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 \

where ,YXY-YX is the commutator and $\=XY+YX$ is the anticommutator. Explicitly, it is given by

:$L_\varrho(A)=2i\sum_ \frac \langle k \vert A \vert l\rangle \vert k\rangle \langle l \vert$

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 _\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\Big exp(-i\theta A)\varrho\exp(+i\theta A)\Bigbigg\vert_ = i(\varrho A-A\varrho) = i varrho,A= \frac\
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
:$\exp(-i\theta A)\varrho\exp(+i\theta A) = \varrho + \underbrace_ + \mathcal(\theta^2)$.

References

{{Reflist

Quantum information science
Quantum optics