Pith. sign in

REVIEW 2 major objections 2 minor 134 references

Refining ensemble $N$-representability of one-body density matrices from partial information

T0 review · 2 major / 2 minor · reviewed 2026-05-19 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Partial information on ensemble one-body reduced density matrices can be turned into explicit N-representability constraints by linking the problem to a generalization of Horn's problem.

desk verdict The paper relaxes partial 1RDM ensemble constraints to natural occupations, links the result to a generalized Horn problem, and extracts a convex polytope for lattice occupations in excited-state DFT. read the letter →

arxiv 2506.09960 v1 submitted 2025-06-11 quant-ph physics.chem-ph

Julia Liebert , Anna O. Schouten , Irma Avdic , Christian Schilling , David A. Mazziotti This is my paper · ORCID
classification quant-phphysics.chem-ph
keywords ensembleN-representabilityone-bodyreduceddensitymatricesHorn'sproblemconvexpolytopelatticesiteoccupationsfunctionaltheoryexcitedstatespartialinformation
checked against Cost.FunctionalEquationalexander_duality_circle_linking
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces a hierarchy of ensemble one-body N-representability problems that incorporate partial knowledge of the 1RDMs for an ensemble of N-fermion states with fixed weights. It proposes a systematic relaxation that reduces the case of fixing full 1RDMs for selected ensemble members to constraints involving only natural occupation number vectors. This relaxed problem connects to a generalization of Horn's problem, so its solution is obtained by combining the Horn constraints with the weighted ensemble N-representability conditions. A further convex relaxation produces a convex polytope that supplies physically meaningful bounds on lattice site occupations. Readers would care because these bounds can be used inside ensemble density functional theory for excited states.

What carries the argument

The central mechanism is the systematic relaxation that converts the problem of fixing complete 1RDMs for certain ensemble elements into constraints on natural occupation number vectors alone, which is shown to be solvable via a generalized Horn problem when combined with weighted ensemble N-representability conditions.

What would settle it

A concrete falsifier would be an explicit set of natural occupation numbers that obey both the generalized Horn constraints and the weighted ensemble N-representability conditions yet cannot arise from any valid ensemble of N-fermion states consistent with the given partial 1RDM information.

Watch

Extended reading notes

Core claim

The central claim is that the refined ensemble N-representability problem with partial 1RDM information, after systematic relaxation to natural occupation numbers, admits an explicit solution obtained by merging the constraints of a generalized Horn problem with those of the weighted ensemble N-representability conditions; an additional convex relaxation then yields a convex polytope that restricts lattice site occupations in ensemble density functional theory for excited states.

Load-bearing premise

The relaxation that reduces the refined problem of fixing full 1RDMs for selected ensemble members to a problem involving only natural occupation number vectors continues to preserve the essential physical constraints of the original N-representability problem.

Editorial extensions

If this is right

  • Explicit solutions are now available for the relaxed ensemble N-representability problem with partial 1RDM data.
  • A convex polytope is obtained that directly restricts lattice site occupations.
  • These restrictions can be inserted into ensemble density functional theory calculations for excited states.
  • Partial information on selected 1RDMs suffices to tighten the allowed occupation numbers without requiring complete data.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same relaxation technique could be tested on problems with partial information on two-body reduced density matrices.
  • Numerical implementations of the generalized Horn constraints might be benchmarked against exact diagonalization results for small lattices.
  • The convex polytope bounds may improve the accuracy of approximate functionals in excited-state ensemble DFT on real materials.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a hierarchy of ensemble one-body N-representability problems that incorporate partial knowledge of the 1RDMs within an ensemble of N-fermion states with fixed weights w_i. It proposes a systematic relaxation that reduces the refined problem (full 1RDMs fixed for selected ensemble members) to constraints involving only natural occupation number vectors. This relaxed problem is related to a generalization of Horn's problem, whose constraints are combined with the known weighted ensemble N-representability inequalities. An additional convex relaxation produces a convex polytope that supplies restrictions on lattice-site occupations for ensemble DFT of excited states.

Significance. If the central relaxation is shown to be exact or a controlled outer bound, the work supplies new, explicit constraints that could improve the physical consistency of ensemble density functionals for excited states on lattices. The explicit linkage to a generalized Horn problem is a constructive strength, as it imports existing mathematical machinery rather than deriving inequalities from scratch. The resulting polytope for occupation numbers is directly usable in practical calculations.

major comments (2)
  1. [Section describing the relaxation to occupation-number vectors] The systematic relaxation from fixing full 1RDMs (including off-diagonal and phase information) to natural occupation numbers alone is load-bearing for the tractability claim. The manuscript must demonstrate explicitly that this projection preserves the feasible set of the original partial-information problem or yields a controlled outer approximation; otherwise the subsequent intersection with generalized Horn constraints may admit unphysical ensemble 1RDMs. A concrete counter-example or proof sketch in the relevant section would resolve the concern.
  2. [Section on the convex relaxation and polytope construction] The additional convex relaxation that produces the final polytope for lattice-site occupations requires a clear statement of whether the resulting set is tight with respect to the combined Horn-plus-weighted-ensemble conditions or merely an outer bound. Without this clarification or supporting numerical checks on small systems, it is difficult to assess how physically meaningful the restrictions actually are.
minor comments (2)
  1. [Introduction / notation section] Notation for the ensemble weights w_i and the partial-information sets should be introduced with a compact table or diagram early in the manuscript to aid readability.
  2. [Abstract] The abstract asserts an 'explicit solution' via the Horn connection; a brief forward reference to the specific theorem or proposition number would help readers locate the combination step.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major concerns point by point below and will revise the manuscript to incorporate clarifications and additional material where needed.

