We derive fundamental bounds on the maximal achievable precision in multiparameter noisy quantum metrology, valid under the most general entanglement-assisted adaptive strategy, which are tighter than the bounds obtained by a direct use of single-parameter results. This allows us to study the issue of the optimal probe incompatibility in the simultaneous estimation of multiple parameters in generic noisy channels, while so far the issue has been studied mostly in effectively noiseless scenarios (where the Heisenberg scaling is possible). We apply our results to the estimation of both unitary and noise parameters, and indicate models where the fundamental probe incompatibility is present. In particular, we show that in lossy multiple arm interferometry the probe incompatibility is as strong as in the noiseless scenario. Finally, going beyond the multiple-parameter estimation paradigm, we introduce the concept of \emph{random quantum sensing} and show how the tools developed may be applied to multiple channel discrimination problems. As an illustration, we provide a simple proof of the loss of the quadratic advantage of time-continuous Grover algorithm in presence of dephasing or erasure noise.
Measurement incompatibility in Bayesian multiparameter quantum estimation
Università di Parma（帕尔马大学）；INFN—Sezione di Milano-Bicocca（意大利国家核物理研究所米兰-比可卡分部）；Università di Firenze（佛罗伦萨大学）；University of Surrey（萨里大学）；University of Exeter（埃克塞特大学）
We identify precision limits for the simultaneous estimation of multiple parameters in multimode interferometers. Quantum strategies to enhance the multiparameter sensitivity are based on entanglement among particles, modes or combining both. The maximum attainable sensitivity of particle-separable states defines the multiparameter shot-noise limit, which can be surpassed without mode entanglement. Further enhancements up to the multiparameter Heisenberg limit are possible by adding mode entanglement. Optimal strategies which saturate the precision bounds are provided.
Interacting quantum systems are attracting increasing interest for developing precise metrology. In particular, the realisation that quantum-correlated states and the dynamics of interacting systems can lead to entirely new and unexpected phenomena have initiated an intense research effort to explore interaction-based metrology both theoretically and experimentally. However, the current framework of interaction-based metrology mainly focuses on single-parameter estimations, a demonstration of multiparameter metrology using interactions as a resource was heretofore lacking. Here we demonstrate …
The quantum Cramér-Rao bound is a cornerstone of modern quantum metrology, as it provides the ultimate precision in parameter estimation. In the multiparameter scenario, this bound becomes a matrix inequality, which can be cast to a scalar form with a properly chosen weight matrix. Multiparameter estimation thus elicits tradeoffs in the precision with which each parameter can be estimated. We show that, if the information is encoded in a unitary transformation, we can naturally choose the weight matrix as the metric tensor linked to the geometry of the underlying algebra $\mathfrak{su}(n)$, wi…
The interest in a system often resides in the interplay among different parameters governing its evolution. It is thus often required to access many of them at once for a complete description. Assessing how quantum enhancement in such multiparameter estimation can be achieved depends on understanding the many subtleties that come into play: establishing solid foundations is key to delivering future technology for this task. In this article we discuss the state of the art of quantum multiparameter estimation, with a particular emphasis on its theoretical tools, on application to imaging problem…
Control incompatibility in multiparameter quantum metrology
Pritzker School of Molecular Engineering, The University of Chicago（芝加哥大学普里茨克分子工程学院）；MIT-Harvard Center for Ultracold Atoms and Research Laboratory of Electronics, Massachusetts Institute of Technology（麻省理工学院哈佛-麻省中心超冷原子与电子学研究室）；Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong（香港中文大学机械与自动化工程系）；Department of Nuclear Science and Engineering, Massachusetts Institute of Technology（麻省理工学院核科学与工程系）；School of Physics, Xi’an Jiaotong University（西安交通大学物理学院）；International Center for Quantum Materials, School of Physics, Peking University（北京大学物理学院量子材料国际中心）
We propose a generalized form of entangled coherent states (ECS) and apply them in a multi-arm optical interferometer to estimate multiple phase shifts. We obtain the quantum Cramer-Rao bounds for both the linear and nonlinear parameterization protocols. Through the analysis, we find that, utilizing the simultaneous estimation, this generalized form of ECS gives a better precision than the generalized NOON states [Phys. Rev. Lett. \textbf{111}, 070403 (2013)]. Moreover, comparing with the independent estimation, both the linear and nonlinear protocols have the same advantage in the relation to…
