Volume-law protection of metrological advantage

TL;DR

The paper addresses the fragility of quantum-enhanced metrology under particle loss by showing that scrambling dynamics disperse the encoded parameter information into many-body correlations, protecting the metrological advantage. It derives exact, Haar-averaged expressions for the quantum Fisher information (QFI) of subsystems after erasing parts of the system, using Page's theorem and Weingarten calculus, and reveals a threshold: any remaining subsystem with more than half the particles ($d_B > d_A$, or loss $K < N/2$) recovers the full QFI. The mechanism is tied to a transition from area-law to volume-law entanglement, which increases the Schmidt rank and locks information into non-local correlations. The study provides two experimentally feasible scrambling protocols (brickwork circuit and chaotic XX-chain) and demonstrates QFI protection for one-axis-twisted probes, pointing toward fault-tolerant quantum sensing in platforms where particle loss is a dominant decoherence channel.

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.

Figures (3)

• Figure 1: Typical value of the QFI for a reduced density matrix $\hat{\varrho}_K^{(U)}$ obtained after tracing-out $K$ out of $N$ qubits, normalized to the pure-state value, $\mathcal{I}_q\left[\hat{\varrho}_K^{(U)}\right]/\mathcal{I}_q\left[\hat{\varrho}(\theta)\right]$. Solid lines correspond to the analytical prediction in Eq. \ref{['eq:qfi_a_specific']} and Eq. \ref{['eq:qfi_b_specific']} for $N=10$ and 26, The squares andcircles represent numerical results. For comparison, we display the analytical curve for $N=1000$ qubits.
• Figure 2: Panels (a)-(b): Bipartite entanglement entropy $S$ at the central bipartition as a function of system size $N$ for various circuit depths $L$ (panel (a)), and analog evolution time $t$ (panel (b)). Circuits with $L\le N/2$ have higher Schmidt rank, converging to the maximal value $\chi_{\rm max}=2^{N/2}$ (dashed line) for $L=N$. Panels (c) - (d): The QFI after tracing-out $K$ qubits for increasing circutis depths $L$, panel (c), and analog evolutions timse $t$, panel (d). The dark regions indicate preserved Heisenberg-limited QFI. As $L$ ($t$) increases, QFI protection extends to larger $k$, demonstrating that volume-law entanglement locks the metrological information against particle loss.
• Figure 3: (a): the QFI for OAT-generated states as a function of time after tracing out $K$ qubits. Blue solid line correspond to pure OAT state without particle loss ($K=0$). The lines (from top to bottom) corresponds to QFI in a reduced state after tracing out $K = 1, 2, N/2, N/2-1$ qubits ($N=20$), indicating diminishing metrological quantum advantage. (b): the QFI for OAT-generated states followed by analog scrambling evolution, resulting in QFI-locking. The QFI is protected against particle loss for $K<N/2$, as illustrated by (c), which plots the QFI ath $\tau=\pi/4$ as a function of the fraction of lost partiles.