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Two Quantum Analogues of the Large Deviation Cramer-Rao Inequality

Hiroshi Nagaoka

IEEE International Symposium on Information Theory (ISIT)Jan 1, 19944 citations

Some remarks on Fisher information, the Cramer–Rao inequality, and their applications to physics

H. G. Miller, Angelo Plastino, Angel Ricardo Plastino

Handbook of statisticsJan 1, 20210 citations

The categorical foundations of quantum information theory: Categories and the Cramer–Rao inequality

Florio M. Ciaglia, Fabio Di Cosmo, L. González-Bravo +2 more

An extension of Cencov’s categorical description of classical inference theory to the domain of quantum systems is presented. It provides a novel categorical foundation to the theory of quantum information that embraces both classical and quantum information theories in a natural way, while also allowing to formalize the notion of quantum environment. A first application of these ideas is provided by extending the notion of statistical manifold to incorporate categories, and investigating a possible, uniparametric Cramer–Rao inequality in this setting.An extension of Cencov’s categorical description of classical inference theory to the domain of quantum systems is presented. It provides a novel categorical foundation to the theory of quantum information that embraces both classical and quantum information theories in a natural way, while also allowing to formalize the notion of quantum environment. A first application of these ideas is provided by extending the notion of statistical manifold to incorporate categories, and investigating a possible, uniparametric Cramer–Rao inequality in this setting.

arXiv:2309.10428Modern Physics Letters AJun 7, 20232 citations

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

J F Bercher

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.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.

arXiv:1211.2008Journal of Physics A Mathematical and TheoreticalFeb 15, 201311 citations

Inequalities for quantum-statistical response functions

С. Окубо, A. Isihara

PhysicaFeb 1, 19723 citations

Quantum Metrology

Reinhold A. Bertlmann, Nicolai Friis

Abstract Chapter 24 provides a compact introduction to the topic of quantum metrology, focusing on Hamiltonian parameter estimation in the frequentist and in the Bayesian paradigms. We first discuss how estimates of non-directly measurable quantities such as phases are obtained from measurement statistics in the frequentist approach. We then discuss the Cramér-Rao bound and the Fisher information and study single-qubit phase estimation in the light of this result. We then turn to the multi-qubit setting and discuss the quantum Cramér-Rao bound, for which the quantum Fisher information is the central quantity of interest. In this context we discuss the Uhlmann fidelity in detail. We further contrast Heisenberg scaling with the standard quantum limit in the phase-estimation scenario. Finally, we analyse phase estimation in the Bayesian-estimation approach and derive the van Trees inequality as a Bayesian version of the Cramér-Rao boundAbstract Chapter 24 provides a compact introduction to the topic of quantum metrology, focusing on Hamiltonian parameter estimation in the frequentist and in the Bayesian paradigms. We first discuss how estimates of non-directly measurable quantities such as phases are obtained from measurement statistics in the frequentist approach. We then discuss the Cramér-Rao bound and the Fisher information and study single-qubit phase estimation in the light of this result. We then turn to the multi-qubit setting and discuss the quantum Cramér-Rao bound, for which the quantum Fisher information is the central quantity of interest. In this context we discuss the Uhlmann fidelity in detail. We further contrast Heisenberg scaling with the standard quantum limit in the phase-estimation scenario. Finally, we analyse phase estimation in the Bayesian-estimation approach and derive the van Trees inequality as a Bayesian version of the Cramér-Rao bound

Sep 7, 20230 citations

On the philosophy of Cramér-Rao-Bhattacharya Inequalities in Quantum Statistics

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