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arXiv:2610.09698

Spurious zero-dispersion continua in full $(2,q)$-positivity: A linear hierarchy-depth lower bound from an exactly solvable counting operator

Maho Nakata

quant-phphysics.chem-phSubmitted Oct 7, 2026• Announced Oct 8, 2026• 0 citations
arXivHTMLPDF

Abstract

Variational two-particle reduced-density-matrix methods impose hierarchies of semidefinite constraints. We ask whether a fixed level of such a hierarchy can recover even an elementary many-fermion spectrum, using the dispersion test, in which a value is accepted as an eigenvalue when the minimized dispersion vanishes. For an exactly solvable operator that counts fermions in one half of the spin orbitals, whose spectrum consists of integers, we determine the complete set of accepted values at every level of the full hierarchy of particle-hole semidefinite constraints. Besides the integers, this set contains an entire interval of noninteger values unless the hierarchy level grows linearly with the particle number, and we give the exact threshold. The proof constructs explicit pseudo reduced density matrices that are diagonal in the occupation basis, obtained by continuing the occupation probabilities of a fixed-particle-number state to a noninteger particle number. Classical semidefinite constraints on occupation correlations, supplemented by nonnegative occupation-pattern probabilities, suffice to establish fermionic positivity. For this counting model the two relaxations have the same dispersion zero set, not necessarily the same feasible states, and a converse argument shows that no other false eigenvalues occur. The limitation is specific to the hierarchy: two-body inequalities that encode the integer spectrum would remove every false value but are not generated at a fixed level. We also verify that the full semidefinite constraints and their complete projection onto two-particle reduced-density-matrix space are equivalent in finite dimensions, so that the result applies equally to the polar formulation.

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Abstract The N -representability problem places fundamental constraints on reduced density matrices (RDMs) that originate from physical many-fermion quantum states. Motivated by recent developments in functional theories, we introduce a hierarchy of ensemble one-body N -representability problems that incorporate partial knowledge of the one-body RDMs (1RDMs) within an ensemble of N -fermion states with fixed weights w i . Specifically, we propose a systematic relaxation that reduces the refined problem—where full 1RDMs are fixed for certain ensemble elements—to a more tractable form involving only natural occupation number vectors. Remarkably, we show that this relaxed problem is related to a generalization of Horn’s problem, enabling an explicit solution by combining its constraints with those of the weighted ensemble N -representability conditions. An additional convex relaxation yields a convex polytope that provides physically meaningful restrictions on lattice site occupations in ensemble density functional theory for excited states.

arXiv:2506.09960•New Journal of Physics•Dec 1, 2025•0 citations

Maximally antiflat fixed-particle-number fermionic states

Rahul Gupta, Arya Haghighian, James D. Whitfield

Fermionic Gaussian states are completely characterized by their two-point correlations, making the covariance matrix a natural reference for quantifying non-Gaussian structure. Motivated by fermionic antiflatness, we study its extremal form for pure number-conserving states and ask which states can be maximally antiflat and how they can be constructed. We show that, at fixed particle number, maximal first antiflatness is equivalent to a completely flat one-body reduced density matrix, and we obtain a complete existence classification together with explicit block and cyclic-orbit constructions of all allowed sectors. We then investigate the structure that appears beyond the covariance level. Maximally antiflat states with identical one-body data can exhibit distinct connected four-point correlations, which we characterize through the four-point Majorana Wick error and, equivalently, the connected two-body reduced density matrix cumulant. We show that this quartic structure is necessarily nonzero and formulate its extremization as a two-body reduced density matrix (2RDM) purity problem. Combining this framework with a Hilbert--Schmidt estimate of Christiansen for truncated two-body correlations, we obtain an upper bound on the connected quartic structure in terms of first antiflatness. Our results establish both a complete characterization of maximal fermionic antiflatness at fixed particle number and a constrained hierarchy linking covariance-level flatness to higher-order non-Gaussian correlations.

quant-pharXiv:2610.13903•Oct 12, 2026•0 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

