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International Symposium on Autonomous Systems (ISAS)

Optimized Quantum Parameter Estimation in Open Quantum Systems

S. Yang, Wei Cui

Published Dec 6, 2020• 1 citation
Publisher

Abstract

Quantum parameter estimation has been long pursued in various physics areas, such as quantum optics, quantum information processing, gravitational wave detection, and so on. Generally, the quantum system dynamics is usually assumed to be fixed and the system is closed, the highest precision can be achieved by preparing the optimal probe state and performing the quantum measurement based on ideal basis that saturates the quantum Cramer-Rao inequality. However, how to extend this problem to a more general open quantum system is still an open question. In this paper, we apply the optimal control theory, Pontrygin optimal principle, to enhance the parameter estimation precision in open quantum systems. The control laws and the control algorithm are exactly presented. Simulation results demonstrated the efficiency of the optimal control laws.

DOI
10.1109/isas49493.2020.9378881

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Quantum parameter estimation has been long pursued in various physics areas, such as quantum optics, quantum information processing, gravitational wave detection, and so on. Generally, the quantum system dynamics is usually assumed to be fixed and the system is closed, the highest precision can be achieved by preparing the optimal probe state and performing the quantum measurement based on ideal basis that saturates the quantum Cramer-Rao inequality. However, how to extend this problem to a more general open quantum system is still an open question. In this paper, we apply the optimal control theory, Pontrygin optimal principle, to enhance the parameter estimation precision in open quantum systems. The control laws and the control algorithm are exactly presented. Simulation results demonstrated the efficiency of the optimal control laws.
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.
Quantum metrology utilizes quantum resources to enhance the precision of parameter estimation. A central objective is to maximize the Quantum Fisher information (QFI), which determines the ultimate estimation precision based on the Cramér-Rao bound. In this work, we formulate QFI optimization as an optimal control problem and apply Pontryagin’s Minimum Principle (PMP) to derive the optimal control protocol. We show that when the control amplitude is constrained below a certain threshold, the optimal solution reduces to a constant control. This strategy enables the QFI to scale quadratically with the total evolution time T, thereby improving estimation precision with longer probe durations.
Estimation of parameters is a pivotal task throughout science and technology. The quantum Cramér-Rao bound provides a fundamental limit of precision allowed to be achieved under quantum theory. For closed quantum systems, it has been shown how the estimation precision depends on the underlying dynamics. Here, we propose a general formulation for metrology scenarios in open quantum systems, aiming to relate the precision more directly to properties of the underlying dynamics. This feature may be employed to enhance an estimation precision, e.g., by quantum control techniques. Specifically, we derive a Cramér-Rao bound for a fairly large class of open system dynamics, which is governed by a (time-dependent) dynamical semigroup map. We illustrate the utility of this scenario through three examples.
Quantum metrology uses the principles of quantum mechanics to improve the accuracy of parameter estimation so that it can surpass the classical limit. However, noise and the challenge of preparing multipartite entangled states hinder practical applications. In this work, we use the Lipkin-Meshkov-Glick model as the experimental platform and the quantum parameter estimation package QuanEstimation as a tool to improve the quantum parameter estimation in many-body systems by using Hamiltonian control optimization. We apply auto-GRAPE, PSO, and DE algorithm to optimize the time-dependent control field. Our results show that the optimal control strategy can significantly enhance the quantum Fisher information and reduce the quantum Cramér-Rao bound even under environmental noise. These findings provide a way to achieve the parameter estimation limit in a noisy environment and promote the development of practical quantum metrology applications.
This is a tutorial aimed at illustrating some recent developments in quantum parameter estimation beyond the Cramèr–Rao bound, as well as their applications in quantum metrology. Our starting point is the observation that there are situations in classical and quantum metrology where the unknown parameter of interest, besides determining the state of the probe, is also influencing the operation of the measuring devices, e.g. the range of possible outcomes. In those cases, nonregular statistical models may appear, for which the Cramèr–Rao theorem does not hold. In turn, the achievable precision may exceed the Cramèr–Rao bound, opening new avenues for enhanced metrology. We focus on quantum estimation of Hamiltonian parameters and show that an achievable bound to precision (beyond the Cramèr–Rao) may be obtained in a closed form for the class of so-called controlled energy measurements. Examples of applications of the new bound to various estimation problems in quantum metrology are worked out in some details.
This is a tutorial aimed at illustrating some recent developments in quantum parameter estimation beyond the Cramèr–Rao bound, as well as their applications in quantum metrology. Our starting point is the observation that there are situations in classical and quantum metrology where the unknown parameter of interest, besides determining the state of the probe, is also influencing the operation of the measuring devices, e.g. the range of possible outcomes. In those cases, nonregular statistical models may appear, for which the Cramèr–Rao theorem does not hold. In turn, the achievable precision may exceed the Cramèr–Rao bound, opening new avenues for enhanced metrology. We focus on quantum estimation of Hamiltonian parameters and show that an achievable bound to precision (beyond the Cramèr–Rao) may be obtained in a closed form for the class of so-called controlled energy measurements. Examples of applications of the new bound to various estimation problems in quantum metrology are worked out in some details.