Optimality of the Heisenberg limit for continuous-variable quantum metrology

Heisenberg-limited metrology via weak-value amplification without using entangled resources

Weak-value amplification (WVA) provides a way for amplified detection of a tiny physical signal at the expense of a lower detection probability. Despite this trade-off, due to its robustness against certain types of noise, WVA has advantages over conventional measurements in precision metrology. Moreover, it has been shown that WVA-based metrology can reach the Heisenberg-limit using entangled resources, but preparing macroscopic entangled resources remains challenging. Here we demonstrate a novel WVA scheme based on iterative interactions, achieving the Heisenberg-limited precision scaling wi…

Heisenberg limited metrology using Quantum Error-Correction Codes

Methods borrowed from the world of quantum information processing have lately been used to enhance the signal-to-noise ratio of quantum detectors. Here we analyze the use of stabilizer quantum error-correction codes for the purpose of signal detection. We show that using quantum error-correction codes a small signal can be measured with Heisenberg limited uncertainty even in the presence of noise. We analyze the limitations to the measurement of signals of interest and discuss two simple examples. The possibility of long coherence times, combined with their Heisenberg limited sensitivity to ce…

Quantum Metrology with Two-Mode Squeezed Vacuum: Parity Detection Beats the Heisenberg Limit

We study the sensitivity and resolution of phase measurement in a Mach-Zehnder interferometer with two-mode squeezed vacuum (<n> photons on average). We show that super-resolution and sub-Heisenberg sensitivity is obtained with parity detection. In particular, in our setup, dependence of the signal on the phase evolves <n> times faster than in traditional schemes, and uncertainty in the phase estimation is better than 1/<n>.

Experimental Adiabatic Quantum Metrology with the Heisenberg scaling

The critical quantum metrology, which exploits the quantum phase transition for high precision measurement, has gained increasing attention recently. The critical quantum metrology with the continuous quantum phase transition, however, is experimentally very challenging since the continuous quantum phase transition only exists at the thermal dynamical limit. Here, we propose an adiabatic scheme on a perturbed Ising spin model with the first order quantum phase transition. By employing the Landau-Zener anticrossing, we can not only encode the unknown parameter in the ground state but also tune …

The elusive Heisenberg limit in quantum enhanced metrology

We provide efficient and intuitive tools for deriving bounds on achievable precision in quantum enhanced metrology based on the geometry of quantum channels and semi-definite programming. We show that when decoherence is taken into account, the maximal possible quantum enhancement amounts generically to a constant factor rather than quadratic improvement. We apply these tools to derive bounds for models of decoherence relevant for metrological applications including: dephasing,depolarization, spontaneous emission and photon loss.

Heisenberg-limited metrology with coherent control on the probes' configuration

A central feature of quantum metrology is the possibility of Heisenberg scaling, a quadratic improvement over the limits of classical statistics. This scaling, however, is notoriously fragile to noise. While for some noise types it can be restored through error correction, for other important types, such as dephasing, the Heisenberg scaling appears to be irremediably lost. Here we show that this limitation can sometimes be lifted if the experimenter has the ability to probe physical processes in a coherent superposition of alternative configurations. As a concrete example, we consider the prob…

Heisenberg-limited quantum metrology using collective dephasing

The goal of quantum metrology is the precise estimation of parameters using quantum properties such as entanglement. This estimation usually consists of three steps: state preparation, time evolution during which information of the parameters is encoded in the state, and readout of the state. Decoherence during the time evolution typically degrades the performance of quantum metrology and is considered to be one of the major obstacles to realizing entanglement-enhanced sensing. We show, however, that under suitable conditions, this decoherence can be exploited to improve the sensitivity. Assum…

Ultimate limits to quantum metrology and the meaning of the Heisenberg limit

For the last 20 years, the question of what are the fundamental capabilities of quantum precision measurements has sparked a lively debate throughout the scientific community. Typically, the ultimate limits in quantum metrology are associated with the notion of the Heisenberg limit expressed in terms of the physical resources used in the measurement procedure. Over the years, a variety of different physical resources were introduced, leading to a confusion about the meaning of the Heisenberg limit. Here, we review the mainstream definitions of the relevant resources and introduce the universal…

