Symmetric logarithmic derivative

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

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

The symmetric logarithmic derivative ${\displaystyle L_{\varrho }(A)}$ is defined implicitly by the equation^[1][2]

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

where ${\displaystyle [X,Y]=XY-YX}$ is the commutator and ${\displaystyle \{X,Y\}=XY+YX}$ is the anticommutator. Explicitly, it is given by^[3]

${\displaystyle L_{\varrho }(A)=2i\sum _{k,l}{\frac {\lambda _{k}-\lambda _{l}}{\lambda _{k}+\lambda _{l}}}\langle k\vert A\vert l\rangle \vert k\rangle \langle l\vert }$

where ${\displaystyle \lambda _{k}}$ and ${\displaystyle \vert k\rangle }$ are the eigenvalues and eigenstates of ${\displaystyle \varrho }$, i.e. ${\displaystyle \varrho \vert k\rangle =\lambda _{k}\vert k\rangle }$ and ${\displaystyle \varrho =\sum _{k}\lambda _{k}\vert k\rangle \langle k\vert }$.

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

The symmetric logarithmic derivative is linear in ${\displaystyle A}$:

${\displaystyle L_{\varrho }(\mu A)=\mu L_{\varrho }(A)}$

${\displaystyle L_{\varrho }(A+B)=L_{\varrho }(A)+L_{\varrho }(B)}$

The symmetric logarithmic derivative is Hermitian if its argument ${\displaystyle A}$ is Hermitian:

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

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

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

where the last equality is per definition of ${\displaystyle L_{\varrho }(A)}$; this relation is the origin of the name "symmetric logarithmic derivative". Further, we obtain the Taylor expansion

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