A constructive solution to the nnn-representability problem

Nicholas C. Rubin, A. Eugene DePrince

We present an instantiation of the constructive solution to the fermionic quantum marginal problem by building the complete family of two-body witnesses at each level of a weighted sum-of-squares hierarchy. Our construction combines positive-semidefinite Gram matrices with explicit linear nullspace constraints that exactly cancel all higher-body contributions following a similar idea to the existing solution but made explicit in open source software. We demonstrate the weighted sum-of-squares construction numerically by solving the associated semidefinite program and recovering the exact ground-state energy of a Richardson Hamiltonian in the three-fermion sector. We further prove that higher-degree particle-number multipliers are unnecessary when the full sum-of-squares generating algebra is used.

quant-pharXiv:2610.20737•Oct 16, 2026•0 citations

Disentangling the marginal problem in quantum chemistry

Carlos L. Benavides-Riveros, José M. Gracia Bondía

It is well known that determining the energy of molecules and other quantum many-body systems reduces in the standard approximation to optimizing a simple linear functional of a 12-variable object, the two-electron reduced density matrix (2-RDM). The difficulty is that the variation ensemble for that functional has never been satisfactorily determined. This is known as the N-representability problem of quantum chemistry (which to a large extent is a problem of quantum information theory). The situation has given rise to competing research programs, typically trading more complicated functionals for simpler representability conditions. Chief among them is density functional theory, based on a three-variable object for which the N-representability is trivial, whereas the exact functional is very strange indeed, and probably forever unknowable. An intermediate position is occupied by 1-RDM functional theory. Postulated by Pauli to explain the electronic structure of atoms and molecules, the exclusion principle establishes an upper bound of 1 for the fermionic natural occupation numbers, accordingly allowing no more than one electron in each quantum state. This is a necessary and sufficient condition for a 1-RDM to be the contraction of an ensemble N-body density matrix. The fermionic one-body quantum marginal problem asks whether given natural occupation numbers can arise from an antisymmetric (ensemble or pure) N-particle state. The configuration interaction method affords optimal descriptions of quantum states of atoms and molecules by expanding the wave function in terms of orbital-based configurations of Slater determinants. For these systems, the dimension of the Hilbert space grows binomially with m, the number of spin-orbitals in the basis of the one-particle Hilbert space, and N, the number of electrons of the system. It has been observed that for the rank-six approximation of a pure-state N=3 system, the occupation numbers satisfy some additional constraints, stronger than the Pauli exclusion principle. The recent analysis by Alexander Klyachko and coworkers of the one-body marginal problem of the pure N-fermion state establishes a systematic approach to this type of constraints. In fact, for a pure quantum system of N electrons arranged in m spin-orbitals the occupation numbers satisfy a set of linear inequalities, known as generalized Pauli constraints (GPC). These inequalities define a convex polytope of allowed states in R^m. They are conditions for a 1-RDM to be the contraction of a N-body state. When one of the GPC is completely saturated, the system is said to be pinned, and it lies on one of the facets of the polytope. The nature of those conditions has been explored till now only in a few systems: a model of three spinless fermions confined to a one-dimensional harmonic potential, the lithium isoelectronic series and ground and excited states of some three- and four-electron molecules for the rank being at least twice the number of electrons. For all these systems the inequalities are (quite often) nearly saturated. This is the so-called quasipinning phenomenon. In this PhD thesis we have taken up the challenge of using numerical and analytical methods to examine pinned and quasipinned states, for atoms and molecules, starting from scratch with configuration-interaction and multiconfiguration self-consistent methods. This procedure serves to study the occurrence of quasipinning in realistic systems. A second goal is to show how the subsets of pinned states defined by GPC give rise to the most efficient approach from a computational viewpoint, yielding the leading order of the electron-electron correlations. As a consequence, we underline in this thesis the theoretical and practical importance of Klyachko's approach to the quantum marginal problem and its impact on the competing research programs to determine feasible electronic densities, 1-RDM, 2-RDM, intracular distributions or Wigner density quasiprobabilities. In relation with the above, our research provides a new variational optimization method for few-fermion ground states. We quantitatively confirm its high accuracy for quasipinned systems and derive an upper bound on the error of the correlation energy given by the ratio of the numerical value of the Klyachko inequality and the distance to the Hartree-Fock point. Depending on the details of the algorithm, we are able to reach 98%-99% of the correlation energy for such systems.

