Coarse Graining Shannon and von Neumann Entropies
The nature of coarse graining is intuitively “obvious”, but it is rather difficult to find explicit and calculable models of the coarse graining process (and the resulting entropy flow) discussed in the literature. What we would like to have at hand is some explicit and calculable process that takes...
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| Format: | Article |
| Language: | English |
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MDPI AG
2017-05-01
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| Series: | Entropy |
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| Online Access: | http://www.mdpi.com/1099-4300/19/5/207 |
| _version_ | 1798043486288609280 |
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| author | Ana Alonso-Serrano Matt Visser |
| author_facet | Ana Alonso-Serrano Matt Visser |
| author_sort | Ana Alonso-Serrano |
| collection | DOAJ |
| description | The nature of coarse graining is intuitively “obvious”, but it is rather difficult to find explicit and calculable models of the coarse graining process (and the resulting entropy flow) discussed in the literature. What we would like to have at hand is some explicit and calculable process that takes an arbitrary system, with specified initial entropy S, and that monotonically and controllably drives the entropy to its maximum value. This does not have to be a physical process, in fact for some purposes it is better to deal with a gedanken-process, since then it is more obvious how the “hidden information” is hiding in the fine-grain correlations that one is simply agreeing not to look at. We shall present several simple mathematically well-defined and easy to work with conceptual models for coarse graining. We shall consider both the classical Shannon and quantum von Neumann entropies, including models based on quantum decoherence, and analyse the entropy flow in some detail. When coarse graining the quantum von Neumann entropy, we find it extremely useful to introduce an adaptation of Hawking’s super-scattering matrix. These explicit models that we shall construct allow us to quantify and keep clear track of the entropy that appears when coarse graining the system and the information that can be hidden in unobserved correlations (while not the focus of the current article, in the long run, these considerations are of interest when addressing the black hole information puzzle). |
| first_indexed | 2024-04-11T22:49:39Z |
| format | Article |
| id | doaj.art-3d89faf8631842c4b6f710509a2dd982 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-11T22:49:39Z |
| publishDate | 2017-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-3d89faf8631842c4b6f710509a2dd9822022-12-22T03:58:37ZengMDPI AGEntropy1099-43002017-05-0119520710.3390/e19050207e19050207Coarse Graining Shannon and von Neumann EntropiesAna Alonso-Serrano0Matt Visser1Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, 18000 Prague, Czech RepublicSchool of Mathematics and Statistics, Victoria University of Wellington, P.O. Box 600, Wellington 6140, New ZealandThe nature of coarse graining is intuitively “obvious”, but it is rather difficult to find explicit and calculable models of the coarse graining process (and the resulting entropy flow) discussed in the literature. What we would like to have at hand is some explicit and calculable process that takes an arbitrary system, with specified initial entropy S, and that monotonically and controllably drives the entropy to its maximum value. This does not have to be a physical process, in fact for some purposes it is better to deal with a gedanken-process, since then it is more obvious how the “hidden information” is hiding in the fine-grain correlations that one is simply agreeing not to look at. We shall present several simple mathematically well-defined and easy to work with conceptual models for coarse graining. We shall consider both the classical Shannon and quantum von Neumann entropies, including models based on quantum decoherence, and analyse the entropy flow in some detail. When coarse graining the quantum von Neumann entropy, we find it extremely useful to introduce an adaptation of Hawking’s super-scattering matrix. These explicit models that we shall construct allow us to quantify and keep clear track of the entropy that appears when coarse graining the system and the information that can be hidden in unobserved correlations (while not the focus of the current article, in the long run, these considerations are of interest when addressing the black hole information puzzle).http://www.mdpi.com/1099-4300/19/5/207coarse grainingentropyinformationShannon entropyvon Neumann entropy |
| spellingShingle | Ana Alonso-Serrano Matt Visser Coarse Graining Shannon and von Neumann Entropies Entropy coarse graining entropy information Shannon entropy von Neumann entropy |
| title | Coarse Graining Shannon and von Neumann Entropies |
| title_full | Coarse Graining Shannon and von Neumann Entropies |
| title_fullStr | Coarse Graining Shannon and von Neumann Entropies |
| title_full_unstemmed | Coarse Graining Shannon and von Neumann Entropies |
| title_short | Coarse Graining Shannon and von Neumann Entropies |
| title_sort | coarse graining shannon and von neumann entropies |
| topic | coarse graining entropy information Shannon entropy von Neumann entropy |
| url | http://www.mdpi.com/1099-4300/19/5/207 |
| work_keys_str_mv | AT anaalonsoserrano coarsegrainingshannonandvonneumannentropies AT mattvisser coarsegrainingshannonandvonneumannentropies |