Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

N-Representability Problem: Particle-Hole Equivalence

Mary Beth Ruskai

Published Nov 1, 1970• 35 citations
Publisher

Abstract

It is shown that every p matrix has a dual matrix which describes the p-hole properties of the system. A general procedure is given for using particle-hole equivalence to obtain new N-representability conditions. In particular, necessary and sufficient conditions are given for both pure and ensemble N-representability of p matrices whose 1-rank is N + p.

DOI
10.1063/1.1665118

Related papers

N -Representability Problem for Fermion Density Matrices. II. The First-Order Density Matrix with N Even

Darwin W. Smith

For a quantum-mechanical system of $N$ identical fermions, the $N$-representability problem is the problem of recognizing whether, for a given pth-order reduced density matrix ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})$, there exists an antisymmetric $N$-particle wave function $\ensuremath{\Psi}(12\ensuremath{\cdots}N)$ such that ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})=({N}{p})\ensuremath{\int}\ensuremath{\Psi}(12\ensuremath{\cdots}N)\ifmmode\times\else\texttimes\fi{}{\ensuremath{\Psi}}^{*}({1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}},p+1\ensuremath{\cdots}N){d}_{{\ensuremath{\tau}}_{p+1}}\ensuremath{\cdots}{d}_{{\ensuremath{\tau}}_{N}}$. It is shown that if the Hamiltonian of a system is time-reversal invariant, and the number of particles, $N$, is even, the necessary and sufficient condition that an approximate first-order density matrix corresponding to a nondegenerate energy eigenstate be $N$-representable is that its natural spin-orbital occupation numbers be equal in pairs.

Physical Review•Jul 29, 1966•67 citations

Grand-canonical-ensemble representability problem for the one-electron reduced density matrix

Diego R. Alcoba, Roberto C. Bochicchio, G. E. Massacessi +2 more

We deal with many-electron systems having a noninteger number of electrons, which cannot be described properly by means of pure states or by canonical statistical ensemble states. The study of the one-electron reduced density matrix for these systems raises the problem of its representability in statistical ensembles of grand canonical type. We derive the necessary and sufficient conditions for the representability of the one-electron reduced density matrix of grand-canonical statistical ensembles.

Physical Review A•Jan 24, 2007•10 citations

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

H. Kummer, Ilyas Absar

Given the set 𝒮N 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 𝒮N 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.

Journal of Mathematical Physics•Feb 1, 1977•12 citations

N -Representability Problem: Conditions on Geminals

Mary Beth Ruskai

The $N$-representability problem is approached by considering geminal product expansions of symmetric and antisymmetric functions. The $N$-completeness problem, i.e., the problem of determining when a set of geminals is a suitable basis for expanding symmetric or antisymmetric $N$-particle functions, is considered. New necessary conditions are given for both $N$-completeness and $N$-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.

Physical Review•Jul 5, 1969•34 citations

Geometry of density matrices. II. Reduced density matrices and N representability

John E. Harriman

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 $p$ particles for each value of $p$. 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 $N$-representability problem is given, and two sufficient conditions and one necessary condition for $N$ representability are given in terms of distances and norms.

Physical Review A•Apr 1, 1978•84 citations

ConnectingN-representability to Weyl's problem: the one-particle density matrix forN= 3 andR= 6

Mary Beth Ruskai

An analytic proof of the necessity of the Borland–Dennis conditions for 3-representability of a one-particle density matrix with rank 6 is given. This may shed some light on Klyachko's recent use of Schubert calculus to find general conditions for N -representability.

arXiv:0706.1855•Journal of Physics A Mathematical and Theoretical•Oct 23, 2007•43 citations
It is shown that every p matrix has a dual matrix which describes the p-hole properties of the system. A general procedure is given for using particle-hole equivalence to obtain new N-representability conditions. In particular, necessary and sufficient conditions are given for both pure and ensemble N-representability of p matrices whose 1-rank is N + p.
For a quantum-mechanical system of NNN identical fermions, the NNN-representability problem is the problem of recognizing whether, for a given pth-order reduced density matrix ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})$, there exists an antisymmetric NNN-particle wave function $\ensuremath{\Psi}(12\ensuremath{\cdots}N)$ such that ${\ensuremath{\Gamma}}^{(p)}(12\ensuremath{\cdots}p|{1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}})=({N}{p})\ensuremath{\int}\ensuremath{\Psi}(12\ensuremath{\cdots}N)\ifmmode\times\else\texttimes\fi{}{\ensuremath{\Psi}}^{*}({1}^{\ensuremath{'}}{2}^{\ensuremath{'}}\ensuremath{\cdots}{p}^{\ensuremath{'}},p+1\ensuremath{\cdots}N){d}_{{\ensuremath{\tau}}_{p+1}}\ensuremath{\cdots}{d}_{{\ensuremath{\tau}}_{N}}$. It is shown that if the Hamiltonian of a system is time-reversal invariant, and the number of particles, NNN, is even, the necessary and sufficient condition that an approximate first-order density matrix corresponding to a nondegenerate energy eigenstate be NNN-representable is that its natural spin-orbital occupation numbers be equal in pairs.
We deal with many-electron systems having a noninteger number of electrons, which cannot be described properly by means of pure states or by canonical statistical ensemble states. The study of the one-electron reduced density matrix for these systems raises the problem of its representability in statistical ensembles of grand canonical type. We derive the necessary and sufficient conditions for the representability of the one-electron reduced density matrix of grand-canonical statistical ensembles.
Given the set 𝒮N 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 𝒮N 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.
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.
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.