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Cambridge University Press eBooks

A Look Back: Early Applications of Maximum Entropy Estimation to Quantum Statistical Mechanics

Douglas J. Scalapino

Published Sep 2, 1993• 0 citations
Publisher

Abstract

. E.T. Jaynes has been the central figure for over three decades in showing how maximum entropy estimation (MEE) provides an extension of logic to cases where one cannot carry out Aristotelian deductive reasoning. Here I will review two early applications of MEE which I hope will provide some insight into how these ideas were being used in quantum and statistical mechanics around 1960. In May of ‘61 I turned in my Ph.D. thesis (Scalapino, 1961) to Stanford University and, following a well-known dictum, went on to work on other problems. Now, three decades later, on this, the occasion of Ed Jaynes’ seventieth birthday, I decided to look back to see what we were doing when I was first getting to know Ed and his special approach to problems. Not surprisingly, my thesis contains several applications of the principle of Maximum Entropy Estimation (MEE). In 1957 Ed had written two seminal articles (Jaynes, 1957(a), 1957(b)) showing how one could use Shannon's (1948) Information Theory to construct density matrices for a variety of different problems in equilibrium statistical mechanics. I had been very much taken with this work and wanted to understand how it could be applied to other systems.

DOI
10.1017/cbo9780511524448.003

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. E.T. Jaynes has been the central figure for over three decades in showing how maximum entropy estimation (MEE) provides an extension of logic to cases where one cannot carry out Aristotelian deductive reasoning. Here I will review two early applications of MEE which I hope will provide some insight into how these ideas were being used in quantum and statistical mechanics around 1960. In May of ‘61 I turned in my Ph.D. thesis (Scalapino, 1961) to Stanford University and, following a well-known dictum, went on to work on other problems. Now, three decades later, on this, the occasion of Ed Jaynes’ seventieth birthday, I decided to look back to see what we were doing when I was first getting to know Ed and his special approach to problems. Not surprisingly, my thesis contains several applications of the principle of Maximum Entropy Estimation (MEE). In 1957 Ed had written two seminal articles (Jaynes, 1957(a), 1957(b)) showing how one could use Shannon's (1948) Information Theory to construct density matrices for a variety of different problems in equilibrium statistical mechanics. I had been very much taken with this work and wanted to understand how it could be applied to other systems.
Preface 1. Recollection of an independent thinker Joel A. Snow 2. A look back: early applications of maximum entropy estimation to quantum statistical mechanics D. J. Scalapino 3. The Jaynes-Cummings revival B. W. Shore and P. L. Knight 4. The Jaynes-Cummings model and the one-atom-master H. Walther 5. The Jaynes-Cummings model is alive and well P. Meystre 6. Self-consistent radiation reaction in quantum optics - Jaynes' influence and a new example in cavity QED J. H. Eberly 7. Enhancing the index of refraction in a nonabsorbing medium: phaseonium versus a mixture of two-level atoms M. O. Scully, T. W. Hansch, M. Fleischhauer, C. H. Keitel and Shi-Yao Zhu 8. Ed Jaynes' steak dinner problem II Michael D. Crisp 9. Source theory of vacuum field effects Peter W. Milonni 10. The natural line shape Edwin A. Power 11. An operational approach to Schrodinger's cat L. Mandel 12. The classical limit of an atom C. R. Stroud, Jr. 13. Mutual radiation reaction in spontaneous emission Richard J. Cook 14. A model of neutron star dynamics F. W. Cummings 15. The kinematic origin of complex wave function David Hestenes 16. On radar target identification C. Ray Smith 17. On the difference in means G. Larry Bretthorst 18. Bayesian analysis, model selection and prediction Arnold Zellner and Chung-ki Min 19. Bayesian numerical analysis John Skilling 20. Quantum statistical inference R. N. Silver 21. Application of the maximum entropy principle to nonlinear systems far from equilibrium H. Haken 22. Nonequilibrium statistical mechanics Baldwin Robertson 23. A backward look to the future E. T. James Appendix. Vita and bibliography of Edwin T. Jaynes Index.
