Volume-law protection of metrological advantage

TL;DR

This paper shows that quantum information scrambling can protect metrological advantage from particle loss by dispersing information into many-body correlations, with a threshold at half the particles for full Fisher information recovery.

Contribution

It introduces a novel approach using scrambling to safeguard quantum metrological precision against particle loss, deriving exact formulas for the quantum Fisher information in this context.

Findings

For Haar-random scrambling, the QFI is preserved if more than half the particles remain.

A threshold at N/2 particles determines whether full QFI can be recovered.

Scrambling induces a transition from area-law to volume-law entanglement, aiding protection.

Abstract

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 exact formulas for the average quantum Fisher information (QFI) of the reduced state after tracing out lost particles. The result exhibits a threshold; any remaining subsystem larger than $N/2$ recovers the full QFI, while smaller subsystems contain negligible information. 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 against the loss of up to half of the particles.