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2011

Asymptotic eigenvalue distributions of block-transposed Wishart matrices

Banica, Teodor, Nechita, Ion

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We study the partial transposition ${W}^\Gamma=(\mathrm{id}\otimes \mathrm{t})W\in M_{dn}(\mathbb C)$ of a Wishart matrix $W\in M_{dn}(\mathbb C)$ of parameters $(dn,dm)$.

Our main result is that, with $d\to\infty$, the law of $m{W}^\Gamma$ is a free difference of free Poisson laws of parameters $m(n\pm 1)/2$.
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.

Asymptotic eigenvalue distributions of block-transposed Wishart matrices · Around

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V.A. Marchenko and L.A. Pastur, Distribution of eigenvalues in certain sets of random matrices, Mat. Sb
1967
Earlier work this paper cites.
D.V. Voiculescu, K.J. Dykema and A. Nica, Free random variables, AMS (1992)
1992
Earlier work this paper cites.

Similar

H. Bercovici and D.V. Voiculescu, Regularity questions for free convolution, Oper. Theory Adv. Appl
1998
Cited alongside, same era.
A. Nica and R. Speicher, Lectures on the combinatorics of free probability, Cambridge Univ. Press (2006)
2006
Cited alongside, same era.
G. Aubrun, Partial transposition of random states and non-centered semicircular distributions, arxiv:1011.0275
Original
Cited in the paper.
T. Banica and I. Nechita, Block-modified Wishart matrices and free Poisson laws, in preparation
Cited in the paper.
R. Lenczewski, Asymptotic properties of random matrices and pseudomatrices, arxiv:1001.0667
Original
Cited in the paper.

Then

T. Banica, S.T. Belinschi, M. Capitaine and B. Collins, Free Bessel laws, Canad. J. Math
2011
Closest in time.

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