Quantum metrology: Heisenberg limit with bound entanglement

Quantum metrology allows for a huge boost in the precision of parameters estimation. However, it seems to be extremely sensitive on the noise. Bound entangled states are states with large amount of noise what makes them unusable for almost all quantum informational tasks. Here we provide a counterintuitive example of a family of bound entangled states which can be used in quantum enhanced metrology. We show that these states give advantage as big as maximally entangled states and asymptotically reach the Heisenberg limit. Moreover, entanglement of the applied states is very weak which is refle…

纠缠增强的量子计量：从标准量子极限到海森堡极限

Entanglement-enhanced quantum metrology: from standard quantum limit to Heisenberg limit

本文介绍纠缠增强量子计量的原理与实验进展，说明利用多粒子纠缠可突破标准量子极限、逼近海森堡极限，还探讨了相关量子传感器的潜在应用。

Non-adaptive Heisenberg-limited metrology with multi-channel homodyne measurements

We show a protocol achieving the ultimate Heisenberg-scaling sensitivity in the estimation of a parameter encoded in a generic linear network, without employing any auxiliary networks, and without the need of any prior information on the parameter nor on the network structure. As a result, this protocol does not require a prior coarse estimation of the parameter, nor an adaptation of the network. The scheme we analyse consists of a single-mode squeezed state and homodyne detectors in each of the $M$ output channels of the network encoding the parameter, making it feasible for experimental appl…

Quantum metrology beyond Heisenberg limit with entangled matter wave solitons

By considering matter wave bright solitons from weakly coupled Bose-Einstein condensates trapped in a double-well potential, we study the formation of macroscopic non-classical states, including Schrödinger-cat superposition states and maximally path entangled $N00N$-states. With these macroscopic states, we examine Mach-Zehnder interferometer in the context of parity measurements, in order to obtain Heisenberg limit accuracy for linear phase shift measurement. We reveal that the ratio between two-body scattering length and intra-well hopping parameter can be measured with the scaling beyond t…

Achieving Heisenberg-limited metrology with spin cat states via interaction-based readout

Spin cat states are promising candidates for quantum-enhanced measurement. Here, we analytically show that the ultimate measurement precision of spin cat states approaches the Heisenberg limit, where the uncertainty is inversely proportional to the total particle number. In order to fully exploit their metrological ability, we propose to use the interaction-based readout for implementing phase estimation. It is demonstrated that the interaction-based readout enables spin cat states to saturate their ultimate precision bounds. The interaction-based readout comprises a one-axis twisting, two $\f…

Experimental super-Heisenberg quantum metrology with indefinite gate order

The precision of quantum metrology is widely believed to be restricted by the Heisenberg limit, corresponding to a root mean square error that is inversely proportional to the number of independent processes probed in an experiment, N. In the past, some proposals have challenged this belief, for example using non-linear interactions among the probes. However, these proposals turned out to still obey the Heisenberg limit with respect to other relevant resources, such as the total energy of the probes. Here, we present a photonic implementation of a quantum metrology protocol surpassing the Heis…

Information Theoretical Analysis of Quantum Metrology

We address the framework of analysing quantum metrology in the information-theoretic picture. Firstly we show how to extract the maximum amount of information in general via suitable state initialization of the probes at the beginning and a quantum measurement at the end. Our analysis can apply to both the single-parameter and the multi-parameter estimation procedures as well as to any other quantum information processing procedures. We then establish a direct connection between the information-theoretic picture of quantum metrology and its conventional variance-covariance picture, by showing …

Achieving the Heisenberg limit in quantum metrology using quantum error correction

Quantum metrology has many important applications in science and technology, ranging from frequency spectroscopy to gravitational wave detection. Quantum mechanics imposes a fundamental limit on measurement precision, called the Heisenberg limit, which can be achieved for noiseless quantum systems, but is not achievable in general for systems subject to noise. Here we study how measurement precision can be enhanced through quantum error correction, a general method for protecting a quantum system from the damaging effects of noise. We find a necessary and sufficient condition for achieving the…

