Quantum Estimation for Quantum Technology

Several quantities of interest in quantum information, including
entanglement and purity, are nonlinear 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.
We review local quantum estimation theory and present explicit formulas
for the symmetric logarithmic derivative and the quantum Fisher
information of relevant families of quantum states. Estimability of a
parameter is defined in terms of the quantum signal-to-noise ratio and
the number of measurements needed to achieve a given relative error.
The connections between the optmization procedure and the geometry
of quantum statistical models are discussed. Our analysis allows to
quantify quantum noise in the measurements of non observable quantities
and provides a tools for the characterization of signals and
devices in quantum technology. We discuss few examples including
estimation of unitary parameters, entanglement and applications
to (quantum) critical systems.