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arXiv:2010.15465 · Quantum

Imaginarity-free quantum multiparameter estimation

Jisho Miyazaki, Keiji Matsumoto

Published Mar 10, 2022• 41 citations
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Abstract

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 matrices, offers compatible multiparameter estimations. The symmetries accompany the target-independent optimal measurements for pure-state models. Based on this finding, we propose methods to implement antiunitary symmetries for quantum metrology schemes. We further introduce a function which measures antiunitary asymmetry of quantum statistical models as a potential tool to characterize quantumness of phase transitions.

DOI
10.22331/q-2022-03-10-665

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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 matrices, offers compatible multiparameter estimations. The symmetries accompany the target-independent optimal measurements for pure-state models. Based on this finding, we propose methods to implement antiunitary symmetries for quantum metrology schemes. We further introduce a function which measures antiunitary asymmetry of quantum statistical models as a potential tool to characterize quantumness of phase transitions.
The estimation of multiple parameters is a ubiquitous requirement in many quantum metrology applications. However, achieving the ultimate precision limit, i.e., the quantum Cramér-Rao bound, becomes challenging in these scenarios compared to single parameter estimation. To address this issue, optimizing the parameters encoding strategies with the aid of antiunitary symmetry is a novel and comprehensive approach. For demonstration, we propose two types of quantum statistical models exhibiting antiunitary symmetry in experiments. The results showcase the simultaneous achievement of ultimate precision for multiple parameters without any trade-off and the precision is improved at least twice compared to conventional encoding strategies. Our work emphasizes the significant potential of antiunitary symmetry in addressing multiparameter estimation problems.
A fundamental challenge in multiparameter quantum estimation arises from the incompatibility of optimal measurements for different parameters, leading to intricate precision trade-offs that obscure the understanding of ultimate quantum limits. Here, we present an approach that precisely quantifies these trade-offs for an arbitrary number of parameters encoded in pure quantum states. Our approach not only derives tight analytical bounds for the trade-offs induced by measurement incompatibility but also provides a systematic methodology to design optimal measurement strategies that saturate these limits. To demonstrate the practical significance of our findings, we apply our framework to quantum radar and obtain a refined Arthurs-Kelly relation that characterizes the ultimate performance for the simultaneous estimation of range and velocity with any given amount of entanglement. This showcases the transformative potential of our findings for a wide range of applications in quantum metrology, sensing, and beyond.
Incompatibility among different physical quantities poses a broad limit on quantum information processing. In multi-parameter quantum estimation, the ultimate precision limit known as quantum Cramer-Rao bound becomes unattainable due to the incompatibility. This paper introduces a prescription to redesign given incompatible quantum statistical models to attain their quantum Cramer-Rao bounds. The prescription leads to a theoretical technique to find compatible models with an aid of anti-unitary symmetries. Our findings also offer a physical interpretation of mean Uhlmann curvature which characterizes quantumness of phase transitions.
Precise estimation of physical parameters underpins both scientific discovery and technological development. A central goal of quantum metrology and sensing is to exploit quantum resources like entanglement to devise optimal strategies for estimating physical parameters as precisely as possible. While substantial progress has been made in single-parameter quantum metrology, the multiparameter scenario remains significantly more challenging due to the issue of parameter incompatibility. In this work, we present a unified and computable approach for the simultaneous estimation of multiple parameters that attains the ultimate precision permitted by quantum mechanics. The core of our approach is to integrate the quantum tester formalism into the recently proposed tight Cramér-Rao type bound. This formulation enables us to figure out the highest achievable precision via upper and lower bounds that are computable via semidefinite programs. More importantly, within this formulation, diverse quantum resources, including entanglement, coherence, quantum control, and indefinite causal order, are treated on equal footing and systematically optimized for the purpose of achieving the ultimate precision in multiparameter estimation. As a result, our approach is applicable to various metrological strategies both in the presence and absence of noise. To demonstrate its utility, we revisit three-dimensional magnetic-field estimation, uncovering the strengths and limitations of existing analytical results and further establishing a strict hierarchy among different types of strategies.
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 parameter alone, assuming all other parameters are perfectly known. We call such models compatible. In establishing a rigorous theoretical framework for investigating compatibility, we clarify some ambiguities and inconsistencies present in the literature and discuss several examples to highlight interesting features of unitary and nonunitary parameter estimation, as well as deriving new bounds for physical problems of interest, such as the simultaneous estimation of phase and local dephasing.
We present a comprehensive and pedagogical formulation of Bayesian multiparameter quantum estimation. Within this framework, we analyse the role of measurement incompatibility and establish its quantitative effect on attainable precision. We achieve this by deriving upper bounds based on the pretty good measurement -- a notion from hypothesis testing -- combined with the evaluation of the Nagaoka-Hayashi lower bound. In general, we prove that, as in the many-copy regime of local estimation theory, incompatibility can at most double the minimum loss relative to the idealised scenario in which individually optimal measurements are assumed jointly implementable. Therefore, in practical situations, the latter may provide a sufficient and computationally efficient benchmark without solving the full optimisation problem. Our results, which we illustrate through applications of discrete phase imaging, phase and dephasing estimation, and qubit sensing, provide analytical and numerical tools for assessing ultimate precision limits and the role of measurement incompatibility in Bayesian multiparameter quantum metrology, including an open-source package for all the bounds discussed here.