Polytopes of Absolutely Wigner Bounded Spin States

Jérôme Denis; Jack Davis; Robert B. Mann; John Martin

doi:10.22331/q-2024-12-04-1550

Polytopes of Absolutely Wigner Bounded Spin States

Jérôme Denis¹, Jack Davis^2,3,4, Robert B. Mann^2,3,5, and John Martin¹

¹Institut de Physique Nucléaire, Atomique et de Spectroscopie, CESAM, University of Liège, B-4000 Liège, Belgium
²Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
³Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
⁴DIENS, École Normale Supérieure, PSL University, CNRS, INRIA, 45 rue d'Ulm, Paris 75005, France
⁵Perimeter Institute for Theoretical Physics, 31 Caroline St N, Waterloo, Ontario, Canada N2L 2Y5

Published:	2024-12-04, volume 8, page 1550
Eprint:	arXiv:2304.09006v3
Doi:	https://doi.org/10.22331/q-2024-12-04-1550
Citation:	Quantum 8, 1550 (2024).

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Abstract

Quasiprobability has become an increasingly popular notion for characterising non-classicality in quantum information, thermodynamics, and metrology. 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-$j$ 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.

Featured image: Polytopes of Absolutely Wigner Positive spin states for $j=1$ (left) and $j=3/2$ (right)

Popular summary

This work explores the relationship between Wigner negativity – a measure of non-classicality in the form of quasiprobability – and state spectra within symmetric qubit systems. We identify and analyze a family of polytopes that characterize density matrices with vanishing Wigner negativity regardless of any unitary evolution they undergo. Our study also draws quantitative parallels to the analogous concept of absolute separability in entanglement theory, providing insight to the structural properties of quantum spin-$j$ systems.

► BibTeX data

@article{Denis2024polytopesof,
  doi = {10.22331/q-2024-12-04-1550},
  url = {https://doi.org/10.22331/q-2024-12-04-1550},
  title = {Polytopes of {A}bsolutely {W}igner {B}ounded {S}pin {S}tates},
  author = {Denis, J{\'{e}}r{\^{o}}me and Davis, Jack and Mann, Robert B. and Martin, John},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1550},
  month = dec,
  year = {2024}
}

► References

[1] G. Champagne, N. Johnston, M. MacDonald, and L. Pipes. ``Spectral properties of symmetric quantum states and symmetric entanglement witnesses''. Linear Algebra Its Appl. 649, 273–300 (2022).
https://doi.org/10.1016/j.laa.2022.05.004

[2] E. Serrano-Ensástiga and J. Martin. ``Maximum entanglement of mixed symmetric states under unitary transformations''. SciPost Phys. 15, 120 (2023).
https://doi.org/10.21468/SciPostPhys.15.3.120

[3] E. Serrano-Ensástiga, J. Denis, and J. Martin. ``Absolute-separability witnesses for symmetric multiqubit states''. Phys. Rev. A 109, 022430 (2024).
https://doi.org/10.1103/PhysRevA.109.022430

[4] A. Acín, N. J. Cerf, A. Ferraro, and J. Niset. ``Tests of multimode quantum nonlocality with homodyne measurements''. Phys. Rev. A 79, 012112 (2009).
https://doi.org/10.1103/physreva.79.012112

[5] V. Veitch, C. Ferrie, D. Gross, and J. Emerson. ``Negative quasi-probability as a resource for quantum computation''. New J. Phys. 14, 113011 (2012).
https://doi.org/10.1088/1367-2630/14/11/113011

[6] A. Mari and J. Eisert. ``Positive Wigner Functions Render Classical Simulation of Quantum Computation Efficient''. Phys. Rev. Lett. 109, 230503 (2012).
https://doi.org/10.1103/PhysRevLett.109.230503

[7] M. Howard, J. Wallman, V. Veitch, and J. Emerson. ``Contextuality supplies the ‘magic’ for quantum computation''. Nature 510, 351–355 (2014).
https://doi.org/10.1038/nature13460

[8] H. Pashayan, J. J. Wallman, and S. D. Bartlett. ``Estimating Outcome Probabilities of Quantum Circuits Using Quasiprobabilities''. Phys. Rev. Lett. 115, 070501 (2015).
https://doi.org/10.1103/PhysRevLett.115.070501

[9] 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).
https://doi.org/10.1088/1367-2630/aa8fe3

[10] D. Schmid, H. Du, J. H. Selby, and M. F. Pusey. ``Uniqueness of Noncontextual Models for Stabilizer Subtheories''. Phys. Rev. Lett. 129, 120403 (2022).
https://doi.org/10.1103/PhysRevLett.129.120403

[11] 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).
https://doi.org/10.1103/PhysRevLett.129.230401

[12] V. Veitch, S. A. Hamed Mousavian, D. Gottesman, and J. Emerson. ``The resource theory of stabilizer quantum computation''. New J. Phys. 16, 013009 (2014).
https://doi.org/10.1088/1367-2630/16/1/013009

[13] 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).
https://doi.org/10.1103/PhysRevA.98.052350

