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arXiv:2102.07079 · Physical Review A

Quantum metrology with precision reaching beyond 1/N -scaling through N -probe entanglement-generating interactions

Xing Deng, Shou-Long Chen, Zhang Mao, Xiao‐Fan Xu, Jing Liu, Zhi Qiang Gao, Xiao‐Chun Duan, Min-Kang Zhou, Lushuai Cao, Zhong-Kun Hu

Published Jul 19, 2021• 9 citations
arXivHTMLPDF

Abstract

Nonlinear interactions are recognized as potential resources for quantum metrology, facilitating parameter estimation precisions that scale as the exponential Heisenberg limit of ${2}^{\ensuremath{-}N}$. We explore such nonlinearity and propose an associated quantum measurement scenario based on the nonlinear interaction of $N$-probe entanglement-generating form. This scenario provides an enhanced precision scaling of ${D}^{\ensuremath{-}N}/(N\ensuremath{-}1)!$ with $D>2$ a tunable parameter. In addition, it can be readily implemented in a variety of experimental platforms and applied to measurements of a wide range of quantities, including local gravitational acceleration $g$, magnetic field, and its higher-order gradients.

DOI
10.1103/physreva.104.012607

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Nonlinear interactions are recognized as potential resources for quantum metrology, facilitating parameter estimation precisions that scale as the exponential Heisenberg limit of 2\ensuremath−N{2}^{\ensuremath{-}N}2\ensuremath−N. We explore such nonlinearity and propose an associated quantum measurement scenario based on the nonlinear interaction of NNN-probe entanglement-generating form. This scenario provides an enhanced precision scaling of D\ensuremath−N/(N\ensuremath−1)!{D}^{\ensuremath{-}N}/(N\ensuremath{-}1)!D\ensuremath−N/(N\ensuremath−1)! with D>2D>2D>2 a tunable parameter. In addition, it can be readily implemented in a variety of experimental platforms and applied to measurements of a wide range of quantities, including local gravitational acceleration ggg, magnetic field, and its higher-order gradients.
Given a physical setup, what is the precision that can be obtained? In this paper, we firstly review the recently theoretical developments about the quantum parameter estimation. The measurement sensitivity can be improved from the Heisenberg limit (HL) to super-Heisenberg limit (SHL) by changing the coupling Hamiltonian from linear to nonlinear. Then we review some techniques which have been used to break through the standard quantum limit (SQL) and reach the HL in optical quantum metrology, and describe the experiment of nonlinear quantum metrology where the interaction among particles as a valuable resource can contribute to measurement sensitivity and give scaling beyond the HL. Finally, we introduce a potential new approach to the future developments of quantum metrology.
The Heisenberg scaling is an ultimate precision limit of parameter estimation allowed by the principles of quantum mechanics, with no counterpart in the classical realm, and has been a long-pursued goal in quantum metrology. It has been known that interactions between the probes can help reach the Heisenberg scaling without entanglement. In this paper, we show that interactions between the probes and the additional dimensions of an ancillary system may also increase the precision of parameter estimation to surpass the standard quantum limit and attain the Heisenberg scaling without entanglement, if the measurement scheme is properly designed. The quantum Fisher information exhibits periodic patterns over the evolution time, implying the existence of optimal time points for measurements that can maximize the quantum Fisher information. By implementing optimizations over the Hamiltonian, the initial states of the probes and the ancillary system, the interaction strength, and the time points for measurements, our protocol achieves the Heisenberg scaling for the parameter of the probe Hamiltonian, in terms of both evolution time and probe number. Our protocol features two aspects: (i) the Heisenberg scaling can be achieved by a product state of the probes and (ii) mere local measurement on the ancilla is sufficient, both of which reduce the quantum resources and the implementation complexity to achieve the Heisenberg scaling. The paper is concluded by the investigation of the effects of noise on this protocol.
