Asymptotic eigenvalue distributions of block-transposed Wishart matrices (Q376256)

Given a \(d \times d\) matrix consisting of \(n \times n\) blocks, one obtains its partial transpose by transposing each individual \(n \times n\) block. It was observed by \textit{G. Aubrun} [Random Matrices Theory Appl. 1, No. 2, 1250001, 29 p. (2012; Zbl 1239.60002)] that, roughly speaking, if the partial transpose is performed on a Wishart matrix, then the Marčenko-Pastur law becomes a shifted semicircle law. In the paper under review, this finding is further elucidated within the framework of free probability, demonstrating that the partial transpose relates the Marčenko-Pastur law (the free Poisson distribution) to free differences of free Poisson laws. Furthermore, a positivity result for these measures is proven, which is of interest to quantum information theory.