Bosonic content of three-fermion highest-spin states
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AbstractA rigorous characterization of the information content of any highest-spin three-fermion wave function is presented. It is based upon a formal decomposition of the wave function into a finite set of fixed invariants, called shapes, whose sole purpose is to satisfy the Pauli principle, and a variable part, constituting the bosonic excitations of these invariants, that provides its physical content. As an example, this decomposition is applied to a benchmark-quality approximate wave function of the lowest-energy quartet electronic state of the lithium atom. This wave function, which comprises hundreds of basis functions, is reduced to eleven shape blocks, only five of which are numerically significant. Such a compact characterization is a generic example of the appearance of superselection rules in configuration space, and provides a qualitative aid in the search for robust few-particle entangled states.Featured image: Three-dimensional histogram of the shape-block contributions to the electronic wave function of the $^4P_u$ quartet state of the lithium atom. The surfaces of the geometric figures are proportional to the probabilities. The yellow cube and green cylinder, the blue squares, and the red cones depict the three-, two-, and one-dimensional shape blocks, respectively.► BibTeX data@article{Cioslowski2026bosoniccontentof, doi = {10.22331/q-2026-05-29-2121}, url = {https://doi.org/10.22331/q-2026-05-29-2121}, title = {Bosonic content of three-fermion highest-spin states}, author = {Cioslowski, Jerzy and Strasburger, Krzysztof and Sunko, Denis K.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2121}, month = may, year = {2026} }► References [1] Pejman Jouzdani, H. Arslan Hashim, and Eduardo R. Mucciolo. Quantum algorithms for state preparation and data classification based on stabilizer codes. Phys. Rev. A, 109:022602, Feb 2024. doi:10.1103/PhysRevA.109.022602. https://doi.org/10.1103/PhysRevA.109.022602 [2] Johannes Herrmann, Sergi Masot Llima, Ants Remm, Petr Zapletal, Nathan A. McMahon, Colin Scarato, François Swiadek, Christian Kraglund Andersen, Christoph Hellings, Sebastian Krinner, Nathan Lacroix, Stefania Lazar, Michael Kerschbaum, Dante Colao Zanuz, Graham J. Norris, Michael J. Hartmann, Andreas Wallraff, and Christopher Eichler. Realizing quantum convolutional neural networks on a superconducting quantum processor to recognize quantum phases. Nature Communications, 13(1):4144, Jul 2022. doi:10.1038/s41467-022-31679-5. https://doi.org/10.1038/s41467-022-31679-5 [3] Eva Pavarini and Erik Koch, editors. Simulating Correlations with Computers, volume 11 of Schriften des Forschungszentrums Jülich Modeling and Simulation. Forschungszentrum Jülich GmbH Zentralbibliothek, Verlag, Jülich, Sep 2021. URL: http://hdl.handle.net/2128/28665. http://hdl.handle.net/2128/28665 [4] Gaurav Saxena, Eric Chitambar, and Gilad Gour. Dynamical resource theory of quantum coherence. Phys. Rev. Res., 2:023298, Jun 2020. doi:10.1103/PhysRevResearch.2.023298. https://doi.org/10.1103/PhysRevResearch.2.023298 [5] D. K. Sunko. Natural generalization of the ground-state Slater determinant to more than one dimension. Phys. Rev. A, 93:062109, 2016. doi:10.1103/PhysRevA.93.062109. https://doi.org/10.1103/PhysRevA.93.062109 [6] Bernd Sturmfels. Algorithms in Invariant Theory. Springer-Verlag, Wien, 2 edition, 2008. doi:10.1007/978-3-211-77417-5. https://doi.org/10.1007/978-3-211-77417-5 [7] Denis K. Sunko. Entropy of pure states: not all wave functions are born equal. 4open, 5:3, 2022. doi:10.1051/fopen/2021006. https://doi.org/10.1051/fopen/2021006 [8] Damiano Aliverti-Piuri, Kaustav Chatterjee, Lexin Ding, Ke Liao, Julia Liebert, and Christian Schilling. What can quantum information theory offer to quantum chemistry? Faraday Discuss., 254:76–106, 2024. doi:10.1039/D4FD00059E. https://doi.org/10.1039/D4FD00059E [9] W. Heisenberg. Mehrkörperproblem und Resonanz in der Quantenmechanik. Zeitschrift für Physik, 38(6):411–426, 1926. doi:10.1007/BF01397160. https://doi.org/10.1007/BF01397160 [10] J. C. Slater. The theory of complex spectra. Phys. Rev., 34:1293–1322, Nov 1929. doi:10.1103/PhysRev.34.1293. https://doi.org/10.1103/PhysRev.34.1293 [11] Erik Koch. Second Quantization and Jordan-Wigner Representations. In E. Pavarini and E. Koch, editors, Simulating Correlations with Computers, Schriften des Forschungszentrums Jülich Modeling and Simulation, pages 1.1–1.29. Forschungszentrum Jülich GmbH Zentralbibliothek, Verlag, 2021. URL: http://hdl.handle.net/2128/28665. http://hdl.handle.net/2128/28665 [12] Katarina Rožman and D. K. Sunko. Generic example of algebraic bosonisation. Eur. Phys. J. Plus, 135:30, 2020. doi:10.1140/epjp/s13360-019-00015-0. https://doi.org/10.1140/epjp/s13360-019-00015-0 [13] D. K. Sunko. Many-fermion wave functions: Structure and examples. In J. Bonča and S. Kruchinin, editors, Advanced Nanomaterials for Detection of CBRN, NATO Science for Peace and Security Series A: Chemistry and Biology., pages 85–99. Springer, 2020. doi:10.1007/978-94-024-2030-2_5. https://doi.org/10.1007/978-94-024-2030-2_5 [14] D K Sunko and J Ciosłowski. The three-dimensional harmonic oscillator and solid harmonics in Bargmann space. European Journal of Physics, 45(5):055401, jul 2024. doi:10.1088/1361-6404/ad61d1. https://doi.org/10.1088/1361-6404/ad61d1 [15] D. K. Sunko. Evaluation and spanning sets of confluent Vandermonde forms. Journal of Mathematical Physics, 63(8):082101, 2022. doi:10.1063/5.0075576. https://doi.org/10.1063/5.0075576 [16] James E. Humphreys. Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics.
Cambridge University Press, 1990. doi:10.1017/CBO9780511623646. https://doi.org/10.1017/CBO9780511623646 [17] R. P. Stanley. Enumerative Combinatorics.
Cambridge University Press, Cambridge, 1999. doi:10.1017/CBO9780511609589. https://doi.org/10.1017/CBO9780511609589 [18] François Bergeron. Multivariate diagonal coinvariant spaces for complex reflection groups. Advances in Mathematics, 239:97–108, 2013. doi:10.1016/j.aim.2013.02.013. https://doi.org/10.1016/j.aim.2013.02.013 [19] Mark Haiman. Combinatorics, symmetric functions, and Hilbert schemes.