Quantum metrology promises measurement precision beyond the classical limit by using suitably tailored quantum states and detection strategies. However, scaling up this advantage is experimentally challenging, due to the difficulty of generating high-quality large-scale probes. Here, we build a photonic setup that achieves enhanced precision scaling by manipulating the probe's dynamics through operations performed in a coherently controlled order. Our setup applies an unknown rotation and a known orbital angular momentum increase in a coherently controlled order, in a way that reproduces a hybrid quantum SWITCH involving gates generated by both discrete and continuous variables. The unknown rotation angle θθθ is measured with precision scaling as 1/4ml1/4ml1/4ml when a photon undergoes a rotation of 2mθ2mθ2mθ and an angular momentum shift of 2lℏ2l \hbar2lℏ. With a practical enhancement factor as high as 2317, the ultimate precision in our experiment is 0.0105′′0.0105^{\prime \prime}0.0105′′ when using 7.16×1077.16\times10^77.16×107 photons, corresponding to a normalized precision of ≈10−4\approx 10^{-4}≈10−4rad per photon. No photon interaction occurs in our experiment, and the precision enhancement consumes only a linearly increasing amount of physical resources while achieving a nonlinear scaling of the precision. We further indicate that this nonlinear enhancement roots in an in-depth exploration of the Heisenberg uncertainty principle (HUP), and our findings not only deepen the understanding of the HUP but also pave a pathway for advancements in quantum metrology.
Abstract The Heisenberg limit is acknowledged as the ultimate precision limit for parameter estimation in quantum metrology. Recent studies have demonstrated super-Heisenberg-scaling limits, where the root mean square errors of estimation decrease faster-than-linearly with the number of probes or evolution processes. This casts doubt on what the ultimate precision limit of quantum metrology really is and, in particular, whether there is a deeper connection with Heisenberg's uncertainty principle. By linking the dynamics of the metrological process with properties in parameter space, we construct a Heisenberg's uncertainty relation between the unknown parameter and its canonical momentum. We propose a novel quantum metrological scheme that utilizes a generating process with indefinite time direction to increase the canonical momentum nonlinearly. This approach achieves a nonlinear-scaling precision limit that improves quadratically with the increase in evolution time length or the number of processes, in accordance with Heisenberg's uncertainty relation. Then we experimentally demonstrate in quantum optical systems that this nonlinear-scaling enhancement can be achieved with a fixed amount of energy in the probes. Consequently, our results provide a deeper insight into the Heisenberg limit in quantum metrology, and shed new light on improving the precision of practical metrological and sensing tasks in realistic quantum systems.
Quantum metrology studies the use of entanglement and other quantum resources to improve precision measurement. An interferometer using N independent particles to measure a parameter χ can achieve at best the "standard quantum limit" (SQL) of sensitivity δχ ∝ N-1/2. The same interferometer using N entangled particles can achieve in principle the "Heisenberg limit" δχ ∝ N-1, using exotic states, e.g., NooN states. Recent theoretical work argues that interactions among particles may be a valuable resource for quantum metrology, allowing scaling beyond the Heisenberg limit. Specifically, a k-particle interaction will produce sensitivity δχ ∝ N-kwith appropriate entangled states and δχ ∝ N-(k-1/2)even without entanglement. Here we demonstrate this "super-Heisenberg" scaling in a nonlinear, non-destructive measurement of the magnetisation of an atomic ensemble.
In quantum precision metrology, the famous result of Heisenberg limit scaling as 1/N1/N1/N (with NNN the number of probes) can be surpassed by considering nonlinear coupling measurement. In this work, we consider the most practice-relevant quadratic nonlinear coupling and show that the metrological precision can be enhanced from the 1/N321/N^{\frac{3}{2}}1/N23​ super-Heisenberg scaling to 1/N21/N^21/N2, by simply employing a pre- and post-selection (PPS) technique, but not using any expensive quantum resources such as quantum entangled state of probes.