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arXiv:2204.14252

Analytical techniques in single and multi-parameter quantum estimation theory: a focused review

Abdallah Slaoui, L.B. Drissi, El Hassan Saidi, R. Ahl Laamara

Published Apr 29, 2022• 2 citations
arXivHTMLPDF

Abstract

As we enter the era of quantum technologies, quantum estimation theory provides an operationally motivating framework for determining high precision devices in modern technological applications. The aim of any estimation process is to extract information from an unknown parameter embedded in a physical system such as the estimation converges to the true value of the parameter. According to the Cramér-Rao inequality in mathematical statistics, the Fisher information in the case of single-parameter estimation procedures, and the Fisher information matrix in the case of multi-parameter estimation, are the key quantities representing the ultimate precision of the parameters specifying a given statistical model. In quantum estimation strategies, it is usually difficult to derive the analytical expressions of such quantities in a given quantum state. This review provides comprehensive techniques on the analytical calculation of the quantum Fisher information as well as the quantum Fisher information matrix in various scenarios and via several methods. Furthermore, it provides a mathematical transition from classical to quantum estimation theory applied to many freedom quantum systems. To clarify these results, we examine these developments using some examples. Other challenges, including their links to quantum correlations and saturating the quantum Cramér-Rao bound, are also addressed.

DOI
10.48550/arxiv.2204.14252

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As we enter the era of quantum technologies, quantum estimation theory provides an operationally motivating framework for determining high precision devices in modern technological applications. The aim of any estimation process is to extract information from an unknown parameter embedded in a physical system such as the estimation converges to the true value of the parameter. According to the Cramér-Rao inequality in mathematical statistics, the Fisher information in the case of single-parameter estimation procedures, and the Fisher information matrix in the case of multi-parameter estimation, are the key quantities representing the ultimate precision of the parameters specifying a given statistical model. In quantum estimation strategies, it is usually difficult to derive the analytical expressions of such quantities in a given quantum state. This review provides comprehensive techniques on the analytical calculation of the quantum Fisher information as well as the quantum Fisher information matrix in various scenarios and via several methods. Furthermore, it provides a mathematical transition from classical to quantum estimation theory applied to many freedom quantum systems. To clarify these results, we examine these developments using some examples. Other challenges, including their links to quantum correlations and saturating the quantum Cramér-Rao bound, are also addressed.
Abstract Multiparameter quantum estimation theory aims to determine simultaneously the ultimate precision of all parameters contained in the state of a given quantum system. Determining this ultimate precision depends on the quantum Fisher information matrix (QFIM) which is essential to obtaining the quantum Cramér–Rao bound. This is the main motivation of this work which concerns the computation of the analytical expression of the QFIM. Inspired by the results reported in J. Phys. A: Math. Theor. 52 035304 (2019), the general formalism of the multiparameter quantum estimation theory of quantum Gaussian states in terms of their first and second moments is given. We give the analytical formulas of right logarithmic derivative and symmetric logarithmic derivative operators. Then we derive the general expressions of the corresponding quantum Fisher information matrices. We also derive an explicit expression of the condition which ensures the saturation of the quantum Cramér–Rao bound in estimating several parameters. Finally, we examine some examples to clarify the use of our results.
Abstract 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 properties and existing calculation techniques of QFIM for various cases, and review the development of QFIM in some aspects of quantum mechanics apart from quantum metrology. On the other hand, as the main application of QFIM, the second part of this paper reviews the quantum multiparameter Cramér–Rao bound, its attainability condition and the associated optimal measurements. Moreover, recent developments in a few typical scenarios of quantum multiparameter estimation and the quantum advantages are also thoroughly discussed in this part.
The aim of this thesis is to develop a theoretical framework to study parameter estimation of quantum channels. We begin by describing the classical task of parameter estimation that we build upon. In its most basic form, parameter estimation is the task of obtaining an estimate of an unknown parameter from some experimental data. This experimental data can be seen as a number of samples of a parameterized probability distribution. In general, the goal of such a task is to obtain an estimate of the unknown parameter while minimizing its error. We study the task of estimating unknown parameters which are encoded in a quantum channel. A quantum channel is a map that describes the evolution of the state of a quantum system. We study this task in the sequential setting. This means that the channel in question is used multiple times, and each channel use happens subsequent to the previous one. A sequential strategy is the most general way to use, or process, a channel multiple times. Our goal is to establish lower bounds on the estimation error in such a task. These bounds are called Cramer--Rao bounds. Quantum channels encompass all possible dynamics allowed by quantum mechanics, and sequential estimation strategies capture the most general way to process multiple uses of a channel. Therefore, the bounds we develop are universally applicable. We consider the use of catalysts to enhance the power of a channel estimation strategy. This is termed amortization. The reason we do so is to investigate if an n-round sequential estimation strategy does better than a simpler parallel strategy. Quantitatively, the power of a channel for a particular estimation task is determined by the channel's Fisher information. Thus, we study how much a catalyst quantum state can enhance the Fisher information of a quantum channel by defining the amortized Fisher information. In the quantum setting, there are many Fisher information quantities that can be defined. We limit our study to two particular ones: the symmetric logarithmic derivative (SLD) Fisher information and the right logarithmic derivative (RLD) Fisher information. We establish our Cramer--Rao bounds by proving that for certain Fisher information quantities, catalyst states do not improve the performance of a sequential estimation protocol. The technical term for this is an amortization collapse. We show how such a collapse leads directly to a corresponding Cramer--Rao bound. We establish bounds both when estimating a single parameter and when
The quantum Fisher information constrains the achievable precision in parameter estimation via the quantum Cramér-Rao bound, which has attracted much attention in Hermitian systems since the 60s of the last century. However, less attention has been paid to non-Hermitian systems. In this Letter, working with different logarithmic operators, we derive two previously unknown expressions for quantum Fisher information, and two Cramér-Rao bounds lower than the well-known one are found for non-Hermitian systems. These lower bounds are due to the merit of non-Hermitian observable and it can be understood as a result of extended regimes of optimization. Two experimentally feasible examples are presented to illustrate the theory, saturation of these bounds and estimation precisions beyond the Heisenberg limit are predicted and discussed. A setup to measure non-Hermitian observable is also proposed.
Abstract We derive several expressions for the quantum Fisher information matrix (QFIM) for the multi-parameter estimation of multi-mode Gaussian quantum states, the corresponding symmetric logarithmic derivatives, and conditions for saturability of the quantum Cramér–Rao bound. This bound determines the ultimate precision with which parameters encoded into quantum states can be estimated. We include expressions for mixed states, for the case when the Williamson decomposition of the covariance matrix is known, expressions in terms of infinite series, and expressions for pure states. We also discuss problematic behavior when some modes are pure, and present a method that allows the use of expressions that are defined only for mixed states, to compute QFIM for states with any number of pure modes.
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.