Bures volume of the set of mixed quantum states

We compute the volume of the (N² − 1)-dimensional set _N of density matrices of size N with respect to the Bures measure and show that it is equal to that of an (N² − 1)-dimensional hyper-hemisphere of radius 1/2. For N = 2 we obtain the volume of the Uhlmann hemisphere, ½S³ ⊂ ⁴. We find also the area of the boundary of the set _N and obtain analogous results for the smaller set of all real density matrices. An explicit formula for the Bures–Hall normalization constants is derived for an arbitrary N.