Limiting Eigenvalue Distribution of Random Matrices Involving Tensor Product

We consider two classes of \(n \times n\) sample covariance matrices arising in quantum informatics. The first class consists of matrices whose data matrix has \(m\) independent columns each of which is the tensor product of \(k\) independent \(d\)-dimensional vectors, thus \(n=d^k\). The matrices of the second class belong to \(\mathcal{M}_n(\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}), \ n=d_1 d_2\) and are obtained from the standard sample covariance matrices by the partial transposition in \(\mathbb{C}^{d_2}\). We find that for the first class the limiting eigenvalue counting measure is the standard MP law despite the strong statistical dependence of the entries while for the second class the limiting eigenvalue counting measure is the shifted semicircle.