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

arXiv:1010.2717 · Physical Review Letters

Quantum Codes Give Counterexamples to the Unique Preimage Conjecture of the N -Representability Problem

Samuel A. Ocko, Xie Chen, Bei Zeng, Beni Yoshida, Zhengfeng Ji, Mary Beth Ruskai, Isaac L. Chuang

Published Mar 14, 2011• 20 citations
arXivHTMLPDF

Abstract

It is well known that the ground state energy of many-particle Hamiltonians involving only 2-body interactions can be obtained using constrained optimizations over density matrices which arise from reducing an N-particle state. While determining which 2-particle density matrices are "N-representable" is a computationally hard problem, all known extreme N-representable 2-particle reduced density matrices arise from a unique N-particle preimage, satisfying a conjecture established in 1972. We present explicit counterexamples to this conjecture through giving Hamiltonians with 2-body interactions which have degenerate ground states that cannot be distinguished by any 2-body operator. We relate the existence of such counterexamples to quantum error correction codes and topologically ordered spin systems.

DOI
10.1103/physrevlett.106.110501

Related papers

Quantum Many-Body Theory from a Solution of the N -Representability Problem

David A. Mazziotti

Here we present a many-body theory based on a solution of the N-representability problem in which the ground-state two-particle reduced density matrix (2-RDM) is determined directly without the many-particle wave function. We derive an equation that re-expresses physical constraints on higher-order RDMs to generate direct constraints on the 2-RDM, which are required for its derivation from an N-particle density matrix, known as N-representability conditions. The approach produces a complete hierarchy of 2-RDM constraints that do not depend explicitly upon the higher RDMs or the wave function. By using the two-particle part of a unitary decomposition of higher-order constraint matrices, we can solve the energy minimization by semidefinite programming in a form where the low-rank structure of these matrices can be potentially exploited. We illustrate by computing the ground-state electronic energy and properties of the H_{8} ring.

arXiv:2304.08570•Physical Review Letters•Apr 12, 2023•41 citations

Reduced Density Matrices and Moduli of Many-Body Eigenstates

Chaoming Song

Many-body wavefunctions usually lie in high-dimensional Hilbert spaces. However, physically relevant states, i.e, the eigenstates of the Schrödinger equation are rare. For many-body systems involving only pairwise interactions, these eigenstates form a low-dimensional subspace of the entire Hilbert space. The geometry of this subspace, which we call the eigenstate moduli problem is parameterized by a set of 2-particle Hamiltonian. This problem is closely related to the $N$-representability conditions for 2-reduced density matrices, a long-standing challenge for quantum many-body systems. Despite its importance, the eigenstate moduli problem remains largely unexplored in the literature. In this Letter, we propose a comprehensive approach to this problem. We discover an explicit set of algebraic equations that fully determine the eigenstate spaces of $m$-interaction systems as projective varieties, which in turn determine the geometry of the spaces for representable reduced density matrices. We investigate the geometrical structure of these spaces, and validate our results numerically using the exact diagonalization method. Finally, we generalize our approach to the moduli problem of the arbitrary family of Hamiltonians parameterized by a set of real variables.

arXiv:2301.01419•Jan 4, 2023•0 citations

Expectation value constraints for the N -representability problem

Neil Shenvi, K. Birgitta Whaley

It has long been known that reduced density matrices (RDM) can be used to obtain the exact ground state energy of many-particle Hamiltonians by performing a constrained minimization over the set of all $N$-representable reduced density matrices. In this paper, we present a set of $N$-representability constraints for both bosons and fermions based on expectation values. These expectation value constraints are derived from a simple yet nontrivial recursive formula, and can easily be included in RDM-based variational schemes. The expectation value constraints are shown to provide additional $N$-representability information when compared to existing $N$-representability constraints.

Physical Review A•Aug 18, 2006•6 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

Direct variational determination of two-particle reduced density matrices for ground and excited states of N-boson systems using the dispersion operator approach

Diego R. Alcoba, P. Capuzzi, Alicia Torre +4 more

Abstract This work deals with the variational determination of two-particle reduced density matrices corresponding to eigenstates of N -boson systems. Stringent N -representability conditions have been imposed in the variational treatment, in which the resulting optimization problem is addressed within a standard semidefinite programming framework. A unified variational treatment, based on the dispersion operator technique, is proposed. This treatment allows to determine ground- and excited-state energies and their corresponding reduced density matrices, which have been evaluated in systems formulated by means of simple Hamiltonians. We report results from different N -representability levels of treatment. These results are compared with those arising from the full configuration interaction procedure, what allows to point out the relevance of the imposed N -representability conditions. We also highlight the differences found in applying this methodology to fermion and boson systems.

