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

Abstract

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