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arXiv:1305.6215

On some interrelations of generalized $q$-entropies and a generalized Fisher information, including a Cramér-Rao inequality

Jean‐François Bercher

Published May 27, 2013• 1 citation
arXivHTMLPDF

Abstract

In this communication, we describe some interrelations between generalized $q$-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 $q$-Gaussian distributions. These generalized $q$-Gaussians are important in several areas of physics and mathematics. They are known to maximize the $q$-entropies subject to a moment constraint. The Cramér-Rao inequality shows that the generalized $q$-Gaussians also minimize the generalized Fisher information among distributions with a fixed moment. Similarly, the generalized $q$-Gaussians also minimize the generalized Fisher information among distributions with a given $q$-entropy.

DOI
10.48550/arxiv.1305.6215

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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.
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.
We derive a parameter-free, falsifiable inequality linking entropy production, Fisher information, and spatial dispersion for systems undergoing free diffusion. The result follows from the matrix Cramér–Rao bound combined with De Bruijn’s identity and holds at all times for probability densities evolving under the diffusion equation ∂ₜρ = TΔρ. Gaussian states saturate the bound, while all other initial conditions obey it strictly. The inequality provides a direct experimental test of the information-geometric structure underlying thermodynamic response and can be validated using reconstructed density profiles in ultracold-atom experiments. This preprint is deposited on Zenodo to establish priority and enable open access prior to journal submission.
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.
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.
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.