Quantum Metrology for Noisy Systems

The estimation of parameters characterizing dynamical processes is a central problem in science and technology. It concerns for instance the evaluation of the duration of some interaction, of the value of a coupling constant, or yet of a frequency in atomic spectroscopy. The estimation error changes with the number N of resources employed in the experiment (which could quantify, for instance, the number of probes or the probing energy). For independent probes, it scales as $1/\sqrt{N}$—the standard limit—a consequence of the central-limit theorem. Quantum strategies, involving for instance entangled or squeezed states, may improve the precision, for noiseless processes, by an extra factor $1/\sqrt{N}$, leading to the so-called Heisenberg limit. For noisy processes, an important question is if and when this improvement can be achieved. Here, we review and detail our recent proposal of a general framework for obtaining attainable and useful lower bounds for the ultimate limit of precision in noisy systems. We apply this bound to lossy optical interferometry and show that, independently of the initial states of the probes, it captures the main features of the transition, as N grows, from the 1/N to the $1/\sqrt{N}$ behavior.