# pr.probability – Maximum Mutual Information of A Matrix Representation

Let $$mathbf{Z}$$ be a $$mtimes n$$ matrix with zero-mean unit-variance i.i.d complex Gaussian entries. What is the maximum value of mutual information $$I(mathbf{X}_1,mathbf{X}_2;mathbf{Y})$$ such that
begin{align} mathbf{Y}=mathbf{X}_1^mathrm{H}mathbf{X}_2+mathbf{X}_1^mathrm{H}mathbf{Z}, end{align}
where $$mathbf{X}_1$$ and $$mathbf{X}_2$$ are two $$mtimes n$$ arbitrary random matrices and $$(.)^mathrm{H}$$ denotes hermitian of matrix.