[14] R. Takagi and Q. Zhuang. ``Convex resource theory of non-Gaussianity''. Phys. Rev. A 97, 062337 (2018).
https://doi.org/10.1103/PhysRevA.97.062337

[15] X. Wang, M. M. Wilde, and Y. Su. ``Quantifying the magic of quantum channels''. New J. Phys. 21, 103002 (2019).
https://doi.org/10.1088/1367-2630/ab451d

[16] R. L. Stratonovich. ``On Distributions in Representation Space''. Journal of Experimental and Theoretical Physics 4, 1012–1020 (1956). url: http://jetp.ras.ru/cgi-bin/e/index/e/4/6/p891?a=list.
http://jetp.ras.ru/cgi-bin/e/index/e/4/6/p891?a=list

[17] 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).
https://doi.org/10.1103/PhysRevResearch.4.043136

[18] C. D. Mink and M. Fleischhauer. ``Collective radiative interactions in the discrete truncated Wigner approximation''. SciPost Phys. 15, 233 (2023).
https://doi.org/10.21468/SciPostPhys.15.6.233

[19] C. Brif and A. Mann. ``Phase-space formulation of quantum mechanics and quantum-state reconstruction for physical systems with Lie-group symmetries''. Phys. Rev. A 59, 971–987 (1999).
https://doi.org/10.1103/PhysRevA.59.971

[20] N. Abbasli, V. Abgaryan, M. Bures, A. Khvedelidze, I. Rogojin, and A. Torosyan. ``On Measures of Classicality/Quantumness in Quasiprobability Representations of Finite-Dimensional Quantum Systems''. Phys. Part. Nuclei 51, 443–447 (2020).
https://doi.org/10.1134/s1063779620040024

[21] V. Abgaryan and A. Khvedelidze. ``On Families of Wigner Functions for $N$-Level Quantum Systems''. Symmetry 13, 1013 (2021).
https://doi.org/10.3390/sym13061013

[22] V. Abgaryan, A. Khvedelidze, and A. Torosyan. ``The Global Indicator of Classicality of an Arbitrary $N$-Level Quantum System''. J. Math. Sci. 251, 301–314 (2020).
https://doi.org/10.1007/s10958-020-05092-6

[23] V. Abgaryan, A. Khvedelidze, and A. Torosyan. ``Kenfack – Życzkowski indicator of nonclassicality for two non-equivalent representations of Wigner function of qutrit''. Phys. Lett. A 412, 127591 (2021).
https://doi.org/10.1016/j.physleta.2021.127591

[24] G. S. Agarwal. ``Relation between atomic coherent-state representation, state multipoles, and generalized phase-space distributions''. Phys. Rev. A 24, 2889–2896 (1981).
https://doi.org/10.1103/PhysRevA.24.2889

[25] 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, 4101–4109 (1994).
https://doi.org/10.1103/PhysRevA.49.4101

[26] M. F. Riedel, P. Böhi, Y. Li, T. W. Hänsch, A. Sinatra, and P. Treutlein. ``Atom-chip-based generation of entanglement for quantum metrology''. Nature 464, 1170–1173 (2010).
https://doi.org/10.1038/nature08988

[27] R. Schmied and P. Treutlein. ``Tomographic reconstruction of the Wigner function on the Bloch sphere''. New J. Phys. 13, 065019 (2011).
https://doi.org/10.1088/1367-2630/13/6/065019

[28] R. McConnell, H. Zhang, J. Hu, S. Ć uk, and V. Vuletić. ``Entanglement with negative Wigner function of almost 3,000 atoms heralded by one photon''. Nature 519, 439–442 (2015).
https://doi.org/10.1038/nature14293

[29] B. Chen, J. Geng, F. Zhou, L. Song, H. Shen, and N. Xu. ``Quantum state tomography of a single electron spin in diamond with Wigner function reconstruction''. Appl. Phys. Lett. 114, 041102 (2019).
https://doi.org/10.1063/1.5082878

[30] A. B. Klimov, J. L. Romero, and H. de Guise. ``Generalized SU(2) covariant Wigner functions and some of their applications''. J. Phys. A 50, 323001 (2017).
https://doi.org/10.1088/1751-8121/50/32/323001

[31] J. C. Várilly and J. M. Gracia-Bondía. ``The Moyal representation for spin''. Ann. Phys. 190, 107–148 (1989).
https://doi.org/10.1016/0003-4916(89)90262-5

[32] 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).
https://doi.org/10.1103/PhysRevA.63.012102

[33] O. Giraud, P. Braun, and D. Braun. ``Classicality of spin states''. Phys. Rev. A 78, 042112 (2008).
https://doi.org/10.1103/PhysRevA.78.042112

[34] F. Bohnet-Waldraff, D. Braun, and O. Giraud. ``Partial transpose criteria for symmetric states''. Phys. Rev. A 94, 042343 (2016).
https://doi.org/10.1103/PhysRevA.94.042343

