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:2609.22010

Selection rules for pinned and quasipinned occupation numbers with degeneracy

Robin Reuvers

quant-phSubmitted Sep 18, 2026• Announced Sep 21, 2026• 0 citations
v1 · Sep 18, 2026v2 · Sep 21, 2026v3 · Sep 25, 2026
arXivHTMLPDF

Abstract

When the natural occupation numbers of a fermionic state saturate a generalized Pauli constraint, the state obeys a selection rule: only those configurations of natural orbitals that saturate the constraint themselves contribute. Pinning thus singles out an active space. The selection rule was proved for non-degenerate occupation numbers; for degenerate ones it was conjectured, and proved for one saturated constraint under an unverified assumption. Here, it is proved for every generalized Pauli constraint, with no assumption, and also in the spin-adapted setting. The rule can only fail for the ordering constraints $λ_j\geλ_{j+1}$. Several saturated constraints are served by one basis whenever their common zero face contains a non-degenerate point. In practice, occupation numbers are quasipinned rather than pinned, and I show what the rule then becomes.

Comments
24 pages, 1 figure

Related papers

Quasipinning and its relevance for N -fermion quantum states

Christian Schilling

Fermionic natural occupation numbers (NON) do not only obey Pauli's famous exclusion principle, but are even further restricted to a polytope by the generalized Pauli constraints, conditions which follow from the fermionic exchange statistics. Whenever given NON are pinned to the polytope's boundary, the corresponding $N$-fermion quantum state $|{\ensuremath{\Psi}}_{N}\ensuremath{\rangle}$ simplifies due to a selection rule. We show analytically and numerically for the most relevant cases that this rule is stable for NON close to the boundary, if the NON are nondegenerate. In case of degeneracy, a modified selection rule is conjectured and its validity is supported. As a consequence, the recently found effect of quasipinning is physically relevant in the sense that its occurrence allows to approximately reconstruct $|{\ensuremath{\Psi}}_{N}\ensuremath{\rangle}$, its entanglement properties, and correlations from one-particle information. Our finding also provides the basis for a generalized Hartree-Fock method by a variational ansatz determined by the selection rule.

arXiv:1409.0019•Physical Review A•Feb 9, 2015•44 citations

Implications of pinned occupation numbers for natural orbital expansions: I. Generalizing the concept of active spaces

Christian Schilling, Carlos L. Benavides-Riveros, Alexandre Lopes +2 more

Abstract The concept of active spaces simplifies the description of interacting quantum many-body systems by restricting to a neighborhood of active orbitals around the Fermi level. The respective wavefunction ansatzes which involve all possible electron configurations of active orbitals can be characterized by the saturation of a certain number of Pauli constraints 0 ≤ n i ≤ 1 , identifying the occupied core orbitals (n i = 1) and the inactive virtual orbitals (n j = 0). In Part I, we generalize this crucial concept of active spaces by referring to the generalized Pauli constraints. To be more specific, we explain and illustrate that the saturation of any such constraint on fermionic occupation numbers characterizes a distinctive set of active electron configurations. A converse form of this selection rule establishes the basis for corresponding multiconfigurational wavefunction ansatzes. In Part II, we provide rigorous derivations of those findings. Moroever, we extend our results to non-fermionic multipartite quantum systems, revealing that extremal single-body information has always strong implications for the multipartite quantum state. In that sense, our work also confirms that pinned quantum systems define new physical entities and the presence of pinnings reflect the existence of (possibly hidden) ground state symmetries.

arXiv:1908.10938•New Journal of Physics•Dec 20, 2019•15 citations

Pinning of Fermionic Occupation Numbers

Christian Schilling, David Gross, Matthias Christandl

The Pauli exclusion principle is a constraint on the natural occupation numbers of fermionic states. It has been suspected since at least the 1970s, and only proved very recently, that there is a multitude of further constraints on these numbers, generalizing the Pauli principle. Here, we provide the first analytic analysis of the physical relevance of these constraints. We compute the natural occupation numbers for the ground states of a family of interacting fermions in a harmonic potential. Intriguingly, we find that the occupation numbers are almost, but not exactly, pinned to the boundary of the allowed region (quasipinned). The result suggests that the physics behind the phenomenon is richer than previously appreciated. In particular, it shows that for some models, the generalized Pauli constraints play a role for the ground state, even though they do not limit the ground-state energy. Our findings suggest a generalization of the Hartree-Fock approximation.

