Journal of Theoretical Probability

, Volume 26, Issue 3, pp 855–869 | Cite as

Asymptotic Eigenvalue Distributions of Block-Transposed Wishart Matrices

Article

Abstract

We study the partial transposition W Γ=(id⊗t)WM dn (ℂ) of a Wishart matrix WM dn (ℂ) of parameters (dn,dm). Our main result is that, with d→∞, the law of mW Γ is a free difference of free Poisson laws of parameters m(n±1)/2. Motivated by questions in quantum information theory, we also derive necessary and sufficient conditions for these measures to be supported on the positive half-line.

Keywords

Wishart matrix Partial transposition Free Poisson law 

Mathematics Subject Classification (2000)

60B20 46L54 81P45 

Notes

Acknowledgements

We would like to thank the ANR projects Galoisint and Granma for their financial support during the accomplishment of the present work. Part of this work was done when I.N. was a postdoctoral fellow at the University of Ottawa where he was supported by NSERC discovery grants and an ERA.

References

  1. 1.
    Aubrun, G.: Partial transposition of random states and non-centered semicircular distributions. arXiv:1011.0275
  2. 2.
    Banica, T., Nechita, I.: Block-modified Wishart matrices and free Poisson laws. arXiv:1201.4792
  3. 3.
    Banica, T., Belinschi, S.T., Capitaine, M., Collins, B.: Free Bessel laws. Can. J. Math. 63, 3–37 (2011) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bercovici, H., Voiculescu, D.V.: Regularity questions for free convolution. Oper. Theory Adv. Appl. 104, 37–47 (1998) MathSciNetGoogle Scholar
  5. 5.
    Lenczewski, R.: Asymptotic properties of random matrices and pseudomatrices. arXiv:1001.0667
  6. 6.
    Marchenko, V.A., Pastur, L.A.: Distribution of eigenvalues in certain sets of random matrices. Mat. Sb. 72, 507–536 (1967) MathSciNetGoogle Scholar
  7. 7.
    Nica, A., Speicher, R.: Lectures on the Combinatorics of Free Probability. Cambridge Univ. Press, Cambridge (2006) MATHCrossRefGoogle Scholar
  8. 8.
    Voiculescu, D.V., Dykema, K.J., Nica, A.: Free Random Variables. AMS, Providence (1992) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsCergy-Pontoise UniversityCergy-PontoiseFrance
  2. 2.CNRS, Laboratoire de Physique Théorique, IRSAMCUniversité de Toulouse, UPSToulouseFrance

Personalised recommendations