Qudit Clauser-Horne-Shimony-Holt Inequality and Nonlocality from Wigner Negativity
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AbstractNonlocality is an essential concept that distinguishes quantum from classical models and has been extensively studied in systems of qubits. For higher-dimensional systems, certain results for their two-level counterpart, like Bell violations with stabilizer states and Clifford operators, do not generalize. On the other hand, similar to continuous variable systems, Wigner negativity is necessary for nonlocality in qudit systems. We propose a new generalization of the CHSH inequality for qudits by inquiring correlations related to the Wigner negativity of stabilizer states under the adjoint action of a generalization of the qubit $\pi/8$-gate. A specified stabilizer state maximally violates the inequality among all qudit states based on its Wigner negativity. The Bell operator not only serves as a measure for the singlet fraction but also quantifies the volume of Wigner negativity. Additionally, we show how a bipartite entangled qudit state can serve as a witness for contextuality when it exhibits Wigner negativity. Furthermore, we identify rational-phase diagonal unitaries as the key resource that exactly reproduce the CGLMP and SATWAP violation with the maximally entangled state through simple phase-difference alignment.► BibTeX data@article{Meyer2026quditclauserhorne, doi = {10.22331/q-2026-06-15-2139}, url = {https://doi.org/10.22331/q-2026-06-15-2139}, title = {Qudit {C}lauser-{H}orne-{S}himony-{H}olt {I}nequality and {N}onlocality from {W}igner {N}egativity}, author = {Meyer, Uta Isabella and {\v{S}}upi{\'{c}}, Ivan and Markham, Damian and Grosshans, Fr{\'{e}}d{\'{e}}ric}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2139}, month = jun, year = {2026} }► References [1] J.S. Shaari, M.R.B. Wahiddin, and S. Mancini. Blind encoding into qudits. Physics Letters A, 372 (12): 1963–1967, 2008. ISSN 0375-9601. https://doi.org/10.1016/j.physleta.2007.08.076. https://doi.org/10.1016/j.physleta.2007.08.076 [2] Xie Chen, Hyeyoun Chung, Andrew W. Cross, Bei Zeng, and Isaac L. Chuang. Subsystem stabilizer codes cannot have a universal set of transversal gates for even one encoded qudit. Phys. Rev. A, 78: 012353, Jul 2008. 10.1103/PhysRevA.78.012353. https://doi.org/10.1103/PhysRevA.78.012353 [3] Julien Niset, Jaromír Fiurášek, and Nicolas J. Cerf. No-go theorem for Gaussian quantum error correction. Phys. Rev. Lett., 102: 120501, Mar 2009. 10.1103/PhysRevLett.102.120501. https://doi.org/10.1103/PhysRevLett.102.120501 [4] Matthias Fitzi, Nicolas Gisin, and Ueli Maurer. Quantum solution to the byzantine agreement problem. Phys. Rev. Lett., 87: 217901, Nov 2001. 10.1103/PhysRevLett.87.217901. https://doi.org/10.1103/PhysRevLett.87.217901 [5] Damian Markham and Barry C. Sanders. Graph states for quantum secret sharing. Phys. Rev. A, 78: 042309, Oct 2008. 10.1103/PhysRevA.78.042309. https://doi.org/10.1103/PhysRevA.78.042309 [6] Elham Kashefi, Damian Markham, Mehdi Mhalla, and Simon Perdrix. Information flow in secret sharing protocols. Electronic Proceedings in Theoretical Computer Science, 9: 87–97, November 2009. ISSN 2075-2180. 10.4204/eptcs.9.10. https://doi.org/10.4204/eptcs.9.10 [7] Adrian Keet, Ben Fortescue, Damian Markham, and Barry C. Sanders. Quantum secret sharing with qudit graph states. Phys. Rev. A, 82: 062315, Dec 2010. 10.1103/PhysRevA.82.062315. https://doi.org/10.1103/PhysRevA.82.062315 [8] Richard Cleve, Daniel Gottesman, and Hoi-Kwong Lo. How to share a quantum secret. Phys. Rev. Lett., 83: 648–651, Jul 1999. 10.1103/PhysRevLett.83.648. https://doi.org/10.1103/PhysRevLett.83.648 [9] J. S. Bell. On the Einstein Podolsky Rosen paradox.
