Leopold Infeld Colloquium

Several quantities of interest in physics are non-linear functions of the density matrix and cannot, even in principle, correspond to proper quantum observables. Any method aimed to determine the value of these quantities should resort to indirect measurements and thus corresponds to a parameter estimation problem whose solution, i.e. the determination of the most precise estimator, unavoidably involves an optimization procedure. In this lecture I review local quantum estimation theory, which allows to quantify quantum noise in the measurements of non observable quantities and provides a tools for the characterization of signals and devices, e.g. in quantum technology. Explicit formulas for the symmetric logarithmic derivative and the quantum Fisher information of relevant families of quantum states are presented, and the connection between the optmization procedure and the geometry of quantum statistical models is discussed in some details. Finally, few applications, ranging from quantum optics to critical systems are illustrated