Symmetric logarithmic derivative
The symmetric logarithmic derivative is an important quantity in quantum metrology, and is related to the quantum Fisher information. Definition. Let formula_1 and formula_2 be two operators, where formula_1 is Hermitian and positive semi-definite. In most applications, formula_1 and formula_2 fulfill further properties, that also formula_2 is Hermitian and formula_1 is a density matrix (which is, in addition, also trace-normalized), but these are not required for the definition. The symmetric logarithmic derivative formula_8 is defined implicitly by the equation where formula_10 is the commutator and formula_11 is the anticommutator. Explicitly, it is given by where  formula_13 and formula_14 are the eigenvalues and eigenstates of formula_15, i.e. formula_16 and formula_17. Formally, the map from operator formula_2 to operator formula_8 is a (linear) superoperator. Properties. The symmetric logarithmic derivative is linear in formula_2: The symmetric logarithmic derivative is Hermitian if its argument formula_2 is Hermitian: The derivative of the expression formula_25 w.r.t. formula_26 at formula_27 reads where the last equality is per definition of formula_8; this relation is the origin of the name "symmetric logarithmic derivative". Further, we obtain the Taylor expansion