About the quantum Fisher information of nearly pure quantum statistical models

2020 ◽  
Vol 18 (01) ◽  
pp. 1941022
Author(s):  
Matteo G. A. Paris
Keyword(s):  
Fisher Information ◽  
Statistical Models ◽  
Logarithmic Derivative ◽  
Quantum Fisher Information ◽  
Independent State ◽  
Approximate Analytic Expression ◽  
Weak Source ◽  
Pure States ◽  
Analytic Expression ◽  
Quantum Statistical

We address nearly pure quantum statistical models, i.e. situations where the information about a parameter is encoded in pure states weakly perturbed by the mixing with a parameter independent state, mimicking a weak source of noise. We show that the symmetric logarithmic derivative is left unchanged, and find an approximate analytic expression for the quantum Fisher information (QFI) which provides bounds on how much a weak source of noise may degrade the QFI.

scholarly journals Taming singularities of the quantum Fisher information

2021 ◽  
Author(s):  
Aaron Z. Goldberg ◽  
José L. Romero ◽  
Ángel S. Sanz ◽  
Luis L. Sánchez-Soto
Keyword(s):  
Density Matrix ◽  
Fisher Information ◽  
Statistical Models ◽  
Probability Distributions ◽  
Quantum Fisher Information ◽  
Quantum Metrology ◽  
Quantum Statistical ◽  
Information Matrices ◽  
Parameter Dependent ◽  
Ultimate Limit

Quantum Fisher information matrices (QFIMs) are fundamental to estimation theory: they encode the ultimate limit for the sensitivity with which a set of parameters can be estimated using a given probe. Since the limit invokes the inverse of a QFIM, an immediate question is what to do with singular QFIMs. Moreover, the QFIM may be discontinuous, forcing one away from the paradigm of regular statistical models. These questions of nonregular quantum statistical models are present in both single- and multiparameter estimation. Geometrically, singular QFIMs occur when the curvature of the metric vanishes in one or more directions in the space of probability distributions, while QFIMs have discontinuities when the density matrix has parameter-dependent rank. We present a nuanced discussion of how to deal with each of these scenarios, stressing the physical implications of singular QFIMs and the ensuing ramifications for quantum metrology.

scholarly journals Asymptotic inference in system identification for the atom maser

2012 ◽  
Vol 370 (1979) ◽  
pp. 5308-5323 ◽  
Author(s):  
Cătălin Cătană ◽  
Merlijn van Horssen ◽  
Mădălin Guţă
Keyword(s):  
System Identification ◽  
Fisher Information ◽  
Statistical Models ◽  
Quantum Fisher Information ◽  
Asymptotic Statistics ◽  
Quantum Dynamical ◽  
Process Tomography ◽  
Quantum Markov Processes ◽  
Set Up ◽  
Quantum Dynamical Systems

System identification is closely related to control theory and plays an increasing role in quantum engineering. In the quantum set-up, system identification is usually equated to process tomography, i.e. estimating a channel by probing it repeatedly with different input states. However, for quantum dynamical systems such as quantum Markov processes, it is more natural to consider the estimation based on continuous measurements of the output, with a given input that may be stationary. We address this problem using asymptotic statistics tools, for the specific example of estimating the Rabi frequency of an atom maser. We compute the Fisher information of different measurement processes as well as the quantum Fisher information of the atom maser, and establish the local asymptotic normality of these statistical models. The statistical notions can be expressed in terms of spectral properties of certain deformed Markov generators, and the connection to large deviations is briefly discussed.

