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Physical Review A

Quantum metrology matrix

Haidong Yuan, Chi‐Hang Fred Fung

Published Jul 7, 2017• 30 citations
Publisher

Abstract

Various strategies exist in quantum metrology, such as with or without ancillary system, with a fixed or optimized measurement, with or without monitoring the environment, etc. Different set of tools are usually needed for different strategies. In this article, we provide a unified framework for these different settings, in particular we introduce a quantum metrology matrix and show that the precision limits of different settings can all be obtained from the trace or the trace norm of the quantum metrology matrix. Furthermore, the probe state enters into the quantum metrology matrix linearly, which makes the identification of the optimal probe states, one of the main quests in quantum metrology, much more efficient than conventional methods.

DOI
10.1103/physreva.96.012310

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Various strategies exist in quantum metrology, such as with or without ancillary system, with a fixed or optimized measurement, with or without monitoring the environment, etc. Different set of tools are usually needed for different strategies. In this article, we provide a unified framework for these different settings, in particular we introduce a quantum metrology matrix and show that the precision limits of different settings can all be obtained from the trace or the trace norm of the quantum metrology matrix. Furthermore, the probe state enters into the quantum metrology matrix linearly, which makes the identification of the optimal probe states, one of the main quests in quantum metrology, much more efficient than conventional methods.
Abstract This tutorial introduces a systematic approach for addressing the key question of quantum metrology: For a generic task of sensing an unknown parameter, what is the ultimate precision given a constrained set of admissible strategies. The approach outputs the maximal attainable precision (in terms of the maximum of quantum Fisher information) as a semidefinite program and optimal strategies as feasible solutions thereof. Remarkably, the approach can identify the optimal precision for different sets of strategies, including parallel, sequential, quantum SWITCH‐enhanced, causally superposed, and generic indefinite‐causal‐order strategies. The tutorial consists of a pedagogic introduction to the background and mathematical tools of optimal quantum metrology, a detailed derivation of the main approach, and various concrete examples. As shown in the tutorial, applications of the approach include, but are not limited to, strict hierarchy of strategies in noisy quantum metrology, memory effect in non‐Markovian metrology, and designing optimal strategies. Compared with traditional approaches, the approach here yields the exact value of the optimal precision, offering more accurate criteria for experiments and practical applications. It also allows for the comparison between conventional strategies and the recently discovered causally‐indefinite strategies, serving as a powerful tool for exploring this new area of quantum metrology.
It can be said that all of humanity's efforts can be understood as a problem of optimization. We each have a natural sense of what is ``good'' or ``bad'' and thus our actions tend towards maximizing -- or optimizing -- some notion of good and minimizing those things we perceive as bad or undesirable. Within the sciences, the greatest form of good is knowledge. It is this pursuit of knowledge that leads to not only life-saving innovations and technology, but also to furthering our understanding of our natural world and driving our philosophical pursuits. The principle method of obtaining knowledge in the sciences is by performing measurements; the simple act of comparing one attribute of a system to a known standard and recording the observed value is how all scientific progress is made. The act of performing measurements is in fact so important that there is an entire field of study surrounding it: metrology. One critical component of metrology is the development of new techniques to perform measurements, or alternative measurement schemes that are more optimal in some way. This is where there is room to exploit quantum physics to improve our techniques \--- we can perform quantum metrology. In quantum mechanics we routinely deal with the smallest, weakest, most delicate of systems. Quantum properties are inherently very sensitive to their environment; this of course makes them highly intolerant of noise but also makes them great resources to perform sensitive measurements. Quantum metrology concerns itself with utilizing quantum phenomena to extract more information from the natural world than is possible by conventional, or classical, means. To perform optimal measurements, these quantum systems must of course be optimal by some metric. Performing the ``optimal'' measurement requires several ingredients. First, we need the optimal tools or instrumentation. In quantum mechanical language, this means we need the optimal probe state. Then, we need to optimize the interaction of our instrumentation with the system we which to interrogate so that we can extract the desired information. This translates to needing the best possible interaction between the probe state and the system in question -- in other words, we need to optimize the evolution of the probe. Finally, we must take care to extract the most information as possible at the output; we must not neglect any information present in the evolved probe state. The entire quantum metrology process can be summarized as thus: probe
Optimal strategies for local quantum metrology -- including the preparation of optimal probe states, implementation of optimal control and measurement strategies, are well established. However, for distributed quantum metrology, where the goal is to estimate global properties of multiple spatially distributed parameters, the optimal scheme -- particularly the role of optimal control -- remains poorly understood. In this work, we address this challenge by developing optimal schemes for distributed quantum metrology that characterize the ultimate precision limits in distributed systems. We derive the optimal probe state, optimal control protocols, and measurement strategies in estimating a linear combination of NNN independent unknown parameters coupled to ddd networked sensors. Crucially, we prove that the optimal control operations can be implemented locally on each sensor, eliminating the need for non-local control operations across distant nodes. This result significantly reduces the complexity of implementing optimal strategies in distributed quantum metrology. To demonstrate the power of our framework, we apply it to several key scenarios.
A pivotal task in quantum metrology, and quantum parameter estimation in general, is to design schemes that achieve the highest precision with the given resources. Standard models of quantum metrology usually assume that the dynamics is fixed and that the highest precision is achieved by preparing the optimal probe states and performing optimal measurements. However, in many practical experimental settings, additional controls are usually available to alter the dynamics. Here we propose to use optimal control methods for further improvement of the precision limit of quantum parameter estimation. We show that, by exploring the additional degree of freedom offered by the controls, a higher-precision limit can be achieved. In particular, we show that the precision limit under the controlled schemes can go beyond the constraints put by the coherent time, which is in contrast with the standard scheme where the precision limit is always bounded by the coherent time.
Abstract Quantum metrology can achieve far better precision than classical metrology, and is one of the most important applications of quantum technologies in the real world. To attain the highest precision promised by quantum metrology, all steps of the schemes need to be optimized, which include the state preparation, parametrization, and measurement. Here the recent progresses on the optimization of these steps, which are essential for the identification and achievement of the ultimate precision limit in quantum metrology, are reviewed. It is hoped this provides a useful reference for the researchers in quantum metrology and related fields.