arXiv:1210.5531•Physical Review Letters•Jan 22, 2013•104 citations

Quasipinning and selection rules for excitations in atoms and molecules

Carlos L. Benavides-Riveros, Michael Springborg

Postulated by Pauli to explain the electronic structure of atoms and molecules, the exclusion principle establishes an upper bound of 1 for fermionic natural occupation numbers ${{n}_{i}}$. A recent analysis of the pure $N$-representability problem provides a wide set of inequalities for the ${{n}_{i}}$, leading to constraints on these numbers. This has a strong potential impact on reduced density matrix functional theory as we know it. In this work we continue our study of the nature of these inequalities for some atomic and molecular systems. The results indicate that (quasi)saturation of some of them leads to selection rules for the dominant configurations in configuration interaction expansions, in favorable cases providing means for significantly reducing their computational requirements.

arXiv:1409.6953•Physical Review A•Jul 16, 2015•33 citations

Toolbox of spin-adapted generalized Pauli constraints

Julia Liebert, Yannick Lemke, Murat Altunbulak +3 more

We establish a toolbox for studying and applying spin-adapted generalized Pauli constraints (GPCs) in few-electron quantum systems. By exploiting the spin symmetry of realistic N -electron wave functions, the underlying one-body pure N -representability problem simplifies, allowing us to calculate the GPCs for larger system sizes than previously accessible. We then uncover and rigorously prove a superselection rule that highlights the significance of GPCs: whenever a spin-adapted GPC is (approximately) saturated—referred to as (quasi)pinning—the corresponding N -electron wave function assumes a simplified structure. Specifically, in a configuration interaction expansion based on natural orbitals only very specific spin configuration state functions may contribute. To assess the nontriviality of (quasi)pinning, we introduce a geometric measure that contrasts it with the (quasi)pinning induced by simple (spin-adapted) Pauli constraints. Applications to few-electron systems suggest that previously observed quasipinning largely stems from spin symmetries.

arXiv:2502.15464•Physical Review Research•Jun 11, 2025•2 citations

Pinning of fermionic occupation numbers: General concepts and one spatial dimension

Felix Tennie, Daniel Ebler, Vlatko Vedral +1 more

Analytical evidence for the physical relevance of generalized Pauli constraints (GPCs) has recently been provided in Schilling et al. [Phys. Rev. Lett. 110, 040404 (2013)]: Natural occupation numbers $\stackrel{P\vec}{\ensuremath{\lambda}}\ensuremath{\equiv}({\ensuremath{\lambda}}_{i})$ of the ground state of a model system in the regime of weak couplings $\ensuremath{\kappa}$ of three spinless fermions in one spatial dimension were found extremely close, at a distance ${D}_{\mathrm{min}}\ensuremath{\sim}{\ensuremath{\kappa}}^{8}$ to the boundary of the allowed region. We provide a self-contained and complete study of this quasipinning phenomenon. In particular, we develop tools for its systematic exploration and quantification. We confirm that quasipinning in one dimension occurs also for larger particle numbers and extends to intermediate coupling strengths, but vanishes for very strong couplings. We further explore the nontriviality of our findings by comparing quasipinning by GPCs to potential quasipinning by the less restrictive Pauli exclusion principle constraints. This allows us to eventually confirm the significance of GPCs beyond Pauli's exclusion principle.

