AbstractLet $$\mathscr {H}$$
H
be a finite-dimensional complex Hilbert space and $$\mathscr {D}$$
D
the set of density matrices on $$\mathscr {H}$$
H
, i.e., the positive operators with trace 1. Our goal in this note is to identify a probability measure u on $$\mathscr {D}$$
D
that can be regarded as the uniform distribution over $$\mathscr {D}$$
D
. We propose a measure on $$\mathscr {D}$$
D
, argue that it can be so regarded, discuss its properties, and compute the joint distribution of the eigenvalues of a random density matrix distributed according to this measure.