Information-Theoretic Models for Physical Observables
This work addresses J.A. Wheeler’s 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 spa...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-10-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/25/10/1448 |
| _version_ | 1797573904463560704 |
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| author | D. Bernal-Casas J. M. Oller |
| author_facet | D. Bernal-Casas J. M. Oller |
| author_sort | D. Bernal-Casas |
| collection | DOAJ |
| description | This work addresses J.A. Wheeler’s 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 Schrödinger’s 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’s variance equals the energy levels of the quantum harmonic oscillator, proving that the estimator’s 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 Cramér–Rao lower bound on the estimator’s 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’ 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’s recent statement that the world is not broken into physical but informational parts. |
| first_indexed | 2024-03-10T21:16:37Z |
| format | Article |
| id | doaj.art-5ca0350cb18846c29b2bf851e73c8253 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T21:16:37Z |
| publishDate | 2023-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-5ca0350cb18846c29b2bf851e73c82532023-11-19T16:25:03ZengMDPI AGEntropy1099-43002023-10-012510144810.3390/e25101448Information-Theoretic Models for Physical ObservablesD. Bernal-Casas0J. M. Oller1Department of Genetics, Microbiology and Statistics, Faculty of Biology, Universitat de Barcelona, 08028 Barcelona, SpainDepartment of Genetics, Microbiology and Statistics, Faculty of Biology, Universitat de Barcelona, 08028 Barcelona, SpainThis work addresses J.A. Wheeler’s 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 Schrödinger’s 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’s variance equals the energy levels of the quantum harmonic oscillator, proving that the estimator’s 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 Cramér–Rao lower bound on the estimator’s 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’ 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’s recent statement that the world is not broken into physical but informational parts.https://www.mdpi.com/1099-4300/25/10/1448information geometryFisher’s informationRiemannian manifoldsSchrödinger’s equationprinciple of minimum Fisher’s informationquantum harmonic oscillator |
| spellingShingle | D. Bernal-Casas J. M. Oller Information-Theoretic Models for Physical Observables Entropy information geometry Fisher’s information Riemannian manifolds Schrödinger’s equation principle of minimum Fisher’s information quantum harmonic oscillator |
| title | Information-Theoretic Models for Physical Observables |
| title_full | Information-Theoretic Models for Physical Observables |
| title_fullStr | Information-Theoretic Models for Physical Observables |
| title_full_unstemmed | Information-Theoretic Models for Physical Observables |
| title_short | Information-Theoretic Models for Physical Observables |
| title_sort | information theoretic models for physical observables |
| topic | information geometry Fisher’s information Riemannian manifolds Schrödinger’s equation principle of minimum Fisher’s information quantum harmonic oscillator |
| url | https://www.mdpi.com/1099-4300/25/10/1448 |
| work_keys_str_mv | AT dbernalcasas informationtheoreticmodelsforphysicalobservables AT jmoller informationtheoreticmodelsforphysicalobservables |