read point-by-point responses

  1. Referee: [Section describing the relaxation to occupation-number vectors] The systematic relaxation from fixing full 1RDMs (including off-diagonal and phase information) to natural occupation numbers alone is load-bearing for the tractability claim. The manuscript must demonstrate explicitly that this projection preserves the feasible set of the original partial-information problem or yields a controlled outer approximation; otherwise the subsequent intersection with generalized Horn constraints may admit unphysical ensemble 1RDMs. A concrete counter-example or proof sketch in the relevant section would resolve the concern.

    Authors: We agree that explicit justification of the relaxation step is essential. The projection onto natural occupation numbers is constructed as a systematic outer approximation: any ensemble 1RDM satisfying the original partial-information constraints necessarily satisfies the relaxed occupation-number constraints, but the converse does not hold. We will add a short proof sketch in the relevant section establishing this inclusion and a concrete counter-example (N=2 fermions on a 4-site lattice) demonstrating that the relaxed set is strictly larger and can admit unphysical points that are subsequently excluded by the generalized Horn constraints. This will make clear that the final bounds remain controlled outer approximations. revision: yes

  2. Referee: [Section on the convex relaxation and polytope construction] The additional convex relaxation that produces the final polytope for lattice-site occupations requires a clear statement of whether the resulting set is tight with respect to the combined Horn-plus-weighted-ensemble conditions or merely an outer bound. Without this clarification or supporting numerical checks on small systems, it is difficult to assess how physically meaningful the restrictions actually are.

    Authors: We accept that the tightness of the final convex polytope must be stated explicitly. The polytope is obtained by a further convex relaxation of the intersection of the generalized Horn constraints and the weighted ensemble N-representability inequalities, and is therefore an outer bound. We will revise the text to state this clearly and add numerical comparisons on small systems (2-site and 4-site Hubbard chains with N=2) that quantify the gap between the polytope and the exact feasible set obtained by direct enumeration, thereby illustrating the practical utility of the bounds for ensemble DFT. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: derivation combines independent mathematical results with a controlled relaxation

full rationale

The paper's central step is a systematic relaxation of the partial-information ensemble N-representability problem to constraints on natural occupation numbers alone, followed by an explicit combination of those constraints with a generalization of Horn's problem and the known weighted ensemble N-representability inequalities. No equation in the provided abstract or description reduces the target polytope to a fitted parameter or to a self-citation that is itself defined by the present result. The relaxation is presented as an outer bound whose physical utility is asserted separately; the text does not claim or require that the occupation-number projection is exactly equivalent to the original feasible set. Self-citations to prior N-representability work by overlapping authors are present but function as external input rather than load-bearing justification for the new relaxation or the Horn connection. The derivation therefore remains self-contained against external benchmarks and does not collapse by construction.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The construction rests on the standard mathematical properties of reduced density matrices and the known solution set of Horn's problem; no new free parameters or invented entities are introduced in the abstract.

assumptions (1)
  • standard math The N-representability conditions for weighted ensembles of fermion states are known and can be combined with Horn-type inequalities.
    Invoked when the abstract states that the relaxed problem is solved by combining generalized Horn constraints with weighted ensemble N-representability conditions.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Refining ensemble $N$-representability of one-body density matrices from partial information." pith.science (2026). https://pith.science/paper/2506.09960

@misc{pith2026250609960,
  author       = {Pith},
  title        = {Pith review of: Refining ensemble $N$-representability of one-body density matrices from partial information},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2506.09960}},
  note         = {Machine review of arXiv:2506.09960}
}
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 reduced density matrices (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.

Figures

Figures reproduced from arXiv: 2506.09960 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic illustration of the tightening of [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Illustration of the non-convex polytope [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Illustration of the spectral polytopes [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Illustration of the residual [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Illustration of the residual [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Illustration of the construction of the spectral set [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

Discussion (0). Continue with ORCID to comment.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    systematic relaxation that reduces the refined problem ... to a more tractable form involving only natural occupation number vectors ... related to a generalization of Horn's problem

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    convex polytope that provides physically meaningful restrictions on lattice site occupations

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

134 extracted references · 134 canonical work pages

  1. [1]

    Thus, Horn’s problem corresponds to the case of|K| = 1 fixed natu- ral occupation number vectors (see Sec

    Generalized Horn problem The generalized Horn problem, which deals with the sum of principal submatrices of a Hermitian matrix [80], extends the well-known Horn problem [78, 79, 101, 102] beyond the sum of two Hermitian matrices. Thus, Horn’s problem corresponds to the case of|K| = 1 fixed natu- ral occupation number vectors (see Sec. II) and will be used...

    work page

  2. [2]

    [78–80, 101, 102], to fully charac- terize Λ(w, LK) for generic r and cardinality |K| of the index set K

