Ultimate precision limits for noisy frequency estimation

How precisely can we estimate the value of an unknown parameter? In classical experiments involving N sensing particles, i.e. N probes, the best estimation strategies lead to an error (as measured by the variance), which scales at most as 1/N, according to the central limit theorem. On the other hand, the use of entangled states can yield a further factor 1/N of improvement, which shows in a paradigmatic way that quantum features can be exploited to get a significant advantage compared to any classical strategy. Such a quantum advantage is nevertheless jeopardized by the interaction of the probing system with the surrounding environment. Previous results showed that the quantum and classical strategies become completely equivalent in the presence of random fluctuations of the parameter to be estimated, due to the influence of a fast-decaying environment.

In this work, we show how the advantage provided by using entangled states can be (partially) re-established, if one deals with a more general and more realistic type of system-environment interactions. Classical strategies can be outperformed if the probes are measured on time-scales short enough, in order to access the universal dynamical regime of open systems, where the survival probabilities decay less than linearly with time. In particular, we derive a lower bound to the estimation error, which holds for a wide and well-defined type of dynamics and we show its attainability, as well as pointing out the crucial dynamical features, which discriminate between classical or super-classical limits to the parameter estimation.

Ultimate precision limits for noisy frequency estimation