Symmetric logarithmic derivative

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

Definition

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

The symmetric logarithmic derivative [math]\displaystyle{ L_\varrho(A) }[/math] is defined implicitly by the equation

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

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

[math]\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 }[/math]

where [math]\displaystyle{ \lambda_k }[/math] and [math]\displaystyle{ \vert k\rangle }[/math] are the eigenvalues and eigenstates of [math]\displaystyle{ \varrho }[/math], i.e. [math]\displaystyle{ \varrho\vert k\rangle=\lambda_k\vert k\rangle }[/math] and [math]\displaystyle{ \varrho=\sum_k \lambda_k \vert k\rangle\langle k\vert }[/math].

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

Properties

The symmetric logarithmic derivative is linear in [math]\displaystyle{ A }[/math]:

[math]\displaystyle{ L_\varrho(\mu A)=\mu L_\varrho(A) }[/math]

[math]\displaystyle{ L_\varrho(A+B)=L_\varrho(A)+L_\varrho(B) }[/math]

The symmetric logarithmic derivative is Hermitian if its argument [math]\displaystyle{ A }[/math] is Hermitian:

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

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

[math]\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)\} }[/math]

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

[math]\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) }[/math].