Comment on "Quantum Fisher information flow and non-Markovian processes of open systems"

Abstract: 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.