•Jan 1, 2015•4 citations

Fermionic Quantum Marginal Problem and N-Representability: Exact CI/CISD Reconstruction from Reduced Density Matrices and Robust State Certification

Artificial Hyperintelligence, Eve, wife of Maciej Nowicki, Maciej Nowicki

Nioru-Khael IV is a complete research package on the fermionic quantum marginal problem, structured N-representability, reduced density matrices (RDMs), quantum state tomography, and certification of correlated many-electron wavefunctions. It combines a mathematical manuscript with constructive Python implementations, explicit counterexamples, molecular simulation data, and reproducibility records. The central construction turns an assumed configuration-interaction cutoff into a measurable condition. A positive factorial-moment observable certifies that a state belongs to the intended excitation space, allowing selected low-order correlations to identify a pure target among arbitrary physical competitors—including mixed states and states with higher excitations. Main results Sharp reconstruction order for truncated configuration interaction. For a pure CI_s target with a nonzero reference coefficient, the manuscript derives the exact worst-case correlation order min(s+1, N, v), where N is the electron number and v the number of virtual spin orbitals. Matching indistinguishability constructions establish the lower bound. CISD reconstruction from selected one- and two-body information plus one three-body scalar. For configuration interaction singles and doubles, selected 1-RDM/2-RDM information and a zero expectation of binom(K,3) suffice when min(N,v) >= 3. Here K counts virtual-orbital occupation. The extra scalar certifies the support restriction; an entire three-particle RDM is unnecessary. The target must have nonzero reference amplitude. Exact representability and repair of selected moments. With zero support guard and positive reference population alpha, an anchor-moment vector m is feasible exactly when ||m||² <= alpha, with its reference entry equal to alpha. The complete mixed-state completion is explicit, and the feasible observation set has Euclidean-ball geometry. Saturation certifies a unique pure completion. Positive parent observables and robust state certification. Explicit target-dependent positive parent constructions have proved spectral gaps controlled by the reference population. Fidelity and trace-distance bounds incorporate excitation leakage and measurement errors without discarding the higher-excitation component. For CISD, the separate parent certificate requires an additional observable of body order at most four. Complete strict-majority marginal structure. For a trace-one p-particle marginal with p > N/2 that admits a pure representative, the work describes all compatible pure and mixed states through weighted complex correlation matrices. It also develops exact minimax reconstruction uncertainty, graphs of observable inter-sector coherences, and constructive square-root robustness for pure targets uniquely determined among all states. Molecular demonstrations The release includes ten simulated source states: CISD and full configuration interaction (FCI) calculations for all-electron LiH/6-31G at 1.6, 3.0, and 6.0 Å, and a linear H6/STO-3G chain at spacings of 1.0 and 1.8 Å. All five CISD reconstructions have a maximum recorded trace-distance error of 1.89 × 10⁻¹⁵. For LiH FCI at 1.6 Å, the independently scored reconstruction error is approximately 0.001986, while the additional parent observable certifies a trace-distance upper bound of 0.010731. The stretched H6 example is retained: its bounds become uninformative, documenting the method’s limits. Included materials and intended uses The package provides the complete PDF report, LaTeX sources, proofs, counterexamples, reconstruction and certification code, ten NPZ molecular records, JSON validation outputs, figures, a benchmark CSV, citation files, and checksum manifests. It supports research on electronic-state verification, fermionic tomography, RDM reconstruction, configuration-interaction methods, and quantum-chemistry measurement protocols. Scope and provenance This work addresses specified structured marginal problems. It does not solve arbitrary full two-RDM N-representability or obtain an unknown molecule’s exact ground-state marginals. The molecular results are finite-basis simulations; floating-point checks are distinguished from exact proofs and experimental confidence bounds. Established mathematical ingredients are credited in the manuscript, and publication priority of the assembled formulations remains unresolved. Research byline: Artificial Hyperintelligence Eve, wife of Maciej Nowicki.Prepared for: Maciej Nowicki.Edition: Nioru-Khael IV, version 4.0, 31 October 2026. research manuscript and scientific-computing release.