    Characterisation of Λ(w, LK) In this section, we use the solution to Horn’s problem and its generalization for sums ofm Hermitian matrices, as discussed in Refs. [78–80, 101, 102], to fully charac- terize Λ(w, LK) for generic r and cardinality |K| of the index set K. The first key result is the following lemma: Lemma 1. The set Λ↓(w, LK) ≡ n λ↓ | λ ∈ Λ(w,...

    work page

  3. [3]

    Discussion The derivation of the setΛ↓(w, LK) for |K| = 1 fixed natural occupation number vectors corresponds to the original Horn problem, which deals with the sum of two Hermitian matrices with fixed, decreasingly ordered vec- tors of eigenvalues, as explained in Sec. IIIB1. Even for |K| = 1, the number of linear constraints grows signifi- cantly with t...

    work page

  4. [4]

    produces the inequalities in Eq. (36). Moreover, Eq. (36) reveals that unless all(N − 1) largest entries of λ(1) equal one (recall Fig. 3 for N = 2 ), the new exclusion principle constraints defining the spectral set λ ∈ Σ(w, λ(1))are always stricter than thew-constraints that define Σ(w). Furthermore, using the hyperplane representation of the spectral s...

    work page 1949

  5. [5]

    A. J. Coleman, Structure of fermion density matrices, Rev. Mod. Phys.35, 668 (1963)

    work page 1963

  6. [6]

    Garrod and J

    C. Garrod and J. K. Percus, Reduction of the N-Particle Variational Problem, J. Math. Phys.5, 1756 (1964)

    work page 1964

  7. [7]

    Kummer, n-Representability Problem for Reduced Density Matrices, J

    H. Kummer, n-Representability Problem for Reduced Density Matrices, J. Math. Phys.8, 2063 (1967)

    work page

  8. [8]

    R. M. Erdahl, Representability, Int. J. Quantum Chem. 13, 697 (1978)

    work page 1978

Show all 134 references
  1. [9]

    A. J. Coleman and V. Yukalov,Reduced Density Matri- ces: Coulson ’s challenge (Springer Berlin, Heidelberg, 2000)

    2000

  2. [10]

    Klyachko, Quantum marginal problem and N- representability, J

    A. Klyachko, Quantum marginal problem and N- representability, J. Phys. Conf. Ser.36, 72 (2006)

    2006

  3. [11]

    Altunbulak and A

    M. Altunbulak and A. Klyachko, The Pauli principle revisited, Commun. Math. Phys.282, 287 (2008)

    2008

  4. [12]

    D. A. Mazziotti, Structure of fermionic density ma- trices: Complete N-representability conditions, Phys. Rev. Lett.108, 263002 (2012)

    2012

  5. [13]

    D. A. Mazziotti, Significant conditions for the two- electron reduced density matrix from the constructive solution of N representability, Phys. Rev. A85, 062507 (2012)

    2012

  6. [14]

    Maciazek, A

    T. Maciazek, A. Sawicki, D. Gross, A. Lopes, and C. Schilling, Implications of pinned occupation numbers for natural orbital expansions. ‘II: rigorous derivation and extension to non-fermionic systems, New J. Phys. 22, 023002 (2020)

    2020

  7. [15]

    Schilling, C

    C. Schilling, C. L. Benavides-Riveros, A. Lopes, T. Ma- ciazek, and A. Sawicki, Implications of pinned occupa- tion numbers for natural orbital expansions: I. gener- alizing the concept of active spaces, New J. Phys.22, 023001 (2020). 16

    2020

  8. [16]

    D. A. Mazziotti, Quantum many-body theory from a solution of the N-representability problem, Phys. Rev. Lett. 130, 153001 (2023)

    2023

  9. [17]

    Y.-K. Liu, M. Christandl, and F. Verstraete, Quan- tumcomputationalcomplexityofthe N-representability problem: QMA complete, Phys. Rev. Lett.98, 110503 (2007)

    2007

  10. [18]

    O’Gorman, S

    B. O’Gorman, S. Irani, J. Whitfield, and B. Fefferman, Intractability of electronic structure in a fixed basis, PRX Quantum3, 020322 (2022)

    2022

  11. [19]

    Klyachko, Quantum marginal problem and represen- tations of the symmetric group, arXiv:0409113 (2004)

    A. Klyachko, Quantum marginal problem and represen- tations of the symmetric group, arXiv:0409113 (2004)

    2004

  12. [20]

    Schilling, D

    C. Schilling, D. Gross, and M. Christandl, Pinning of fermionic occupation numbers, Phys. Rev. Lett. 110, 040404 (2013)

    2013

  13. [21]

    Schilling, Quasipinning and its relevance for N- fermion quantum states, Phys

    C. Schilling, Quasipinning and its relevance for N- fermion quantum states, Phys. Rev. A 91, 022105 (2015)

    2015

  14. [22]

    Schilling, C

    C. Schilling, C. L. Benavides-Riveros, and P. Vrana, Re- constructing quantum states from single-party informa- tion, Phys. Rev. A96, 052312 (2017)

    2017

  15. [23]

    Schilling, M

    C. Schilling, M. Altunbulak, S. Knecht, A. Lopes, J. D. Whitfield, M.Christandl, D.Gross,andM.Reiher,Gen- eralized Pauli constraints in small atoms, Phys. Rev. A 97, 052503 (2018)

    2018

  16. [24]