无辅助量子比特的海森堡极限量子计量

Heisenberg-Limited Quantum Metrology without Ancillae

该研究针对实际场景中无无噪声辅助量子比特的量子计量性能不明的问题，提出衔接量子计量与量子信道渐近理论的新框架，证明了重复应用待估信道时有无交错幺正控制下达到海森堡极限的充分条件，并设计了显式识别控制操作的算法。

迈向量子机制下非厄米计量学中的海森堡标度

Toward Heisenberg Scaling in Non-Hermitian Metrology at the Quantum Regime

研究量子机制下的非厄米参数估计，提出适用于一般非厄米哈密顿量的量子Fisher信息表达式，推导最优测量条件并实验验证了估计精度中海森堡标度的实现。

在任意双通道线性干涉仪中利用 squeezed 光实现 Heisenberg 缩放的多参数量子计量

Multiparameter quantum metrology at Heisenberg scaling for an arbitrary two-channel linear interferometer with squeezed light

利用 squeezed 光在任意双通道线性干涉仪中实现 Heisenberg 缩放的多参数量子计量，为高精度光学干涉计量提供实验可行的设计原则。

迪克超位置探针用于抗噪的海森堡和超海森堡计量

Dicke superposition probes for noise-resilient Heisenberg and super-Heisenberg Metrology

本文研究了在噪声环境下使用迪克态超位置探针进行量子相位传感，探讨了一种在单体和双体相互作用哈密顿量下生成参数编码的迪克态超位置状态，展示了其在相位阻尼噪声下的增强鲁棒性及良好的计量性能。

通过工程化 dressed 状态保护 Heisenberg 缩放在量子计量学中

Protecting Heisenberg scaling in quantum metrology via engineered dressed states

本文提出利用静态场生成的 dressed 状态来抑制噪声并保持信号灵敏度，证明在低温噪声下，若信号生成器不在系统-环境耦合算符线性空间中，可实现 Heisenberg 缩放。

通过不确定演化实现同时海森堡极限多参数计量学

Simultaneous Heisenberg-Limited Multiparameter Metrology via Indefinite Evolution

该研究提出基于不确定演化的多参数计量学框架，解决多参数计量中参数编码与测量不相容问题，实现同时海森堡极限精度，为干涉传感开辟新路径。

微秒可编程合成量子物质：绕过绝热瓶颈实现海森堡极限计量

Microsecond-Programmable Synthetic Quantum Matter: Circumventing the Adiabatic Bottleneck for Heisenberg-Limited Metrology

提出非绝热Floquet协议，在单腔BEC中动态合成TAT相互作用，快速产生海森堡极限自旋压缩，绕过绝热瓶颈，实现微秒级计量。

Quantum metrology in the presence of spatially correlated noise: Restoring Heisenberg scaling

Environmental noise can hinder the metrological capabilities of entangled states. While the use of entanglement allows for Heisenberg-limited resolution, the largest permitted by quantum mechanics, deviations from strictly unitary dynamics quickly restore the standard scaling dictated by the central limit theorem. Product and maximally entangled states become asymptotically equivalent when the noisy evolution is both local and strictly Markovian. However, temporal correlations in the noise have been shown to lift this equivalence while fully (spatially) correlated noise allows for the identifi…

Limited-control metrology approaching the Heisenberg limit without entanglement preparation

Current metrological bounds typically assume full control over all particles that are involved in the protocol. Relaxing this assumption we study metrological performance when only limited control is available. As an example, we measure a static magnetic field when a fully controlled quantum sensor is supplemented by particles over which only global control is possible. We show that even for a noisy quantum sensor, a protocol that maps the magnetic field to a precession frequency can achieve transient super-Heisenberg scaling in measurement time and Heisenberg scaling in the particle number. T…

Quantum correlations in optical metrology: Heisenberg-limited phase estimation without mode entanglement

The quantum fisher information and quantum correlation parameters are employed to study the application of non-classical light to the problem of parameter estimation. It is shown that the optimal measurement sensitivity of a quantum state is determined by its inter-mode correlations (which depends of path-entanglement) and intra-mode correlations (which depends of the photon statistics). In light of these results, we consider the performance of quantum-enhanced optical interferometers. Furthermore, we propose a Heisenberg-limited metrology protocol involving standard elements from passive and …

