TY - JOUR T1 - Comment on some results of Erdahl and the convex structure of reduced density matrices JF - Journal of Mathematical Physics Y1 - 2012 A1 - Jianxin Chen A1 - Zhengfeng Ji A1 - Mary Beth Ruskai A1 - Bei Zeng A1 - Duan-Lu Zhou AB - 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. VL - 53 U4 - 072203 UR - http://arxiv.org/abs/1205.3682v1 CP - 7 J1 - J. Math. Phys. U5 - 10.1063/1.4736842 ER - TY - JOUR T1 - Principle of Maximum Entropy and Ground Spaces of Local Hamiltonians Y1 - 2010 A1 - Jianxin Chen A1 - Zhengfeng Ji A1 - Mary Beth Ruskai A1 - Bei Zeng A1 - Duanlu Zhou AB - The structure of the ground spaces of quantum systems consisting of local interactions is of fundamental importance to different areas of physics. In this Letter, we present a necessary and sufficient condition for a subspace to be the ground space of a k-local Hamiltonian. Our analysis are motivated by the concept of irreducible correlations studied by [Linden et al., PRL 89, 277906] and [Zhou, PRL 101, 180505], which is in turn based on the principle of maximum entropy. It establishes a better understanding of the ground spaces of local Hamiltonians and builds an intimate link of ground spaces to the correlations of quantum states. UR - http://arxiv.org/abs/1010.2739v4 ER -