symmetric logarithmic derivative

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

The symmetric logarithmic derivative $L\_\backslash varrho(A)$ is defined implicitly by the equation{{cite journal | last1=Braunstein | first1=Samuel L. | last2=Caves | first2=Carlton M. | authorlink2=Carlton Caves |title=Statistical distance and the geometry of quantum states | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=72 | issue=22 | date=1994-05-30 | issn=0031-9007 | doi=10.1103/physrevlett.72.3439 | pmid=10056200 | pages=3439–3443| bibcode=1994PhRvL..72.3439B }}{{cite journal |last1=Braunstein |first1=Samuel L. |last2=Caves |first2=Carlton M. |last3=Milburn |first3=G.J. |authorlink2=Carlton Caves |title=Generalized Uncertainty Relations: Theory, Examples, and Lorentz Invariance |journal=Annals of Physics |date=April 1996 |volume=247 |issue=1 |pages=135–173 | arxiv=quant-ph/9507004 | doi=10.1006/aphy.1996.0040|bibcode=1996AnPhy.247..135B |s2cid=358923 }}

:$i[\backslash varrho,A]=\backslash frac\{1\}\{2\}\; \backslash \{\backslash varrho,\; L\_\backslash varrho(A)\backslash \}$

where $[X,Y]=XY-YX$ is the commutator and $\backslash \{X,Y\backslash \}=XY+YX$ is the anticommutator. Explicitly, it is given by{{cite journal |last1=Paris |first1=Matteo G. A. |title=Quantum Estimation for Quantum Technology|journal=International Journal of Quantum Information |date=21 November 2011 |volume=07 |issue=supp01 |pages=125–137 |doi=10.1142/S0219749909004839|arxiv=0804.2981 |s2cid=2365312 }}

:$L\_\backslash varrho(A)=2i\backslash sum\_\{k,l\}\; \backslash frac\{\backslash lambda\_k-\backslash lambda\_l\}\{\backslash lambda\_k+\backslash lambda\_l\}\; \backslash langle\; k\; \backslash vert\; A\; \backslash vert\; l\backslash rangle\; \backslash vert\; k\backslash rangle\; \backslash langle\; l\; \backslash vert$

where $\backslash lambda\_k$ and $\backslash vert\; k\backslash rangle$ are the eigenvalues and eigenstates of $\backslash varrho$, i.e. $\backslash varrho\backslash vert\; k\backslash rangle=\backslash lambda\_k\backslash vert\; k\backslash rangle$ and $\backslash varrho=\backslash sum\_k\; \backslash lambda\_k\; \backslash vert\; k\backslash rangle\backslash langle\; k\backslash vert$.

Formally, the map from operator $A$ to operator $L\_\backslash varrho(A)$ is a (linear) superoperator.

The symmetric logarithmic derivative is linear in $A$:

:$L\_\backslash varrho(\backslash mu\; A)=\backslash mu\; L\_\backslash varrho(A)$

:$L\_\backslash varrho(A+B)=L\_\backslash varrho(A)+L\_\backslash varrho(B)$

The symmetric logarithmic derivative is Hermitian if its argument $A$ is Hermitian:

:$A=A^\backslash dagger\backslash Rightarrow[L\_\backslash varrho(A)]^\backslash dagger=L\_\backslash varrho(A)$

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

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

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

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

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