= Symmetric logarithmic derivative
{title2=$L_ heta$}
For a differentiable family of <density matrices> $\rho_\theta$, the <symmetric logarithmic derivative> is a Hermitian operator $L_\theta$ satisfying
$$
\partial_\theta\rho_\theta=\tfrac12(\rho_\theta L_\theta+L_\theta\rho_\theta).
$$
If $\rho_\theta=\sum_jp_j|j\rangle\langle j|$, its determined matrix entries are $(L_\theta)_{jk}=2\langle j|\partial_\theta\rho_\theta|k\rangle/(p_j+p_k)$ whenever $p_j+p_k>0$. Entries wholly in the kernel are undetermined when a solution exists and do not affect $\operatorname{Tr}(\rho_\theta L_\theta^2)$. Rank-changing boundary points need extra care.
For $\rho_\theta=\operatorname{diag}(\theta,1-\theta)$, $0<\theta<1$, one gets $L_\theta=\operatorname{diag}(1/\theta,-1/(1-\theta))$ and <Quantum Fisher information> $F_Q=1/[\theta(1-\theta)]$. This recovers ordinary <Fisher information> for a binary observation. In noncommuting families the symmetrized equation supplies the operator version needed to quantify local sensitivity.<Quantum estimation for quantum technology by Matteo G. A. Paris>[Section 2, symmetric logarithmic derivative and spectral formula]{ref}
Back to article page