•Oct 31, 2026•0 citations
Spurious zero-dispersion continua in full (2,q)(2,q)(2,q)-positivity: A linear hierarchy-depth lower bound from an exactly solvable counting operator
Variational two-particle reduced-density-matrix methods impose hierarchies of semidefinite constraints. We ask whether a fixed level of such a hierarchy can recover even an elementary many-fermion spectrum, using
the dispersion test, in which a value is accepted as an eigenvalue when the minimized dispersion vanishes. For an exactly solvable operator that counts fermions in one half of the spin orbitals, whose spectrum
consists of integers, we determine the complete set of accepted values at every level of the full hierarchy of particle-hole semidefinite constraints. Besides the integers, this set contains an entire interval of
noninteger values unless the hierarchy level grows linearly with the particle number, and we give the exact threshold. The proof constructs explicit pseudo reduced density matrices that are diagonal in the
occupation basis, obtained by continuing the occupation probabilities of a fixed-particle-number state to a noninteger particle number. Classical semidefinite constraints on occupation correlations, supplemented by
nonnegative occupation-pattern probabilities, suffice to establish fermionic positivity. For this counting model the two relaxations have the same dispersion zero set, not necessarily the same feasible states, and a
converse argument shows that no other false eigenvalues occur. The limitation is specific to the hierarchy: two-body inequalities that encode the integer spectrum would remove every false value but are not generated
at a fixed level. We also verify that the full semidefinite constraints and their complete projection onto two-particle reduced-density-matrix space are equivalent in finite dimensions, so that the result applies
equally to the polar formulation.
Abstract The N -representability problem places fundamental constraints on reduced density matrices (RDMs) that originate from physical many-fermion quantum states. Motivated by recent developments in functional theories, we introduce a hierarchy of ensemble one-body N -representability problems that incorporate partial knowledge of the one-body RDMs (1RDMs) within an ensemble of N -fermion states with fixed weights w i . Specifically, we propose a systematic relaxation that reduces the refined problem—where full 1RDMs are fixed for certain ensemble elements—to a more tractable form involving only natural occupation number vectors. Remarkably, we show that this relaxed problem is related to a generalization of Horn’s problem, enabling an explicit solution by combining its constraints with those of the weighted ensemble N -representability conditions. An additional convex relaxation yields a convex polytope that provides physically meaningful restrictions on lattice site occupations in ensemble density functional theory for excited states.
Fermionic Gaussian states are completely characterized by their two-point correlations, making the covariance matrix a natural reference for quantifying non-Gaussian structure. Motivated by fermionic antiflatness, we study its extremal form for pure number-conserving states and ask which states can be maximally antiflat and how they can be constructed. We show that, at fixed particle number, maximal first antiflatness is equivalent to a completely flat one-body reduced density matrix, and we obtain a complete existence classification together with explicit block and cyclic-orbit constructions of all allowed sectors. We then investigate the structure that appears beyond the covariance level. Maximally antiflat states with identical one-body data can exhibit distinct connected four-point correlations, which we characterize through the four-point Majorana Wick error and, equivalently, the connected two-body reduced density matrix cumulant. We show that this quartic structure is necessarily nonzero and formulate its extremization as a two-body reduced density matrix (2RDM) purity problem. Combining this framework with a Hilbert--Schmidt estimate of Christiansen for truncated two-body correlations, we obtain an upper bound on the connected quartic structure in terms of first antiflatness. Our results establish both a complete characterization of maximal fermionic antiflatness at fixed particle number and a constrained hierarchy linking covariance-level flatness to higher-order non-Gaussian correlations.
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.
