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 and independent by von Neumann which deals with doubly stochastic matrices.

I do not work in probability theory. I have heard that the above result is the 'only' result in this direction. 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 also.

I have asked this question already math.stackexchange; but did not get any reply. Hence I decided to ask it here. I My motivation is to look at the analogues problem in a quantum set up. In particular see paper 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.


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