Asymptotic free independence and entry permutations for Gaussian random matrices

Popa, Mihai

doi:10.1090/proc/15783

Asymptotic free independence and entry permutations for Gaussian random matrices
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by Mihai Popa

Proc. Amer. Math. Soc. 150 (2022), 3379-3394

DOI: https://doi.org/10.1090/proc/15783

Published electronically: May 13, 2022

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Abstract:

The paper presents conditions on entry permutations that induce asymptotic freeness when acting on Gaussian random matrices. The class of permutations described includes the matrix transpose, as well as entry permutations relevant in Quantum Information Theory and Quantum Physics.

References

Octavio Arizmendi, Ion Nechita, and Carlos Vargas, On the asymptotic distribution of block-modified random matrices, J. Math. Phys. 57 (2016), no. 1, 015216, 25. MR 3432744, DOI 10.1063/1.4936925
Guillaume Aubrun, Stanisław Szarek, and Elisabeth Werner, Hastings’s additivity counterexample via Dvoretzky’s theorem, Comm. Math. Phys. 305 (2011), no. 1, 85–97. MR 2802300, DOI 10.1007/s00220-010-1172-y
Teodor Banica and Ion Nechita, Asymptotic eigenvalue distributions of block-transposed Wishart matrices, J. Theoret. Probab. 26 (2013), no. 3, 855–869. MR 3090554, DOI 10.1007/s10959-012-0409-4
Benoît Collins and Ion Nechita, Random matrix techniques in quantum information theory, J. Math. Phys. 57 (2016), no. 1, 015215, 34. MR 3432743, DOI 10.1063/1.4936880
Thibault Damour, Sophie de Buyl, Marc Henneaux, and Christiane Schomblond, Einstein billiards and overextensions of finite-dimensional simple Lie algebras, J. High Energy Phys. 8 (2002), no. 30, 28. MR 1942156, DOI 10.1088/1126-6708/2002/08/030
Edward G. Effros and Mihai Popa, Feynman diagrams and Wick products associated with $q$-Fock space, Proc. Natl. Acad. Sci. USA 100 (2003), no. 15, 8629–8633. MR 1994546, DOI 10.1073/pnas.1531460100
MichałHorodecki, PawełHorodecki, and Ryszard Horodecki, Separability of mixed states: necessary and sufficient conditions, Phys. Lett. A 223 (1996), no. 1-2, 1–8. MR 1421501, DOI 10.1016/S0375-9601(96)00706-2
Svante Janson, Gaussian Hilbert spaces, Cambridge Tracts in Mathematics, vol. 129, Cambridge University Press, Cambridge, 1997. MR 1474726, DOI 10.1017/CBO9780511526169
A. Mandarino, T. Linowski, and K. Życzowski, Bipartite unitary gates and billiard dynamics in the Weyl chamber, Phys. Rev. A 98 (2018), 012335, arXiv:1710.10983 [quant-ph].
James A. Mingo and Mihai Popa, Freeness and the transposes of unitarily invariant random matrices, J. Funct. Anal. 271 (2016), no. 4, 883–921. MR 3507993, DOI 10.1016/j.jfa.2016.05.006
James A. Mingo and Mihai Popa, Freeness and the partial transposes of Wishart random matrices, Canad. J. Math. 71 (2019), no. 3, 659–681. MR 3952494, DOI 10.4153/cjm-2018-002-2
J. A. Mingo and M. Popa, Freeness and the partial transpose of Wishart random matrices: Part II, Preprint, arXiv:2005.04348, 2020.
James A. Mingo and Roland Speicher, Free probability and random matrices, Fields Institute Monographs, vol. 35, Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2017. MR 3585560, DOI 10.1007/978-1-4939-6942-5
Alexandru Nica and Roland Speicher, Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, vol. 335, Cambridge University Press, Cambridge, 2006. MR 2266879, DOI 10.1017/CBO9780511735127
Zhiwei Hao and Mihai Popa, A combinatorial result on asymptotic independence relations for random matrices with non-commutative entries, J. Operator Theory 80 (2018), no. 1, 47–76. MR 3835448, DOI 10.7900/jot.2017may21.2170
Mihai Popa and Zhiwei Hao, An asymptotic property of large matrices with identically distributed Boolean independent entries, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 22 (2019), no. 4, 1950024, 22. MR 4065491, DOI 10.1142/S0219025719500243