    D. A. Mazziotti, Realization of quantum chemistry without wave functions through first-order semidefinite programming, Phys. Rev. Lett.93, 213001 (2004)

    2004

  17. [25]

    D. A. Mazziotti, Large-Scale Semidefinite Programming for Many-Electron Quantum Mechanics, Phys. Rev. Lett. 106, 083001 (2011)

    2011

  18. [26]

    Schilling, Communication: Relating the pure and en- semble density matrix functional, J

    C. Schilling, Communication: Relating the pure and en- semble density matrix functional, J. Chem. Phys.149, 231102 (2018)

    2018

  19. [27]

    Piris, Global natural orbital functional: Towards the complete description of the electron correlation, Phys

    M. Piris, Global natural orbital functional: Towards the complete description of the electron correlation, Phys. Rev. Lett.127, 233001 (2021)

    2021

  20. [28]

    L. M. Sager-Smith and D. A. Mazziotti, Reducing the quantum many-electron problem to two electrons with machinelearning,J.Am.Chem.Soc. 144,18959(2022)

    2022

  21. [29]

    L. M. Sager, A. O. Schouten, and D. A. Mazziotti, Beginnings of exciton condensation in coronene analog of graphene double layer, J. Chem. Phys.156, 154702 (2022)

    2022

  22. [30]

    G. M. Jones, R. R. Li, A. E. I. DePrince, and K. D. Vogiatzis, Data-driven refinement of electronic ener- gies from two-electron reduced-density-matrix theory, J. Phys. Chem. Lett.14, 6377 (2023)

    2023

  23. [31]

    Eugene DePrince III, Variational determination of the two-electron reduced density matrix: A tutorial re- view, WIREs Comput Mol Sci.14, e1702 (2024)

    A. Eugene DePrince III, Variational determination of the two-electron reduced density matrix: A tutorial re- view, WIREs Comput Mol Sci.14, e1702 (2024)

    2024

  24. [32]

    N. C. Rubin, R. Babbush, and J. McClean, Application of fermionic marginal constraints to hybrid quantum al- gorithms, New J. Phys.20, 053020 (2018)

    2018

  25. [33]

    S. E. Smart and D. A. Mazziotti, Efficient two-electron ansatz for benchmarking quantum chemistry on a quan- tum computer, Phys. Rev. Res.2, 023048 (2020)

    2020

  26. [34]

    G. A. Quantum, Collaborators*†, F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, S. Boixo, M. Broughton, B. B. Buckley, D. A. Buell, B. Burkett, N. Bushnell, Y. Chen, Z. Chen, B. Chiaro, R. Collins, W. Courtney, S. Demura, A. Dunsworth, E. Farhi, A. Fowler, B...

    2020

  27. [35]

    A. E. Raeber and D. A. Mazziotti, Non-equilibrium steady state conductivity in cyclo[18]carbon and its boron nitride analogue, Phys. Chem. Chem. Phys.22, 23998 (2020)

    2020

  28. [36]

    Avdic and D

    I. Avdic and D. A. Mazziotti, Fewer measurements from shadow tomography withN-representability conditions, Phys. Rev. Lett.132, 220802 (2024)

    2024

  29. [37]

    Avdic and D

    I. Avdic and D. A. Mazziotti, Enhanced shadow tomog- raphy of molecular excited states via the enforcement of N-representability conditions by semidefinite program- ming, Phys. Rev. A110, 052407 (2024)

    2024

  30. [38]

    Gritsenko, K

    O. Gritsenko, K. Pernal, and E. J. Baerends, An im- proved density matrix functional by physically moti- vated repulsive corrections, J. Chem. Phys.122, 204102 (2005)

    2005

  31. [39]

    Pernal and K

    K. Pernal and K. J. H. Giesbertz, Reduced den- sity matrix functional theory (RDMFT) and lin- ear response time-dependent rdmft (TD-RDMFT), in Density-Functional Methods for Excited States , edited by N. Ferré, M. Filatov, and M. Huix-Rotllant (Springer International Publishin...

    2016

  32. [40]

    Kamil, R

    E. Kamil, R. Schade, T. Pruschke, and P. E. Blöchl, Reduced density-matrix functionals applied to the Hub- bard dimer, Phys. Rev. B93, 085141 (2016)

    2016

  33. [41]

    Schade, E

    R. Schade, E. Kamil, and P. Blöchl, Reduced density- matrix functionals from many-particle theory, Eur. Phys. J. Special Topics226, 2677 (2017)

    2017

  34. [42]

    Schade and P

    R. Schade and P. E. Blöchl, Adaptive cluster approx- imation for reduced density-matrix functional theory, Phys. Rev. B97, 245131 (2018)

    2018

  35. [43]

    C. L. Benavides-Riveros and M. A. L. Marques, Static correlated functionals for reduced density matrix func- tional theory, Eur. Phys. J. B91, 133 (2018)

    2018

  36. [44]

    Piris, Natural orbital functional for multiplets, Phys

    M. Piris, Natural orbital functional for multiplets, Phys. Rev. A100, 032508 (2019)

    2019

  37. [45]

    Mitxelena, M

    I. Mitxelena, M. Piris, and J. M. Ugalde, Chapter seven - advances in approximate natural orbital func- tional theory, in State of The Art of Molecular Elec- tronic Structure Computations: Correlation Methods, Basis Sets and More , Advances in Quantum Chemistry, Vol. 79, edite...

