Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace

Akemann G, Vivo P (2011)
J.Stat.Mech. 2011(05).

Journal Article | Published | English

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Abstract
The degree of entanglement of random pure states in bipartite quantum systemscan be estimated from the distribution of the extreme Schmidt eigenvalues. Fora bipartition of size M\geq N, these are distributed according to aWishart-Laguerre ensemble (WL) of random matrices of size N x M, with afixed-trace constraint. We first compute the distribution and moments of thesmallest eigenvalue in the fixed trace orthogonal WL ensemble for arbitraryM\geq N. Our method is based on a Laplace inversion of the recursive resultsfor the corresponding orthogonal WL ensemble by Edelman. Explicit examples aregiven for fixed N and M, generalizing and simplifying earlier results. In themicroscopic large-N limit with M-N fixed, the orthogonal and unitary WLdistributions exhibit universality after a suitable rescaling and are thereforeindependent of the constraint. We prove that very recent results given in termsof hypergeometric functions of matrix argument are equivalent to more explicitexpressions in terms of a Pfaffian or determinant of Bessel functions. Whilethe latter were mostly known from the random matrix literature on the QCD Diracoperator spectrum, we also derive some new results in the orthogonal symmetryclass.
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Akemann G, Vivo P. Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace. J.Stat.Mech. 2011;2011(05).
Akemann, G., & Vivo, P. (2011). Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace. J.Stat.Mech., 2011(05).
Akemann, G., and Vivo, P. (2011). Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace. J.Stat.Mech. 2011.
Akemann, G., & Vivo, P., 2011. Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace. J.Stat.Mech., 2011(05).
G. Akemann and P. Vivo, “Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace”, J.Stat.Mech., vol. 2011, 2011.
Akemann, G., Vivo, P.: Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace. J.Stat.Mech. 2011, (2011).
Akemann, Gernot, and Vivo, Pierpaolo. “Compact smallest eigenvalue expressions in Wishart-Laguerre ensembles with or without fixed-trace”. J.Stat.Mech. 2011.05 (2011).
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