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Asymptotic Eigenvalue Distributions of Block-Transposed Wishart Matrices

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Abstract

We study the partial transposition W Γ=(id⊗t)W∈M dn (ℂ) of a Wishart matrix W∈M 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.

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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.

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Authors and Affiliations

  1. Department of Mathematics, Cergy-Pontoise University, 95000, Cergy-Pontoise, France

    Teodor Banica

  2. CNRS, Laboratoire de Physique Théorique, IRSAMC, Université de Toulouse, UPS, 31062, Toulouse, France

    Ion Nechita

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  1. Teodor Banica
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  2. Ion Nechita
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Correspondence to Ion Nechita.

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Banica, T., Nechita, I. Asymptotic Eigenvalue Distributions of Block-Transposed Wishart Matrices. J Theor Probab 26, 855–869 (2013). https://doi.org/10.1007/s10959-012-0409-4

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  • DOI: https://doi.org/10.1007/s10959-012-0409-4

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