Heisenberg Scaling in Many-Body Kinetic Uncertainty Relation via Quantum Feedback

In quantum metrology, Heisenberg scaling—where estimation variance decreases as 1/N^2—is a gold standard for precision. While many-body effects like superradiance can significantly increase the activity of a system, this enhancement is typically transient, causing the system to drift away from high-activity states. The authors investigate whether quantum feedback can stabilize these states to achieve Heisenberg-like scaling for counting observables, which are governed by Kinetic Uncertainty Relations (KURs).

The researchers utilize the Dicke superradiance model, consisting of N identical two-level systems. They analyze the system's dynamics using a Lindblad master equation and quantum jump trajectories. To maintain the system in the high-activity superradiant region (near the equator of the Bloch sphere), they implement a Markovian feedback protocol where each detected quantum jump triggers an immediate unitary rotation. They derive a many-body KUR to establish theoretical precision bounds and use mean-field equations to determine the optimal feedback strength required to counteract the natural downward drift of the magnetization.

The study establishes that without feedback, superradiance provides only a transient boost to precision that fails to maintain Heisenberg scaling as the system size N increases. By applying the proposed feedback protocol, the system is successfully steered to remain in the high-activity region. Analytical derivations and numerical simulations confirm that this feedback-stabilized dynamics allows both the KUR lower bound and the actual relative fluctuation of the counting observable to scale as 1/N^2. This demonstrates that collective dissipation, when controlled, can be transformed into a valuable resource for high-precision quantum devices.