  In [Phys. Rev. A 82, 042103 (2010)], the authors showed that "for a class of
the non-Markovian master equations in time-local forms", the quantum Fisher
information (QFI) flow can be decomposed into additive subflows corresponding
to different dissipative channels. However, the paper does not specify the
class of non-Markovian time-local master equations for which their analytic
decomposition of the QFI flow is valid. Here we show that several suppositions
have to be made in order to reach the central result of Ref. \cite{luwsun10},
which appears to be valid for a narrow class of density operators $\rho
(\theta;t)$ and quantum Fisher information $\mathcal{F}(\theta;t)$, and under
strict conditions on the time-local master equation. More precisely, the
decomposition of the QFI flow obtained in Ref. \cite{luwsun10} is valid under
two conditions not mentioned in the paper: (i) $\frac{d}{dt} \left(
\frac{\partial \rho}{\partial \theta} \right)=$ $\frac{\partial}{\partial
\theta} \left( \frac{d \rho}{dt} \right)$; (ii) $\frac{\partial H}{\partial
\theta}=0$, $\frac{\partial \gamma_i}{\partial \theta}=0$, $\frac{\partial
A_i}{\partial \theta}=0$, meaning that the Hamiltonian $H(t)$, the decay rates
$\gamma_i(t)$, and the Lindblad operators $A_i(t)$ appearing in the
non-Markovian time-local master equation have to not depend on the parameter
$\theta$ about which the quantum Fisher information is defined.