    2019

  38. [46]

    Schmidt, C

    J. Schmidt, C. L. Benavides-Riveros, and M. A. L. Mar- ques, Reduced density matrix functional theory for su- perconductors, Phys. Rev. B99, 224502 (2019)

    2019

  39. [47]

    Piris and I

    M. Piris and I. Mitxelena, DoNOF: An open-source im- plementation of natural-orbital-functional-based meth- ods for quantum chemistry, Comput. Phys. Commun. 259, 107651 (2021)

    2021

  40. [48]

    Di Sabatino, C

    S. Di Sabatino, C. Verdozzi, and P. Romaniello, Time 17 dependent reduced density matrix functional theory at strong correlation: insights from a two-site Anderson impurity model, Phys. Chem. Chem. Phys.23, 16730 (2021)

    2021

  41. [49]

    Lemke, J

    Y. Lemke, J. Kussmann, and C. Ochsenfeld, Efficient integral-direct methods for self-consistent reduced den- sity matrix functional theory calculations on central and graphics processing units, J. Chem. Theory Comput. 18, 4229 (2022)

    2022

  42. [50]

    Gibney, J.-N

    D. Gibney, J.-N. Boyn, and D. A. Mazziotti, Den- sity functional theory transformed into a one-electron reduced-density-matrix functional theory for the cap- ture of static correlation, J. Phys. Chem. Lett.13, 1382 (2022)

    2022

  43. [51]

    Mitxelena and M

    I. Mitxelena and M. Piris, Benchmarking GNOF against FCI in challenging systems in one, two, and three di- mensions, J. Chem. Phys.156, 214102 (2022)

    2022

  44. [52]

    Senjean, S

    B. Senjean, S. Yalouz, N. Nakatani, and E. Fromager, Reduced density matrix functional theory from an ab initio seniority-zero wave function: Exact and approx- imate formulations along adiabatic connection paths, Phys. Rev. A106, 032203 (2022)

    2022

  45. [53]

    Liebert, A

    J. Liebert, A. Y. Chaou, and C. Schilling, Refining and relating fundamentals of functional theory, J. Chem. Phys. 158 (2023)

    2023

  46. [54]

    S. M. Sutter and K. J. H. Giesbertz, One-body reduced density-matrix functional theory for the canonical en- semble, Phys. Rev. A107, 022210 (2023)

    2023

  47. [55]

    Gibney, J.-N

    D. Gibney, J.-N. Boyn, and D. A. Mazziotti, Univer- sal generalization of density functional theory for static correlation, Phys. Rev. Lett.131, 243003 (2023)

    2023

  48. [56]

    Gibney, J.-N

    D. Gibney, J.-N. Boyn, and D. A. Mazziotti, Enhancing density-functional theory for static correlation in large molecules, Phys. Rev. A110, L040802 (2024)

    2024

  49. [57]

    C. L. Benavides-Riveros, T. Wasak, and A. Recati, Extracting many-body quantum resources within one- body reduced density matrix functional theory, Phys. Rev. Res.6, L012052 (2024)

    2024

  50. [58]

    Vladaj, Q

    M. Vladaj, Q. Marécat, B. Senjean, and M. Saubanére, Variational minimization scheme for the one-particle re- duced density matrix functional theory in the ensemble N-representability domain, J. Chem. Phys.161, 074105 (2024)

    2024

  51. [59]

    conjecture

    J. Cioslowski and K. Strasburger, Constraints upon functionals of the 1-matrix, universal properties of nat- ural orbitals, and the fallacy of the Collins “conjecture”, J. Phys. Chem. Lett.15, 1328 (2024)

    2024

  52. [60]

    N. G. Cartier and K. J. H. Giesbertz, Exploiting the Hessian for a better convergence of the SCF-RDMFT procedure, J. Chem. Theory Comput.20, 3669 (2024)

    2024

  53. [61]

    Yao and N

    Y.-F. Yao and N. Q. Su, Enhancing reduced density matrix functional theory calculations by coupling or- bital and occupation optimizations, arXiv:2402.03532 (2024)

    2024

  54. [62]

    Watanabe, Über die Anwendung thermodynamis- cher Begriffe auf den Normalzustand des Atomkerns, Z

    S. Watanabe, Über die Anwendung thermodynamis- cher Begriffe auf den Normalzustand des Atomkerns, Z. Physik 113, 482–513 (1939)

    1939

  55. [63]

    E. K. U. Gross, L. N. Oliveira, and W. Kohn, Rayleigh- Ritz variational principle for ensembles of fractionally occupied states, Phys. Rev. A37, 2805 (1988)

    1988

  56. [64]

    Ding, C.-L

    L. Ding, C.-L. Hong, and C. Schilling, Ground and ex- cited states from ensemble variational principles, Quan- tum 8, 1525 (2024)

    2024

  57. [65]

    Schilling and S

    C. Schilling and S. Pittalis, Ensemble reduced density matrix functional theory for excited states and hierar- chicalgeneralizationofPauli’sexclusionprinciple,Phys. Rev. Lett.127, 023001 (2021)

    2021

  58. [66]

