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arXiv:2506.20632 · Physical Review Applied

Nonlinear enhancement of measurement precision via a hybrid quantum switch

Anonymous, Yuxiang Yang, Gongchu Li, Xu-Song Hong, Si-Qi Zhang, H. Xu, Yuancheng Liu, Giulio Chiribella, Zhibo Hou, Geng Chen, Chuan-Feng Li, Guang‐Can Guo

Published Jun 29, 2026• 1 citation
arXivHTMLPDF

Abstract

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 $θ$ is measured with precision scaling as $1/4ml$ when a photon undergoes a rotation of $2mθ$ and an angular momentum shift of $2l \hbar$. With a practical enhancement factor as high as 2317, the ultimate precision in our experiment is $0.0105^{\prime \prime}$ when using $7.16\times10^7$ photons, corresponding to a normalized precision of $\approx 10^{-4}$rad 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.

DOI
10.1103/d5bb-5vmv

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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 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 θ is measured with precision scaling as 1/4ml when a photon undergoes a rotation of 2mθ and an angular momentum shift of 2lℏ. With a practical enhancement factor as high as 2317, the ultimate precision in our experiment is 0.0105′′ when using 7.16 × 107 photons, corresponding to a normalized precision of ≈ 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.
The precision limit in quantum metrology, quantified by the root-mean-square error of parameter estimation, is conventionally expected to improve at most linearly with the total interrogation time T and with the number N of queried quantum gates. Although several metrological schemes have been shown to achieve precision scaling faster than linear in T and N, they typically rely on unbounded probe-side information resources, usually qualified by an increasingly large variance of the parameter generator. This requirement complicates the interpretation of the resulting scaling advantage and poses substantial technical challenges. In this work, we employ an indefinite-time-direction encoding process to achieve a nonlinear-scaling enhancement of the precision limit. Rather than relying on increasingly informative probe states, our method converts controllable noncommuting encoding operations into metrological gain. Experimentally, we implement this protocol for angular-rotation measurement in a quantum optical system and demonstrate a nonlinear-scaling improvement in practical precision without using probe-side information resources. These results establish a practical framework for surpassing conventional linear-scaling precision limits in quantum metrology and provide new insights into precision enhancement in realistic quantum metrological and sensing applications.
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 employs quantum resources to achieve measurement precision beyond classical limits. This work investigates a Mach--Zehnder interferometer incorporating a Kerr nonlinear phase shifter, with photon-added two-mode squeezed coherent states generated via four-wave mixing as input. We demonstrate that increasing both the photon-addition order and the input resource strength systematically enhances phase sensitivity, quantum Fisher information, and the corresponding quantum Cramér--Rao bound. The proposed system not only surpasses the standard quantum limit but also approaches or exceeds the Heisenberg limit for linear phase shifts, while Kerr nonlinearity enables surpassing the super-Heisenberg limit. Furthermore, the scheme exhibits enhanced robustness against photon loss, providing a promising pathway toward practical high-precision quantum metrology applications.
Abstract This study demonstrates that quantum switch can be applied to an entangled system comprising a control qubit and a probe state for parameter estimation, providing additional resources and degrees of freedom to enhance precision. Using a photon probe state as an example, we show that while this approach does not surpass the Heisenberg scaling, it can exceed the Heisenberg bound compared to a definite causal order scheme. For the quantum harmonic oscillator state, when the product of its mass and angular frequency mω is extremely large or extremely small, a suitable encoding method can always be selected to improve the precision. Finally, we demonstrate that if certain operators with specific commutation relations can be realized, the measurement precision has the potential to be further improved.
Abstract The Heisenberg limit, corresponding to a root mean square error vanishing as 1/N with the number N of independent processes probed in an experiment, is widely believed to be an ultimate limit to the precision of quantum metrology.In this work, we experimentally demonstrate a quantum metrology protocol surpassing Heisenberg limit by implementing indefinite (a superposition of) orders of two groups of independent processes. Each process creates a phase space displacement, and the precision to estimate the geometric phase introduced by a total number of 2N processes approaches the super-Heisenberg limit of 1/N2. In our setup, the polarization of a single photon coherently controls the order of displacements on the transverse modes of the radiation field, resulting into indefinite order and allowing us to outperform every setup where the displacements are probed in a definite order. Our experiment features a realization of coherent control over the order in a continuous-variable system, and can be applied to the measurement of various important parameters.