固有退相干下各向异性海森堡自旋相互作用的受控量子计量学

Controlled Quantum Metrology with Anisotropic Heisenberg Spin Interactions under Intrinsic Decoherence

研究Milburn模型固有退相干下两量子比特各向异性海森堡自旋系统（含DM相互作用）的量子参数估计，利用量子Fisher信息分析磁场和DM相互作用强度的估计精度，发现各向异性交换耦合和初始纠缠态可显著提升性能。

超海森堡量子计量学中灵敏度标度的噪声约束

Noise constraints on sensitivity scaling in super-Heisenberg quantum metrology

研究超海森堡量子计量学中非线性增强在噪声下的限制，发现非线性相互作用会引发并放大涌现误差，导致精度增益消失，并确定了避免该误差的参数临界值。

量子比特辅助量子计量接近海森堡标度的判据

Criterion for qubit-assisted quantum metrology approaching Heisenberg scaling

研究探针系统与辅助量子比特耦合时的计量精度，发现限制探针-量子比特耦合方向是达到海森堡极限的充分判据，并在玻色子和自旋系综系统中验证了量子Fisher信息的海森堡标度行为。

编码量子信号处理用于海森堡极限计量

Encoded Quantum Signal Processing for Heisenberg-Limited Metrology

本文提出编码量子信号处理框架，通过单量子比特模型实现抗噪且灵敏的逻辑传感器，利用综合征测量恢复海森堡极限精度，通过纠缠或信号放大克服标准量子限。

Typicality of Heisenberg scaling precision in multi-mode quantum metrology

We propose a measurement setup reaching Heisenberg scaling precision for the estimation of any distributed parameter $φ$ (not necessarily a phase) encoded into a generic $M$-port linear network composed only of passive elements. The scheme proposed can be easily implemented from an experimental point of view since it employs only Gaussian states and Gaussian measurements. Due to the complete generality of the estimation problem considered, it was predicted that one would need to carry out an adaptive procedure which involves both the input states employed and the measurement performed at the o…

Multiparameter quantum metrology in the Heisenberg Limit regime: many repetition scenario vs. full optimization

We discuss the Heisenberg limit in the multiparameter metrology within two different paradigms -- the one, where the measurement is repeated many times (so the Cramér-Rao bound is guaranteed to be asymptotically saturable) and the second one, where all the resources are allocated into one experimental realization (analyzed with the mimimax approach). We investigate the potential advantage of measuring all the parameter simultaneously compared to estimating them individually, while spending the same total amount of resources. We show that in general the existence of such an advantage, its magni…

从各向异性铁磁体的量子淬火动力学实现海森堡极限计量

Heisenberg-limited metrology from the quantum-quench dynamics of an anisotropic ferromagnet

通过各向异性铁磁体的量子淬火动力学实现海森堡极限参数估计，利用磁极子压缩的量子相关性。

通过量子计量中探针-辅助系统相互作用实现海森堡标度

Achieving Heisenberg scaling by probe-ancilla interaction in quantum metrology

针对量子计量中实现海森堡标度的目标，提出通过优化探针-辅助系统相互作用、初始态等参数，在无纠缠且仅局域测量辅助系统的情况下，实现探针哈密顿量参数的海森堡标度，减少资源与复杂度。

非渐近海森堡标度：适用于宽资源范围的实验计量学

Non-asymptotic Heisenberg scaling: experimental metrology for a wide resources range

本文提出了一种无需先验信息即可实现海森堡标度的量子参数估计方法，并通过测量旋转角实验验证了在宽范围内资源的优越性能。

Enhanced nonlinear quantum metrology with weakly coupled solitons and particle losses

The estimation of physical parameters with Heisenberg sensitivity and beyond is one of the crucial problems for current quantum metrology. However, unavoidable lossy effect is commonly believed to be the main obstacle when applying fragile quantum states. To utilize the lossy quantum metrology, we offer an interferometric procedure for phase parameters estimation at the Heisenberg (up to 1/N) and super-Heisenberg (up to 1/N^3) scaling levels in the framework of the linear and nonlinear metrology approaches, respectively. The heart of our setup is the novel soliton Josephson Junction (SJJ) syst…