arXiv:1602.05198•Physical Review A•Apr 29, 2016•24 citations
When the natural occupation numbers of a fermionic state saturate a generalized Pauli constraint, the state obeys a selection rule: only those configurations of natural orbitals that saturate the constraint themselves contribute. Pinning thus singles out an active space. The selection rule was proved for non-degenerate occupation numbers; for degenerate ones it was conjectured, and proved for one saturated constraint under an unverified assumption. Here, it is proved for every generalized Pauli constraint, with no assumption, and also in the spin-adapted setting. The rule can only fail for the ordering constraints λj≥λj+1λ_j\geλ_{j+1}λj​≥λj+1​. Several saturated constraints are served by one basis whenever their common zero face contains a non-degenerate point. In practice, occupation numbers are quasipinned rather than pinned, and I show what the rule then becomes.
Fermionic natural occupation numbers (NON) do not only obey Pauli's famous exclusion principle, but are even further restricted to a polytope by the generalized Pauli constraints, conditions which follow from the fermionic exchange statistics. Whenever given NON are pinned to the polytope's boundary, the corresponding NNN-fermion quantum state $|{\ensuremath{\Psi}}_{N}\ensuremath{\rangle}$ simplifies due to a selection rule. We show analytically and numerically for the most relevant cases that this rule is stable for NON close to the boundary, if the NON are nondegenerate. In case of degeneracy, a modified selection rule is conjectured and its validity is supported. As a consequence, the recently found effect of quasipinning is physically relevant in the sense that its occurrence allows to approximately reconstruct $|{\ensuremath{\Psi}}_{N}\ensuremath{\rangle}$, its entanglement properties, and correlations from one-particle information. Our finding also provides the basis for a generalized Hartree-Fock method by a variational ansatz determined by the selection rule.
Abstract The concept of active spaces simplifies the description of interacting quantum many-body systems by restricting to a neighborhood of active orbitals around the Fermi level. The respective wavefunction ansatzes which involve all possible electron configurations of active orbitals can be characterized by the saturation of a certain number of Pauli constraints 0 ≤ n i ≤ 1 , identifying the occupied core orbitals (n i = 1) and the inactive virtual orbitals (n j = 0). In Part I, we generalize this crucial concept of active spaces by referring to the generalized Pauli constraints. To be more specific, we explain and illustrate that the saturation of any such constraint on fermionic occupation numbers characterizes a distinctive set of active electron configurations. A converse form of this selection rule establishes the basis for corresponding multiconfigurational wavefunction ansatzes. In Part II, we provide rigorous derivations of those findings. Moroever, we extend our results to non-fermionic multipartite quantum systems, revealing that extremal single-body information has always strong implications for the multipartite quantum state. In that sense, our work also confirms that pinned quantum systems define new physical entities and the presence of pinnings reflect the existence of (possibly hidden) ground state symmetries.
The Pauli exclusion principle is a constraint on the natural occupation numbers of fermionic states. It has been suspected since at least the 1970s, and only proved very recently, that there is a multitude of further constraints on these numbers, generalizing the Pauli principle. Here, we provide the first analytic analysis of the physical relevance of these constraints. We compute the natural occupation numbers for the ground states of a family of interacting fermions in a harmonic potential. Intriguingly, we find that the occupation numbers are almost, but not exactly, pinned to the boundary of the allowed region (quasipinned). The result suggests that the physics behind the phenomenon is richer than previously appreciated. In particular, it shows that for some models, the generalized Pauli constraints play a role for the ground state, even though they do not limit the ground-state energy. Our findings suggest a generalization of the Hartree-Fock approximation.
Postulated by Pauli to explain the electronic structure of atoms and molecules, the exclusion principle establishes an upper bound of 1 for fermionic natural occupation numbers ni{{n}_{i}}ni​. A recent analysis of the pure NNN-representability problem provides a wide set of inequalities for the ni{{n}_{i}}ni​, leading to constraints on these numbers. This has a strong potential impact on reduced density matrix functional theory as we know it. In this work we continue our study of the nature of these inequalities for some atomic and molecular systems. The results indicate that (quasi)saturation of some of them leads to selection rules for the dominant configurations in configuration interaction expansions, in favorable cases providing means for significantly reducing their computational requirements.
We establish a toolbox for studying and applying spin-adapted generalized Pauli constraints (GPCs) in few-electron quantum systems. By exploiting the spin symmetry of realistic N -electron wave functions, the underlying one-body pure N -representability problem simplifies, allowing us to calculate the GPCs for larger system sizes than previously accessible. We then uncover and rigorously prove a superselection rule that highlights the significance of GPCs: whenever a spin-adapted GPC is (approximately) saturated—referred to as (quasi)pinning—the corresponding N -electron wave function assumes a simplified structure. Specifically, in a configuration interaction expansion based on natural orbitals only very specific spin configuration state functions may contribute. To assess the nontriviality of (quasi)pinning, we introduce a geometric measure that contrasts it with the (quasi)pinning induced by simple (spin-adapted) Pauli constraints. Applications to few-electron systems suggest that previously observed quasipinning largely stems from spin symmetries.
Analytical evidence for the physical relevance of generalized Pauli constraints (GPCs) has recently been provided in Schilling et al. [Phys. Rev. Lett. 110, 040404 (2013)]: Natural occupation numbers $\stackrel{P\vec}{\ensuremath{\lambda}}\ensuremath{\equiv}({\ensuremath{\lambda}}_{i})$ of the ground state of a model system in the regime of weak couplings $\ensuremath{\kappa}$ of three spinless fermions in one spatial dimension were found extremely close, at a distance ${D}_{\mathrm{min}}\ensuremath{\sim}{\ensuremath{\kappa}}^{8}$ to the boundary of the allowed region. We provide a self-contained and complete study of this quasipinning phenomenon. In particular, we develop tools for its systematic exploration and quantification. We confirm that quasipinning in one dimension occurs also for larger particle numbers and extends to intermediate coupling strengths, but vanishes for very strong couplings. We further explore the nontriviality of our findings by comparing quasipinning by GPCs to potential quasipinning by the less restrictive Pauli exclusion principle constraints. This allows us to eventually confirm the significance of GPCs beyond Pauli's exclusion principle.