Interplay between nonequilibrium and non-Markovianity in controlling the entropic uncertainty bound

The entropic uncertainty relation imposes a limit on the accuracy of measurement outcomes of two conjugate observables, which can be reduced in the presence of quantum memory. We theoretically study the dynamical behaviors of the quantum-memory-assisted entropic uncertainty relation for a bipartite system under local nonequilibrium dephasing environments and a common nonequilibrium dephasing environment. Particularly, we study the influence of the competition between nonequilibrium and non-Markovianity in controlling the entropic uncertainty bound (EUB) in weak- and strong-coupling regimes. We find that the nonequilibrium nature renders a promising protocol to efficiently harness the EUB in both regimes without requiring any operations on the main system. Further, we reveal that the EUB can be significantly reduced in a common bath case compared to the case of local baths. Moreover, we elucidate the primary factor responsible for governing the dynamical behaviors of EUB by examining the interplay between quantum discord and the minimal missing information about the particle to be measured. In addition, we investigate the state preparation and measurement choice (SPMC) condition in both environmental setups with weak- and strong-coupling regimes. Remarkably, we show that the SPMC condition does not hold in the common environment scenario.