Quantum probes for fractional Gaussian processes

We address the characterization of classical fractional random noise via quantum probes. In particular, we focus on estimation and discrimination problems involving the fractal dimension of the trajectories of a system subject to fractional Brownian noise. We assume that the classical degree of freedom exposed to the environmental noise is coupled to a quantum degree of freedom of the same system, e.g. its spin, and exploit quantum limited measurements on the spin part to characterize the classical fractional noise. More generally, our approach may be applied to any two-level system subject to dephasing perturbations described by fractional Brownian noise, in order to assess the precision of quantum limited measurements in the characterization of the external noise. In order to assess the performances of quantum probes we evaluate the Bures metric, as well as the Helstrom and the Chernoff bound, and optimize their values over the interaction time. We find that quantum probes may be successfully employed to obtain a reliable characterization of fractional Gaussian process when the coupling with the environment is weak or strong. In the first case decoherence is not much detrimental and for long interaction times the probe acquires information about the environmental parameters without being too much mixed. Conversely, for strong coupling information is quickly impinged on the quantum probe and can be effectively retrieved by measurements performed in the early stage of the evolution. In the intermediate situation, none of the two above effects take place: information is flowing from the environment to the probe too slowly compared to decoherence, and no measurements can be effectively employed to extract it from the quantum probe. The two regimes of weak and strong coupling are defined in terms of a threshold value of the coupling, which itself increases with the fractional dimension.