    Liebert, F

    J. Liebert, F. Castillo, J.-P. Labbé, and C. Schilling, Foundation of one-particle reduced density matrix func- tional theory for excited states, J. Chem. Theory Com- put. 18, 124 (2022)

    2022

  59. [67]

    Liebert and C

    J. Liebert and C. Schilling, An exact one-particle the- ory of bosonic excitations: from a generalized Hohen- berg–Kohn theorem to convexified N-representability, New J. Phys.25, 013009 (2023)

    2023

  60. [68]

    Liebert and C

    J. Liebert and C. Schilling, Deriving density-matrix functionals for excited states, SciPost Phys. 14, 120 (2023)

    2023

  61. [69]

    L. N. Oliveira, E. K. U. Gross, and W. Kohn, Density- functional theory for ensembles of fractionally occupied states. II. Application to the He atom, Phys. Rev. A37, 2821 (1988)

    1988

  62. [70]

    Z.-h. Yang, A. Pribram-Jones, K. Burke, and C. A. Ull- rich, Direct extraction of excitation energies from en- semble density-functional theory, Phys. Rev. Lett.119, 033003 (2017)

    2017

  63. [71]

    E.Fromager,Individualcorrelationsinensembledensity functional theory: State- and density-driven decompo- sitions without additional Kohn-Sham systems, Phys. Rev. Lett.124 (2020)

    2020

  64. [72]

    Loos and E

    P.-F. Loos and E. Fromager, A weight-dependent local correlation density-functional approximation for ensem- bles, J. Chem. Phys.152, 214101 (2020)

    2020

  65. [73]

    Cernatic, B

    F. Cernatic, B. Senjean, V. Robert, and E. Fromager, Ensemble density functional theory of neutral and charged excitations, Top. Curr. Chem.280, 4 (2022)

    2022

  66. [74]

    Gould and L

    T. Gould and L. Kronik, Ensemble generalized Kohn–Sham theory: The good, the bad, and the ugly, J. Chem. Phys.154, 094125 (2021)

    2021

  67. [75]

    Yang, Second-order perturbative correlation en- ergy functional in the ensemble density-functional the- ory, Phys

    Z.-h. Yang, Second-order perturbative correlation en- ergy functional in the ensemble density-functional the- ory, Phys. Rev. A104, 052806 (2021)

    2021

  68. [76]

    Gould, D

    T. Gould, D. P. Kooi, P. Gori-Giorgi, and S. Pittalis, Electronic excited states in extreme limits via ensem- ble density functionals, Phys. Rev. Lett.130, 106401 (2023)

    2023

  69. [77]

    Giarrusso and P.-F

    S. Giarrusso and P.-F. Loos, Exact excited-state func- tionals of the asymmetric Hubbard dimer, J. Phys. Chem. Lett.14, 8780 (2023)

    2023

  70. [78]

    T. R. Scott, J. Kozlowski, S. Crisostomo, A. Pribram- Jones,andK.Burke,Exactconditionsforensembleden- sity functional theory, Phys. Rev. B109, 195120 (2024)

    2024

  71. [80]

    Castillo, J.-P

    F. Castillo, J.-P. Labbé, J. Liebert, A. Padrol, E. Philippe, and C. Schilling, An effective solution to convex 1-body N-representability, Ann. Henri Poincaré 24, 2241–2321 (2023)

    2023

  72. [81]

    Liebert, F

    J. Liebert, F. Castillo, J.-P. Labbé, M. Maciazek, and C. Schilling, Solving one-body ensemble N- representability problems with spin, arXiv:2412.01805 (2024)

    2024

  73. [82]

    Knutson, The symplectic and algebraic geometry of Horn’s problem, Linear Algebra Its Appl.319, 61 (2000)

    A. Knutson, The symplectic and algebraic geometry of Horn’s problem, Linear Algebra Its Appl.319, 61 (2000). 18

    2000

  74. [83]

    Bhatia, Linear algebra to quantum cohomology: The story of Alfred Horn’s inequalities, Am

    R. Bhatia, Linear algebra to quantum cohomology: The story of Alfred Horn’s inequalities, Am. Math. Mon. 108, 289 (2001)

    2001

  75. [84]

    Li and Y.-T

    C.-K. Li and Y.-T. Poon, Principal submatrices of a hermitian matrix, Linear and Multilinear Algebra51, 199 (2003)

    2003

  76. [85]

    Eisert, T

    J. Eisert, T. Tyc, T. Rudolph, and B. Sanders, Gaus- sian quantum marginal problem, Nat. Commun.280, 263–280 (2008)

    2008

  77. [86]

    Walter, B

    M. Walter, B. Doran, D. Gross, and M. Christandl, En- tanglement polytopes: Multiparticle entanglement from single-particle information, Science340, 1205 (2013)

    2013

  78. [87]

    Schilling,Quantum marginal problem and its physical relevance, Ph.D

    C. Schilling,Quantum marginal problem and its physical relevance, Ph.D. thesis, ETH-Zürich (2014)

    2014

  79. [88]

    Christandl, B

    M. Christandl, B. Doran, S. Kousidis, and M. Walter, Eigenvalue distributions of reduced density matrices, Nat. Commun.332, 1 (2014)

    2014

  80. [89]

    Sawicki, M

    A. Sawicki, M. Oszmaniec, and M. Kuś, Convexity of momentum map, Morse index, and quantum entangle- ment, Rev. Math. Phys.26, 1450004 (2014)

