Optimal observables for (non-)equilibrium quantum metrology from the master equation

We demonstrate how observables with optimal sensitivity to environmental properties can be constructed explicitly from the master equation of an open-quantum system. Our approach does not rely on the explicit solution of the master equation. This makes the symmetric logarithmic derivative (SLD), the operator of optimal sensitivity and key quantity in quantum metrology, available to a large class of systems of interest, both in and out-of-equilibrium. We validate our approach by reproducing the SLD for temperature in quantum Brownian motion and demonstrate its versatility by constructing the optimal observable for the non-equilibrium relaxation rate.