In Current Developments in Mathematics, volume 2002, pages 39–111. International Press, Somerville, MA, 2003. URL: https://projecteuclid.org/ebooks/current-developments-in-mathematics/Current-Developments-in-Mathematics-2002/Chapter/Combinatorics-symmetric-functions-and-Hilbert-schemes/cdm/1088530398. https://projecteuclid.org/ebooks/current-developments-in-mathematics/Current-Developments-in-Mathematics-2002/Chapter/Combinatorics-symmetric-functions-and-Hilbert-schemes/cdm/1088530398 [20] Brendon Rhoades and Andrew Timothy Wilson. Set superpartitions and superspace duality modules. Forum Math. Sigma, 10:e105, 2022. Zbl 1504.05303. doi:10.1017/fms.2022.90. https://doi.org/10.1017/fms.2022.90 [21] J. E. Hirsch. Two-dimensional Hubbard model: Numerical simulation study. Phys. Rev. B, 31:4403–4419, Apr 1985. doi:10.1103/PhysRevB.31.4403. https://doi.org/10.1103/PhysRevB.31.4403 [22] Pierre-Loïc Méliot. Representation theory of symmetric groups. Discrete Mathematics and Its Applications. CRC Press, Boca Raton, 2017. doi:10.1201/9781315371016. https://doi.org/10.1201/9781315371016 [23] Jim Mitroy, Sergiy Bubin, Wataru Horiuchi, Yasuyuki Suzuki, Ludwik Adamowicz, Wojciech Cencek, Krzysztof Szalewicz, Jacek Komasa, D. Blume, and Kálmán Varga. Theory and application of explicitly correlated Gaussians. Rev. Mod. Phys., 85:693–749, 2013. doi:10.1103/RevModPhys.85.693. https://doi.org/10.1103/RevModPhys.85.693 [24] Sergiy Bubin, Michele Pavanello, Wei-Cheng Tung, Keeper L. Sharkey, and Ludwik Adamowicz. Born–Oppenheimer and Non-Born–Oppenheimer, Atomic and Molecular Calculations with Explicitly Correlated Gaussians. Chemical Reviews, 113(1):36–79, Jan 2013. doi:10.1021/cr200419d. https://doi.org/10.1021/cr200419d [25] Aage Bohr. Rotational motion in nuclei. Rev. Mod. Phys., 48:365–374, Jul 1976. doi:10.1103/RevModPhys.48.365. https://doi.org/10.1103/RevModPhys.48.365 [26] Ben Mottelson. Elementary modes of excitation in the nucleus. Rev. Mod. Phys., 48:375–383, Jul 1976. doi:10.1103/RevModPhys.48.375. https://doi.org/10.1103/RevModPhys.48.375 [27] D. Janssen, R.V. Jolos, and F. Dönau. An algebraic treatment of the nuclear quadrupole degree of freedom. Nuclear Physics A, 224(1):93–115, 1974. doi:10.1016/0375-9474(74)90165-1. https://doi.org/10.1016/0375-9474(74)90165-1 [28] Francesco Iachello. Algebraic models of many-body systems and their dynamic symmetries and supersymmetries. Journal of Physics: Conference Series, 1194(1):012048, apr 2019. doi:10.1088/1742-6596/1194/1/012048. https://doi.org/10.1088/1742-6596/1194/1/012048 [29] Michiel A. Bakker, Sebastian Mehl, Tuukka Hiltunen, Ari Harju, and David P. DiVincenzo. Validity of the single-particle description and charge noise resilience for multielectron quantum dots. Phys. Rev. B, 91:155425, Apr 2015. doi:10.1103/PhysRevB.91.155425. https://doi.org/10.1103/PhysRevB.91.155425 [30] Xiaoling Wu, Xinhui Liang, Yaoqi Tian, Fan Yang, Cheng Chen, Yong-Chun Liu, Meng Khoon Tey, and Li You. A concise review of Rydberg atom based quantum computation and quantum simulation. Chinese Physics B, 30(2):020305, feb 2021. doi:10.1088/1674-1056/abd76f. https://doi.org/10.1088/1674-1056/abd76f [31] Paul Breiding, Türkü Özlüm Çelik, Timothy Duff, Alexander Heaton, Aida Maraj, Anna-Laura Sattelberger, Lorenzo Venturello, and Oǧuzhan Yürük. Nonlinear algebra and applications. Numerical Algebra, Control and Optimization, 13(1):81–116, 2023. doi:10.3934/naco.2021045. https://doi.org/10.3934/naco.2021045 [32] Mateusz Michałek and Bernd Sturmfels. Invitation to Nonlinear Algebra. In Graduate studies in mathematics, vol. 211.
