Influence of weak measurement on uncertainty relations in a quantum dissipative system

We investigate the influence of weak measurement on the quantum-memory-assisted entropic uncertainty relation and quantum speed limit time in a quantum dissipative system by making use of the hierarchical equations of motion method. It is demonstrated that the weak measurement and measurement reversal can suppress the entropic uncertainty during the evolution of the system, and we find a periodical crossover of the quantum speed limit time for different weak measurement strengths, which disappears when increasing the coupling strength. Similar influence of the weak measurement on uncertainty relations is then explored at the finite temperature. We also consider the effect of the counter-rotating-wave term and show that the rotating-wave approximation is an applicable approximation when studying quantum speed limit time but not appropriate when investigating the quantum-memory-assisted entropic uncertainty relation.