One of the well-known problems of classical probability theory is the determination of the set of all extreme points in the convex set of all probability distributions in a product Borel space $\left (X \times Y, \:\:{\cal F} \times {\cal G} \right )$ with fixed marginal distributions $\mu$ and $\nu$ on $\left (X, {\cal F} \right )$ and $\left (Y, {\cal G} \right )$ respectively. Denote this convex set by $C(\mu, \nu).$ 

When $X=Y=${$1,2,\ldots,n$} and  ${\cal F} = {\cal G}$ is the field of all subsets of $X$ and $\mu = \nu$ is the uniform distribution then the problem is answered by the Birkhoff.

I do not work in probability theory. I have heard that the above result is the 'only' result in this direction. Is it true? Can someone please tell me the present status of the above problem. It is nice, if anybody can point out survey article(s) on this matter. 

I have asked this question already [here][1] but did not get any reply. Hence decided to ask  it here. I My motivation is to look at the analogues  problem in a quantum set up. In particular see [paper][2] and its followup(s). Birkhoff's theorem and methods are used in some recent works regarding quantum channels as well. Hence I want to know the present status of the classical problem. Advanced thanks for all the helps.  

  [1]: http://math.stackexchange.com/q/167864/28724
  [2]: http://dx.doi.org/10.1016/j.anihpb.2003.10.009