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Freeness and The Partial Transposes of Wishart Random Matrices

  • James A. Mingo (a1) and Mihai Popa (a2) (a3)

Abstract

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.

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Copyright

Footnotes

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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.

Footnotes

References

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[1] 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.
[2] 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.
[3] Banica, T. and Nechita, I., Asymptotic eigenvalue distributions of block-transposed Wishart matrices . J. Theor. Probab. 26(2013), 855869. https://doi.org/10.1007/s10959-012-0409-4.
[4] Biane, P., Some properties of crossings and partitions . Discrete Mathematics 175(1997), 4153. https://doi.org/10.1016/S0012-365X(96)00139-2.
[5] Cori, R., Un code pour les graphes planaires et ses applications. Astérisque, 27, Société Mathématique de France, Paris, 1975.
[6] 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.
[7] Janson, S., Gaussian Hilbert spaces. Cambridge Tracts in Mathematics, 129, Cambridge University Press, Cambridge, 1997.
[8] 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.
[9] Mingo, J. A. and Popa, M., Freeness and the transposes of unitarily invariant random matrices . J. Funct. Anal. 271(2016), 883921. https://doi.org/10.1016/j.jfa.2016.05.006.
[10] 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.
[11] Nica, A. and Speicher, R., Lectures on the combinatorics of free probability. Cambridge University Press, Cambridge, 2006.
[12] Redelmeier, C. E. I., Genus expansion for real Wishart matrices . J. Theoret. Probab. 24(2011), 10441062. https://doi.org/10.1007/s10959-010-0278-7.
[13] 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.
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