Symmetric logarithmic derivative

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

Definition

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.

Properties

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

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

References

1. Braunstein, Samuel L.; Caves, Carlton M. (1994-05-30). "Statistical distance and the geometry of quantum states". Physical Review Letters . American Physical Society (APS). 72 (22): 3439–3443. Bibcode:1994PhRvL..72.3439B. doi:10.1103/physrevlett.72.3439. ISSN   0031-9007. PMID   10056200.
2. Braunstein, Samuel L.; Caves, Carlton M.; Milburn, G.J. (April 1996). "Generalized Uncertainty Relations: Theory, Examples, and Lorentz Invariance". Annals of Physics . 247 (1): 135–173. arXiv: quant-ph/9507004 . Bibcode:1996AnPhy.247..135B. doi:10.1006/aphy.1996.0040. S2CID   358923.
3. Paris, Matteo G. A. (21 November 2011). "Quantum Estimation for Quantum Technology". International Journal of Quantum Information . 07 (supp01): 125–137. arXiv: 0804.2981 . doi:10.1142/S0219749909004839. S2CID   2365312.