Information Geometric Measures of Complexity with Applications to Classical and Quantum Physical Settings
Abstract
:1. Theoretical Background
2. Indicators of Complexity
2.1. Information Geometric Entropy
2.2. Curvature
2.3. Jacobi Fields
3. Applications
3.1. Uncorrelated Gaussian Statistical Models
3.2. Correlated Gaussian Statistical Models
3.3. Inverted Harmonic Oscillators
3.4. Quantum Spin Chains
3.5. Statistical Embedding and Complexity Reduction
3.6. Entanglement Induced via Scattering
3.7. Softening of Classical Chaos by Quantization
3.8. Topologically Distinct Correlational Structures
4. Final Remarks
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Caticha, A. Entropic Inference and the Foundations of Physics; USP Press: São Paulo, Brazil, 2012; Available online: http://www.albany.edu/physics/ACaticha-EIFP-book.pdf (accessed on 20 July 2021).
- Amari, S.; Nagaoka, H. Methods of Information Geometry; Oxford University Press: Oxford, UK, 2000. [Google Scholar]
- Cafaro, C. The Information Geometry of Chaos. Ph.D. Thesis, State University of New York, Albany, NY, USA, 2008. [Google Scholar]
- Ali, S.A.; Cafaro, C.; Gassner, S.; Giffin, A. An information geometric perspective on the complexity of macroscopic predictions arising from incomplete information. Adv. Math. Phys. 2018, 2018, 2048521. [Google Scholar] [CrossRef] [Green Version]
- Felice, D.; Cafaro, C.; Mancini, S. Information geometric methods for complexity. Chaos 2018, 28, 032101. [Google Scholar] [CrossRef]
- Caticha, A. Entropic Dynamics. AIP Conf. Proc. 2002, 617, 302. [Google Scholar]
- Cafaro, C.; Ali, S.A. Maximum caliber inference and the stochastic Ising model. Phys. Rev. 2016, E94, 052145. [Google Scholar] [CrossRef] [Green Version]
- Ali, S.A.; Cafaro, C. Theoretical investigations of an information geometric approach to complexity. Rev. Math. Phys. 2017, 29, 1730002. [Google Scholar] [CrossRef] [Green Version]
- Cafaro, C.; Ali, S.A. The spacetime algebra approach to massive classical electrodynamics with magnetic monopoles. Adv. Appl. Clifford Algebr. 2007, 17, 23. [Google Scholar] [CrossRef] [Green Version]
- Cafaro, C.; Giffin, A.; Ali, S.A.; Kim, D.-H. Reexamination of an information geometric construction of entropic indicators of complexity. Appl. Math. Comput. 2010, 217, 2944. [Google Scholar] [CrossRef]
- Kittel, C. Elementary Statistical Physics; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 1958. [Google Scholar]
- Ito, S.; Oizumi, M.; Amari, S. Unified framework for the entropy production and the stochastic interaction based on information geometry. Phys. Rev. Res. 2020, 2, 033048. [Google Scholar] [CrossRef]
- Kaniadakis, G. Statistical mechanics in the context of special relativity. Phys. Rev. 2002, E66, 056125. [Google Scholar] [CrossRef] [Green Version]
- Fisher, R.A. Theory of statistical estimation. Proc. Cambridge Philos. Soc. 1925, 122, 700. [Google Scholar] [CrossRef] [Green Version]
- Rao, C.R. Information and accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 1945, 37, 81. [Google Scholar]
- Cencov, N.N. Statistical decision rules and optimal inference. Transl. Math. Monogr. Amer. Math. Soc. 1981, 53. [Google Scholar]
- Campbell, L.L. An extended Cencov characterization of the information metric. Proc. Am. Math. Soc. 1986, 98, 135. [Google Scholar]
- Weinberg, S. Gravitation and Cosmology; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 1972. [Google Scholar]
- Lee, J.M. Riemannian Manifolds: An Introduction to Curvature; Springer: Berlin, Germany, 1997. [Google Scholar]
- Cafaro, C.; Ali, S.A. Jacobi fields on statistical manifolds of negative curvature. Physica 2007, D70, 234. [Google Scholar] [CrossRef] [Green Version]
