Stochastic Time Evolution, Information Geometry, and the Cramér-Rao Bound

We investigate the connection between the time evolution of averages of stochastic quantities and the Fisher information and its induced statistical length. As a consequence of the Cramér-Rao bound, we find that the rate of change of the average of any observable is bounded from above by its varianc...

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Auteurs principaux:	Sosuke Ito, Andreas Dechant
Format:	article
Langue:	EN
Publié:	American Physical Society 2020
Sujets:	Physics QC1-999
Accès en ligne:	https://doaj.org/article/be8bd99c38d2461abb8242d2d12606af
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spelling	oai:doaj.org-article:be8bd99c38d2461abb8242d2d12606af2021-12-02T10:47:06ZStochastic Time Evolution, Information Geometry, and the Cramér-Rao Bound10.1103/PhysRevX.10.0210562160-3308https://doaj.org/article/be8bd99c38d2461abb8242d2d12606af2020-06-01T00:00:00Zhttp://doi.org/10.1103/PhysRevX.10.021056http://doi.org/10.1103/PhysRevX.10.021056https://doaj.org/toc/2160-3308We investigate the connection between the time evolution of averages of stochastic quantities and the Fisher information and its induced statistical length. As a consequence of the Cramér-Rao bound, we find that the rate of change of the average of any observable is bounded from above by its variance times the temporal Fisher information. As a consequence of this bound, we obtain a speed limit on the evolution of stochastic observables: Changing the average of an observable requires a minimum amount of time given by the change in the average squared, divided by the fluctuations of the observable times the thermodynamic cost of the transformation. In particular, for relaxation dynamics, which do not depend on time explicitly, we show that the Fisher information is a monotonically decreasing function of time and that the minimal required time is determined by the initial preparation of the system. We further show that the monotonicity of the Fisher information can be used to detect hidden variables in the system and demonstrate our findings for simple examples of continuous and discrete random processes.Sosuke ItoAndreas DechantAmerican Physical SocietyarticlePhysicsQC1-999ENPhysical Review X, Vol 10, Iss 2, p 021056 (2020)
institution	DOAJ
collection	DOAJ
language	EN
topic	Physics QC1-999
spellingShingle	Physics QC1-999 Sosuke Ito Andreas Dechant Stochastic Time Evolution, Information Geometry, and the Cramér-Rao Bound
description	We investigate the connection between the time evolution of averages of stochastic quantities and the Fisher information and its induced statistical length. As a consequence of the Cramér-Rao bound, we find that the rate of change of the average of any observable is bounded from above by its variance times the temporal Fisher information. As a consequence of this bound, we obtain a speed limit on the evolution of stochastic observables: Changing the average of an observable requires a minimum amount of time given by the change in the average squared, divided by the fluctuations of the observable times the thermodynamic cost of the transformation. In particular, for relaxation dynamics, which do not depend on time explicitly, we show that the Fisher information is a monotonically decreasing function of time and that the minimal required time is determined by the initial preparation of the system. We further show that the monotonicity of the Fisher information can be used to detect hidden variables in the system and demonstrate our findings for simple examples of continuous and discrete random processes.
format	article
author	Sosuke Ito Andreas Dechant
author_facet	Sosuke Ito Andreas Dechant
author_sort	Sosuke Ito
title	Stochastic Time Evolution, Information Geometry, and the Cramér-Rao Bound
title_short	Stochastic Time Evolution, Information Geometry, and the Cramér-Rao Bound
title_full	Stochastic Time Evolution, Information Geometry, and the Cramér-Rao Bound
title_fullStr	Stochastic Time Evolution, Information Geometry, and the Cramér-Rao Bound
title_full_unstemmed	Stochastic Time Evolution, Information Geometry, and the Cramér-Rao Bound
title_sort	stochastic time evolution, information geometry, and the cramér-rao bound
publisher	American Physical Society
publishDate	2020
url	https://doaj.org/article/be8bd99c38d2461abb8242d2d12606af
work_keys_str_mv	AT sosukeito stochastictimeevolutioninformationgeometryandthecramerraobound AT andreasdechant stochastictimeevolutioninformationgeometryandthecramerraobound
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Stochastic Time Evolution, Information Geometry, and the Cramér-Rao Bound

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