I am currently facing the problem of calculating integrals that take the general form
where $P(\sigma)$ is a probability density over the space of mixed quantum states, $d\sigma$ is the Hilbert-Schmidt measure and $R$ is some subregion of state space, which in general can be quite complicated.
Effectively, this can be thought of as a multivariate integral for which Monte Carlo integration techniques are particularly well suited. However, I am new to this numerical technique and would like to have a better understanding of progress in this field before jumping in. So my question is:
Are there any algorithms for Monte Carlo integration that have been specifically constructed for functions of mixed quantum states? Ideally, have integrals of this form been studied before in any other context?