Characterization of dynamical measurement's uncertainty in a two-qubit system coupled with bosonic reservoirs

Highlights: • The dynamics of the entropic uncertainty is investigated under the crossover of Markovian and non-Markovian regimes. • The trace distance and non-Markovianity are revealed. • The strategy to reduce the measurement's uncertainty is proposed. -- Abstract: We study the dynamical characteristics of the entropy-based uncertainty with regard to a pair of incompatible measurements under a bipartite qubit-system suffering from quantum decoherence induced by hierarchical environments. How non-Markovian and Markovian environments affect the dynamical behaviors of the measurement's uncertainty is revealed. We prove that the measured uncertainty of interest demonstrates a non-monotonic behavior, viz., the amount will increase initially and subsequently oscillate periodically with the growth of time in a non-Markovian regime; On the contrary, the uncertainty will inflate firstly and monotonically decrease in a Markovian regime. Noteworthily, we put forward a simple and feasible strategy to suppress the damping of the system and hence be good for decreasing the magnitude of the uncertainty, by virtue of optimal combination of pre-weak measurements and post-filtering operations. Furthermore, we explore the applications of the uncertainty relation investigated on entanglement witness and channel capacity.