Comment on some results of Erdahl and the convex structure of reduced density matrices

In J. Math. Phys. 13, 1608-1621 (1972), Erdahl considered the convex
structure of the set of $N$-representable 2-body reduced density matrices in
the case of fermions. Some of these results have a straightforward extension to
the $m$-body setting and to the more general quantum marginal problem. We
describe these extensions, but can not resolve a problem in the proof of
Erdahl's claim that every extreme point is exposed in finite dimensions.
Nevertheless, we can show that when $2m \geq N$ every extreme point of the set
of $N$-representable $m$-body reduced density matrices has a unique pre-image
in both the symmetric and anti-symmetric setting. Moreover, this extends to the
quantum marginal setting for a pair of complementary $m$-body and $(N-m)$-body
reduced density matrices.