在资源受限的光学计量中多参量海森堡标度的几何界限

Geometric bounds on multiparameter Heisenberg scaling in optical metrology with limited squeezed resources

研究多参量同时估计中，在多模高斯网络里给定压缩探测器时，有多少参数组合能继承海森堡标度。通过分析量子费希尔信息矩阵，得出海森堡标度系数秩的界限，确定饱和条件并构建干涉仪族。

基于群体编码的海森堡标度

Heisenberg scaling based on population coding

针对Fisher信息无法反映全局结构的问题，提出以互信息作为群体编码视角下的替代品质因数，并证明若干酉模型可实现海森堡标度。

Heisenberg limited single-mode quantum metrology

Two-mode interferometers, such as Michelson interferometer based on two spatial optical modes, lay the foundations for quantum metrology. Instead of exploring quantum entanglement in the two-mode interferometers, a single bosonic mode also promises a measurement precision beyond the shot-noise limit (SNL) by taking advantage of the infinite-dimensional Hilbert space of Fock states. However, the experimental demonstration still remains elusive. Here, we demonstrate a single-mode phase estimation that approaches the Heisenberg limit (HL) unconditionally. Due to the strong dispersive nonlinearity…

Quantum metrology enhanced by coherence-induced-driving in a cavity QED setup

We propose a quantum metrology scheme in a cavity QED setup to achieve the Heisenberg limit. In our scheme, a series of identical two-level atoms randomly pass through and interact with a dissipative single-mode cavity. Different from the entanglement based Heisenberg limit metrology scheme, we do not need to prepare the atomic entangled states before they enter into the cavity. We show that the initial atomic coherence will induce an effective driving to the cavity field, whose steady state is an incoherent superposition of orthogonal states, with the superposition probabilities being depende…

Restoring Heisenberg scaling in noisy quantum metrology by monitoring the environment

We study quantum frequency estimation for $N$ qubits subjected to independent Markovian noise, via strategies based on time-continuous monitoring of the environment. Both physical intuition and an extended convexity property of the quantum Fisher information (QFI) suggest that these strategies are more effective than the standard ones based on the measurement of the unconditional state after the noisy evolution. Here we focus on initial GHZ states and on parallel or transverse noise. For parallel noise, i.e. dephasing, we show that perfectly efficient time-continuous photo-detection allows to …

具有任意权重的海森堡极限连续变量分布式量子计量

Heisenberg-limited continuous-variable distributed quantum metrology with arbitrary weights

本文阐明双非真空输入线性连续变量DQM网络结构，给出任意权重线性组合及全局函数估计的严格灵敏度界、所需非经典态条件、最优探测方案与两种增强机制。

Practical Limits of Error Correction for Quantum Metrology

Noise is the greatest obstacle in quantum metrology that limits it achievable precision and sensitivity. There are many techniques to mitigate the effect of noise, but this can never be done completely. One commonly proposed technique is to repeatedly apply quantum error correction. Unfortunately, the required repetition frequency needed to recover the Heisenberg limit is unachievable with the existing quantum technologies. In this article we explore the discrete application of quantum error correction with current technological limitations in mind. We establish that quantum error correction c…

在含噪量子计量中利用 Dicke 态实现 Heisenberg 极限

Achieving the Heisenberg limit with Dicke states in noisy quantum metrology

针对开放量子系统中的含噪参数估计，本文分析对称耦合量子比特—谐振器模型的含时量子 Fisher 信息，证明最优激发数 Dicke 态可在有限衰减下达到 Heisenberg 极限，并优于 GHZ 态。

Heisenberg scaling precision in multi-mode distributed quantum metrology

We propose an $N$-photon Gaussian measurement scheme which allows the estimation of a parameter $φ$ encoded into a multi-port interferometer with a Heisenberg scaling precision (i.e. of order $1/N$). In this protocol, no restrictions on the structure of the interferometer are imposed other than linearity and passivity, allowing the parameter $φ$ to be distributed over several components. In all previous proposals Heisenberg scaling has been obtained provided that both the input state and the measurement at the output are suitably adapted to the unknown parameter $φ$. This is a serious drawback…

