# At the limits of criticality-based quantum metrology: apparent super-Heisenberg scaling revisited

## TL;DR

This paper investigates the limits of super-Heisenberg scaling in quantum metrology, demonstrating that apparent super-scaling at criticality is constrained by adiabatic time requirements, and provides a practical experimental proposal.

## Contribution

It unifies two approaches to quantum metrology scaling, clarifies the conditions under which super-Heisenberg scaling appears, and shows how the Heisenberg limit is restored when considering state preparation time.

## Key findings

- Elementary estimators can outperform Heisenberg scaling, achieving N^{-1.5}.
- Fisher information indicates ultimate scaling of N^{-1.75} for certain measurements.
- Adiabatic evolution time constraints restore the Heisenberg limit in critical quantum systems.

## Abstract

We address the question whether the super-Heisenberg scaling for quantum estimation is realizable. We unify the results of two approaches. In the first one, the original system is compared with its copy rotated by the parameter dependent dynamics. If the parameter is coupled to the one-body part of the Hamiltonian the precision of its estimation is known to scale at most as $N^{-1}$ (Heisenberg scaling) in terms of the number of elementary subsystems used, $N$. The second approach considers fidelity at criticality often leading to an apparent super-Heisenberg scaling. However, scaling of time needed to ensure adiabaticity of the evolution brings back the the Heisenberg limit. We illustrate the general theory on a ferromagnetic Heisenberg spin chain which exhibits such super-Heisenberg scaling of fidelity around the critical value of the magnetic field. Even an elementary estimator represented by a single-site magnetization already outperforms the Heisenberg behavior providing the $N^{-1.5}$ scaling. In this case Fisher information sets the ultimate scaling as $N^{-1.75}$ which can be saturated by measuring magnetization on all sites simultaneously. We discuss universal scaling predictions of the estimation precision offered by such observables, both at zero and finite temperatures, and support them with numerical simulations in the model. We provide an experimental proposal of realization of the considered model via mapping the system to ultra-cold bosons in periodically shaken optical lattice. We explicitly derive that the Heisenberg limit is recovered when time needed for preparation of quantum states involved is taken into acocunt.