[35] F. Bohnet-Waldraff, O. Giraud, and D. Braun. ``Absolutely classical spin states''. Phys. Rev. A 95, 012318 (2017).
https://doi.org/10.1103/PhysRevA.95.012318

[36] K. E. Cahill and R. J. Glauber. ``Density Operators and Quasiprobability Distributions''. Phys. Rev. 177, 1882–1902 (1969).
https://doi.org/10.1103/PhysRev.177.1882

[37] C. T. Lee. ``Measure of the nonclassicality of nonclassical states''. Phys. Rev. A 44, R2775–R2778 (1991).
https://doi.org/10.1103/PhysRevA.44.R2775

[38] F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas. ``Atomic coherent states in quantum optics''. Phys. Rev. A 6, 2211–2237 (1972).
https://doi.org/10.1103/PhysRevA.6.2211

[39] B. 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).
https://doi.org/10.1103/PhysRevA.101.022318

[40] 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).
https://doi.org/10.1002/qute.202100016

[41] A. Grossmann. ``Parity operator and quantization of delta-functions''. Commun. Math. Phys. 48, 191–194 (1976).
https://doi.org/10.1007/BF01617867

[42] A. Royer. ``Wigner function as the expectation value of a parity operator''. Phys. Rev. A 15, 449–450 (1977).
https://doi.org/10.1103/PhysRevA.15.449

[43] D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii. ``Quantum Theory of Angular Momentum''. World Scientific. (1988).
https://doi.org/10.1142/0270

[44] J. Davis, M. Kumari, R. B. Mann, and S. Ghose. ``Wigner negativity in spin-$j$ systems''. Phys. Rev. Research 3, 033134 (2021).
https://doi.org/10.1103/PhysRevResearch.3.033134

[45] S. Heiss and S. Weigert. ``Discrete Moyal-type representations for a spin''. Phys. Rev. A 63, 012105 (2000).
https://doi.org/10.1103/PhysRevA.63.012105

[46] C. Brif and A. Mann. ``A general theory of phase-space quasiprobability distributions''. J. Phys. A: Math. Gen. 31, L9–L17 (1998).
https://doi.org/10.1088/0305-4470/31/1/002

[47] W. Dür, G. Vidal, and J. I. Cirac. ``Three qubits can be entangled in two inequivalent ways''. Phys. Rev. A 62, 062314 (2000).
https://doi.org/10.1103/PhysRevA.62.062314

[48] F. J. Narcowich. ``Conditions for the convolution of two Wigner distributions to be itself a Wigner distribution''. J. Math. Phys 29, 2036–2041 (1988).
https://doi.org/10.1063/1.527861

[49] J. M. Gracia-Bondía and J. C. Várilly. ``Non-negative mixed states in Weyl-Wigner-Moyal theory''. Phys. Lett. A 128, 20–24 (1988).
https://doi.org/10.1016/0375-9601(88)91035-3

[50] T. Bröcker and R. F. Werner. ``Mixed states with positive Wigner functions''. J. Math. Phys 36, 62–75 (1995).
https://doi.org/10.1063/1.531326

[51] A. Mandilara, E. Karpov, and N. J. Cerf. ``Gaussianity bounds for quantum mixed states with a positive Wigner function''. J. Phys. Conf. Ser 254, 012011 (2010).
https://doi.org/10.1088/1742-6596/254/1/012011

[52] J. Huber, P. Kirton, and P. Rabl. ``Phase-space methods for simulating the dissipative many-body dynamics of collective spin systems''. SciPost Phys. 10, 045 (2021).
https://doi.org/10.21468/SciPostPhys.10.2.045

[53] S. Gherardini and G. De Chiara. ``Quasiprobabilities in quantum thermodynamics and many-body systems''. PRX Quantum 5, 030201 (2024).
https://doi.org/10.1103/PRXQuantum.5.030201

[54] A. B. Klimov and S. M. Chumakov. ``Quasi-probability distributions for the simplest dynamical groups''. J. Opt. Soc. Am. 17, 2315 (2000).
https://doi.org/10.1364/JOSAA.17.002315

[55] Blender Online Community. ``Blender - a 3D modelling and rendering package''. Blender Foundation. Stichting Blender Foundation, Amsterdam. (2018). url: http://www.blender.org.
http://www.blender.org

[56] S. Danisch and J. Krumbiegel. ``Makie.jl: Flexible high-performance data visualization for Julia''. J. Open Source Softw. 6, 3349 (2021).
https://doi.org/10.21105/joss.03349

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[3] Carlo Marconi, Guillem Müller-Rigat, Jordi Romero-Pallejà, Jordi Tura, and Anna Sanpera, "Symmetric quantum states: a review of recent progress", Reports on Progress in Physics 89 2, 024001 (2026).

[4] Eduardo Serrano-Ensástiga, Jérôme Denis, and John Martin, "Absolute-separability witnesses for symmetric multiqubit states", Physical Review A 109 2, 022430 (2024).

[5] Michael Zurel and Jack Davis, "Basis-independent stabilizerness and maximally noisy magic states", arXiv:2602.22336, (2026).

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