# Monte Carlo integration over space of quantum states

+ 9 like - 0 dislike
17 views

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?

This post has been migrated from (A51.SE)
retagged Apr 19, 2014
Juan, welcome here and thanks for asking. However, one sentence (that with However,) seems to be broken. Could you fix it?

This post has been migrated from (A51.SE)
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.

This post has been migrated from (A51.SE)
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.

This post has been migrated from (A51.SE)

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.
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOve$\varnothing$flowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.