The symmetric logarithmic derivative is an important quantity in quantum metrology, and is related to the quantum Fisher information.
is Hermitian and positive semi-definite.
fulfill further properties, that also
is Hermitian and
is a density matrix (which is also trace-normalized), but these are not required for the definition.
The symmetric logarithmic derivative
is defined implicitly by the equation[1][2] where
are the eigenvalues and eigenstates of
Formally, the map from operator
is a (linear) superoperator.
The symmetric logarithmic derivative is linear in
: The symmetric logarithmic derivative is Hermitian if its argument
is Hermitian: The derivative of the expression
exp ( − i θ
) ϱ exp ( + i θ
reads where the last equality is per definition of
; this relation is the origin of the name "symmetric logarithmic derivative".
Further, we obtain the Taylor expansion