  We study the time-dependent quantum Fisher information (QFI) in an open
quantum system satisfying the Gorini-Kossakowski-Sudarshan-Lindblad master
equation. We also study the dynamics of the system from an effective
non-Hermitian dynamics standpoint and use it to understand the scaling of the
QFI when multiple probes are used. A focus of our work is how the QFI is
maximized at certain times suggesting that the best precision in parameter
estimation can be achieved by focusing on these times. The propagation of
errors analysis allows us to confirm and better understand this idea. We also
propose a parameter estimation procedure involving relatively low resource
consuming measurements followed by higher resource consuming measurements and
demonstrate it in simulation.
