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Two important distributions with non-positive quasiprobability are the Wigner function and the Glauber-Sudarshan function. Here we study properties of the spin Wigner function for finite-dimensional quantum systems and draw comparisons with its infinite-dimensional analog, focusing in particular on the relation to the Glauber-Sudarshan function and the existence of absolutely Wigner-bounded states. More precisely, we investigate unitary orbits of mixed spin states that are characterized by Wigner functions lower-bounded by a specified value. To this end, we extend a characterization of the set of absolutely Wigner positive states as a set of linear eigenvalue constraints, which together define a polytope centred on the maximally mixed state in the simplex of spin-j<\/mml:mi><\/mml:math> states. The lower bound determines the relative size of such absolutely Wigner bounded (AWB) polytopes and we study their geometric characteristics. In each dimension a Hilbert-Schmidt ball representing a tight purity-based sufficient condition to be AWB is exactly determined, while another ball representing a necessary condition to be AWB is conjectured. Special attention is given to the case where the polytope separates orbits containing only positive Wigner functions from other orbits because of the use of Wigner negativity as a witness of non-classicality. Comparisons are made to absolute symmetric state separability and spin Glauber-Sudarshan positivity, with additional details given for low spin quantum numbers.<\/jats:p>","DOI":"10.22331\/q-2024-12-04-1550","type":"journal-article","created":{"date-parts":[[2024,12,4]],"date-time":"2024-12-04T16:41:06Z","timestamp":1733330466000},"page":"1550","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":3,"title":["Polytopes of Absolutely Wigner Bounded Spin States"],"prefix":"10.22331","volume":"8","author":[{"given":"J\u00e9r\u00f4me","family":"Denis","sequence":"first","affiliation":[{"name":"Institut de Physique Nucl\u00e9aire, Atomique et de Spectroscopie, CESAM, University of Li\u00e8ge, B-4000 Li\u00e8ge, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jack","family":"Davis","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1"},{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1"},{"name":"DIENS, \u00c9cole Normale Sup\u00e9rieure, PSL University, CNRS, INRIA, 45 rue d'Ulm, Paris 75005, France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Robert B.","family":"Mann","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1"},{"name":"Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1"},{"name":"Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo, Ontario, Canada N2L 2Y5"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"John","family":"Martin","sequence":"additional","affiliation":[{"name":"Institut de Physique Nucl\u00e9aire, Atomique et de Spectroscopie, CESAM, University of Li\u00e8ge, B-4000 Li\u00e8ge, Belgium"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2024,12,4]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"G. Champagne, N. Johnston, M. MacDonald, and L. Pipes. ``Spectral properties of symmetric quantum states and symmetric entanglement witnesses''. Linear Algebra Its Appl. 649, 273\u2013300 (2022).","DOI":"10.1016\/j.laa.2022.05.004"},{"key":"1","doi-asserted-by":"publisher","unstructured":"E. Serrano-Ens\u00e1stiga and J. Martin. ``Maximum entanglement of mixed symmetric states under unitary transformations''. SciPost Phys. 15, 120 (2023).","DOI":"10.21468\/SciPostPhys.15.3.120"},{"key":"2","doi-asserted-by":"publisher","unstructured":"E. Serrano-Ens\u00e1stiga, J. Denis, and J. Martin. ``Absolute-separability witnesses for symmetric multiqubit states''. Phys. Rev. A 109, 022430 (2024).","DOI":"10.1103\/PhysRevA.109.022430"},{"key":"3","doi-asserted-by":"publisher","unstructured":"A. Ac\u00edn, N. J. Cerf, A. Ferraro, and J. Niset. ``Tests of multimode quantum