n-Representability Problem for Reduced Density Matrices

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 (function) and form the corresponding reduced p-particle operator. Tnp¯ 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.

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