This article provides insufficient context for those unfamiliar with the subject.(December 2019) |
The symmetric logarithmic derivative is an important quantity in quantum metrology, and is related to the quantum Fisher information.
Let and be two operators, where is Hermitian and positive semi-definite. In most applications, and 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 is the commutator and is the anticommutator. Explicitly, it is given by [3]
where and are the eigenvalues and eigenstates of , i.e. and .
Formally, the map from operator to 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 w.r.t. at 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
This article provides insufficient context for those unfamiliar with the subject.(December 2019) |
The symmetric logarithmic derivative is an important quantity in quantum metrology, and is related to the quantum Fisher information.
Let and be two operators, where is Hermitian and positive semi-definite. In most applications, and 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 is the commutator and is the anticommutator. Explicitly, it is given by [3]
where and are the eigenvalues and eigenstates of , i.e. and .
Formally, the map from operator to 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 w.r.t. at 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