  Recently, a widely-used computation expression for quantum Fisher information
was shown to be discontinuous at the parameter points where the rank of the
parametric density operator changes. The quantum Cram\'er-Rao bound can be
violated on such singular parameter points if one uses this computation
expression for quantum Fisher information. We point out that the discontinuity
of the computation expression of quantum Fisher information is accompanied with
the unboundedness of the symmetric logarithmic derivation operators, based on
which the quantum Fisher information is formally defined and the quantum
Cram\'er-Rao bound is originally proved. We argue that the limiting version of
quantum Cram\'er-Rao bound still holds when the parametric density operator
changes its rank by closing the potential loophole of involving an unbounded
SLD operator in the proof of the bound. Moreover, we analyze a typical example
of the quantum statistical models with parameter-dependent rank.