- Ohanian, H.C.; Ruffini, R. Gravitation and Spacetime; W. W. Norton & Company: New York, NY, USA, 1994. [Google Scholar]
- Carmo, M.P.D. Riemannian Geometry; Birkhauser: Basel, Switzerland, 1992. [Google Scholar]
- Cafaro, C.; Ali, S.A.; Giffin, A. An application of reversible entropic dynamics on curved statistical manifolds. AIP Conf. Proc. 2006, 872, 243. [Google Scholar]
- Cafaro, C. Information geometry and chaos on negatively curved statistical manifolds. AIP Conf. Proc. 2007, 954, 175. [Google Scholar]
- Cafaro, C. Recent theoretical progress on an information geometrodynamical approach to chaos. AIP Conf. Proc. 2008, 1073, 16. [Google Scholar]
- Ali, S.A.; Cafaro, C.; Giffin, A.; Kim, D.-H. Complexity characterization in a probabilistic approach to dynamical systems through information geometry and inductive inference. Phys. Scr. 2012, 85, 025009. [Google Scholar] [CrossRef] [Green Version]
- Cafaro, C. Information geometric complexity of entropic motion on curved statistical manifolds. In Proceedings of the 12th Joint European Thermodynamics Conference, Brescia, Italy, 1–5 July 2013; Pilotelli, M., Beretta, G.P., Eds.; Cartolibreria Snoopy: Brescia, Italy, 2013; pp. 110–118. [Google Scholar]
- Cafaro, C. Information-geometric indicators of chaos in Gaussian models on statistical manifolds of negative Ricci curvature. Int. J. Theor. Phys. 2008, 47, 2924. [Google Scholar] [CrossRef] [Green Version]
- Ali, S.A.; Cafaro, C.; Kim, D.-H.; Mancini, S. The effect of microscopic correlations on the information geometric complexity of Gaussian statistical models. Physica 2010, A389, 3117. [Google Scholar] [CrossRef] [Green Version]
- Caticha, A.; Cafaro, C. From information geometry to Newtonian dynamics. AIP Conf. Proc. 2007, 954, 165. [Google Scholar]
- Zurek, W.H.; Paz, J.P. Decoherence, chaos, and the second law. Phys. Rev. Lett. 1994, 72, 2508. [Google Scholar] [CrossRef] [Green Version]
- Zurek, W.H.; Paz, J.P. Quantum chaos: A decoherent definition. Physica 1995, D83, 300. [Google Scholar] [CrossRef] [Green Version]
- Cafaro, C.; Ali, S.A. Geometrodynamics of information on curved statistical manifolds and its applications to chaos. EJTP 2008, 5, 139. [Google Scholar]
- Cafaro, C. Works on an information geometrodynamical approach to chaos. Chaos Solitons Fractals 2009, 41, 886. [Google Scholar] [CrossRef] [Green Version]
- Cafaro, C. Information geometry, inference methods and chaotic energy levels statistics. Mod. Phys. Lett. 2008, B22, 1879. [Google Scholar] [CrossRef] [Green Version]
- Cafaro, C.; Ali, S.A. Can chaotic quantum energy levels statistics be characterized using information geometry and inference methods? Physica 2008, A387, 6876. [Google Scholar] [CrossRef] [Green Version]
- Prosen, T.; Znidaric, M. Is the efficiency of classical simulations of quantum dynamics related to integrability? Phys. Rev. 2007, E75, 015202. [Google Scholar] [CrossRef] [Green Version]
- Prosen, T.; Pizorn, I. Operator space entanglement entropy in transverse Ising chain. Phys. Rev. 2007, A76, 032316. [Google Scholar] [CrossRef] [Green Version]
- Cafaro, C.; Mancini, S. On the complexity of statistical models admitting correlations. Phys. Scr. 2010, 82, 035007. [Google Scholar] [CrossRef]
- Cafaro, C.; Mancini, S. Quantifying the complexity of geodesic paths on curved statistical manifolds through information geometric entropies and Jacobi fields. Phys. D Nonlinear Phenom. 2011, 240, 607. [Google Scholar] [CrossRef] [Green Version]
- Kim, D.-H.; Ali, S.A.; Cafaro, C.; Mancini, S. An information geometric analysis of entangled continuous variable quantum systems. J. Phys. Conf. Ser. 2011, 306, 012063. [Google Scholar] [CrossRef]
