Hilbert--Schmidt volume of the set of mixed quantum states
Abstract
We compute the volume of the convex N2-1 dimensional set MN 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 an information about the complex structure of MN. Similar investigations are also performed for the smaller set of all real density matrices. As an intermediate step we analyze volumes of the unitary and orthogonal groups and of the flag manifolds.
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