  High-quality random samples of quantum states are needed for a variety of
tasks in quantum information and quantum computation. Searching the
high-dimensional quantum state space for a global maximum of an objective
function with many local maxima or evaluating an integral over a region in the
quantum state space are but two exemplary applications of many. These tasks can
only be performed reliably and efficiently with Monte Carlo methods, which
involve good samplings of the parameter space in accordance with the relevant
target distribution. We show how the standard strategies of rejection sampling,
importance sampling, and Markov-chain sampling can be adapted to this context,
where the samples must obey the constraints imposed by the positivity of the
statistical operator. For a comparison of these sampling methods, we generate
sample points in the probability space for two-qubit states probed with a
tomographically incomplete measurement, and then use the sample for the
calculation of the size and credibility of the recently-introduced optimal
error regions [see New J. Phys. 15 (2013) 123026]. Another illustration is the
computation of the fractional volume of separable two-qubit states.
