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On the geometry of mixed states and the Fisher information tensor


Contreras, Ivan; Ercolessi, Elisa; Schiavina, Michele (2016). On the geometry of mixed states and the Fisher information tensor. Journal of Mathematical Physics, 57(6):online.

Abstract

In this paper, we will review the co-adjoint orbit formulation of finite dimensional quantum mechanics, and in this framework, we will interpret the notion of quantum Fisher information index (and metric). Following previous work of part of the authors, who introduced the definition of Fisher information tensor, we will show how its antisymmetric part is the pullback of the natural Kostant–Kirillov–Souriau symplectic form along some natural diffeomorphism. In order to do this, we will need to understand the symmetric logarithmic derivative as a proper 1-form, settling the issues about its very definition and explicit computation. Moreover, the fibration of co-adjoint orbits, seen as spaces of mixed states, is also discussed.

Abstract

In this paper, we will review the co-adjoint orbit formulation of finite dimensional quantum mechanics, and in this framework, we will interpret the notion of quantum Fisher information index (and metric). Following previous work of part of the authors, who introduced the definition of Fisher information tensor, we will show how its antisymmetric part is the pullback of the natural Kostant–Kirillov–Souriau symplectic form along some natural diffeomorphism. In order to do this, we will need to understand the symmetric logarithmic derivative as a proper 1-form, settling the issues about its very definition and explicit computation. Moreover, the fibration of co-adjoint orbits, seen as spaces of mixed states, is also discussed.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Scopus Subject Areas:Physical Sciences > Statistical and Nonlinear Physics
Physical Sciences > Mathematical Physics
Language:English
Date:June 2016
Deposited On:10 Aug 2016 07:30
Last Modified:16 Nov 2023 08:02
Publisher:American Institute of Physics
ISSN:0022-2488
OA Status:Green
Publisher DOI:https://doi.org/10.1063/1.4954328
Project Information:
  • : FunderSNSF
  • : Grant IDP300P2_154552
  • : Project TitleFollow up of 'Geometric aspects of classical and quantum gauge theories with boundary'
  • Content: Accepted Version
  • Language: English