Bounding the precision of frequency estimation subject to generic non-Markovian noise.

The quantum Fisher information (QFI) is a measure of the best attainable estimation precision in terms of the interrogation time, t, of frequency estimation protocols. Quantum mechanics imposes a fundamental limit on how the QFI scales with time. In particular, the maximum scaling is always upper bounded by O(t²). In noiseless systems, it is well-established that this O(t²) scaling can be achieved. However, for systems subjected to Markovian noise, the scaling is typically reduced, and even with the inclusion of error-correcting protocols, recovering the O(t²) scaling is often not possible. In this work, we investigate broad classes of systems subjected to generic noise allowing for full and fast controls. We model the noise as an interaction between the system and its environment mediated by a system-environment Hamiltonian that couples the two subsystems. To reduce the noise, we allow for perfect quantum control operations. Under these conditions, we study the maximum achievable QFI and elucidate the general limits of its scaling.