We derive an explicit condition that determines whether in a noisy quantum frequency estimation problem
the estimation precision of the most general adaptive quantum metrological protocol cannot reach the
Heisenberg-like scaling. The condition is a simple algebraic statement on a relation between the Hamiltonian operator representing the unitary part of the dynamics and the noise operators appearing in the quantum Master equation, and does not require any finite-time integration of the dynamics. In particular these results allow us to understand when application of quantum error correction protocols in order to recover the Heisenberg scaling in quantum metrology is not possible. Additionally, we provide methods to obtain quantitative bounds on achievable precision in the most general adaptive quantum metrological models. Finally, we apply the newly developed tools to prove fundamental bounds in atomic interferometry with many-body effects such as many body losses as well as models involving many-body terms in the Hamiltonian part of the dynamics commonly referred to as non-linear quantum metrology.
Rafal Demkowicz-Dobrzanski, Pavel Sekatski, and Jan Czajkowski, "Adaptive quantum metrology under general Markovian dynamics (Conference Presentation)," Proc. SPIE 10442, Quantum Information Science and Technology III, 104420I (Presented at SPIE Security + Defence: September 14, 2017; Published: 19 October 2017); https://doi.org/10.1117/12.2279695.5618105385001.
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