Asymptotic eigenvalue distributions of block-transposed Wishart matrices

Abstract

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.

References in corpus (1)

• Asymptotic properties of random matrices and pseudomatrices

Cited by in corpus (1)

• Negativity Spectra in Random Tensor Networks and Holography