- Kim, D.-H.; Ali, S.A.; Cafaro, C.; Mancini, S. Information geometric modeling of scattering induced quantum entanglement. Phys. Lett. 2011, A375, 2868. [Google Scholar] [CrossRef] [Green Version]
- Kim, D.-H.; Ali, S.A.; Cafaro, C.; Mancini, S. Information geometry of quantum entangled wave-packets. Physica 2012, A391, 4517. [Google Scholar] [CrossRef] [Green Version]
- Cafaro, C.; Giffin, A.; Lupo, C.; Mancini, S. Insights into the softening of chaotic statistical models by quantum considerations. AIP Conf. Proc. 2012, 1443, 366. [Google Scholar]
- Ali, S.A.; Cafaro, C.; Giffin, A.; Lupo, C.; Mancini, S. On a differential geometric viewpoint of Jaynes’ MaxEnt method and its quantum extension. AIP Conf. Proc. 2012, 1443, 120. [Google Scholar]
- Giffin, A.; Ali, S.A.; Cafaro, C. Local softening of chaotic statistical models with quantum consideration. AIP Conf. Proc. 2013, 1553, 238. [Google Scholar]
- Cafaro, C.; Giffin, A.; Lupo, C.; Mancini, S. Softening the complexity of entropic motion on curved statistical manifolds. Open Syst. Inf. Dyn. 2012, 19, 1250001. [Google Scholar] [CrossRef] [Green Version]
- Giffin, A.; Ali, S.A.; Cafaro, C. Local softening of information geometric indicators of chaos in statistical modeling in the presence of quantum-like considerations. Entropy 2013, 15, 4622. [Google Scholar] [CrossRef] [Green Version]
- Felice, D.; Cafaro, C.; Mancini, S. Information geometric complexity of a trivariate Gaussian statistical model. Entropy 2014, 16, 2944. [Google Scholar] [CrossRef] [Green Version]
- Sadoc, J.F.; Mosseri, R. Geometrical Frustration; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Greven, A.; Keller, G.; Warnecke, G. Entropy; Princeton University Press: Princeton, NJ, USA, 2003. [Google Scholar]
- Peres, A. Quantum Theory: Concepts and Methods; Kluwer Academic Publishers: London, UK, 1995. [Google Scholar]
- Peng, L.; Sun, H.; Xu, G. Information geometric characterization of the complexity of fractional Brownian motion. J. Math. Phys. 2012, 53, 123305. [Google Scholar] [CrossRef]
- Peng, L.; Sun, H.; Sun, D.; Yi, J. The geometric structures and instability of entropic dynamical models. Adv. Math. 2011, 227, 459. [Google Scholar] [CrossRef] [Green Version]
- Semarak, O.; Sukova, P. Free motion around black holes with discs or rings: Between integrability and chaos-I. Mon. Not. R. Astron. Soc. 2010, 404, 545. [Google Scholar] [CrossRef] [Green Version]
- Li, C.; Sun, H.; Zhang, S. Characterization of the complexity of an ED model via information geometry. Eur. Phys. J. Plus 2013, 128, 70. [Google Scholar] [CrossRef]
- Cao, L.; Li, D.; Zhang, E.; Sun, H. A statistical cohomogeneity one metric on the upper plane with constant negative curvature. Adv. Math. Phys. 2014, 2014, 832683. [Google Scholar] [CrossRef]
- Felice, D.; Mancini, S.; Pettini, M. Quantifying networks complexity from information geometry viewpoint. J. Math. Phys. 2014, 55, 043505. [Google Scholar] [CrossRef] [Green Version]
- Abtahi, S.M.; Sadati, S.H.; Salarieh, H. Ricci-based chaos analysis for roto-translatory motion of a Kelvin-type gyrostat satellite. J.-Multi-Body Dyn. 2014, 228, 34. [Google Scholar] [CrossRef]
- Mikes, J.; Stepanova, E. A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold. Ann. Glob. Anal. Geom. 2014, 45, 111. [Google Scholar] [CrossRef] [Green Version]
- Weis, S. Continuity of the maximum-entropy inference. Commun. Math. Phys. 2014, 330, 1263. [Google Scholar]
- Li, C.; Peng, L.; Sun, H. Entropic dynamical models with unstable Jacobi fields. Rom. Journ. Phys. 2015, 60, 1249. [Google Scholar]
- Itoh, M.; Satoh, H. Geometry of Fisher information metric and the barycenter map. Entropy 2015, 17, 1814. [Google Scholar] [CrossRef] [Green Version]
