Abstract
We address the question of whether the super-Heisenberg scaling for quantum estimation is indeed 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 (Heisenberg scaling) in terms of the number of elementary subsystems used . The second approach compares the overlap between the ground states of the parameter-dependent Hamiltonian in critical systems, often leading to an apparent super-Heisenberg scaling. However, we point out that if one takes into account the scaling of time needed to perform the necessary operations, i.e., ensuring adiabaticity of the evolution, the Heisenberg limit given by the rotation scenario is recovered. We illustrate the general theory on a ferromagnetic Heisenberg spin chain example and show that it exhibits such super-Heisenberg scaling of ground-state fidelity around the critical value of the parameter (magnetic field) governing the one-body part of the Hamiltonian. Even an elementary estimator represented by a single-site magnetization already outperforms the Heisenberg behavior providing the scaling. In this case, Fisher information sets the ultimate scaling as , 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 ultracold bosons in a periodically shaken optical lattice. We explicitly derive that the Heisenberg limit is recovered when the time needed for preparation of quantum states involved is taken into account.
- Received 6 May 2017
- Revised 19 February 2018
DOI:https://doi.org/10.1103/PhysRevX.8.021022
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
Quantum metrology is the science of taking into account quantum effects to make ultraprecise physical measurements. The detection of gravitational waves from deep space, for example, relies on measuring distances between mirrors to a precision of roughly one-thousandth the diameter of a proton. Quantum mechanics gives a fundamental limit to the accuracy of such measurements (known as the Heisenberg limit). However, it has been argued that this limit can be overcome in some settings. Using mathematical arguments, we show that such claims must be supplemented by a careful analysis of the timescale required for performing the necessary operations.
It has been argued that many-body quantum systems near their critical point are potentially useful probes for quantum metrology. The state of such systems is highly sensitive to external parameters. There are examples in the literature where the scaling of the precision in terms of the number of elementary subsystems used apparently breaks the Heisenberg limit.
We show, however, that if the time needed for preparing the operations is properly taken into account, then the Heisenberg limit is recovered. On the other hand, we also prove that systems near their critical point do still have metrological potential, and we derive what the ultimate precision is regardless of the time required. Indeed, such a precision depends only on the number of elementary subsystems and the universality class of the model at the criticality.
Our results reconcile two approaches to quantum metrology discussed in the literature and highlight that time is both an important factor and resource.