The maximal evolution speed of any quantum system can be expressed by the quantum speed limit time. In this paper, we consider a model in which the system has a correlation with the environment. The influence of the initial correlation between the system and environment on the quantum speed limit is investigated. It is shown that the appearance of non-Markovianity effects causes the speedup of quantum evolution. Moreover, we demonstrate the dependence of quantum dynamical speedup on the quantum coherence of the correlated initial state.

A novel approach is introduced to assess one-way Normalized Entropic Uncertainty Relations (NEUR)-steering in a two-qubit system by utilizing an average of conditional entropy squeezing. The mathematical expressions of conditional entropy squeezing and NEUR-steering are derived and presented. To gain a better understanding of the relationship between the two measures, a comparative analysis is conducted on a set of two-qubit states. Our results reveal that the two measures exhibit complete similarity when applied to a maximally entangled state, while they display comparable behavior with minor deviations for partially entangled states. Additionally, it is observed that the two measures are proportionally affected by some quantum processes such as acceleration, noisy channels, and swapping. As a result, the average of conditional entropy squeezing proves to be an effective indicator of NEUR-steering.