n‐Representability Problem for Reduced Density Matrices

TLDR: In this article, it was shown that a density operator Dp belongs to Tnp¯ (the bar indicates the closure with respect to a certain topology) if and only if Tr (DpBp) ≥ 0 for all bounded self-adjoint p-particle operators Bp, such that their n−expansion (pn)ΓpnBp≡ ∑ i1<…<ipBp(i1…ip) is a positive operator in n-particles space.

Abstract: In this paper we prove some theorems about the n‐representability problem for reduced density operators. The first theorem (Theorem 6) sharpens a theorem proved by Garrod and Percus. Let Tnp be the set of all n‐representable p‐density operators. Then a density operator Dp belongs to Tnp¯ (the bar indicates the closure with respect to a certain topology) if and only if Tr (DpBp) ≥ 0 for all bounded self‐adjoint p‐particle operators Bp, such that their n‐expansion (pn)ΓpnBp≡ ∑ i1<…<ipBp(i1…ip) is a positive operator in n‐particle space. Moreover, it is shown that Tnp¯ is the closed convex hull of the exposed points of Tnp of finite one‐rank (Theorem 9). A more practical version of this theorem may be formulated in the following manner (cf. Theorem 8).Consider the set γp of subspaces of the n‐particle space, occurring as an eigenspace to the deepest eigenvalue of a bounded n‐particle operator which is the n expansion of some p‐particle operator. Choose from every element of γp one (and only one) vector (func...