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Journal of Mathematical Physics

Some aspects of the N-representability problem in finite dimensions. II. Operator endomorphisms which induce sufficient conditions

H. Kummer, Ilyas Absar

Published Feb 1, 1977ā€¢ 12 citations
Publisher

Abstract

Given the set š’®N of all operator endomorphisms T with the property that whenever D is any 2-density operator, then T (D) is N-representable is considered. We specify a pair {Q2,PN} of extreme points of š’®N which are related to each other via the particle-hole duality (Theorem 1.2). The rest of the paper is devoted to a complete characterization of the set ŜN of all operator endomorphisms T with the property that whenever D is any 2-density operator then T (D) is quasi-N-representable, i.e., satisfies the necessary conditions Id, QN, BN, and CN.

DOI
10.1063/1.523273

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Approach to N representability

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Abstract In this paper, some operator endomorphisms which give rise to conditions for N representability of pth order density matrices are presented.

International Journal of Quantum Chemistryā€¢Feb 1, 1983ā€¢3 citations
Given the set š’®N of all operator endomorphisms T with the property that whenever D is any 2-density operator, then T (D) is N-representable is considered. We specify a pair {Q2,PN} of extreme points of š’®N which are related to each other via the particle-hole duality (Theorem 1.2). The rest of the paper is devoted to a complete characterization of the set ŜN of all operator endomorphisms T with the property that whenever D is any 2-density operator then T (D) is quasi-N-representable, i.e., satisfies the necessary conditions Id, QN, BN, and CN.
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 TnpĀÆ (the bar indicates the closure with respect to a certain topology) if and only if Tr (DpBp) ā‰„ 0 for all bounded self-adjoint p-particle operators Bp, such that their n-expansion (pn)Ī“pnBpā‰” āˆ‘ i1<ā€¦<ipBp(i1ā€¦ip)is a positive operator in n-particle space. Moreover, it is shown that TnpĀÆ 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 Ī³p 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 Ī³p one (and only one) vector (function) and form the corresponding reduced p-particle operator. TnpĀÆ 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.
The reduction of density matrices defined with respect to a finite basis set is considered. A basis set can be introduced into the vector space of Hermitian matrices acting on functions of the coordinates of ppp particles for each value of ppp. An analogy with the construction of spin eigenfunctions is used to obtain a particular basis set called the reducing basis. Reduction of one of these basis elements is either one to one or maps into the origin. Any element in the preimage of a density matrix can thus be resolved into two components, one uniquely determined by the density matrix and the other arbitrary within a certin subset. A restatement of the NNN-representability problem is given, and two sufficient conditions and one necessary condition for NNN representability are given in terms of distances and norms.
The NNN-representability problem is approached by considering geminal product expansions of symmetric and antisymmetric functions. The NNN-completeness problem, i.e., the problem of determining when a set of geminals is a suitable basis for expanding symmetric or antisymmetric NNN-particle functions, is considered. New necessary conditions are given for both NNN-completeness and NNN-representability. In some cases, one can also obtain sufficient conditions; examples of such cases are discussed. The circumstances under which a density matrix can be derived from two or more different functions are also treated. Finally, extensions to higher order are also mentioned.