Random matrix theory of singular values of rectangular complex matrices I: Exact formula of onebody distribution function in fixedtrace ensemble
Abstract
The fixedtrace 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 fixedtrace 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 onebody distribution function of singular values of random complex matrices in the fixedtrace 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 PetkovsekWilfZeilberger theory that calculates definite hypergeometric sums in a closed form.
 Authors:
 Department of Physics, Faculty of Science, Tokyo Institute of Technology, Meguro 1528550 (Japan), Email: adachi@aa.ap.titech.ac.jp
 Department of Physics, Faculty of Science, Nara Womens University, Nara 6308506 (Japan), Email: toda@kirin.phys.narawu.ac.jp
 Institute of Physics, Faculty of Engineering, Kanagawa University, Yokohama 2218686 (Japan), Email: kubotani@yukawa.kyotou.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: S00034916(09)00089X; 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
Citation Formats
Adachi, Satoshi, Toda, Mikito, and Kubotani, Hiroto. Random matrix theory of singular values of rectangular complex matrices I: Exact formula of onebody distribution function in fixedtrace ensemble. United States: N. p., 2009.
Web. doi:10.1016/j.aop.2009.04.007.
Adachi, Satoshi, Toda, Mikito, & Kubotani, Hiroto. Random matrix theory of singular values of rectangular complex matrices I: Exact formula of onebody distribution function in fixedtrace ensemble. United States. doi:10.1016/j.aop.2009.04.007.
Adachi, Satoshi, Toda, Mikito, and Kubotani, Hiroto. 2009.
"Random matrix theory of singular values of rectangular complex matrices I: Exact formula of onebody distribution function in fixedtrace ensemble". United States.
doi:10.1016/j.aop.2009.04.007.
@article{osti_21308043,
title = {Random matrix theory of singular values of rectangular complex matrices I: Exact formula of onebody distribution function in fixedtrace ensemble},
author = {Adachi, Satoshi and Toda, Mikito and Kubotani, Hiroto},
abstractNote = {The fixedtrace 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 fixedtrace 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 onebody distribution function of singular values of random complex matrices in the fixedtrace 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 PetkovsekWilfZeilberger theory that calculates definite hypergeometric sums in a closed form.},
doi = {10.1016/j.aop.2009.04.007},
journal = {Annals of Physics (New York)},
number = 11,
volume = 324,
place = {United States},
year = 2009,
month =
}

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