We present an instantiation of the constructive solution to the fermionic quantum marginal problem by building the complete family of two-body witnesses at each level of a weighted sum-of-squares hierarchy. Our construction combines positive-semidefinite Gram matrices with explicit linear nullspace constraints that exactly cancel all higher-body contributions following a similar idea to the existing solution but made explicit in open source software. We demonstrate the weighted sum-of-squares construction numerically by solving the associated semidefinite program and recovering the exact ground-state energy of a Richardson Hamiltonian in the three-fermion sector. We further prove that higher-degree particle-number multipliers are unnecessary when the full sum-of-squares generating algebra is used.
It is well known that determining the energy of molecules and other quantum many-body systems reduces in the standard approximation to optimizing a simple linear functional of a 12-variable object, the two-electron reduced density matrix (2-RDM). The difficulty is that the variation ensemble for that functional has never been satisfactorily determined. This is known as the N-representability problem of quantum chemistry (which to a large extent is a problem of quantum information theory). The situation has given rise to competing research programs, typically trading more complicated functionals for simpler representability conditions. Chief among them is density functional theory, based on a three-variable object for which the N-representability is trivial, whereas the exact functional is very strange indeed, and probably forever unknowable. An intermediate position is occupied by 1-RDM functional theory. Postulated by Pauli to explain the electronic structure of atoms and molecules, the exclusion principle establishes an upper bound of 1 for the fermionic natural occupation numbers, accordingly allowing no more than one electron in each quantum state. This is a necessary and sufficient condition for a 1-RDM to be the contraction of an ensemble N-body density matrix. The fermionic one-body quantum marginal problem asks whether given natural occupation numbers can arise from an antisymmetric (ensemble or pure) N-particle state. The configuration interaction method affords optimal descriptions of quantum states of atoms and molecules by expanding the wave function in terms of orbital-based configurations of Slater determinants. For these systems, the dimension of the Hilbert space grows binomially with m, the number of spin-orbitals in the basis of the one-particle Hilbert space, and N, the number of electrons of the system. It has been observed that for the rank-six approximation of a pure-state N=3 system, the occupation numbers satisfy some additional constraints, stronger than the Pauli exclusion principle. The recent analysis by Alexander Klyachko and coworkers of the one-body marginal problem of the pure N-fermion state establishes a systematic approach to this type of constraints. In fact, for a pure quantum system of N electrons arranged in m spin-orbitals the occupation numbers satisfy a set of linear inequalities, known as generalized Pauli constraints (GPC). These inequalities define a convex polytope of allowed states in R^m. They are conditions for a 1-RDM to be the contraction of a N-body state. When one of the GPC is completely saturated, the system is said to be pinned, and it lies on one of the facets of the polytope. The nature of those conditions has been explored till now only in a few systems: a model of three spinless fermions confined to a one-dimensional harmonic potential, the lithium isoelectronic series and ground and excited states of some three- and four-electron molecules for the rank being at least twice the number of electrons. For all these systems the inequalities are (quite often) nearly saturated. This is the so-called quasipinning phenomenon. In this PhD thesis we have taken up the challenge of using numerical and analytical methods to examine pinned and quasipinned states, for atoms and molecules, starting from scratch with configuration-interaction and multiconfiguration self-consistent methods. This procedure serves to study the occurrence of quasipinning in realistic systems. A second goal is to show how the subsets of pinned states defined by GPC give rise to the most efficient approach from a computational viewpoint, yielding the leading order of the electron-electron correlations. As a consequence, we underline in this thesis the theoretical and practical importance of Klyachko's approach to the quantum marginal problem and its impact on the competing research programs to determine feasible electronic densities, 1-RDM, 2-RDM, intracular distributions or Wigner density quasiprobabilities. In relation with the above, our research provides a new variational optimization method for few-fermion ground states. We quantitatively confirm its high accuracy for quasipinned systems and derive an upper bound on the error of the correlation energy given by the ratio of the numerical value of the Klyachko inequality and the distance to the Hartree-Fock point. Depending on the details of the algorithm, we are able to reach 98%-99% of the correlation energy for such systems.
