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Freeness and The Partial Transposes of Wishart Random Matrices

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

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 )

Information

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.

References

Arizmendi, O., Nechita, I., and Vargas, C., On the asymptotic distribution of block-modified random matrices . J. Math. Phys. 57(2016), no. 1, 015216. https://doi.org/10.1063/1.4936925.Google Scholar
Aubrun, G., Partial transposition of random states and non-centered semicircular distributions . Random Matrices Theory Appl. 1(2012), no. 2, 1250001. https://doi.org/10.1142/S2010326312500013.Google Scholar
Banica, T. and Nechita, I., Asymptotic eigenvalue distributions of block-transposed Wishart matrices . J. Theor. Probab. 26(2013), 855–869. https://doi.org/10.1007/s10959-012-0409-4.Google Scholar
Biane, P., Some properties of crossings and partitions . Discrete Mathematics 175(1997), 41–53. https://doi.org/10.1016/S0012-365X(96)00139-2.Google Scholar
Cori, R., Un code pour les graphes planaires et ses applications. Astérisque, 27, Société Mathématique de France, Paris, 1975.Google Scholar
Fukuda, M. and Śniady, P., Partial transpose of random quantum states: exact formulas and meanders . J. Math. Phys. 54(2013), no. 4, 042202. https://doi.org/10.1063/1.4799440.Google Scholar
Janson, S., Gaussian Hilbert spaces. Cambridge Tracts in Mathematics, 129, Cambridge University Press, Cambridge, 1997.Google Scholar
Mingo, J. A. and Popa, M., Real second order freeness and Haar orthogonal matrices . J. Math. Phys. 54(2013), no. 5, 051701. https://doi.org/10.1063/1.4804168.Google Scholar
Mingo, J. A. and Popa, M., Freeness and the transposes of unitarily invariant random matrices . J. Funct. Anal. 271(2016), 883–921. https://doi.org/10.1016/j.jfa.2016.05.006.Google Scholar
Mingo, J. A. and Speicher, R., Free probability and random matrices. Fields Institute Monographs, 35, Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2017.Google Scholar
Nica, A. and Speicher, R., Lectures on the combinatorics of free probability. Cambridge University Press, Cambridge, 2006.Google Scholar
Redelmeier, C. E. I., Genus expansion for real Wishart matrices . J. Theoret. Probab. 24(2011), 1044–1062. https://doi.org/10.1007/s10959-010-0278-7.Google Scholar
Redelmeier, C. E. I., Real second-order freeness and the asymptotic real second-order freeness of several real matrix models. Int. Math. Res. Not. IMRN 2014, no. 12, 3353–3395.Google Scholar