Symmetric logarithmic derivative

The symmetric logarithmic derivative is an important quantity in quantum metrology, and is related to the quantum Fisher information.

Definition
Let $$\rho$$ and $$A$$ be two operators, where $$\rho$$ is Hermitian and positive semi-definite. In most applications, $$\rho$$ and $$A$$ fulfill further properties, that also $$A$$ is Hermitian and $$\rho$$ is a density matrix (which is also trace-normalized), but these are not required for the definition.

The symmetric logarithmic derivative $$L_\varrho(A)$$ is defined implicitly by the equation


 * $$i[\varrho,A]=\frac{1}{2} \{\varrho, L_\varrho(A)\}$$

where $$[X,Y]=XY-YX$$ is the commutator and $$\{X,Y\}=XY+YX$$ is the anticommutator. Explicitly, it is given by


 * $$L_\varrho(A)=2i\sum_{k,l} \frac{\lambda_k-\lambda_l}{\lambda_k+\lambda_l} \langle k \vert A \vert l\rangle \vert k\rangle \langle l \vert$$

where $$\lambda_k$$ and $$\vert k\rangle$$ are the eigenvalues and eigenstates of $$\varrho$$, i.e. $$\varrho\vert k\rangle=\lambda_k\vert k\rangle$$ and $$\varrho=\sum_k \lambda_k \vert k\rangle\langle k\vert$$.

Formally, the map from operator $$A$$ to operator $$L_\varrho(A)$$ is a (linear) superoperator.

Properties
The symmetric logarithmic derivative is linear in $$A$$:
 * $$L_\varrho(\mu A)=\mu L_\varrho(A)$$
 * $$L_\varrho(A+B)=L_\varrho(A)+L_\varrho(B)$$

The symmetric logarithmic derivative is Hermitian if its argument $$A$$ is Hermitian:
 * $$A=A^\dagger\Rightarrow[L_\varrho(A)]^\dagger=L_\varrho(A)$$

The derivative of the expression $$\exp(-i\theta A)\varrho\exp(+i\theta A)$$ w.r.t. $$\theta$$ at $$\theta=0$$ reads
 * $$\frac{\partial}{\partial\theta}\Big[\exp(-i\theta A)\varrho\exp(+i\theta A)\Big]\bigg\vert_{\theta=0} = i(\varrho A-A\varrho) = i[\varrho,A] = \frac{1}{2}\{\varrho, L_\varrho(A)\}$$

where the last equality is per definition of $$L_\varrho(A)$$; this relation is the origin of the name "symmetric logarithmic derivative". Further, we obtain the Taylor expansion
 * $$\exp(-i\theta A)\varrho\exp(+i\theta A) = \varrho + \underbrace{\frac{1}{2}\theta\{\varrho, L_\varrho(A)\}}_{=i\theta[\varrho,A]} + \mathcal{O}(\theta^2)$$.