At the Limits of Criticality-Based Quantum Metrology: Apparent Super-Heisenberg Scaling Revisited

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At the Limits of Criticality-Based Quantum Metrology: Apparent Super-Heisenberg Scaling Revisited

Marek M. Rams, Piotr Sierant, Omyoti Dutta, Paweł Horodecki, and Jakub Zakrzewski

Phys. Rev. X 8, 021022 – Published 19 April 2018

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 $N^{- 1}$ (Heisenberg scaling) in terms of the number of elementary subsystems used $N$ . 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 $N^{- 1.5}$ 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 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)

Cold atoms & matter waves Cold gases in optical lattices Critical phenomena Entanglement in quantum gases Quantum metrology Quantum phase transitions Quantum simulation Resource theories

Quantum Information, Science & TechnologyAtomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Marek M. Rams^1,*, Piotr Sierant^1,†, Omyoti Dutta^1,2, Paweł Horodecki^3,‡, and Jakub Zakrzewski^1,4,§

¹Instytut Fizyki im. Mariana Smoluchowskiego, Uniwersytet Jagielloński, Łojasiewicza 11, 30-348 Kraków, Poland
²Donostia International Physics Center DIPC, Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastián, Spain
³Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Gabriela Narutowicza 11/12, 80-233 Gdańsk, Poland and National Quantum Information Center of Gdańsk, Władysława Andersa 27, 81-824 Sopot, Poland
⁴Mark Kac Complex Systems Research Center, Uniwersytet Jagielloński, Kraków 30-348, Poland

^*marek.rams@uj.edu.pl
^†piotr.sierant@uj.edu.pl
^‡pawel@mif.pg.gda.pl
^§jakub.zakrzewski@uj.edu.pl

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.

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Vol. 8, Iss. 2 — April - June 2018

