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

$L_{\varrho }(A)=2i\sum _{k,l}\frac{{\lambda }_k-{\lambda }_l}{{\lambda }_k+{\lambda }_l}⟨k|A|l⟩|k⟩⟨l|$

where ${\lambda }_k$ and $|k⟩$ are the eigenvalues and eigenstates of $\varrho$, i.e. $\varrho |k⟩={\lambda }_k|k⟩$ and $\varrho =\sum _k{\lambda }_k|k⟩⟨k|$. 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:

$A=A^{†}\Rightarrow [L_{\varrho }(A){]}^{†}=L_{\varrho }(A)$

The derivative of the expression $\exp (-i\theta A)\varrho \exp (+i\theta A)$ w.r.t. $\theta$ at $\theta =0$ reads

$\frac{\partial }{\partial \theta }[\exp (-i\theta A)\varrho \exp (+i\theta A)]{|}_{\theta =0}=i(\varrho A-A\varrho )=i[\varrho ,A]=\frac{1}{2}\{\varrho ,L_{\varrho }(A)\}$

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

$\exp (-i\theta A)\varrho \exp (+i\theta A)=\varrho +\underset{=i\theta [\varrho ,A]}{\underbrace{\frac{1}{2}\theta \{\varrho ,L_{\varrho }(A)\}}}+{𝒪}({\theta }^2)$.