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...

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Vydáno v:Annals of physics Ročník 324; číslo 11; s. 2278 - 2358
Hlavní autoři: Adachi, Satoshi, Toda, Mikito, Kubotani, Hiroto
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Elsevier Inc 01.11.2009
Elsevier BV
Témata:
CHAOS THEORY
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
DISTRIBUTION FUNCTIONS
EIGENVALUES
ENERGY LEVELS
Fixed-trace ensemble
INTEGRALS
MATHEMATICAL EVOLUTION
MATRICES
Matrix
Physics
PROBABILITY
Quantum chaos
QUANTUM ENTANGLEMENT
QUANTUM MECHANICS
Quantum theory
Random matrix theory
RANDOMNESS
Schmidt decomposition
Selberg integral
Selberg–Kaneko integral
Selberg integral
03.65.Yz
Random matrix theory
Quantum entanglement
03.67.Mn
Fixed-trace ensemble
03.65.Ud
Quantum chaos
05.45.Mt
Schmidt decomposition
Selberg–Kaneko integral
ISSN:0003-4916, 1096-035X
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Shrnutí: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 Petkovšek–Wilf–Zeilberger theory that calculates definite hypergeometric sums in a closed form.
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content type line 14
ISSN:0003-4916
1096-035X
DOI:10.1016/j.aop.2009.04.007