  In a number of previous studies, we have investigated the use of the volume
element of the Bures (minimal monotone) metric -- identically, one-fourth of
the statistical distinguishability (SD) metric -- as a natural measure over the
(n^2-1)-dimensional convex set of n x n density matrices. This has led us for
the cases n = 4 and 6 to estimates of the prior (Bures/SD) probabilities that
qubit-qubit and qubit-qutrit pairs are separable. Here, we extend this work
from such bipartite systems to the tripartite "laboratory'' quantum systems
possessing U x U x U symmetry recently constructed by Eggeling and Werner
(Phys. Rev. A 63 [2001], 042324). We derive the associated SD metric tensors
for the three-qubit and three-qutrit cases, and then obtain estimates of the
various related Bures/SD probabilities using Monte Carlo methods.