Physics Physique Fizika, 1: 195–200, Nov 1964. 10.1103/PhysicsPhysiqueFizika.1.195. https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195 [10] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. Bell nonlocality. Rev. Mod. Phys., 86: 419–478, Apr 2014. 10.1103/RevModPhys.86.419. https://doi.org/10.1103/RevModPhys.86.419 [11] Daniel Collins, Nicolas Gisin, Noah Linden, Serge Massar, and Sandu Popescu. Bell inequalities for arbitrarily high-dimensional systems. Phys. Rev. Lett., 88: 040404, Jan 2002. 10.1103/PhysRevLett.88.040404. https://doi.org/10.1103/PhysRevLett.88.040404 [12] Yeong-Cherng Liang, Chu-Wee Lim, and Dong-Ling Deng. Reexamination of a multisetting Bell inequality for qudits. Phys. Rev. A, 80: 052116, Nov 2009. 10.1103/PhysRevA.80.052116. https://doi.org/10.1103/PhysRevA.80.052116 [13] Se-Wan Ji, Jinhyoung Lee, James Lim, Koji Nagata, and Hai-Woong Lee. Multisetting Bell inequality for qudits. Phys. Rev. A, 78: 052103, Nov 2008. 10.1103/PhysRevA.78.052103. https://doi.org/10.1103/PhysRevA.78.052103 [14] Alexia Salavrakos, Remigiusz Augusiak, Jordi Tura, Peter Wittek, Antonio Acín, and Stefano Pironio. Bell inequalities tailored to maximally entangled states. Phys. Rev. Lett., 119: 040402, Jul 2017. 10.1103/PhysRevLett.119.040402. https://doi.org/10.1103/PhysRevLett.119.040402 [15] Nicolas J. Cerf, Serge Massar, and Stefano Pironio. Greenberger-Horne-Zeilinger paradoxes for many qudits. Phys. Rev. Lett., 89: 080402, Aug 2002. 10.1103/PhysRevLett.89.080402. https://doi.org/10.1103/PhysRevLett.89.080402 [16] Weidong Tang, Sixia Yu, and C. H. Oh. Greenberger-Horne-Zeilinger paradoxes from qudit graph states. Phys. Rev. Lett., 110: 100403, Mar 2013. 10.1103/PhysRevLett.110.100403. https://doi.org/10.1103/PhysRevLett.110.100403 [17] Jay Lawrence. Mermin inequalities for perfect correlations in many-qutrit systems. Phys. Rev. A, 95: 042123, Apr 2017. 10.1103/PhysRevA.95.042123. https://doi.org/10.1103/PhysRevA.95.042123 [18] Dagomir Kaszlikowski, L. C. Kwek, Jing-Ling Chen, Marek Żukowski, and C. H. Oh. Clauser-Horne inequality for three-state systems. Phys. Rev. A, 65: 032118, Feb 2002a. 10.1103/PhysRevA.65.032118. https://doi.org/10.1103/PhysRevA.65.032118 [19] Dagomir Kaszlikowski, Darwin Gosal, E. J. Ling, L. C. Kwek, Marek Żukowski, and C. H. Oh. Three-qutrit correlations violate local realism more strongly than those of three qubits. Phys. Rev. A, 66: 032103, Sep 2002b. 10.1103/PhysRevA.66.032103. https://doi.org/10.1103/PhysRevA.66.032103 [20] A. Acín, J. L. Chen, N. Gisin, D. Kaszlikowski, L. C. Kwek, C. H. Oh, and M. Żukowski. Coincidence Bell inequality for three three-dimensional systems. Phys. Rev. Lett., 92: 250404, Jun 2004. 10.1103/PhysRevLett.92.250404. https://doi.org/10.1103/PhysRevLett.92.250404 [21] Jacek Gruca, Wiesław Laskowski, and Marek Żukowski. Nonclassicality of pure two-qutrit entangled states. Phys. Rev. A, 85: 022118, Feb 2012. 