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Title: Random matrix theory of singular values of rectangular complex matrices I: Exact formula of one-body distribution function in fixed-trace ensemble

The fixed-trace ensemble of random complex matrices is the fundamental model that excellently describes the entanglement in the quantum states realized in a coupled system by its strongly chaotic dynamical evolution [see H. Kubotani, S. Adachi, M. Toda, Phys. Rev. Lett. 100 (2008) 240501]. The fixed-trace ensemble fully takes into account the conservation of probability for quantum states. The present paper derives for the first time the exact analytical formula of the one-body distribution function of singular values of random complex matrices in the fixed-trace ensemble. The distribution function of singular values (i.e. Schmidt eigenvalues) of a quantum state is so important since it describes characteristics of the entanglement in the state. The derivation of the exact analytical formula utilizes two recent achievements in mathematics, which appeared in 1990s. The first is the Kaneko theory that extends the famous Selberg integral by inserting a hypergeometric type weight factor into the integrand to obtain an analytical formula for the extended integral. The second is the Petkovsek-Wilf-Zeilberger theory that calculates definite hypergeometric sums in a closed form.
Authors:
 [1] ;  [2] ;  [3]
  1. Department of Physics, Faculty of Science, Tokyo Institute of Technology, Meguro 152-8550 (Japan), E-mail: adachi@aa.ap.titech.ac.jp
  2. Department of Physics, Faculty of Science, Nara Womens University, Nara 630-8506 (Japan), E-mail: toda@ki-rin.phys.nara-wu.ac.jp
  3. Institute of Physics, Faculty of Engineering, Kanagawa University, Yokohama 221-8686 (Japan), E-mail: kubotani@yukawa.kyoto-u.ac.jp
Publication Date:
OSTI Identifier:
21308043
Resource Type:
Journal Article
Resource Relation:
Journal Name: Annals of Physics (New York); Journal Volume: 324; Journal Issue: 11; Other Information: DOI: 10.1016/j.aop.2009.04.007; PII: S0003-4916(09)00089-X; Copyright (c) 2009 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CHAOS THEORY; DISTRIBUTION FUNCTIONS; EIGENVALUES; ENERGY LEVELS; INTEGRALS; MATHEMATICAL EVOLUTION; MATRICES; PROBABILITY; QUANTUM ENTANGLEMENT; QUANTUM MECHANICS; RANDOMNESS