Hilbert–Schmidt volume of the set of mixed quantum states

We compute the volume of the convex ( N 2 − 1)-dimensional set ℳ N of density matrices of size N with respect to the Hilbert–Schmidt measure. The hyper-area of the boundary of this set is also found and its ratio to the volume provides information about the structure of ℳ N . Similar investigations are also performed for the smaller set of all real density matrices. As an intermediate step, we analyse volumes of the unitary and orthogonal groups and of the flag manifolds.