Abstract
Maximum Quantum Entropy (MQE), recently introduced by Richard Silver and also known as Quantum Statistical Inference (QSI), is a method of estimating smooth, non-quantummechanical (“classical”) densities, given information about those densities. It is formally analogous to Maximum Entropy (ME), the difference being that the Shannon entropy is replaced by the quantum entropy. We present a concise description of MQE from a mathematical perspective, not relying on physical analogy. We introduce density matrices and the quantum entropy, compare MQE with ME, discuss the nature of constraints in MQE and show how these constraints influence the density estimate. We conclude with a discussion of the status of MQE as a maximum entropy method.
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References
Shun-ichi Amari. Differential-Geometrical Methods in Statistics. Springer-Verlag, Berlin, 1985.
R. Balian and N. L. Balazs. `Equiprobability, inference, and entropy in quantum theory’. Ann. Phys., 179: 97–144, 1987.
Roger Balian, Yoram Alhassid, and Hugo Reinhardt. `Dissipation in many-body systems: A geometric approach based on information theory.’ Phys. Rep.131:1–146 1986.
Richard P. Feynman. `An operator calculus having applications in quantum electrodynamics.’ Phys. Rev., 84: 108–128, 1951.
Richard P. Feynman. Statistical Mechanics. Benjamin/Cummings, Reading 1972.
Paul F. Fougère, editor. Maximum Entropy and Bayesian Methods: Dartmouth U.S.A. 1989. Kluwer, Dordrecht 1990.
Stephen F. Gull. `Developments in maximum entropy data analysis.’ In J. Skilling, editor, Maximum Entropy and Bayesian Methods, pages 53–71, Dordrecht, 1989. Kluwer.
E. T. Jaynes. `Where do we stand on maximum entropy?’ In R. D. Levine and M. Tribus, editors The Maximum Entropy Formalism. M.I.T. press, Cambridge 1978.
Edwin T. Jaynes. ‘Information theory and statistical mechanics, I.’ 106:620–630 1957.
Edwin T. Jaynes. ‘Information theory and statistical mechanics, II.’ 108:171–190 1957.
Robert Karplus and Julian Schwinger. `A note on saturation in microwave spectroscopy.’ Phys. Rev. 73:1020–1026 1948.
D. W. Scott, R. A. Tapia, and J. R. Thompson. `Nonparametric probability density estimation by discrete maximum penalized-likelihood criteria.’ Ann. Statist. 8:820–832 1980.
John E. Shore and Rodney W. Johnson. `Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy.’ IEEE Trans. Info. Th., pages 26–37, 1980.
Richard Silver and Harry Martz. `Quantum statistical inference.’ to be published, 1993.
Richard Silver, Timothy C. Wallstrom, and Harry Martz. ‘Maximum quantum entropy for non-parametric density function estimation.’ to be published 1993.
Richard N. Silver. `Quantum statistical inference.’ In M. Djafari and G. Demoment, editors Maximum Entropy and Bayesian Methodspages 167–182. Kluwer, Dordrecht 1993.
Richard N. Silver. `Quantum statistical inference.’ In Physics and Probability: Essays in Honor of E. T. Jaynespages 223–238. Cambridge University Press, Cambridge 1993.
John Skilling. `Classic maximum entropy.’ In J. Skilling, editor, Maximum Entropy and Bayesian Methods, pages 45–52, Dordrecht, 1989. Kluwer.
John Skilling, editor. Maximum Entropy and Bayesian Methods: Cambridge, England, 1988. Kluwer, Dordrecht, 1989.
John Skilling. `Quantified maximum entropy.’ In P. F. Fougére, editor, Maximum Entropy and Bayesian Methods, pages 341–350, Dordrecht, 1990. Kluwer.
Y.Tikochinsky, N. Z. Tishby, and R. D. Levine. `Alternative approach to maximum-entropy inference.’ Phys. Rev. A, 30(5):2638–2644, 1984.
Timothy C. Wallstrom. `Smoothing in maximum quantum entropy.’ These proceedings.
Alfred Wehrl. `General properties of entropy.’ Rev. Mod. Phys. 50:221–260 1978.
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© 1996 Springer Science+Business Media Dordrecht
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Wallstrom, T.C. (1996). Maximum Quantum Entropy for Classical Density Functions. In: Heidbreder, G.R. (eds) Maximum Entropy and Bayesian Methods. Fundamental Theories of Physics, vol 62. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8729-7_10
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DOI: https://doi.org/10.1007/978-94-015-8729-7_10
Publisher Name: Springer, Dordrecht
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