Compressed quantum metrology for the Ising Hamiltonian

We show how quantum metrology protocols that seek to estimate the parameters of a Hamiltonian that exhibits a quantum phase transition can be efficiently simulated on an exponentially smaller quantum computer. Specifically, by exploiting the fact that the ground state of such a Hamiltonian changes drastically around its phase transition point, we construct a suitable observable from which one can estimate the relevant parameters of the Hamiltonian with Heisenberg scaling precision. We then show how, for the one-dimensional Ising Hamiltonian with transverse magnetic field acting on N spins, suc…

Beyond Heisenberg Limit Quantum Metrology through Quantum Signal Processing

Leveraging quantum effects in metrology such as entanglement and coherence allows one to measure parameters with enhanced sensitivity. However, time-dependent noise can disrupt such Heisenberg-limited amplification. We propose a quantum-metrology method based on the quantum-signal-processing framework to overcome these realistic noise-induced limitations in practical quantum metrology. Our algorithm separates the gate parameter $φ$~(single-qubit Z phase) that is susceptible to time-dependent error from the target gate parameter $θ$~(swap-angle between |10> and |01> states) that is largely free…

通过辅助量子比特上的测量实现计量学的海森堡极限

Achieving the Heisenberg limit of metrology via measurement on an ancillary qubit

该研究提出一种通过辅助量子比特测量实现海森堡极限标度的量子计量协议，利用平行演化路径将自旋系综制备为叠加态，并证明热态也能达到二次标度，无需GHZ态或非线性哈密顿量。

Interaction-based quantum metrology showing scaling beyond the Heisenberg limit

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 X can achieve at best the "standard quantum limit" (SQL) of sensitivity δX \propto N^{-1/2}. The same interferometer using N entangled particles can achieve in principle the "Heisenberg limit" δX \propto N^{-1}, using exotic 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-parti…

Distributed Quantum Metrology and the Entangling Power of Linear Networks

We derive a bound on the ability of a linear optical network to estimate a linear combination of independent phase shifts by using an arbitrary non-classical but unentangled input state, thereby elucidating the quantum resources required to obtain the Heisenberg limit with a multi-port interferometer. Our bound reveals that while linear networks can generate highly entangled states, they cannot effectively combine quantum resources that are well distributed across multiple modes for the purposes of metrology: in this sense linear networks endowed with well-distributed quantum resources behave …

蝴蝶回声协议用于轴无关的海森堡极限计量

Butterfly Echo Protocol for Axis-Agnostic Heisenberg-Limited Metrology

提出一种基于混沌动力学的单次回声协议，利用随机对称探针态估计未知轴的小旋转，实现海森堡标度，并分析退相干影响。

Squeezing metrology: a unified framework

Quantum metrology theory has up to now focused on the resolution gains obtainable thanks to the entanglement among N probes. Typically, a quadratic gain in resolution is achievable, going from the 1/sqrt(N) of the central limit theorem to the 1/N of the Heisenberg bound. Here we focus instead on quantum squeezing and provide a unified framework for metrology with squeezing, showing that, similarly, one can generally attain a quadratic gain when comparing the resolution achievable by a squeezed probe to the best N-probe classical strategy achievable with the same energy. Namely, here we give a …

Heisenberg scaling in Gaussian quantum metrology

We address the issue of precisely estimating small parameters encoded in a general linear transformation of the modes of a bosonic quantum field. Such Bogoliubov transformations frequently appear in the context of quantum optics. We provide a set of instructions for computing the quantum Fisher information for arbitrary pure initial states. We show that the maximally achievable precision of estimation is inversely proportional to the squared average particle number and that such Heisenberg scaling requires non-classical, but not necessarily entangled states. Our method further allows us to qua…

Quantum Metrology with Indefinite Causal Order

We address the study of quantum metrology enhanced by indefinite causal order, demonstrating a quadratic advantage in the estimation of the product of two average displacements in a continuous variable system. We prove that no setup where the displacements are probed in a fixed order can have root-mean-square error vanishing faster than the Heisenberg limit 1/N, where N is the number of displacements contributing to the average. In stark contrast, we show that a setup that probes the displacements in a superposition of two alternative orders yields a root-mean-square error vanishing with super…