- Franzosi, R.; Felice, D.; Mancini, S.; Pettini, M. A geometric entropy detecting the Erdös-Rényi phase transition. Eur. Phys. Lett. 2015, 111, 20001. [Google Scholar] [CrossRef] [Green Version]
- Martins, A.C.R. Opinion particles: Classical physics and opinion dynamics. Phys. Lett. 2015, A379, 89. [Google Scholar] [CrossRef] [Green Version]
- Muhammad, S.A.; Zhang, E.; Sun, H. Jacobi fields on the manifold of Freund. Ital. J. Pure Appl. Math. 2015, 34, 181. [Google Scholar]
- Felice, D.; Mancini, S. Gaussian network’s dynamics reflected into geometric entropy. Entropy 2015, 17, 5660. [Google Scholar] [CrossRef] [Green Version]
- Wen-Haw, C. A review of geometric mean of positive definite matrices. Br. J. Math. Comput. 2015, 5, 1. [Google Scholar]
- Weis, S.; Knauf, A.; Ay, N.; Zhao, M.-J. Maximizing the divergence from a hierarchical model of quantum states. Open Syst. Inf. Dyn. 2015, 22, 1550006. [Google Scholar] [CrossRef] [Green Version]
- Weis, S. Maximum-entropy inference and inverse continuity of the numerical range. Rep. Math. Phys. 2015, 77, 251. [Google Scholar] [CrossRef] [Green Version]
- Shalymov, D.S.; Fradkov, A.L. Dynamics of non-stationary processes that follow the maximum of the Rényi entropy principle. Proc. R. Soc. 2016, A472, 20150324. [Google Scholar] [CrossRef] [PubMed]
- Henry, G.; Rodriguez, D. On the instability of two entropic dynamical models. Chaos Solitons Fractals 2016, 91, 604. [Google Scholar] [CrossRef] [Green Version]
- Gomez, I.S.; Portesi, M. Ergodic statistical models: Entropic dynamics and chaos. AIP Conf. Proc. 2017, 1853, 100001. [Google Scholar]
- Gomez, I.S. Notions of the ergodic hierarchy for curved statistical manifolds. Physica 2017, A484, 117. [Google Scholar] [CrossRef] [Green Version]
- Gassner, S.; Cafaro, C. Information geometric complexity of entropic motion on curved statistical manifolds under different metrizations of probability spaces. Int. J. Geom. Methods Mod. 2019, 16, 1950082. [Google Scholar] [CrossRef] [Green Version]
- Gomez, I.S.; Portesi, M.; Borges, E.P. Universality classes for the Fisher metric derived from relative group entropy. Physica 2020, A547, 123827. [Google Scholar] [CrossRef]
- Summers, R.L. Experiences in the Biocontinuum: A New Foundation for Living Systems; Cambridge Scholar Publishing: Cambridge, UK, 2020. [Google Scholar]
- Deshmukh, S.; Ishan, A.; Al-Shaik, S.B.; Özgür, C. A note on Killing calculus on Riemannian manifolds. Mathematics 2021, 9, 307. [Google Scholar] [CrossRef]
Surface | Curvature | Jacobi Field Behavior | IGE Behavior |
---|---|---|---|
sphere | positive | oscillatory | sublogarithmic |
cylinder | zero | linear | logarithmic |
hyperboloid | negative | exponential | linear |
Math & IGAC | Classical & IGAC | Quantum & IGAC |
---|---|---|
Micro and macro correlations | Geometrization of Newtonian mechanics | Spin chains and energy levels statistics |
Statistical embeddings | Inverted harmonic oscillators | Scattering induced entanglement |
Topology and correlational structures | Macro effects from micro information | Softening chaoticity by quantization |
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Cafaro, C.; Ali, S.A. Information Geometric Measures of Complexity with Applications to Classical and Quantum Physical Settings. Foundations 2021, 1, 45-62. https://doi.org/10.3390/foundations1010006
Cafaro C, Ali SA. Information Geometric Measures of Complexity with Applications to Classical and Quantum Physical Settings. Foundations. 2021; 1(1):45-62. https://doi.org/10.3390/foundations1010006
Chicago/Turabian StyleCafaro, Carlo, and Sean A. Ali. 2021. "Information Geometric Measures of Complexity with Applications to Classical and Quantum Physical Settings" Foundations 1, no. 1: 45-62. https://doi.org/10.3390/foundations1010006