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Precision measurements with quantum systems rely on our ability to trace the differences between experimental signals to variations in unknown physical parameters. In this Letter we derive the Fisher information and the ensuing Cramer-Rao…

Quantum Physics · Physics 2015-06-17 Søren Gammelmark , Klaus Mølmer

The Holevo Cram\'er Rao bound is a lower bound on the sum of the mean square error of estimates for parameters of a state. We provide a method for calculating the Holevo Cram\'er-Rao bound for estimation of quadrature mean parameters of a…

Quantum Physics · Physics 2018-01-22 Mark Bradshaw , Ping Koy Lam , Syed M. Assad

In recent times, distributed sensing has been extensively studied using squeezed states. While this is an excellent development, it is desirable to investigate the use of other quantum probes, such as entangled states of light. In this…

Quantum Physics · Physics 2025-01-14 Jiaxuan Wang , Girish Agarwal

Estimation of physical parameters encoded in a Hamiltonian is a central task in quantum sensing and learning. While the ultimate precision limit for estimating a single parameter coupled to a single generator is well established, the…

This paper explores as didactically as possible the fundamental principles of both classical and quantum metrology, focusing on the Cram\'er-Rao Bound and how it defines the maximum precision in parameter estimation, taking into account…

Quantum Physics · Physics 2024-12-13 Leonardo A. M. Souza

We derive explicit expressions for the quantum Fisher information and the symmetric logarithmic derivative (SLD) of a quantum state in the exponential form; the SLD is expressed in terms of the generator. Applications include…

Quantum Physics · Physics 2014-04-23 Zhang Jiang

Precise estimation of physical parameters underpins both scientific discovery and technological development. A central goal of quantum metrology and sensing is to exploit quantum resources like entanglement to devise optimal strategies for…

Quantum Physics · Physics 2026-03-09 Zhao-Yi Zhou , Da-Jian Zhang

Quantum state estimation is a fundamental task in quantum information theory, where one estimates real parameters continuously embedded in a family of quantum states. In the theory of quantum state estimation, the widely used Cram\'er Rao…

Quantum Physics · Physics 2025-07-23 Masahito Hayashi , Yingkai Ouyang

Advanced super-resolution imaging techniques require specific approaches for accurate and consistent estimation of the achievable spatial resolution. Fisher information supplied to Cramer-Rao bound (CRB) has proved to be a powerful and…

The widely used quantum Cramer-Rao bound (QCRB) sets a lower bound for the mean square error of unbiased estimators in quantum parameter estimation, however, in general QCRB is only tight in the asymptotical limit. With a limited number of…

Quantum Physics · Physics 2016-09-07 Jing Liu , Haidong Yuan

The aim of this thesis is to develop a theoretical framework to study parameter estimation of quantum channels. We study the task of estimating unknown parameters encoded in a channel in the sequential setting. A sequential strategy is the…

Quantum Physics · Physics 2022-01-06 Vishal Katariya

We consider two variants of a quantum-statistical generalization of the Cramer-Rao inequality that establishes an invariant lower bound on the mean square error of a generalized quantum measurement. The proposed complex variant of this…

Quantum Physics · Physics 2007-05-23 V. P. Belavkin

We determine the quantum Cram\'er-Rao bound for the precision with which the oscillator frequency and damping constant of a damped quantum harmonic oscillator in an arbitrary Gaussian state can be estimated. This goes beyond standard…

Quantum Physics · Physics 2020-08-05 Patrick Binder , Daniel Braun

The Cramer-Rao bound, satisfied by classical Fisher information, a key quantity in information theory, has been shown in different contexts to give rise to the Heisenberg uncertainty principle of quantum mechanics. In this paper, we show…

Quantum Physics · Physics 2022-11-23 Yakov Bloch , Eliahu Cohen

Uncertainty relations in quantum mechanics express bounds on our ability to simultaneously obtain knowledge about expectation values of non-commuting observables of a quantum system. They quantify trade-offs in accuracy between…

Quantum Physics · Physics 2020-03-16 Ilya Kull , Philippe Allard Guérin , Frank Verstraete

Bayesian analysis is a framework for parameter estimation that applies even in uncertainty regimes where the commonly used local (frequentist) analysis based on the Cram\'er-Rao bound is not well defined. In particular, it applies when no…

Quantum Physics · Physics 2021-03-17 Simon Morelli , Ayaka Usui , Elizabeth Agudelo , Nicolai Friis

Quantum estimation theory provides optimal observations for various estimation problems for unknown parameters in the state of the system under investigation. However, the theory has been developed under the assumption that every observable…

Quantum Physics · Physics 2009-11-10 M. Hotta , M. Ozawa

We experimentally analyzed the statistical errors in quantum-state estimation and examined whether their lower bound, which is derived from the Cramer-Rao inequality, can be truly attained or not. In the experiments, polarization states of…

Quantum Physics · Physics 2013-05-29 Koji Usami , Yoshihiro Nambu , Yoshiyuki Tsuda , Keiji Matsumoto , Kazuo Nakamura

In the present work, we show how the generalized Cram\'er-Rao inequality for the estimation of a parameter, presented in a recent paper, can be extended to the mutidimensional case with general norms on $\mathbb{R}^{n}$, and to a wider…

Mathematical Physics · Physics 2013-02-26 J. -F. Bercher

We address several estimation problems in quantum optics by means of the maximum-likelihood principle. We consider Gaussian state estimation and the determination of the coupling parameters of quadratic Hamiltonians. Moreover, we analyze…

Quantum Physics · Physics 2009-11-06 G. Mauro D'Ariano , Matteo G. A. Paris , Massimiliano F. Sacchi