nonlocality with homodyne measurements''. Phys. Rev. A 79, 012112 (2009).","DOI":"10.1103\/physreva.79.012112"},{"key":"4","doi-asserted-by":"publisher","unstructured":"V. Veitch, C. Ferrie, D. Gross, and J. Emerson. ``Negative quasi-probability as a resource for quantum computation''. New J. Phys. 14, 113011 (2012).","DOI":"10.1088\/1367-2630\/14\/11\/113011"},{"key":"5","doi-asserted-by":"publisher","unstructured":"A. Mari and J. Eisert. ``Positive Wigner Functions Render Classical Simulation of Quantum Computation Efficient''. Phys. Rev. Lett. 109, 230503 (2012).","DOI":"10.1103\/PhysRevLett.109.230503"},{"key":"6","doi-asserted-by":"publisher","unstructured":"M. Howard, J. Wallman, V. Veitch, and J. Emerson. ``Contextuality supplies the \u2018magic\u2019 for quantum computation''. Nature 510, 351\u2013355 (2014).","DOI":"10.1038\/nature13460"},{"key":"7","doi-asserted-by":"publisher","unstructured":"H. Pashayan, J. J. Wallman, and S. D. Bartlett. ``Estimating Outcome Probabilities of Quantum Circuits Using Quasiprobabilities''. Phys. Rev. Lett. 115, 070501 (2015).","DOI":"10.1103\/PhysRevLett.115.070501"},{"key":"8","doi-asserted-by":"publisher","unstructured":"N. Delfosse, C. Okay, J. Bermejo-Vega, D. E. Browne, and R. Raussendorf. ``Equivalence between contextuality and negativity of the Wigner function for qudits''. New J. Phys. 19, 123024 (2017).","DOI":"10.1088\/1367-2630\/aa8fe3"},{"key":"9","doi-asserted-by":"publisher","unstructured":"D. Schmid, H. Du, J. H. Selby, and M. F. Pusey. ``Uniqueness of Noncontextual Models for Stabilizer Subtheories''. Phys. Rev. Lett. 129, 120403 (2022).","DOI":"10.1103\/PhysRevLett.129.120403"},{"key":"10","doi-asserted-by":"publisher","unstructured":"R. I. Booth, U. Chabaud, and P.-E. Emeriau. ``Contextuality and Wigner Negativity Are Equivalent for Continuous-Variable Quantum Measurements''. Phys. Rev. Lett. 129, 230401 (2022).","DOI":"10.1103\/PhysRevLett.129.230401"},{"key":"11","doi-asserted-by":"publisher","unstructured":"V. Veitch, S. A. Hamed Mousavian, D. Gottesman, and J. Emerson. ``The resource theory of stabilizer quantum computation''. New J. Phys. 16, 013009 (2014).","DOI":"10.1088\/1367-2630\/16\/1\/013009"},{"key":"12","doi-asserted-by":"publisher","unstructured":"F. Albarelli, M. G. Genoni, M. G. A. Paris, and A. Ferraro. ``Resource theory of quantum non-Gaussianity and Wigner negativity''. Phys. Rev. A 98, 052350 (2018).","DOI":"10.1103\/PhysRevA.98.052350"},{"key":"13","doi-asserted-by":"publisher","unstructured":"R. Takagi and Q. Zhuang. ``Convex resource theory of non-Gaussianity''. Phys. Rev. A 97, 062337 (2018).","DOI":"10.1103\/PhysRevA.97.062337"},{"key":"14","doi-asserted-by":"publisher","unstructured":"X. Wang, M. M. Wilde, and Y. Su. ``Quantifying the magic of quantum channels''. New J. Phys. 21, 103002 (2019).","DOI":"10.1088\/1367-2630\/ab451d"},{"key":"15","unstructured":"R. L. Stratonovich. ``On Distributions in Representation Space''. Journal of Experimental and Theoretical Physics 4, 1012\u20131020 (1956). url: http:\/\/jetp.ras.ru\/cgi-bin\/e\/index\/e\/4\/6\/p891?a=list."},{"key":"16","doi-asserted-by":"publisher","unstructured":"C. D. Mink, D. Petrosyan, and M. Fleischhauer. ``Hybrid discrete-continuous truncated wigner approximation for driven, dissipative spin systems''. Phys. Rev. Res. 4, 043136 (2022).","DOI":"10.1103\/PhysRevResearch.4.043136"},{"key":"17","doi-asserted-by":"publisher","unstructured":"C. D. Mink and M. Fleischhauer. ``Collective radiative interactions in the discrete truncated Wigner approximation''. SciPost Phys. 15, 233 (2023).","DOI":"10.21468\/SciPostPhys.15.6.233"},{"key":"18","doi-asserted-by":"publisher","unstructured":"C. Brif and A. Mann. ``Phase-space formulation of quantum mechanics and quantum-state reconstruction for physical systems with Lie-group symmetries''. Phys. Rev. 