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Transactions of the American Mathematical Society

A combinatorial approach to the diagonal 𝑁-representability problem

Mark Laurance Yoseloff

Published Jan 1, 1974• 14 citations
Publisher

Abstract

The problem considered is that of the diagonal N -representability of a p th-order reduced density matrix, p ≥ 2 p \geq 2 , for a system of N identical fermions or bosons. A finite number M of allowable single particle states is assumed. The problem is divided into three cases, namely: Case I. M = N + p M = N + p ; Case II. M > N + p M > N + p ; Case III. M > N + p M > N + p . Using the theory of polyhedral convex cones, a complete set of necessary and sufficient conditions is first found for Case I. This solution is then employed to find such conditions for Case II. For Case III, two algorithms are developed to generate solutions for the problem, and examples of the usage of these algorithms are given.

DOI
10.1090/s0002-9947-1974-0337218-1

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The problem considered is that of the diagonal N -representability of a p th-order reduced density matrix, p ≥ 2 p \geq 2 , for a system of N identical fermions or bosons. A finite number M of allowable single particle states is assumed. The problem is divided into three cases, namely: Case I. M = N + p M = N + p ; Case II. M > N + p M > N + p ; Case III. M > N + p M > N + p . Using the theory of polyhedral convex cones, a complete set of necessary and sufficient conditions is first found for Case I. This solution is then employed to find such conditions for Case II. For Case III, two algorithms are developed to generate solutions for the problem, and examples of the usage of these algorithms are given.
This paper considers the determination of N-representability (for diagonal elements) of p-density matrices restricted to certain finite-dimensional subspaces of l2 of the configuration space of N identical antisymmetric particles. In particular, an arbitrary set of N + p spin orbitals is selected and one considers the (N+pp)-dimensional subspace generated by all possible Slater determinants of the spin orbitals being considered. Applying a combinatorial approach to the problem, a necessary and sufficient set of conditions is determined; previous work has dealt only with necessary conditions, except in the 1-matrix case. The paper concludes by presenting a probabilistic interpretation of these conditions which seems of particular interest for the 2-matrix case. The conditions presented here in combination with the Pauli principle give a probabilistic view of the expected occupation of p-tuples of spin orbitals in terms of the expected occupations of lower-order-tuples of spin orbitals.
The latest results of density matrix theory are summarized in terms of analysis of off-diagonal components. For a given reduced density matrix (RDM) of arbitrary order, the full density matrix is explicitly constructed, from which the RDM is derived by means of the corresponding reduction procedure. Both the matrices satisfy all the necessary conditions, such antisymmetry, normalization, hermiticity, except that the non-negative definiteness of the full DM is not guaranteed. If an RDM satisfies the necessary inequalities with the same number of indices as it has itself, then, as proved here, there is an additive correction to the full DM matrix that turns it into a non-negative definite matrix leading to the same RDM and satisfying the other requirements. This establishes a fundamental fact that these inequalities are sufficient as well, providing the N-representability of the given RDM. The construction is based on the classification, introduced in the article, of the multi-index elements of the RDM and full DM according to their degree of off-diagonality. RDM elements with a predetermined degree of off-diagonality are expressed through full DM elements with the same off-diagonality. This shows that the conventional variational approach to calculating the energy of the system by convex programming methods under the conditions of N-representability (before and even after reaching the minimum energy) generally does not lead to constructing a mixed quantum state composed of solutions to the Schrödinger equation. Therefore, except for the energy, it does not guarantee a correct description of the other properties so long as the conditions of correctness for each particular property are not expressed in terms of the RDM and not incorporated into the specific conditions of N-representability (regardless of the requirement of energy minimization).
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.
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.
Abstract A partial trace over the occupation numbers of all but k states in the density matrix of an ensemble with an arbitrary number of single‐particle states is defined as the (reduced) k‐state density matrix. This matrix is used to obtain a complete, practical solution to the problem of determining the representability of the diagonal elements of the one‐ and two‐particle (reduced) density matrices. This solution is expressed as a series of linear inequalities involving the density‐matrix elements; the inequalities are identical with those derived previously by Davidson and McCrae by a different method. In addition, our method is used to obtain nonlinear, matrix inequalities on the off‐diagonal elements of the density matrices.
Representability determines when a two-particle reduced density matrix (2-RDM) corresponds to a physical quantum state, enabling many-particle quantum calculations with 2-RDMs rather than the wave function. In this Letter, we present a solution of the representability problem for quantum systems without particle-number conservation. The physically allowed set of 2-RDMs can be characterized from a geometrically `orthogonal' set, the polar cone. We derive explicit linear equations for the two-body operators in the polar cone -- the intersection of the ppp-positive cone with the two-body operator space -- to obtain a systematic hierarchy of representability conditions that do not depend on higher RDMs or the wave function. Moreover, by augmenting these conditions with the particle-number variance, we obtain a unified framework for treating both particle-number-conserving and nonconserving systems. We illustrate with a spin system and molecular H4_44​.