  We investigate a symmetric logarithmic derivative (SLD) Fisher information
for kinetic uncertainty relations (KURs) of open quantum systems described by
the GKSL quantum master equation with and without the detailed balance
condition. In a quantum kinetic uncertainty relation derived by Vu and Saito
[Phys. Rev. Lett. 128, 140602 (2022)], the Fisher information of probability of
quantum trajectory with a time-rescaling parameter plays an essential role.
This Fisher information is upper bounded by the SLD Fisher information. For a
finite time and arbitrary initial state, we derive a concise expression of the
SLD Fisher information, which is a double time integral and can be calculated
by solving coupled first-order differential equations. We also derive a simple
lower bound of the Fisher information of quantum trajectory. We point out that
the SLD Fisher information also appears in the speed limit based on the
Mandelstam-Tamm relation by Hasegawa [Nat. Commun. 14, 2828 (2023)]. When the
jump operators connect eigenstates of the system Hamiltonian, we show that the
Bures angle in the interaction picture is upper bounded by the square root of
the dynamical activity at short times, which contrasts with the classical
counterpart.
