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 WΓ=(id⊗t)WMdn(ℂ) of a Wishart matrix WMdn(ℂ) of parameters (dn,dm). Our main result is that, with d→∞, the law of mWΓ 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 

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

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