  We consider an ensemble of random density matrices distributed according to
the Bures measure. The corresponding joint probability density of eigenvalues
is described by the fixed trace Bures-Hall ensemble of random matrices which,
in turn, is related to its unrestricted trace counterpart via a Laplace
transform. We investigate the spectral statistics of both these ensembles and,
in particular, focus on the level density, for which we obtain exact
closed-form results involving Pfaffians. In the fixed trace case, the level
density expression is used to obtain an exact result for the average
Havrda-Charv\'at-Tsallis (HCT) entropy as a finite sum. Averages of von Neumann
entropy, linear entropy and purity follow by considering appropriate limits in
the average HCT expression. Based on exact evaluations of the average von
Neumann entropy and the average purity, we also conjecture very simple formulae
for these, which are similar to those in the Hilbert-Schmidt ensemble.
