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Freeness and The Partial Transposes of Wishart Random Matrices

Part of: Special matrices Probability theory on algebraic and topological structures Selfadjoint operator algebras

Published online by Cambridge University Press:  09 January 2019

James A. Mingo  and
Mihai Popa
James A. Mingo
Affiliation:
Department of Mathematics and Statistics, Queen’s University, Jeffery Hall, Kingston, Ontario K7L 3N6 Email: mingo@mast.queensu.ca
Mihai Popa
Affiliation:
Department of Mathematics, The University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249, USA Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, Bucharest, RO-70700, Romania Email: mihai.popa@utsa.edu
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Abstract

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We show that the partial transposes of complex Wishart random matrices are asymptotically free. We also investigate regimes where the number of blocks is fixed but the size of the blocks increases. This gives an example where the partial transpose produces freeness at the operator level. Finally, we investigate the case of real Wishart matrices.

Keywords

free probabilityrandom matrixpartial transposequantum information theory

MSC classification

Primary: 15B52: Random matrices
Secondary: 46L54: Free probability and free operator algebras 60B20: Random matrices (probabilistic aspects; for algebraic aspects see )
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 3 , June 2019 , pp. 659 - 681
Copyright
© Canadian Mathematical Society 2018 

Footnotes

Research of both authors was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. Research of author M.P. was also supported by the Simons Foundation, grant No. 360242.

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