In multiparameter quantum metrology, the weighted-arithmetic-mean error of estimation is often used as a scalar cost function to be minimized during design optimization. However, other types of mean error can reveal different facets of permissible error combination. By introducing the weighted $f$-mean of estimation error and quantum Fisher information, we derive various quantum Cramér-Rao bounds on mean error in a very general form and also give their refined versions with complex quantum Fisher information matrices. We show that the geometric- and harmonic-mean quantum Cramér-Rao bounds can …
Single parameter estimation is known to benefit from extreme sensitivity to parameter changes in quantum critical systems. However, the simultaneous estimation of multiple parameters is generally limited due to the incompatibility arising from the quantum nature of the underlying system. A key question is whether quantum criticality may also play a positive role in reducing the incompatibility in the simultaneous estimation of multiple parameters. We argue that this is generally the case and verify this prediction in paradigmatic quantum many-body systems close to first and second order phase …
We establish the multiparameter quantum Cramér-Rao bound for simultaneously estimating the centroid, the separation, and the relative intensities of two incoherent optical point sources using alinear imaging system. For equally bright sources, the Cramér-Rao bound is independent of the source separation, which confirms that the Rayleigh resolution limit is just an artifact of the conventional direct imaging and can be overcome with an adequate strategy. For the general case of unequally bright sources, the amount of information one can gain about the separation falls to zero, but we show that …
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…
We address the use of asymptotic incompatibility (AI) to assess the quantumness of a multiparameter quantum statistical model. AI is a recently introduced measure which quantifies the difference between the Holevo and the SLD scalar bounds, and can be evaluated using only the symmetric logarithmic derivative (SLD) operators of the model. At first, we evaluate analytically the AI of the most general quantum statistical models involving two-level (qubit) and single-mode Gaussian continuous-variable quantum systems, and prove that AI is a simple monotonous function of the state purity. Then, we n…
Simultaneous estimation of multiple parameters in quantum metrological models is complicated by factors relating to the (i) existence of a single probe state allowing for optimal sensitivity for all parameters of interest, (ii) existence of a single measurement optimally extracting information from the probe state on all the parameters, and (iii) statistical independence of the estimated parameters. We consider the situation when these concerns present no obstacle and for every estimated parameter the variance obtained in the multiparameter scheme is equal to that of an optimal scheme for that…
Hierarchy of saturation conditions for multiparameter quantum metrology bounds
Institute of Systems and Information Engineering, University of Tsukuba（筑波大学系统信息工程学院）；Center for Artificial Intelligence Research (C-AIR), University of Tsukuba（筑波大学人工智能研究中心）；Institute of Fundamental and Transdisciplinary Research, Institute of Quantum Sensing, and Institute for Advanced Study in Physics, Zhejiang University（浙江大学基础与跨学科研究院、量子传感研究所、物理高等研究院）；Nordita, KTH Royal Institute of Technology and Stockholm University（诺迪塔，皇家理工学院和斯德哥尔摩大学）；Istituto Nazionale di Ottica del Consiglio Nazionale delle Ricerche (CNR-INO)（意大利国家光学研究所（CNR-INO））；European Laboratory for Nonlinear Spectroscopy (LENS)（欧洲非线性光谱学实验室）
Quantum Fisher information matrix (QFIM) is a core concept in theoretical quantum metrology due to the significant importance of quantum Cramér-Rao bound in quantum parameter estimation. However, studies in recent years have revealed wide connections between QFIM and other aspects of quantum mechanics, including quantum thermodynamics, quantum phase transition, entanglement witness, quantum speed limit and non-Markovianity. These connections indicate that QFIM is more than a concept in quantum metrology, but rather a fundamental quantity in quantum mechanics. In this paper, we summarize the pr…
Quantum metrology aims to exploit quantum phenomena to overcome classical limitations in the estimation of relevant parameters. We consider a probe undergoing a phase shift $φ$ whose generator is randomly sampled according to a distribution with unknown concentration $κ$, which introduces a physical source of noise. We then investigate strategies for the joint estimation of the two parameters $φ$ and $κ$ given a finite number $N$ of interactions with the phase imprinting channel. We consider both single qubit and multipartite entangled probes, and identify regions of the parameters where simul…
