Asymptotic Eigenvalue Distributions of Block-Transposed Wishart Matrices

We study the partial transposition W (I)=(idaSut)WaM (dn) (a,) of a Wishart matrix WaM (dn) (a,) of parameters (dn,dm). Our main result is that, with d -> a, the law of mW (I) 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.