Physica Scripta•Mar 19, 2026•0 citations

Necessary conditions for the N -representability of the second-order reduced density matrix: Upper bounds on the P and Q matrices

Dimitri Van Neck, Paul W. Ayers

Usually, variational calculations for the second-order reduced density matrix are performed subject to the constraint that the ``$P$'' and ``$Q$'' matrices are positive semidefinite, which only constrains the lowest eigenvalue of these matrices. We characterize the highest eigenvalue of these matrices and discuss how the associated constraint (which is related to the ground-state energy of the Hamiltonians $H=\ensuremath{-}P$ and $H=\ensuremath{-}Q$) can be implemented in practical calculations. This necessary condition for $N$-representability should help ensure that the second-order reduced density matrix is not ``overcorrelated.''

Physical Review A•Mar 5, 2007•13 citations
It is well known that the ground state energy of many-particle Hamiltonians involving only 2-body interactions can be obtained using constrained optimizations over density matrices which arise from reducing an N-particle state. While determining which 2-particle density matrices are "N-representable" is a computationally hard problem, all known extreme N-representable 2-particle reduced density matrices arise from a unique N-particle preimage, satisfying a conjecture established in 1972. We present explicit counterexamples to this conjecture through giving Hamiltonians with 2-body interactions which have degenerate ground states that cannot be distinguished by any 2-body operator. We relate the existence of such counterexamples to quantum error correction codes and topologically ordered spin systems.
Here we present a many-body theory based on a solution of the N-representability problem in which the ground-state two-particle reduced density matrix (2-RDM) is determined directly without the many-particle wave function. We derive an equation that re-expresses physical constraints on higher-order RDMs to generate direct constraints on the 2-RDM, which are required for its derivation from an N-particle density matrix, known as N-representability conditions. The approach produces a complete hierarchy of 2-RDM constraints that do not depend explicitly upon the higher RDMs or the wave function. By using the two-particle part of a unitary decomposition of higher-order constraint matrices, we can solve the energy minimization by semidefinite programming in a form where the low-rank structure of these matrices can be potentially exploited. We illustrate by computing the ground-state electronic energy and properties of the H_{8} ring.
Many-body wavefunctions usually lie in high-dimensional Hilbert spaces. However, physically relevant states, i.e, the eigenstates of the Schrödinger equation are rare. For many-body systems involving only pairwise interactions, these eigenstates form a low-dimensional subspace of the entire Hilbert space. The geometry of this subspace, which we call the eigenstate moduli problem is parameterized by a set of 2-particle Hamiltonian. This problem is closely related to the NNN-representability conditions for 2-reduced density matrices, a long-standing challenge for quantum many-body systems. Despite its importance, the eigenstate moduli problem remains largely unexplored in the literature. In this Letter, we propose a comprehensive approach to this problem. We discover an explicit set of algebraic equations that fully determine the eigenstate spaces of mmm-interaction systems as projective varieties, which in turn determine the geometry of the spaces for representable reduced density matrices. We investigate the geometrical structure of these spaces, and validate our results numerically using the exact diagonalization method. Finally, we generalize our approach to the moduli problem of the arbitrary family of Hamiltonians parameterized by a set of real variables.
It has long been known that reduced density matrices (RDM) can be used to obtain the exact ground state energy of many-particle Hamiltonians by performing a constrained minimization over the set of all NNN-representable reduced density matrices. In this paper, we present a set of NNN-representability constraints for both bosons and fermions based on expectation values. These expectation value constraints are derived from a simple yet nontrivial recursive formula, and can easily be included in RDM-based variational schemes. The expectation value constraints are shown to provide additional NNN-representability information when compared to existing NNN-representability constraints.
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 This work deals with the variational determination of two-particle reduced density matrices corresponding to eigenstates of N -boson systems. Stringent N -representability conditions have been imposed in the variational treatment, in which the resulting optimization problem is addressed within a standard semidefinite programming framework. A unified variational treatment, based on the dispersion operator technique, is proposed. This treatment allows to determine ground- and excited-state energies and their corresponding reduced density matrices, which have been evaluated in systems formulated by means of simple Hamiltonians. We report results from different N -representability levels of treatment. These results are compared with those arising from the full configuration interaction procedure, what allows to point out the relevance of the imposed N -representability conditions. We also highlight the differences found in applying this methodology to fermion and boson systems.
Usually, variational calculations for the second-order reduced density matrix are performed subject to the constraint that the ``PPP'' and ``QQQ'' matrices are positive semidefinite, which only constrains the lowest eigenvalue of these matrices. We characterize the highest eigenvalue of these matrices and discuss how the associated constraint (which is related to the ground-state energy of the Hamiltonians $H=\ensuremath{-}P$ and $H=\ensuremath{-}Q$) can be implemented in practical calculations. This necessary condition for NNN-representability should help ensure that the second-order reduced density matrix is not ``overcorrelated.''