scholarly journals Quantum Fisher Information in Two-Qubit Pure States

2010 ◽  
Vol 49 (10) ◽  
pp. 2463-2475 ◽  
Author(s):  
Wan-Fang Liu ◽  
Li-Hua Zhang ◽  
Chun-Jie Li
Keyword(s):  
Fisher Information ◽  
Quantum Fisher Information ◽  
Pure States

scholarly journals Symmetric Logarithmic Derivative of Fermionic Gaussian States

2018 ◽  
Author(s):  
Angelo Carollo ◽  
Bernardo Spagnolo ◽  
Davide Valenti
Keyword(s):  
Fisher Information ◽  
Logarithmic Derivative ◽  
Quantum Fisher Information ◽  
Closed Form Expression ◽  
Quantum Metrology ◽  
Many Body ◽  
Form Expression ◽  
Gaussian States ◽  
Many Body Systems ◽  
Thermal States

In this article we derive a closed form expression for the symmetric logarithmic derivative of Fermionic Gaussian states. This provides a direct way of computing the quantum Fisher Information for Fermionic Gaussian states. Applications range from quantum Metrology with thermal states to non-equilibrium steady states with Fermionic many-body systems.

scholarly journals On the properties of the asymptotic incompatibility measure in multiparameter quantum estimation

2021 ◽  
Vol 54 (48) ◽  
pp. 485301
Author(s):  
Alessandro Candeloro ◽  
Matteo G A Paris ◽  
Marco G Genoni
Keyword(s):  
Statistical Model ◽  
Statistical Models ◽  
Quantum Dynamics ◽  
Logarithmic Derivative ◽  
Single Mode ◽  
Continuous Variable ◽  
Quantum Systems ◽  
Unknown Parameters ◽  
Quantum Statistical ◽  
The Difference

Abstract We address the use of asymptotic incompatibility (AI) to assess the quantumness of a multiparameter quantum statistical model. AI is a recently introduced measure which quantifies the difference between the Holevo and the symmetric logarithmic derivative (SLD) scalar bounds, and can be evaluated using only the SLD operators of the model. At first, we evaluate analytically the AI of the most general quantum statistical models involving two-level (qubit) and single-mode Gaussian continuous-variable quantum systems, and prove that AI is a simple monotonous function of the state purity. Then, we numerically investigate the same problem for qudits (d-dimensional quantum systems, with 2 < d ⩽ 4), showing that, while in general AI is not in general a function of purity, we have enough numerical evidence to conclude that the maximum amount of AI is attainable only for quantum statistical models characterized by a purity larger than μ min = 1 / ( d − 1 ) . In addition, by parametrizing qudit states as thermal (Gibbs) states, numerical results suggest that, once the spectrum of the Hamiltonian is fixed, the AI measure is in one-to-one correspondence with the fictitious temperature parameter β characterizing the family of density operators. Finally, by studying in detail the definition and properties of the AI measure we find that: (i) given a quantum statistical model, one can readily identify the maximum number of asymptotically compatible parameters; (ii) the AI of a quantum statistical model bounds from above the AI of any sub-model that can be defined by fixing one or more of the original unknown parameters (or functions thereof), leading to possibly useful bounds on the AI of models involving noisy quantum dynamics.

scholarly journals SUFFICIENCY IN QUANTUM STATISTICAL INFERENCE: A SURVEY WITH EXAMPLES

2006 ◽  
Vol 09 (03) ◽  
pp. 331-351 ◽  
Author(s):  
ANNA JENČOVÁ ◽  
DÉNES PETZ
Keyword(s):  
Statistical Inference ◽  
Fisher Information ◽  
Exponential Family ◽  
Weyl Algebra ◽  
Quantum Fisher Information ◽  
Quantum Statistics ◽  
Quantum Statistical ◽  
Basic Concepts ◽  
Noncommutative Analogue

This paper attempts to give an overview about sufficiency in the setting of quantum statistics. The basic concepts are treated in parallel to the measure theoretic case. It turns out that several classical examples and results have a noncommutative analogue. Some of the results are presented without proof (but with exact references) and the presentation is intended to be self-contained. The main examples discussed in the paper are related to the Weyl algebra and to the exponential family of states. The characterization of sufficiency in terms of quantum Fisher information is a new result.

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