For a differentiable family of density matrices , the symmetric logarithmic derivative is a Hermitian operator satisfyingIf , its determined matrix entries are whenever . Entries wholly in the kernel are undetermined when a solution exists and do not affect . Rank-changing boundary points need extra care.
For , , one gets and Quantum Fisher information . This recovers ordinary Fisher information for a binary observation. In noncommuting families the symmetrized equation supplies the operator version needed to quantify local sensitivity.[ref]
Quantum estimation for quantum technology by Matteo G. A. Paris. Section 2, symmetric logarithmic derivative and spectral formula
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The symmetric logarithmic derivative (SLD) is a concept from the field of quantum information theory and quantum mechanics, particularly in the context of density matrices and quantum statistical mechanics. It is used to describe how a quantum state evolves and how it interacts with measurements. For a quantum system described by a density operator \( \rho \), the symmetric logarithmic derivative is defined in relation to a measurement or an observable \( A \).