  Quantum Fisher information, as an intrinsic quantity for quantum states, is a
central concept in quantum detection and estimation. When quantum measurements
are performed on quantum states, classical probability distributions arise,
which in turn lead to classical Fisher information. In this article, we exploit
the classical Fisher information induced by quantum measurements, and reveal a
rich hierarchical structure of such measurement-induced Fisher information. We
establish a general framework for the distribution and transfer of the Fisher
information. In particular, we illustrate three extremal distribution types of
the Fisher information: the locally owned type, the locally inaccessible type,
and the fully shared type. Furthermore, we indicate the significant role played
by the distribution and flow of the Fisher information in some physical
problems, e.g., the non-Markovianity of open quantum processes, the
environment-assisted metrology, the cloning and broadcasting, etc.