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Torosyan. ``Kenfack \u2013 \u017byczkowski indicator of nonclassicality for two non-equivalent representations of Wigner function of qutrit''. Phys. Lett. A 412, 127591 (2021).","DOI":"10.1016\/j.physleta.2021.127591"},{"key":"23","doi-asserted-by":"publisher","unstructured":"G. S. Agarwal. ``Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions''. Phys. Rev. A 24, 2889\u20132896 (1981).","DOI":"10.1103\/PhysRevA.24.2889"},{"key":"24","doi-asserted-by":"publisher","unstructured":"J. P. Dowling, G. S. Agarwal, and W. P. Schleich. ``Wigner distribution of a general angular-momentum state: Applications to a collection of two-level atoms''. Phys. Rev. A 49, 4101\u20134109 (1994).","DOI":"10.1103\/PhysRevA.49.4101"},{"key":"25","doi-asserted-by":"publisher","unstructured":"M. F. Riedel, P. B\u00f6hi, Y. Li, T. W. H\u00e4nsch, A. Sinatra, and P. Treutlein. ``Atom-chip-based generation of entanglement for quantum metrology''. 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Romero, and H. de Guise. ``Generalized SU(2) covariant Wigner functions and some of their applications''. J. Phys. A 50, 323001 (2017).","DOI":"10.1088\/1751-8121\/50\/32\/323001"},{"key":"30","doi-asserted-by":"publisher","unstructured":"J. C. V\u00e1rilly and J. M. Gracia-Bond\u00eda. ``The Moyal representation for spin''. Ann. Phys. 190, 107\u2013148 (1989).","DOI":"10.1016\/0003-4916(89)90262-5"},{"key":"31","doi-asserted-by":"publisher","unstructured":"J.-P. Amiet and S. Weigert. ``Contracting the Wigner kernel of a spin to the Wigner kernel of a particle''. Phys. Rev. A 63, 012102 (2000).","DOI":"10.1103\/PhysRevA.63.012102"},{"key":"32","doi-asserted-by":"publisher","unstructured":"O. Giraud, P. Braun, and D. Braun. ``Classicality of spin states''. Phys. Rev. A 78, 042112 (2008).","DOI":"10.1103\/PhysRevA.78.042112"},{"key":"33","doi-asserted-by":"publisher","unstructured":"F. Bohnet-Waldraff, D. Braun, and O. Giraud. ``Partial transpose criteria for symmetric states''. Phys. 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Koczor, R. Zeier, and S. J. Glaser. ``Continuous phase-space representations for finite-dimensional quantum states and their tomography''. Phys. Rev. A 101, 022318 (2020).","DOI":"10.1103\/PhysRevA.101.022318"},{"key":"39","doi-asserted-by":"publisher","unstructured":"R. P. Rundle and M. J. Everitt. ``Overview of the Phase space Formulation of Quantum Mechanics with Application to Quantum Technologies''. Adv. Quantum Technol. 4, 2100016 (2021).","DOI":"10.1002\/qute.202100016"},{"key":"40","doi-asserted-by":"publisher","unstructured":"A. Grossmann. ``Parity operator and quantization of delta-functions''. Commun. Math. Phys. 48, 191\u2013194 (1976).","DOI":"10.1007\/BF01617867"},{"key":"41","doi-asserted-by":"publisher","unstructured":"A. Royer. ``Wigner function as the expectation value of a parity operator''. Phys. Rev. A 15, 449\u2013450 (1977).","DOI":"10.1103\/PhysRevA.15.449"},{"key":"42","doi-asserted-by":"publisher","unstructured":"D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii. ``Quantum Theory of Angular Momentum''. World Scientific. (1988).","DOI":"10.1142\/0270"},{"key":"43","doi-asserted-by":"publisher","unstructured":"J. Davis, M. Kumari, R. B. Mann, and S. Ghose. ``Wigner negativity in spin-$j$ systems''. Phys. Rev. Research 3, 033134 (2021).","DOI":"10.1103\/PhysRevResearch.3.033134"},{"key":"44","doi-asserted-by":"publisher","unstructured":"S. Heiss and S. Weigert. ``Discrete Moyal-type representations for a spin''. Phys. Rev. A 63, 012105 (2000).","DOI":"10.1103\/PhysRevA.63.012105"},{"key":"45","doi-asserted-by":"publisher","unstructured":"C. Brif and A. Mann. ``A general theory of phase-space quasiprobability distributions''. J. Phys. A: Math. Gen. 31, L9\u2013L17 (1998).","DOI":"10.1088\/0305-4470\/31\/1\/002"},{"key":"46","doi-asserted-by":"publisher","unstructured":"W. D\u00fcr, G. Vidal, and J. I. Cirac. ``Three qubits can be entangled in two inequivalent ways''. Phys. Rev. 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