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Cubic symmetry and magnetic frustration on the fcc spin lattice in ${\mathrm{K}}_{2}{\mathrm{IrCl}}_{6}$. Phys. Rev. B, 99:144425, Apr 2019. doi:10.1103/PhysRevB.99.144425. https://doi.org/10.1103/PhysRevB.99.144425Cited byCould not fetch Crossref cited-by data during last attempt 2026-05-29 10:32:13: Could not fetch cited-by data for 10.22331/q-2026-05-29-2121 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-05-29 10:32:14: Cannot retrieve data from ADS due to rate limitations.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractA rigorous characterization of the information content of any highest-spin three-fermion wave function is presented. It is based upon a formal decomposition of the wave function into a finite set of fixed invariants, called shapes, whose sole purpose is to satisfy the Pauli principle, and a variable part, constituting the bosonic excitations of these invariants, that provides its physical content. As an example, this decomposition is applied to a benchmark-quality approximate wave function of the lowest-energy quartet electronic state of the lithium atom. This wave function, which comprises hundreds of basis functions, is reduced to eleven shape blocks, only five of which are numerically significant. Such a compact characterization is a generic example of the appearance of superselection rules in configuration space, and provides a qualitative aid in the search for robust few-particle entangled states.Featured image: Three-dimensional histogram of the shape-block contributions to the electronic wave function of the $^4P_u$ quartet state of the lithium atom. The surfaces of the geometric figures are proportional to the probabilities. The yellow cube and green cylinder, the blue squares, and the red cones depict the three-, two-, and one-dimensional shape blocks, respectively.► BibTeX data@article{Cioslowski2026bosoniccontentof, doi = {10.22331/q-2026-05-29-2121}, url = {https://doi.org/10.22331/q-2026-05-29-2121}, title = {Bosonic content of three-fermion highest-spin states}, author = {Cioslowski, Jerzy and Strasburger, Krzysztof and Sunko, Denis K.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2121}, month = may, year = {2026} }► References [1] Pejman Jouzdani, H. Arslan Hashim, and Eduardo R. Mucciolo. Quantum algorithms for state preparation and data classification based on stabilizer codes. Phys. Rev. A, 109:022602, Feb 2024. doi:10.1103/PhysRevA.109.022602. https://doi.org/10.1103/PhysRevA.109.022602 [2] Johannes Herrmann, Sergi Masot Llima, Ants Remm, Petr Zapletal, Nathan A. McMahon, Colin Scarato, François Swiadek, Christian Kraglund Andersen, Christoph Hellings, Sebastian Krinner, Nathan Lacroix, Stefania Lazar, Michael Kerschbaum, Dante Colao Zanuz, Graham J. Norris, Michael J. Hartmann, Andreas Wallraff, and Christopher Eichler. Realizing quantum convolutional neural networks on a superconducting quantum processor to recognize quantum phases. Nature Communications, 13(1):4144, Jul 2022. doi:10.1038/s41467-022-31679-5. https://doi.org/10.1038/s41467-022-31679-5 [3] Eva Pavarini and Erik Koch, editors. Simulating