Quantum Fisher information for states in exponential form

We derive explicit expressions for the quantum Fisher information and the symmetric logarithmic derivative (SLD) of a quantum state in the exponential form $\rho =exp\left(G\right)$; the SLD is expressed in terms of the generator $G$. Applications include quantum-metrology problems with Gaussian states and general thermal states. Specifically, we give the SLD for a Gaussian state in two forms, in terms of its generator and its moments; the Fisher information is also calculated for both forms. Special cases are discussed, including pure, degenerate, and very noisy Gaussian states.