General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology

The estimation of parameters characterizing dynamical processes is central to science and technology. 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). Typically, it scales as . Quantum strategies may improve the precision, for noiseless processes, by an extra factor . For noisy processes, it is not known in general if and when this improvement can be achieved. Here we propose 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 atomic spectroscopy in the presence of dephasing, showing that it captures the main features of the transition from the 1/ N to the behaviour as N increases, independently of the initial state of the probes, and even with use of adaptive feedback. Quantum strategies can help to make parameter-estimation schemes more precise, but for noisy processes it is typically not known how large that improvement may be. Here, a universal quantum bound is derived for the error in the estimation of parameters that characterize dynamical processes.