Correlations with Computers, volume 11 of Schriften des Forschungszentrums Jülich Modeling and Simulation. Forschungszentrum Jülich GmbH Zentralbibliothek, Verlag, Jülich, Sep 2021. URL: http://hdl.handle.net/2128/28665. http://hdl.handle.net/2128/28665 [4] Gaurav Saxena, Eric Chitambar, and Gilad Gour. Dynamical resource theory of quantum coherence. Phys. Rev. Res., 2:023298, Jun 2020. doi:10.1103/PhysRevResearch.2.023298. https://doi.org/10.1103/PhysRevResearch.2.023298 [5] D. K. Sunko. Natural generalization of the ground-state Slater determinant to more than one dimension. Phys. Rev. A, 93:062109, 2016. doi:10.1103/PhysRevA.93.062109. https://doi.org/10.1103/PhysRevA.93.062109 [6] Bernd Sturmfels. Algorithms in Invariant Theory. Springer-Verlag, Wien, 2 edition, 2008. doi:10.1007/978-3-211-77417-5. https://doi.org/10.1007/978-3-211-77417-5 [7] Denis K. Sunko. Entropy of pure states: not all wave functions are born equal. 4open, 5:3, 2022. doi:10.1051/fopen/2021006. https://doi.org/10.1051/fopen/2021006 [8] Damiano Aliverti-Piuri, Kaustav Chatterjee, Lexin Ding, Ke Liao, Julia Liebert, and Christian Schilling. What can quantum information theory offer to quantum chemistry? Faraday Discuss., 254:76–106, 2024. doi:10.1039/D4FD00059E. https://doi.org/10.1039/D4FD00059E [9] W. Heisenberg. Mehrkörperproblem und Resonanz in der Quantenmechanik. Zeitschrift für Physik, 38(6):411–426, 1926. doi:10.1007/BF01397160. https://doi.org/10.1007/BF01397160 [10] J. C. Slater. The theory of complex spectra. Phys. Rev., 34:1293–1322, Nov 1929. doi:10.1103/PhysRev.34.1293. https://doi.org/10.1103/PhysRev.34.1293 [11] Erik Koch. Second Quantization and Jordan-Wigner Representations. In E. Pavarini and E. Koch, editors, Simulating Correlations with Computers, Schriften des Forschungszentrums Jülich Modeling and Simulation, pages 1.1–1.29. Forschungszentrum Jülich GmbH Zentralbibliothek, Verlag, 2021. URL: http://hdl.handle.net/2128/28665. http://hdl.handle.net/2128/28665 [12] Katarina Rožman and D. K. Sunko. Generic example of algebraic bosonisation. Eur. Phys. J. Plus, 135:30, 2020. doi:10.1140/epjp/s13360-019-00015-0. https://doi.org/10.1140/epjp/s13360-019-00015-0 [13] D. K. Sunko. Many-fermion wave functions: Structure and examples. In J. Bonča and S. Kruchinin, editors, Advanced Nanomaterials for Detection of CBRN, NATO Science for Peace and Security Series A: Chemistry and Biology., pages 85–99. Springer, 2020. doi:10.1007/978-94-024-2030-2_5. https://doi.org/10.1007/978-94-024-2030-2_5 [14] D K Sunko and J Ciosłowski. The three-dimensional harmonic oscillator and solid harmonics in Bargmann space. European Journal of Physics, 45(5):055401, jul 2024. doi:10.1088/1361-6404/ad61d1. https://doi.org/10.1088/1361-6404/ad61d1 [15] D. K. Sunko. Evaluation and spanning sets of confluent Vandermonde forms. Journal of Mathematical Physics, 63(8):082101, 2022. doi:10.1063/5.0075576. https://doi.org/10.1063/5.0075576 [16] James E. Humphreys. Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics.
Cambridge University Press, 1990. doi:10.1017/CBO9780511623646. https://doi.org/10.1017/CBO9780511623646 [17] R. P. Stanley. Enumerative Combinatorics.
