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

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