Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction
Jordan algebras naturally arise in (quantum) information geometry, and we aim to understand their role and structure within this framework. Inspired by Kirillov's discussion of the symplectic structure on coadjoint orbits, we provide a similar construction in the case of real Jordan algebras. G...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2023 |
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2023
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/212006 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction. Florio M. Ciaglia, Jürgen Jost and Lorenz J. Schwachhöfer. SIGMA 19 (2023), 078, 27 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862667882642538496 |
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| author | Ciaglia, Florio M. Jost, Jürgen Schwachhöfer, Lorenz J. |
| author_facet | Ciaglia, Florio M. Jost, Jürgen Schwachhöfer, Lorenz J. |
| citation_txt | Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction. Florio M. Ciaglia, Jürgen Jost and Lorenz J. Schwachhöfer. SIGMA 19 (2023), 078, 27 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Jordan algebras naturally arise in (quantum) information geometry, and we aim to understand their role and structure within this framework. Inspired by Kirillov's discussion of the symplectic structure on coadjoint orbits, we provide a similar construction in the case of real Jordan algebras. Given a real, finite-dimensional, formally real Jordan algebra , we exploit the generalized distribution determined by the Jordan product on the dual * to induce a pseudo-Riemannian metric tensor on the leaves of the distribution. In particular, these leaves are the orbits of a Lie group, which is the structure group of , in clear analogy with what happens for coadjoint orbits. However, this time, in contrast with the Lie-algebraic case, we prove that not all points in * lie on a leaf of the canonical Jordan distribution. When the leaves are contained in the cone of positive linear functionals on , the pseudo-Riemannian structure becomes Riemannian and, for appropriate choices of , it coincides with the Fisher-Rao metric on non-normalized probability distributions on a finite sample space, or with the Bures-Helstrom metric for non-normalized, faithful quantum states of a finite-level quantum system, thus showing a direct link between the mathematics of Jordan algebras and both classical and quantum information geometry.
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| first_indexed | 2026-03-16T11:30:41Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-212006 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-16T11:30:41Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Ciaglia, Florio M. Jost, Jürgen Schwachhöfer, Lorenz J. 2026-01-22T09:15:22Z 2023 Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction. Florio M. Ciaglia, Jürgen Jost and Lorenz J. Schwachhöfer. SIGMA 19 (2023), 078, 27 pages 1815-0659 2020 Mathematics Subject Classification: 17C20; 17C27; 17B60; 53B12 arXiv:2112.09781 https://nasplib.isofts.kiev.ua/handle/123456789/212006 https://doi.org/10.3842/SIGMA.2023.078 Jordan algebras naturally arise in (quantum) information geometry, and we aim to understand their role and structure within this framework. Inspired by Kirillov's discussion of the symplectic structure on coadjoint orbits, we provide a similar construction in the case of real Jordan algebras. Given a real, finite-dimensional, formally real Jordan algebra , we exploit the generalized distribution determined by the Jordan product on the dual * to induce a pseudo-Riemannian metric tensor on the leaves of the distribution. In particular, these leaves are the orbits of a Lie group, which is the structure group of , in clear analogy with what happens for coadjoint orbits. However, this time, in contrast with the Lie-algebraic case, we prove that not all points in * lie on a leaf of the canonical Jordan distribution. When the leaves are contained in the cone of positive linear functionals on , the pseudo-Riemannian structure becomes Riemannian and, for appropriate choices of , it coincides with the Fisher-Rao metric on non-normalized probability distributions on a finite sample space, or with the Bures-Helstrom metric for non-normalized, faithful quantum states of a finite-level quantum system, thus showing a direct link between the mathematics of Jordan algebras and both classical and quantum information geometry. F.M.C. acknowledges that this work has been supported by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of “Research Funds for Beatriz Galindo Fellowships” (C&QIG-BG-CM-UC3M), and in the context of the V PRICIT (Regional Program of Research and Technological Innovation). He also wants to thank the incredible support of the Max Planck Institute for Mathematics in the Sciences in Leipzig, where he was formerly employed when this work was initially started and developed. L.S. acknowledges partial support by grant SCHW893/5-1 of the Deutsche Forschungsgemeinschaft, and also expresses his gratitude for the hospitality of the Max Planck Institute for Mathematics in the Sciences in Leipzig during numerous visits. This publication is based upon work from COST Action CaLISTA CA21109 supported by COST (European Cooperation in Science and Technology, www.cost.eu). We also thank the anonymous referees for their valuable comments and suggestions, which enabled us to improve this manuscript significantly. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction Article published earlier |
| spellingShingle | Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction Ciaglia, Florio M. Jost, Jürgen Schwachhöfer, Lorenz J. |
| title | Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction |
| title_full | Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction |
| title_fullStr | Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction |
| title_full_unstemmed | Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction |
| title_short | Information Geometry, Jordan Algebras, and a Coadjoint Orbit-Like Construction |
| title_sort | information geometry, jordan algebras, and a coadjoint orbit-like construction |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212006 |
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