n-Representability Problem for Reduced Density Matrices

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 ${\mathcal{T}}_n^p$ be the set of all n‐representable p‐density operators. Then a density operator D^p belongs to $\overline{{\mathcal{T}}_n^p}$ (the bar indicates the closure with respect to a certain topology) if and only if Tr (D^pB^p) ≥ 0 for all bounded self‐adjoint p‐particle operators B^p, such that their n‐expansion $${(}_p^n{\mathrm{)\Gamma }}_p^n{B}^p\equiv {\displaystyle \sum _{i_1{\mathrm{<\dots <i}}_p}}{B}^p{\mathrm{(i}}_1{\mathrm{\dots i}}_p)$$ is a positive operator in n‐particle space. Moreover, it is shown that $\overline{{\mathcal{T}}_n^p}$ is the closed convex hull of the exposed points of ${\mathcal{T}}_n^p$ 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 γ^p 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 γ^p one (and only one) vector (function) and form the corresponding reduced p‐particle operator. $\overline{{\mathcal{T}}_n^p}$ is the closed convex hull of all these p‐density operators (cf. Theorem 9). For p = 1, this theorem reduces to Coleman's theorem about the n representability of the 1 matrix.