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

TL;DR

This paper extends Erdahl's results on the convex structure of N-representable reduced density matrices to more general quantum marginal problems, establishing uniqueness of pre-images under certain conditions.

Contribution

It generalizes Erdahl's convex structure results to m-body and quantum marginal settings, proving uniqueness of extreme points' pre-images when 2m ≥ N.

Findings

When 2m ≥ N, every extreme point has a unique pre-image.

Extensions to quantum marginal problems for complementary m- and (N-m)-body reductions.

Partial resolution of Erdahl's claim regarding exposed points in finite dimensions.

Abstract

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.