  In this thesis, first, we investigate the metrological usefulness of a family
of states known as unpolarized Dicke states, which turn to be very sensitive to
the magnetic field. Quantum mechanics plays a central role in achieving such a
high precision. Second, we investigate possible lower bounds on the quantum
Fisher information, a quantity that characterizes the usefulness of a state for
quantum metrology, using the theory of Legendre transforms such that we obtain
tight lower bounds based on few measurements of the initial quantum state that
will be used for metrology. And last but not least, we investigate gradient
magnetometry, i.e., we develop a theory to study the sensitivity of some states
on the change in space of the magnetic field.
