{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:33:48Z","timestamp":1760150028358,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,10,14]],"date-time":"2023-10-14T00:00:00Z","timestamp":1697241600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>This work addresses J.A. Wheeler\u2019s critical idea that all things physical are information-theoretic in origin. In this paper, we introduce a novel mathematical framework based on information geometry, using the Fisher information metric as a particular Riemannian metric, defined in the parameter space of a smooth statistical manifold of normal probability distributions. Following this approach, we study the stationary states with the time-independent Schr\u00f6dinger\u2019s equation to discover that the information could be represented and distributed over a set of quantum harmonic oscillators, one for each independent source of data, whose coordinate for each oscillator is a parameter of the smooth statistical manifold to estimate. We observe that the estimator\u2019s variance equals the energy levels of the quantum harmonic oscillator, proving that the estimator\u2019s variance is definitively quantized, being the minimum variance at the minimum energy level of the oscillator. Interestingly, we demonstrate that quantum harmonic oscillators reach the Cram\u00e9r\u2013Rao lower bound on the estimator\u2019s variance at the lowest energy level. In parallel, we find that the global probability density function of the collective mode of a set of quantum harmonic oscillators at the lowest energy level equals the posterior probability distribution calculated using Bayes\u2019 theorem from the sources of information for all data values, taking as a prior the Riemannian volume of the informative metric. Interestingly, the opposite is also true, as the prior is constant. Altogether, these results suggest that we can break the sources of information into little elements: quantum harmonic oscillators, with the square modulus of the collective mode at the lowest energy representing the most likely reality, supporting A. Zeilinger\u2019s recent statement that the world is not broken into physical but informational parts.<\/jats:p>","DOI":"10.3390\/e25101448","type":"journal-article","created":{"date-parts":[[2023,10,14]],"date-time":"2023-10-14T14:38:42Z","timestamp":1697294322000},"page":"1448","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Information-Theoretic Models for Physical Observables"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4623-5935","authenticated-orcid":false,"given":"D.","family":"Bernal-Casas","sequence":"first","affiliation":[{"name":"Department of Genetics, Microbiology and Statistics, Faculty of Biology, Universitat de Barcelona, 08028 Barcelona, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9643-4406","authenticated-orcid":false,"given":"J. M.","family":"Oller","sequence":"additional","affiliation":[{"name":"Department of Genetics, Microbiology and Statistics, Faculty of Biology, Universitat de Barcelona, 08028 Barcelona, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1080\/00963402.1963.11454520","article-title":"The Philosophy of Niels Bohr","volume":"19","author":"Petersen","year":"1963","journal-title":"Bull. At. Sci."},{"key":"ref_2","first-page":"95","article-title":"The Representation of Nature in Contemporary Physics","volume":"87","author":"Heisenberg","year":"1958","journal-title":"Daedalus"},{"key":"ref_3","unstructured":"Wheeler, J. (1989, January 28\u201331). Information, Physics, Quantum: The search for links. Proceedings of the III International Symposium on Foundations of Quantum Mechanics, Tokyo, Japan."},{"key":"ref_4","unstructured":"Ansede, M. (2023, July 01). 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Rev."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1004","DOI":"10.1119\/1.15810","article-title":"Fisher information as the basis for the Schr\u00f6dinger wave equation","volume":"57","author":"Frieden","year":"1989","journal-title":"Am. J. Phys."},{"key":"ref_12","first-page":"489","article-title":"Quantisierung als Eigenwertproblem","volume":"79","year":"1926","journal-title":"Ann. der Phys."},{"key":"ref_13","unstructured":"Laplace, P. (1811). M\u00e9moires de la Classe des Sciences Math\u00e9matiques et Physiques de L\u2019institut Imp\u00e9rial de France, L\u2019institut Imp\u00e9rial de France."},{"key":"ref_14","unstructured":"Hermite, C. (1864). Sur un Nouveau D\u00e9veloppement en S\u00e9rie de Fonctions, Acad\u00e9mie des Sciences and Centre National de la Recherche Scientifique de France."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Cram\u00e9r, H. (1946). Mathematical Methods of Statistics, Princeton University Press. 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Sci."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/10\/1448\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T21:06:55Z","timestamp":1760130415000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/10\/1448"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,14]]},"references-count":18,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2023,10]]}},"alternative-id":["e25101448"],"URL":"https:\/\/doi.org\/10.3390\/e25101448","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2023,10,14]]}}}