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Quantum Statistical Manifolds
Quantum information geometry studies families of quantum states by means of differential geometry. A new approach is followed with the intention to facilitate the introduction of a more general theory in subsequent work. To this purpose, the emphasis is shifted from a manifold of strictly positive d...
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Formato: | Online Artículo Texto |
Lenguaje: | English |
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MDPI
2018
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Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7512990/ https://www.ncbi.nlm.nih.gov/pubmed/33265562 http://dx.doi.org/10.3390/e20060472 |
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author | Naudts, Jan |
author_facet | Naudts, Jan |
author_sort | Naudts, Jan |
collection | PubMed |
description | Quantum information geometry studies families of quantum states by means of differential geometry. A new approach is followed with the intention to facilitate the introduction of a more general theory in subsequent work. To this purpose, the emphasis is shifted from a manifold of strictly positive density matrices to a manifold of faithful quantum states on the C*-algebra of bounded linear operators. In addition, ideas from the parameter-free approach to information geometry are adopted. The underlying Hilbert space is assumed to be finite-dimensional. In this way, technicalities are avoided so that strong results are obtained, which one can hope to prove later on in a more general context. Two different atlases are introduced, one in which it is straightforward to show that the quantum states form a Banach manifold, the other which is compatible with the inner product of Bogoliubov and which yields affine coordinates for the exponential connection. |
format | Online Article Text |
id | pubmed-7512990 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2018 |
publisher | MDPI |
record_format | MEDLINE/PubMed |
spelling | pubmed-75129902020-11-09 Quantum Statistical Manifolds Naudts, Jan Entropy (Basel) Article Quantum information geometry studies families of quantum states by means of differential geometry. A new approach is followed with the intention to facilitate the introduction of a more general theory in subsequent work. To this purpose, the emphasis is shifted from a manifold of strictly positive density matrices to a manifold of faithful quantum states on the C*-algebra of bounded linear operators. In addition, ideas from the parameter-free approach to information geometry are adopted. The underlying Hilbert space is assumed to be finite-dimensional. In this way, technicalities are avoided so that strong results are obtained, which one can hope to prove later on in a more general context. Two different atlases are introduced, one in which it is straightforward to show that the quantum states form a Banach manifold, the other which is compatible with the inner product of Bogoliubov and which yields affine coordinates for the exponential connection. MDPI 2018-06-17 /pmc/articles/PMC7512990/ /pubmed/33265562 http://dx.doi.org/10.3390/e20060472 Text en © 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). |
spellingShingle | Article Naudts, Jan Quantum Statistical Manifolds |
title | Quantum Statistical Manifolds |
title_full | Quantum Statistical Manifolds |
title_fullStr | Quantum Statistical Manifolds |
title_full_unstemmed | Quantum Statistical Manifolds |
title_short | Quantum Statistical Manifolds |
title_sort | quantum statistical manifolds |
topic | Article |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7512990/ https://www.ncbi.nlm.nih.gov/pubmed/33265562 http://dx.doi.org/10.3390/e20060472 |
work_keys_str_mv | AT naudtsjan quantumstatisticalmanifolds |