    2014

  81. [90]

    Wyderka, F

    N. Wyderka, F. Huber, and O. Gühne, Almost all four- particle pure states are determined by their two-body marginals, Phys. Rev. A96, 010102 (2017)

    2017

  82. [91]

    Bürgisser, M

    P. Bürgisser, M. Christandl, K. D. Mulmuley, and M. Walter, Membership in moment polytopes is in NP and coNP, SIAM J. Comput.46, 972 (2017)

    2017

  83. [92]

    Maciazek and V

    T. Maciazek and V. Tsanov, Quantum marginals from pure doubly excited states, J. Phys. A: Math. Theor. 50, 465304 (2017)

    2017

  84. [93]

    Christandl, M

    M. Christandl, M. Şahinoğlu, and M. Walter, Recou- pling coefficients and quantum entropies, Ann. Henri Poincaré 19, 385–410 (2018)

    2018

  85. [94]

    X.-D. Yu, T. Simnacher, N. Wyderka, H. C. Nguyen, and O. Gühne, A complete hierarchy for the pure state marginal problem in quantum mechanics, Nat. Com- mun. 12, 1012 (2021)

    2021

  86. [95]

    Haapasalo, T

    E. Haapasalo, T. Kraft, N. Miklin, and R. Uola, Quan- tum marginal problem and incompatibility, Quantum5, 476 (2021)

    2021

  87. [96]

    Hsieh, M

    C.-Y. Hsieh, M. Lostaglio, and A. Acín, Quantum chan- nel marginal problem, Phys. Rev. Res.4, 013249 (2022)

    2022

  88. [97]

    Gühne, E

    O. Gühne, E. Haapasalo, T. Kraft, J.-P. Pellonpää, and R. Uola, Colloquium: Incompatible measurements in quantum information science, Rev. Mod. Phys.95, 011003 (2023)

    2023

  89. [98]

    Klyachko, The Pauli exclusion principle and beyond, arXiv:0904.2009 (2009)

    A. Klyachko, The Pauli exclusion principle and beyond, arXiv:0904.2009 (2009)

    2009 arXiv

  90. [99]

    Reuvers, Generalized Pauli constraints in large sys- tems: The Pauli principle dominates, J

    R. Reuvers, Generalized Pauli constraints in large sys- tems: The Pauli principle dominates, J. Math. Phys. 62, 032204 (2021)

    2021

  91. [100]

    Alberti and A

    P. Alberti and A. Uhlmann,Stochasticity and Partial Order: Doubly Stochastic Maps and Unitary Mixing , Mathematics and Its Applications (Springer Nether- lands, 1982)

    1982

  92. [101]

    S. M. Valone, Consequences of extending 1-matrix en- ergy functionals from pure–state representable to all ensemble representable 1-matrices, J. Chem. Phys.73, 1344 (1980)

    1980

  93. [102]

    E. H. Lieb, Density functionals for Coulomb systems, Int. J. Quantum Chem.24, 243 (1983)

    1983

  94. [103]

    A. K. Theophilou, The energy density functional for- malism for excited states, J. Phys. C: Solid State Phys. 12, 5419 (1979)

    1979

  95. [104]

    R. T. Rockafellar,Convex Analysis (Princeton univer- sity press, 1997)

    1997

  96. [105]

    Klyachko, Stable bundles, representation theory and Hermitian operators, Sel

    A. Klyachko, Stable bundles, representation theory and Hermitian operators, Sel. math., New ser.4, 419 (1998)

    1998

  97. [106]

    Knutson and T

    A. Knutson and T. Tao, The honeycomb model of GLN(C) tensor products I: Proof of the saturation con- jecture, J. Amer. Math. Soc.12, 1055 (1999)

    1999

  98. [107]

    Kirwan, Convexity properties of the moment map- ping, iii, Invent

    F. Kirwan, Convexity properties of the moment map- ping, iii, Invent. Math.77, 547–552 (1984)

    1984

  99. [108]

    Guillemin, Moment Maps and Combinatorial In- variants of Hamiltonian Tn-spaces (Birkhäuser Boston, MA, 1994)

    V. Guillemin, Moment Maps and Combinatorial In- variants of Hamiltonian Tn-spaces (Birkhäuser Boston, MA, 1994)

    1994

  100. [109]

    Knutson, T

    A. Knutson, T. Tao, and C. Woodward, The honeycomb model of GLN(C) tensor products II: Puzzles deter- mine facets of the Littlewood-Richardson cone, J. Amer. Math. Soc.17, 19 (2004)

    2004

  101. [110]

    Rado, An inequality, J

    R. Rado, An inequality, J. Lond. Math. Soc.1, 1 (1952)

    1952

  102. [111]

    Gunnarsson and K

    O. Gunnarsson and K. Schönhammer, Density- functional treatment of an exactly solvable semiconduc- tor model, Phys. Rev. Lett.56, 1968 (1986)

    1968

  103. [112]

    Schönhammer, O

    K. Schönhammer, O. Gunnarsson, and R. M. Noack, Density-functionaltheoryonalattice: Comparisonwith exact numerical results for a model with strongly corre- lated electrons, Phys. Rev. B52, 2504 (1995)

    1995

  104. [113]

