Volume-Law Protection of Metrological Advantage

Although entanglement can boost metrological precision beyond the standard quantum limit, the advantage often disappears with particle loss. We demonstrate that scrambling safeguards precision by dispersing information about the encoded parameter into many-body correlations. For Haar-random scrambling unitaries, we derive closed-form analytical expressions for the average quantum Fisher information of the reduced state after tracing out lost particles. The result exhibits a sharp threshold: majority subsystems retain essentially the full quantum Fisher information, while minority subsystems contain negligible information in the large-system limit. We link this threshold to the scrambling-induced transition from area-law to volume-law entanglement and the associated growth of the Schmidt rank. We outline two realizations-a brickwork circuit and chaotic XX-chain evolution-and demonstrate the protection of one-axis-twisted probes for loss of fewer than half the particles.