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arXiv:1211.2008 · Journal of Physics A Mathematical and Theoretical

On multidimensional generalized Cramér–Rao inequalities, uncertainty relations and characterizations of generalizedq-Gaussian distributions

J F Bercher

Published Feb 15, 2013• 11 citations
arXivHTMLPDF

Abstract

In the present work, we show how the generalized Cramér–Rao inequality for the estimation of a parameter, presented in a recent paper, can be extended to the multidimensional case with general norms on , and to a wider context. As a particular case, we obtain a new multidimensional Cramér–Rao inequality which is saturated by generalized q -Gaussian distributions. We also give another related Cramér–Rao inequality, for a general norm, which is saturated as well by these distributions. Finally, we derive uncertainty relations from these Cramér–Rao inequalities. These uncertainty relations involve moments computed with respect to escort distributions, and we show that some of these relations are saturated by generalized q -Gaussian distributions. These results introduce extended versions of Fisher information, new Cramér–Rao inequalities, and new characterizations of generalized q -Gaussian distributions which are important in several areas of physics and mathematics.

DOI
10.1088/1751-8113/46/9/095303

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In the present work, we show how the generalized Cramér–Rao inequality for the estimation of a parameter, presented in a recent paper, can be extended to the multidimensional case with general norms on , and to a wider context. As a particular case, we obtain a new multidimensional Cramér–Rao inequality which is saturated by generalized q -Gaussian distributions. We also give another related Cramér–Rao inequality, for a general norm, which is saturated as well by these distributions. Finally, we derive uncertainty relations from these Cramér–Rao inequalities. These uncertainty relations involve moments computed with respect to escort distributions, and we show that some of these relations are saturated by generalized q -Gaussian distributions. These results introduce extended versions of Fisher information, new Cramér–Rao inequalities, and new characterizations of generalized q -Gaussian distributions which are important in several areas of physics and mathematics.
On some interrelations of generalized qqq-entropies and a generalized Fisher information, including a Cramér-Rao inequality
In this communication, we describe some interrelations between generalized qqq-entropies and a generalized version of Fisher information. In information theory, the de Bruijn identity links the Fisher information and the derivative of the entropy. We show that this identity can be extended to generalized versions of entropy and Fisher information. More precisely, a generalized Fisher information naturally pops up in the expression of the derivative of the Tsallis entropy. This generalized Fisher information also appears as a special case of a generalized Fisher information for estimation problems. Indeed, we derive here a new Cramér-Rao inequality for the estimation of a parameter, which involves a generalized form of Fisher information. This generalized Fisher information reduces to the standard Fisher information as a particular case. In the case of a translation parameter, the general Cramér-Rao inequality leads to an inequality for distributions which is saturated by generalized qqq-Gaussian distributions. These generalized qqq-Gaussians are important in several areas of physics and mathematics. They are known to maximize the qqq-entropies subject to a moment constraint. The Cramér-Rao inequality shows that the generalized qqq-Gaussians also minimize the generalized Fisher information among distributions with a fixed moment. Similarly, the generalized qqq-Gaussians also minimize the generalized Fisher information among distributions with a given qqq-entropy.
Measurement estimation bounds for local quantum multiparameter estimation, which provide lower bounds on possible measurement uncertainties, have so far been formulated in two ways: by extending the classical Cramér-Rao bound (e.g., the quantum Cramér-Rao bound and the Nagaoka Cramér-Rao bound) and by incorporating the parameter estimation framework with the uncertainty principle, as in the Lu-Wang uncertainty relation. In this work, we present a general framework that allows a direct comparison between these different types of estimation limits. Specifically, we compare the attainability of the Nagaoka Cramér-Rao bound and the Lu-Wang uncertainty relation, using analytical and numerical techniques. We show that these two limits can provide different information about the physically attainable precision. We present an example where both limits provide the same attainable precision and an example where the Lu-Wang uncertainty relation is not attainable even for pure states. We further demonstrate that the unattainability in the latter case arises because the figure of merit underpinning the Lu-Wang uncertainty relation (the difference between the quantum and classical Fisher information matrices) does not necessarily agree with the conventionally used figure of merit (mean-squared error). The results offer insights into the general attainability and applicability of the Lu-Wang uncertainty relation. Furthermore, our proposed framework for comparing bounds of different types may prove useful in other settings. Published by the American Physical Society 2024
Fisher information plays a central role in statistics and quantum metrology, providing the basis for the celebrated Cramér-Rao bound. In this work, we introduce a new information measure based on higher-order Fisher information and show that it naturally leads to a generalized uncertainty relation for parameter estimation, which can be regarded as an extension of the Cramér-Rao bound. As an application, we analyze the case of quantum phase estimation with a single qubit and compare our theoretical bounds with the well-known established hierarchical bounds. Finally, we experimentally validate the proposed framework using a photonic platform.
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.
We introduce a generalized Wigner-Yanase skew information and then derive the trace inequality related to the uncertainty relation. This inequality is a non-trivial generalization of the uncertainty relation derived by S.Luo for the quantum uncertainty quantity excluding the classical mixture. And we introduce a generalized Fisher information and then derive a generalized Cramer-Rao inequality. We also give an example for our generalized Fisher information and then derive the uncertainty relation for two observables.