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The symmetric logarithmic derivative $L_i$ is defined as the operator solutions to equations $$\frac{1}{2}\left(L_i\rho_{\lambda}+\rho_{\lambda}L_i\right)=\partial_i\rho_{\lambda}$$ The quantum Fisher information metric is then a symmetric positive or a positive semi-definite metric defined by $$H_{ij}(\lambda)=\frac{1}{2}Tr\left[\left(L_iL_j+L_jLi\right)\rho_{\lambda}\right]$$ Using the spectral decomposition of the density matrix as $\rho_{\lambda}=\sum_{k}p_k\mid k><k\mid$ we can write the quantum Fisher information metric as $$H_{ij}(\lambda)=2\sum_{p_k+p_l>0}\frac{Re\left(<k\mid\partial_i\rho_{\lambda}\mid l><l\mid\partial_j\rho_{\lambda}\mid k>\right)}{p_k+p_l}$$ Where $Re$ denotes the real part. Now the relative entropy measures the distinguishability of a density matrix $\rho$ from that of a reference density matrix $\sigma$. It is defined as $$S(\rho\mid\mid\sigma)=Tr(\rho\log\rho)-Tr(\rho\log\sigma)$$ In this paper they calculated the change in the quantum relative entropy for a one parameter family of nearby states $\rho(\lambda)=\rho_0+\lambda\rho_1+\lambda^2\rho_2$ from the reference state $\rho_0=\sigma$. At second order the change in the quantum relative entropy gives the quantum Fisher information metric as $$<\delta\rho,\delta\rho>=\frac{1}{2}Tr\left(\delta\rho\frac{d}{d\lambda}log(\sigma+\lambda\delta\rho)\mid_{\lambda=0}\right)$$ Where $\rho_1=\delta\rho=\rho(\lambda)\partial_{\lambda}log\rho(\lambda)$. My query is whether these two expression for the quantum Fisher information metric are same and how to obtain one from another?

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