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Figure 1
$X X Z$ model in the external field, Eq. (21). (a) Scaling of the error propagation formula for operators ${\hat{M}}_{x}$ and ${\hat{m}}_{x}$ , and the ultimate bound given by inverse of QFI, as a function of the system size. The error propagation formula for ${\hat{M}}_{x}$ closely follows the ultimate bound. Dashed lines indicate the slopes corresponding to the expected scaling and serve as guidance for the eye. The fits give $G (λ_{c})^{- 1 / 2} \sim N^{- 1.74}$ , $Δ_{δ_{λ}} ({\hat{M}}_{x}, λ_{c}) \sim N^{- 1.73}$ , and $Δ_{δ_{λ}} ({\hat{m}}_{x}, λ_{c}) \sim N^{- 1.49}$ , where the expected exponents are 1.75, 1.75, and 1.5, respectively. Here, $J_{z} = 0$ and the fits were done for $N = 128 - 256$ [76]. (b) In the considered model the scaling exponents in Eqs. (11), (17), and (18) depend continuously on the value of parameter $J_{z}$ , following Eq. (22). We compare those predictions with numerical results obtained similarly as in (a). The exponent associated with the standard Heisenberg limit is marked with the dashed line for comparison. Its apparent breaking is the main subject of this article.
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Figure 2
Time dependence of QFI at the critical point in the $X X Z$ model. Results from a small external magnetic field change as in Eq. (13). The time is rescaled by the system size according to Eq. (15). QFI (blue symbols) is rescaled corresponding to the adiabatic limit given by Eq. (11). Here, for $J_{z} = 0$ , $z / d = 1$ and $d ν = 4 / 7$ . For evolution times $t ≫ N^{z / d}$ we recover the adiabatic limit. For short times $G^{1 / 2} \sim t N^{1 - [h] / d}$ , which is marked with dashed line. In our case, $1 - [h] / d = 3 / 4$ . We also show the time dependence of the precision allowed by ${\hat{M}}_{x}$ , as described by the error propagation formula (red symbols). For long time, in the adiabatic limit, it is almost optimal and nearly saturates QFI. For short times it is suboptimal. In that case $Δ_{δ_{λ}} ({\hat{M}}_{x}, λ_{c}, t) \sim t^{- 1.5} N^{- 0.25}$ is below the shot-noise limit as a function of $N$ , see Eq. (25), a result of strong fluctuation at the critical point.
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Figure 3
Time dependence of QFI at the critical point in the $X X Z$ model, which results from a small instantaneous shift of the external magnetic field, i.e., the Loschmidt echo. The time is rescaled by the system size according to Eq. (15). The maximal value which can be reached by QFI is bounded by the ground-state fidelity susceptibility (dashed line). We rescale QFI (blue lines) according to Eq. (11). Here, for $J_{z} = 0$ , $z / d = 1$ and $d ν = 4 / 7$ . For short times, $G^{1 / 2} \sim t N^{1 - [h] / d}$ , marked with dashed line, with $1 - [h] / d = 3 / 4$ in our case. For evolution time $t \sim N^{z / d}$ , QFI almost reaches it maximal value and later exhibits periodic revivals characteristic for the Loschmidt echo. We also show the time dependence of the precision allowed by ${\hat{M}}_{x}$ (red lines). For long time it is close to optimal, mimicking the behavior of QFI. For short times it is suboptimal. In that case, $Δ_{δ_{λ}} ({\hat{M}}_{x}, λ_{c}, t) \sim t^{- 1.5} N^{- 0.25}$ is below the shot-noise limit as a function of $N$ ; see Eq. (25).
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Figure 4
The crossover between different scaling limits when $λ$ is not tuned exactly to the critical point. Results correspond to the adiabatic limit of the evolution. For given deviation $λ - λ_{c}$ we recover the apparent super-Heisenberg scaling when $N$ is small enough. When the system size is further increased and $N / ξ^{d} \sim N | λ - λ_{c} |^{d ν} ≫ 1$ , $G (λ)^{- 1 / 2}$ and $Δ_{δ_{λ}} ({\hat{M}}_{x}, λ)$ have a crossover to the classical $N^{- 1 / 2}$ dependence. On the other hand, $Δ_{δ_{λ}} ({\hat{m}}_{x}, λ)$ saturates and becomes xindependent on the system size in that limit. Nevertheless, notice that even in this case the prefactors in front of $N^{- 1 / 2 (0)}$ are enhanced by the vicinity of the critical point. $Δ_{δ_{λ}} ({\hat{M}}_{x}, λ)$ is closely following the optimal bound set by QFI for all values of the parameters. Dashed lines indicate various scalings and serve as guidance for the eye. Results for the $X X Z$ model in the external field in Eq. (21) with $J_{z} = 0$ .
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Figure 5
The crossover between different scaling limits when the temperature $T$ is nonzero. Results obtained by comparing states at equilibrium. For given small $T$ we recover the apparent super-Heisenberg scaling when $N$ is small enough, or alternatively, for $T ≪ N^{- z / d}$ . In the opposite limit, when the system size is further increased and $N T^{d / z} ≫ 1$ , $\tilde{G} (λ_{c}, T)^{- 1 / 2}$ and $Δ_{δ_{λ}} ({\hat{M}}_{x}, λ_{c}, T)$ have a crossover to the classical $N^{- 1 / 2}$ dependence, however, with the prefactors that are enhanced by the presence of a quantum critical point at $T = 0$ . $Δ_{δ_{λ}} ({\hat{m}}_{x}, λ, T)$ saturates and becomes independent on the system size in that limit. Similarly as in Fig. 4, we observe that $Δ_{δ_{λ}} ({\hat{H}}_{1}, λ, T)$ is closely following the optimal bound set by QFI even at nonzero temperatures. $\tilde{G} (λ_{c}, T)^{- 1 / 2}$ plotted here is a lower bound of the (inverse of the square root of) Fisher information, which lies between $\tilde{G} (λ_{c}, T)^{- 1 / 2}$ and $(\tilde{G} (λ_{c}, T) / 2)^{- 1 / 2}$ ; see text for discussion. Solid lines show the corresponding ultimate bound for $T = 0$ from Fig. 1. Dashed lines indicate various scalings and serve as guidance for the eye. Results for $J_{z} = 0$ .
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