Asymptotic eigenvalue distributions of block-transposed Wishart matrices

We study the partial transposition ${W}^\Gamma=($\mathrm{i}\mathrm{d}$\otimes $\mathrm{t}$)W\in M_{dn}(\mathbb C)$ofaWishartmatrix$W\in M_{dn}(\mathbb C)$ofparameters$(dn,dm)$\mathrm{.}Ourmainresultisthat,with$d\to\infty$,thelawof$m{W}^\Gamma$isafreedifferenceoffreePoissonlawsofparameters$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.