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In Current Developments in Mathematics, volume 2002, pages 39–111. International Press, Somerville, MA, 2003. URL: https://projecteuclid.org/ebooks/current-developments-in-mathematics/Current-Developments-in-Mathematics-2002/Chapter/Combinatorics-symmetric-functions-and-Hilbert-schemes/cdm/1088530398. https://projecteuclid.org/ebooks/current-developments-in-mathematics/Current-Developments-in-Mathematics-2002/Chapter/Combinatorics-symmetric-functions-and-Hilbert-schemes/cdm/1088530398 [20] Brendon Rhoades and Andrew Timothy Wilson. Set superpartitions and superspace duality modules. Forum Math. Sigma, 10:e105, 2022. Zbl 1504.05303. doi:10.1017/fms.2022.90. https://doi.org/10.1017/fms.2022.90 [21] J. E. Hirsch. Two-dimensional Hubbard model: Numerical simulation study. Phys. Rev. B, 31:4403–4419, Apr 1985. doi:10.1103/PhysRevB.31.4403. https://doi.org/10.1103/PhysRevB.31.4403 [22] Pierre-Loïc Méliot. Representation theory of symmetric groups. Discrete Mathematics and Its Applications. CRC Press, Boca Raton, 2017. doi:10.1201/9781315371016. https://doi.org/10.1201/9781315371016 [23] Jim Mitroy, Sergiy Bubin, Wataru Horiuchi, Yasuyuki Suzuki, Ludwik Adamowicz, Wojciech Cencek, Krzysztof Szalewicz, Jacek Komasa, D. Blume, and Kálmán Varga. Theory and application of explicitly correlated Gaussians. Rev. Mod. Phys., 85:693–749, 2013. doi:10.1103/RevModPhys.85.693. https://doi.org/10.1103/RevModPhys.85.693 [24] Sergiy Bubin, Michele Pavanello, Wei-Cheng Tung, Keeper L. Sharkey, and Ludwik Adamowicz. Born–Oppenheimer and Non-Born–Oppenheimer, Atomic and Molecular Calculations with Explicitly Correlated Gaussians. Chemical Reviews, 113(1):36–79, Jan 2013. doi:10.1021/cr200419d. https://doi.org/10.1021/cr200419d [25] Aage Bohr. Rotational motion in nuclei. Rev. Mod. Phys., 48:365–374, Jul 1976. doi:10.1103/RevModPhys.48.365. https://doi.org/10.1103/RevModPhys.48.365 [26] Ben Mottelson. Elementary modes of excitation in the nucleus. Rev. Mod. Phys., 48:375–383, Jul 1976. doi:10.1103/RevModPhys.48.375. https://doi.org/10.1103/RevModPhys.48.375 [27] D. Janssen, R.V. Jolos, and F. Dönau. An algebraic treatment of the nuclear quadrupole degree of freedom. Nuclear Physics A, 224(1):93–115, 1974. doi:10.1016/0375-9474(74)90165-1. https://doi.org/10.1016/0375-9474(74)90165-1 [28] Francesco Iachello. Algebraic models of many-body systems and their dynamic symmetries and supersymmetries. Journal of Physics: Conference Series, 1194(1):012048, apr 2019. doi:10.1088/1742-6596/1194/1/012048. https://doi.org/10.1088/1742-6596/1194/1/012048 [29] Michiel A. Bakker, Sebastian Mehl, Tuukka Hiltunen, Ari Harju, and David P. DiVincenzo. Validity of the single-particle description and charge noise resilience for multielectron quantum dots. Phys. Rev. B, 91:155425, Apr 2015. doi:10.1103/PhysRevB.91.155425. https://doi.org/10.1103/PhysRevB.91.155425 [30] Xiaoling Wu, Xinhui Liang, Yaoqi Tian, Fan Yang, Cheng Chen, Yong-Chun Liu, Meng Khoon Tey, and Li You. A concise review of Rydberg atom based quantum computation and quantum simulation. Chinese Physics B, 30(2):020305, feb 2021. doi:10.1088/1674-1056/abd76f. https://doi.org/10.1088/1674-1056/abd76f [31] Paul Breiding, Türkü Özlüm Çelik, Timothy Duff, Alexander Heaton, Aida Maraj, Anna-Laura Sattelberger, Lorenzo Venturello, and Oǧuzhan Yürük. Nonlinear algebra and applications. Numerical Algebra, Control and Optimization, 13(1):81–116, 2023. doi:10.3934/naco.2021045. https://doi.org/10.3934/naco.2021045 [32] Mateusz Michałek and Bernd Sturmfels. Invitation to Nonlinear Algebra. In Graduate studies in mathematics, vol. 211.
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