1These lectures will focus on quantum metrology, which aims to determine the ultimate precision limits in the estimation of parameters. It also helps to understand some basic problems in quantum physics, like for instance the energy-time uncertainty and its extension to noisy systems. Applications to several systems will be discussed, including phase estimation in optical interferometers, frequency estimation in atomic spectroscopy, force estimation, weak-value amplification, and the quantum speed limit.

2The lectures will be organized as follows:

Lecture I

3After a general introduction, this lecture will focus on optical interferometers, discussing several possible strategies for phase estimation, and demonstrating that states with special quantum features may lead to bounds on precision that overcome the shot-noise limit, reaching the so-called Heisenberg limit.

Lecture II

4The general theory of parameter estimation is introduced. The Cramér-Rao bound is demonstrated, and the Fisher information, as well as the maximum likelihood estimator, are introduced. The role of entanglement in quantum-enhanced metrology is clarified. These results are applied to optical interferometry, atomic spectroscopy, and weak-value amplification.

Lecture III

5Recent results concerning noisy quantum-enhanced metrology are presented, and a general framework for evaluating the ultimate precision limit in the estimation of parameters for open systems is developed. This method is applied to optical interferometry, atomic spectroscopy, and force estimation.

Lecture IV

6Different possible interpretations of the time-energy uncertainty relation are reviewed. The techniques developed in the previous lecture are used to extend the time-energy uncertainty relation to open systems, with applications to atomic physics.