  We present marginal cumulative distribution functions (CDF) for density
matrices $\rho$ of fixed purity
$\tfrac{1}{N}\le\mu_N(\rho)=\textrm{Tr}[\rho^2]\le 1$ for arbitrary dimension
$N$. We give closed form analytic formulas for the cases $N=2$ (trivial), $N=3$
and $N=4$, and present a prescription for CDFs of higher arbitrary dimensions.
These formulas allows one to uniformly sample density matrices at a user
selected, fixed constant purity, and also detail how these density matrices are
distributed nonlinearly in the range $\mu_N(\rho)\in[\tfrac{1}{N}, 1]$. As an
illustration of these formulas, we compare the logarithmic negativity and
quantum discord to the (Wootter's) concurrence spanning a range of fixed purity
values in $\mu_4(\rho)\in[\tfrac{1}{4}, 1]$ for the case of $N=4$ (two qubits).
We also investigate the distribution of eigenvalues of a reduced
$N$-dimensional obtained by tracing out the reservoir of its higher-dimensional
purification. Lastly, we numerically investigate a recently proposed
complementary-quantum correlation conjecture which lower bounds the quantum
mutual information of a bipartite system by the sum of classical mutual
informations obtained from two pairs of mutually unbiased measurements.
Finally, numerical implementation issues for the computation of the CDFs and
inverse CDFs necessary for uniform sampling $\rho$ for fixed purity at very
high dimension are briefly discussed.
