We study various methods to generate ensembles of random density matrices of a fixed size N, obtained by partial trace of pure states on composite systems. Structured ensembles of random pure states, invariant with respect to local unitary transformations are introduced. To analyze statistical properties of quantum entanglement in bi-partite systems we analyze the distribution of Schmidt coefficients of random pure states. Such a distribution is derived in the case of a superposition of k random maximally entangled states. For another ensemble, obtained by performing selective measurements in a maximally entangled basis on a multi-partite system, we show that this distribution is given by the Fuss-Catalan law and find the average entanglement entropy. A more general class of structured ensembles proposed, containing also the case of Bures, forms an extension of the standard ensemble of structureless random pure states, described asymptotically, as N → ∞, by the Marchenko-Pastur distribution.
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June 2011
Research Article|
June 09 2011
Generating random density matrices
Karol Życzkowski;
Karol Życzkowski
a)
1Institute of Physics,
Jagiellonian University
, ul. Reymonta 4, 30-059 Kraków, Poland
2Centrum Fizyki Teoretycznej,
Polska Akademia Nauk
, Al. Lotników 32/44, 02-668 Warszawa, Poland
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Karol A. Penson;
Karol A. Penson
b)
3
Université Paris VI
, Laboratoire de Physique de la Matière Condensée (LPTMC), CNRS UMR 7600, t.13, 5ème ét. BC.121, 4, pl. Jussieu, F 75252 Paris Cedex 05, France
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Ion Nechita;
Ion Nechita
c)
4Department of Mathematics and Statistics,
University of Ottawa
, Ontario K1N6N5, Canada
5Laboratoire de Physique Théorique du CNRS, IRSAMC,
Université de Toulouse
, UPS, F-31062 Toulouse, France
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Benoît Collins
Benoît Collins
d)
4Department of Mathematics and Statistics,
University of Ottawa
, Ontario K1N6N5, Canada
6CNRS,
Institut Camille Jordan Université Lyon 1
, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne, France
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a)
Electronic mail: karol@tatry.if.uj.edu.pl.
b)
Electronic mail: penson@lptl.jussieu.fr.
c)
Electronic mail: inechita@uottawa.ca.
d)
Electronic mail: bcollins@uottawa.ca.
J. Math. Phys. 52, 062201 (2011)
Article history
Received:
December 09 2010
Accepted:
May 07 2011
Citation
Karol Życzkowski, Karol A. Penson, Ion Nechita, Benoît Collins; Generating random density matrices. J. Math. Phys. 1 June 2011; 52 (6): 062201. https://doi.org/10.1063/1.3595693
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