggett-Garg inequalities (LGIs) for a two level system (TLS) undergoing Markovian dynamics described by unital maps. We find the 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 all possible unital maps can never exceed Luders bound of K-3(Luders ) = 3/2. Our findings establish a threshold for decoherence which will allow for K-3 = K-3(Luders ) and demonstrate the importance of temporal sequencing of the exposure of a TLS to Markovian baths in obtaining Luders bound.