Quantum control can be employed in quantum metrology to improve the precision limit for the estimation of unknown parameters. The optimal control, however, typically depends on the actual values of the parameters and thus needs to be designed adaptively with the updated estimations of those parameters. Traditional methods, such as gradient ascent pulse engineering (GRAPE), need to be rerun for each new set of parameters encountered, making the optimization costly, especially when many parameters are involved. Here we study the generalizability of optimal control, namely, optimal controls that …
Weight-dependent and weight-independent measures of quantum incompatibility in multiparameter estimation
QTF Centre of Excellence, Department of Physics, University of Helsinki（赫尔辛基大学）；Dipartimento di Fisica Aldo Pontremoli, Università degli Studi di Milano（米兰大学）
Minimal Trade-off and Optimal Measurement for Multiparameter Quantum Estimation
The Chinese University of Hong Kong（香港中文大学）；The Hong Kong Institute of Quantum Information Science and Technology（香港量子信息科学与技术研究所）；State Key Laboratory of Quantum Information Technologies and Materials（量子信息技术与材料国家重点实验室）；Shenzhen University（深圳大学）
Multiparameter quantum estimation is made difficult by the following three obstacles. First, incompatibility among different physical quantities poses a limit on the attainable precision. Second, the ultimate precision is not saturated until you discover the optimal measurement. Third, the optimal measurement may generally depend on the target values of parameters, and thus may be impossible to perform for unknown target states. We present a method to circumvent these three obstacles. A class of quantum statistical models, which utilizes antiunitary symmetries or, equivalently, real density ma…
Geometric bounds on multiparameter Heisenberg scaling in optical metrology with limited squeezed resources
University of Portsmouth（朴茨茅斯大学）；Quantum Science and Technology Hub, University of Portsmouth（朴茨茅斯大学量子科学与技术中心）；Università di Bari（巴里大学）；INFN, Sezione di Bari（意大利国家核物理研究所巴里分部）；Institute of Cosmology and Gravitation, University of Portsmouth（朴茨茅斯大学宇宙学与引力研究所）
Dynamical, thermal, and ground-state multiparameter quantum metrology: Fundamental limits and their attainability in magnetometry
Universitat Autònoma de Barcelona（巴塞罗那自治大学）；Università di Bologna（博洛尼亚大学）；BITS Pilani KK Birla Goa Campus（比拉印度理工学院果阿校区）；Harish-Chandra Research Institute（哈里什-钱德拉研究所）；Institute for Quantum Optics and Quantum Information - IQOQI Vienna, Austrian Academy of Sciences（奥地利科学院维也纳量子光学与量子信息研究所）
We investigate the ultimate precision achievable in Gaussian quantum metrology. We derive general analytical expressions for the quantum Fisher information matrix and for the measurement compatibility condition, ensuring asymptotic saturability of the quantum Cramér-Rao bound, for the estimation of multiple parameters encoded in multimode Gaussian states. We then apply our results to the joint estimation of a phase shift and two parameters characterizing Gaussian phase covariant noise in optical interferometry. In such a scheme, we show that two-mode displaced squeezed input probes with optima…
Quantifying Uhlmann curvature from Yang-Mills action and its implications in quantum multiparameter estimation
School of Sciences, Hangzhou Dianzi University（杭州电子科技大学理学院）；Zhejiang Key Laboratory of Quantum State Control and Optical Field Manipulation, Hangzhou Dianzi University（杭州电子科技大学量子态调控与光场操纵浙江省重点实验室）
Quantum Estimation under Decoherence in Neutrino Oscillations: Quantum Resources, Flavor Accessibility, and Multiparameter Incompatibility
LPMS, Faculty of Sciences, Ibn Tofail University（伊本·图费利大学理学院）；LPHE–Modeling and Simulation, Faculty of Sciences, Mohammed V University in Rabat（拉巴特穆罕默德五世大学理学院）；Centre of Physics and Mathematics, CPM, Faculty of Sciences, Mohammed V University in Rabat（拉巴特穆罕默德五世大学理学院物理与数学中心）；College of Computing and Mathematical Sciences, Department of Applied Mathematics and Sciences and Center for Cyber-Physical Systems (C2PS), Khalifa University of Science and Technology（阿联酋哈利法大学科学技术学院计算与数理科学学院）
Most studies in multiparameter estimation assume the dynamics is fixed and focus on identifying the optimal probe state and the optimal measurements. In practice, however, controls are usually available to alter the dynamics, which provides another degree of freedom. In this paper we employ optimal control methods, particularly the gradient ascent pulse engineering (GRAPE), to design optimal controls for the improvement of the precision limit in multiparameter estimation. We show that the controlled schemes not only capable to provide a higher precision limit, but also have a higher stability …
Squeezing currently represents the leading strategy for quantum enhanced precision measurements of a single parameter in a variety of continuous- and discrete-variable settings and technological applications. However, many important physical problems including imaging and field sensing require the simultaneous measurement of multiple unknown parameters. The development of multiparameter quantum metrology is yet hindered by the intrinsic difficulty in finding saturable sensitivity bounds and feasible estimation strategies. Here, we derive the general operational concept of multiparameter squeez…