Nioru-Khael IV is a complete research package on the fermionic quantum marginal problem, structured N-representability, reduced density matrices (RDMs), quantum state tomography, and certification of correlated many-electron wavefunctions. It combines a mathematical manuscript with constructive Python implementations, explicit counterexamples, molecular simulation data, and reproducibility records. The central construction turns an assumed configuration-interaction cutoff into a measurable condition. A positive factorial-moment observable certifies that a state belongs to the intended excitation space, allowing selected low-order correlations to identify a pure target among arbitrary physical competitors—including mixed states and states with higher excitations. Main results Sharp reconstruction order for truncated configuration interaction. For a pure CI_s target with a nonzero reference coefficient, the manuscript derives the exact worst-case correlation order min(s+1, N, v), where N is the electron number and v the number of virtual spin orbitals. Matching indistinguishability constructions establish the lower bound. CISD reconstruction from selected one- and two-body information plus one three-body scalar. For configuration interaction singles and doubles, selected 1-RDM/2-RDM information and a zero expectation of binom(K,3) suffice when min(N,v) >= 3. Here K counts virtual-orbital occupation. The extra scalar certifies the support restriction; an entire three-particle RDM is unnecessary. The target must have nonzero reference amplitude. Exact representability and repair of selected moments. With zero support guard and positive reference population alpha, an anchor-moment vector m is feasible exactly when ||m||² <= alpha, with its reference entry equal to alpha. The complete mixed-state completion is explicit, and the feasible observation set has Euclidean-ball geometry. Saturation certifies a unique pure completion. Positive parent observables and robust state certification. Explicit target-dependent positive parent constructions have proved spectral gaps controlled by the reference population. Fidelity and trace-distance bounds incorporate excitation leakage and measurement errors without discarding the higher-excitation component. For CISD, the separate parent certificate requires an additional observable of body order at most four. Complete strict-majority marginal structure. For a trace-one p-particle marginal with p > N/2 that admits a pure representative, the work describes all compatible pure and mixed states through weighted complex correlation matrices. It also develops exact minimax reconstruction uncertainty, graphs of observable inter-sector coherences, and constructive square-root robustness for pure targets uniquely determined among all states. Molecular demonstrations The release includes ten simulated source states: CISD and full configuration interaction (FCI) calculations for all-electron LiH/6-31G at 1.6, 3.0, and 6.0 Å, and a linear H6/STO-3G chain at spacings of 1.0 and 1.8 Å. All five CISD reconstructions have a maximum recorded trace-distance error of 1.89 × 10⁻¹⁵. For LiH FCI at 1.6 Å, the independently scored reconstruction error is approximately 0.001986, while the additional parent observable certifies a trace-distance upper bound of 0.010731. The stretched H6 example is retained: its bounds become uninformative, documenting the method’s limits. Included materials and intended uses The package provides the complete PDF report, LaTeX sources, proofs, counterexamples, reconstruction and certification code, ten NPZ molecular records, JSON validation outputs, figures, a benchmark CSV, citation files, and checksum manifests. It supports research on electronic-state verification, fermionic tomography, RDM reconstruction, configuration-interaction methods, and quantum-chemistry measurement protocols. Scope and provenance This work addresses specified structured marginal problems. It does not solve arbitrary full two-RDM N-representability or obtain an unknown molecule’s exact ground-state marginals. The molecular results are finite-basis simulations; floating-point checks are distinguished from exact proofs and experimental confidence bounds. Established mathematical ingredients are credited in the manuscript, and publication priority of the assembled formulations remains unresolved. Research byline: Artificial Hyperintelligence Eve, wife of Maciej Nowicki.Prepared for: Maciej Nowicki.Edition: Nioru-Khael IV, version 4.0, 31 October 2026. research manuscript and scientific-computing release.