Geodesic
Freeness and The Partial Transposes of Wishart Random Matrices
Canadian journal of mathematics, Tome 71 (2019) no. 3, pp. 659-681

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CJM-2018-002-2
Mots-clés : free probability, random matrix, partial transpose, quantum information theory 
Mingo, James A.; Popa, Mihai. Freeness and The Partial Transposes of Wishart Random Matrices. Canadian journal of mathematics, Tome 71 (2019) no. 3, pp. 659-681. doi: 10.4153/CJM-2018-002-2
@article{10_4153_CJM_2018_002_2,
     author = {Mingo, James A. and Popa, Mihai},
     title = {Freeness and {The} {Partial} {Transposes} of {Wishart} {Random} {Matrices}},
     journal = {Canadian journal of mathematics},
     pages = {659--681},
     year = {2019},
     volume = {71},
     number = {3},
     doi = {10.4153/CJM-2018-002-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-2018-002-2/}
}
TY  - JOUR
AU  - Mingo, James A.
AU  - Popa, Mihai
TI  - Freeness and The Partial Transposes of Wishart Random Matrices
JO  - Canadian journal of mathematics
PY  - 2019
SP  - 659
EP  - 681
VL  - 71
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-2018-002-2/
DO  - 10.4153/CJM-2018-002-2
ID  - 10_4153_CJM_2018_002_2
ER  - 
%0 Journal Article
%A Mingo, James A.
%A Popa, Mihai
%T Freeness and The Partial Transposes of Wishart Random Matrices
%J Canadian journal of mathematics
%D 2019
%P 659-681
%V 71
%N 3
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-2018-002-2/
%R 10.4153/CJM-2018-002-2
%F 10_4153_CJM_2018_002_2

[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. . Google Scholar | DOI

[2] Aubrun, G., Partial transposition of random states and non-centered semicircular distributions . Random Matrices Theory Appl. 1(2012), no. 2, 1250001. . Google Scholar | DOI

[3] Banica, T. and Nechita, I., Asymptotic eigenvalue distributions of block-transposed Wishart matrices . J. Theor. Probab. 26(2013), 855–869. . Google Scholar | DOI

[4] Biane, P., Some properties of crossings and partitions . Discrete Mathematics 175(1997), 41–53. . Google Scholar | DOI

[5] Cori, R., Un code pour les graphes planaires et ses applications. Astérisque, 27, Société Mathématique de France, Paris, 1975. Google Scholar

[6] Fukuda, M. and Śniady, P., Partial transpose of random quantum states: exact formulas and meanders . J. Math. Phys. 54(2013), no. 4, 042202. . Google Scholar | DOI

[7] Janson, S., Gaussian Hilbert spaces. Cambridge Tracts in Mathematics, 129, Cambridge University Press, Cambridge, 1997. Google Scholar

[8] Mingo, J. A. and Popa, M., Real second order freeness and Haar orthogonal matrices . J. Math. Phys. 54(2013), no. 5, 051701. . Google Scholar | DOI

[9] Mingo, J. A. and Popa, M., Freeness and the transposes of unitarily invariant random matrices . J. Funct. Anal. 271(2016), 883–921. . Google Scholar | DOI

[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. Google Scholar

[11] Nica, A. and Speicher, R., Lectures on the combinatorics of free probability. Cambridge University Press, Cambridge, 2006. Google Scholar

[12] Redelmeier, C. E. I., Genus expansion for real Wishart matrices . J. Theoret. Probab. 24(2011), 1044–1062. . Google Scholar | DOI

[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. Google Scholar

Cité par Sources :