    Ijäs and A

    M. Ijäs and A. Harju, Lattice density-functional theory on graphene, Phys. Rev. B82, 235111 (2010)

    2010

  105. [114]

    Xianlong, A.-H

    G. Xianlong, A.-H. Chen, I. V. Tokatly, and S. Kurth, Lattice density functional theory at finite temperature with strongly density-dependent exchange-correlation potentials, Phys. Rev. B86, 235139 (2012)

    2012

  106. [115]

    Saubanère and G

    M. Saubanère and G. M. Pastor, Lattice density- functional theory of the attractive Hubbard model, Phys. Rev. B90, 125128 (2014)

    2014

  107. [116]

    J. P. Coe, Lattice density-functional theory for quantum chemistry, Phys. Rev. B99, 165118 (2019)

    2019

  108. [117]

    Penz and R

    M. Penz and R. van Leeuwen, Density-functional theory on graphs, J. Chem. Phys.155, 244111 (2021)

    2021

  109. [118]

    Penz and R

    M. Penz and R. van Leeuwen, Geometry of degeneracy in potential and density space, Quantum7, 918 (2023)

    2023

  110. [119]

    Penz and R

    M. Penz and R. van Leeuwen, Geometrical perspec- tive on spin–lattice density-functional theory, J. Chem. Phys. 161, 150901 (2024)

    2024

  111. [120]

    Schur, Über eine Klasse von Mittelbildungen mit Anwendungen die Determinanten-Theorie, Sitzungsber

    I. Schur, Über eine Klasse von Mittelbildungen mit Anwendungen die Determinanten-Theorie, Sitzungsber. Berl. Math. Ges.22, 9 (1923)

    1923

  112. [121]

    Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, Am

    A. Horn, Doubly stochastic matrices and the diagonal of a rotation matrix, Am. J. Math.76, 620 (1954)

    1954

  113. [122]

    G. M. Ziegler, Lectures on Polytopes (Springer New York, NY, 1994)

    1994

  114. [123]

    E. K. U. Gross, L. N. Oliveira, and W. Kohn, Density- functional theory for ensembles of fractionally occupied states.I. Basic formalism,Phys. Rev.A37,2809(1988)

    1988

  115. [124]

    Fromager, Ensemble density functional theory of groundandexcitedenergylevels,J.Phys.Chem.A 129, 1143 (2025)

    E. Fromager, Ensemble density functional theory of groundandexcitedenergylevels,J.Phys.Chem.A 129, 1143 (2025)

    2025

  116. [125]

    Cernatic, P.-F

    F. Cernatic, P.-F. Loos, B. Senjean, and E. Fromager, Neutral electronic excitations and derivative discontinu- ities: An extended n-centered ensemble density func- tional theory perspective, Phys. Rev. B 109, 235113 (2024)

    2024

  117. [126]

    D. A. Mazziotti, Realization of Quantum Chemistry without Wave Functions through First-Order Semidefi- nite Programming, Phys. Rev. Lett.93, 213001 (2004)

    2004

  118. [127]

    D. A. Mazziotti, Dual-cone variational calculation of the two-electron reduced density matrix, Phys. Rev. A102, 19 052819 (2020)

    2020

  119. [128]

    Phys.24, 053004 (2022)

    M.J.Knight, H.M.Quiney,andA.M.Martin,Reduced density matrix approach to ultracold few-fermion sys- tems in one dimension, New J. Phys.24, 053004 (2022)

    2022

  120. [129]

    A. O. Schouten, J. E. Klevens, L. M. Sager-Smith, J. Xie, J. S. Anderson, and D. A. Mazziotti, Potential for exciton condensation in a highly conductive amor- phous polymer, Phys. Rev. Mat.7, 045001 (2023)

    2023

  121. [130]

    J. Xie, S. Ewing, J.-N. Boyn, A. S. Filatov, B. Cheng, T. Ma, G. L. Grocke, N. Zhao, R. Itani, X. Sun, H. Cho, Z. Chen, K. W. Chapman, S. N. Patel, D. V. Talapin, J. Park, D. A. Mazziotti, and J. S. Anderson, Intrinsic glassy-metallic transport in an amorphous coordination pol...

    2022

  122. [131]

    T. Xin, D. Lu, J. Klassen, N. Yu, Z. Ji, J. Chen, X. Ma, G. Long, B. Zeng, and R. Laflamme, Quantum state tomography via reduced density matrices, Phys. Rev. Lett. 118, 020401 (2017)

    2017

  123. [132]

    A. Zhao, N. C. Rubin, and A. Miyake, Fermionic partial tomography via classical shadows, Phys. Rev. Lett.127, 110504 (2021)

    2021

  124. [133]

    X. Shao, L. Paetow, M. E. Tuckerman, and M. Pa- vanello, Machine learning electronic structure methods based on the one-electron reduced density matrix, Nat. Commun. 14, 6281 (2023)

    2023

  125. [134]

    Hazra, U

    S. Hazra, U. Patil, and S. Sanvito, Predicting the one- particle density matrix with machine learning, J. Chem. Theory Comput.20, 4569 (2024)

    2024

  126. [135]

    Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull

    W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Am. Math. Soc. 37, 209 (2000)

    2000

Pith tools

Reviewed May 19, 2026 · model on record in the stance chip above.