The problem of constructing models for the statistical dynamics of finite-level quantum mechanical systems is considered. The maximum entropy principle formulated by E.T. Jaynes in 1957 and asserting that the entropy of any physical system increases until it attains its maximum value under constraints imposed by other physical laws is applied. In accordance with this principle, the von Neumann entropy is taken for the objective function; a dynamical equation describing the evolution of the density operator in finite-level systems is derived by using the speed gradient principle. In this case, physical constraints are the mass conservation law and the energy conservation law. The stability of the equilibrium points of the system thus obtained is investigated. By using LaSalle’s theorem, it is shown that the density function tends to a Gibbs distribution, under which the entropy attains its maximum. The method is exemplified by analyzing a finite system of identical particles distributed between cells. Results of numerical simulation are presented.
. Can quantum probability theory be applied, beyond the microscopic scale of atoms and quarks, to the human problem of reasoning from incomplete and uncertain data? A unified theory of quantum statistical mechanics and Bayesian statistical inference is proposed. QSI is applied to ordinary data analysis problems such as the interpolation and deconvolution of continuous density functions from both exact and noisy data. The information measure has a classical limit of negative entropy and a quantum limit of Fisher information (kinetic energy). A smoothing parameter analogous to a de Broglie wavelength is determined by Bayesian methods. There is no statistical regularization parameter. A priori criteria are developed for good and bad measurements in an experimental design. The optimal image is estimated along with statistical and incompleteness errors. QSI yields significantly better images than the maximum entropy method, because it explicitly accounts for image continuity. Introduction Jaynes has been an eloquent advocate for a compelling hypothesis: Probability Theory as Logic . That is, probabilities represent degrees of belief; probability theory develops and applies universal principles of logical inference from incomplete information. Two of his primary interests have been Bayesian probability theory and the interpretation of quantum mechanics. Bayesian probability theory yields inferences by systematically and consistently combining new data with prior knowledge. Jaynes pioneered the maximum entropy (ME) class of Bayesian methods for density function estimation (Jaynes, 1983), which has been applied successfully to numerous data analysis problems.
We derive Bose-Einstein statistics and Fermi-Dirac statistics by Principle of Maximum Entropy applied to two families of entropy functions different from the Boltzmann-Gibbs-Shannon entropy. These entropy functions are identified with special cases of modified Naudts' ϕϕϕ-entropy.
The pioneering work of Edwin T. Jaynes in the field of statistical physics, quantum optics, and probability theory has had a significant and lasting effect on the study of many physical problems, ranging from fundamental theoretical questions through to practical applications such as optical image restoration. Physics and Probability is a collection of papers in these areas by some of his many colleagues and former students, based largely on lectures given at a symposium celebrating Jaynes' contributions, on the occasion of his seventieth birthday and retirement as Wayman Crow Professor of Physics at Washington University. The collection contains several authoritative overviews of current research on maximum entropy and quantum optics, where Jaynes' work has been particularly influential, as well as reports on a number of related topics. In the concluding paper, Jaynes looks back over his career, and gives encouragement and sound advice to young scientists. All those engaged in research on any of the topics discussed in these papers will find this a useful and fascinating collection, and a fitting tribute to an outstanding and innovative scientist.
For an unknown quantum state ρ\rhoρ, a set of measurement outcomes provides partial information. The principe of maximum entropy states that the density matrix which best represents the current knowlege is the one with the maximum entropy. However, the maximum entropy estimation (MEE) is hard to calculate in general. In this work, we develop a method that outputs an approximation to the MEE from limited measurement results by using supervised learning techniques. The method estimates the MEE of theoretical data of several different quantum systems with high fidelities. We also test our method experimentally with a photonic set-up, which exhibits high fidelities (all greater than 99.7%) and robustness against experimental errors. In comparison with previous known optimization algorithms for MEE, our method provides adequate fidelities with high efficiency.