  The dissipative quantum Fisher information (DQFI) for a dynamic map with a
general parameter in an open quantum system is investigated, which can be
regarded as an analog of the quantum Fisher information (QFI) in the Liouville
space. We first derive a general dissipative generator in the Liouville space,
and based on its decomposition form, find the DQFI stems from two parts. One is
the dependence of eigenvalues of the Liouvillian supermatrix on the estimated
parameter, which shows a linear dependence on time. The other is the variation
of the eigenvectors with the estimated parameter. The relationship between this
part and time presents rich characteristics, including harmonic oscillation,
pure exponential gain and attenuation, as well as exponential gain and
attenuation of oscillatory type, which depend specifically on the properties of
the Liouville spectrum. This is in contrast to that of the conventional
generator, where only oscillatory dependencies are seen. Further, we illustrate
the theory through a toy model: a two-level system with spin-flip noise.
Especially, by using the DQFI, we demonstrated that the exceptional estimation
precision cannot be obtained at the Liouvillian exceptional point.
