Monte Carlo integration over space of quantum states

Question

I am currently facing the problem of calculating integrals that take the general form

$\int_{R} P(\sigma)d\sigma$

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?

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Comments

Juan, welcome here and thanks for asking. However, one sentence (that with `However,`) seems to be broken. Could you fix it?

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Do you want something simple like the mean of $P(\sigma)$ or the mean of some function $f(\sigma)$ with respect to $P(\sigma)$. As it is written now, the value of the integral you wrote is just 1.

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Piotr: Thanks for your suggestion, I have amended the text. Chris: Roughly, my goal is to compute the probability that a state lies in a subregion $R$ of all possible quantum states e.g. the set of entangled states. So the integral is not taken over the entire state space, but one can easily see how it can in general be very difficult to calculate analytically.

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Answer 1

There are two that I know of in the context of state estimation. The first is for estimating the mean of $P$ and is a Metropolis-Hasting MCMC algorithm here: Optimal, reliable estimation of quantum states. The second is also mainly for computing the mean (but can do other functions -- including the characteristic function of the region you are interested in). It is a Sequential Monte Carlo algorithm and is here: Adaptive Bayesian Quantum Tomography.

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