10.1103/PhysRevA.85.022118. https://doi.org/10.1103/PhysRevA.85.022118 [22] Hai Tao Li and Xiao Yu Chen. Entanglement of graph qutrit states. In 2011 International Conference on Intelligence Science and Information Engineering, pages 61–64, 2011. ISBN 9780769544809. 10.1109/ISIE.2011.10. https://doi.org/10.1109/ISIE.2011.10 [23] Jelena Mackeprang, Daniel Bhatti, Matty J. Hoban, and Stefanie Barz. The power of qutrits for non-adaptive measurement-based quantum computing. New Journal of Physics, 25 (7): 073007, jul 2023. 10.1088/1367-2630/acdf77. https://doi.org/10.1088/1367-2630/acdf77 [24] Jędrzej Kaniewski, Ivan Šupić, Jordi Tura, Flavio Baccari, Alexia Salavrakos, and Remigiusz Augusiak. Maximal nonlocality from maximal entanglement and mutually unbiased bases, and self-testing of two-qutrit quantum systems. Quantum, 3: 198, October 2019. ISSN 2521-327X. 10.22331/q-2019-10-24-198. https://doi.org/10.22331/q-2019-10-24-198 [25] Mark Howard. Maximum nonlocality and minimum uncertainty using magic states. Phys. Rev. A, 91: 042103, Apr 2015. 10.1103/PhysRevA.91.042103. https://doi.org/10.1103/PhysRevA.91.042103 [26] Mark Howard and Jiri Vala. Qudit versions of the qubit ${\pi}/8$ gate. Phys. Rev. A, 86: 022316, Aug 2012. 10.1103/PhysRevA.86.022316. https://doi.org/10.1103/PhysRevA.86.022316 [27] D. Gross. Hudson's theorem for finite-dimensional quantum systems. Journal of Mathematical Physics, 47, 2006. ISSN 00222488. 10.1063/1.2393152. https://doi.org/10.1063/1.2393152 [28] Mark Howard, Joel Wallman, Victor Veitch, and Joseph Emerson. Contextuality supplies the 'magic' for quantum computation. Nature, 510, 2014. ISSN 14764687. 10.1038/nature13460. https://doi.org/10.1038/nature13460 [29] Nicolas Delfosse, Cihan Okay, Juan Bermejo-Vega, Dan E. Browne, and Robert Raussendorf. Equivalence between contextuality and negativity of the Wigner function for qudits. New Journal of Physics, 19, 12 2017. ISSN 13672630. 10.1088/1367-2630/aa8fe3. https://doi.org/10.1088/1367-2630/aa8fe3 [30] Robert I. Booth, Ulysse Chabaud, and Pierre-Emmanuel Emeriau. Contextuality and Wigner negativity are equivalent for continuous-variable quantum measurements. Phys. Rev. Lett., 129: 230401, Nov 2022. 10.1103/PhysRevLett.129.230401. https://doi.org/10.1103/PhysRevLett.129.230401 [31] Otfried Gühne and Géza Tóth. Entanglement detection. Physics Reports, 474 (1): 1–75, 2009. ISSN 0370-1573. https://doi.org/10.1016/j.physrep.2009.02.004. https://doi.org/10.1016/j.physrep.2009.02.004 [32] Rudolf Lidl and Harald Niederreiter. Finite Fields.
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[3] Huan Liu, Zu-wu Chen, Xue-feng Zhan, Hong-chun Yuan, and Xue-xiang Xu, "Exploring entanglement, Wigner negativity and Bell nonlocality for anisotropic two-qutrit states", arXiv:2506.03879, (2025). [4] R. A. Macêdo, P. Andriolo, S. Zamora, D. Poderini, and R. Chaves, "Witnessing nonstabilizerness with Bell inequalities", Physical Review A 112 5, L050401 (2025). [5] Uta Isabella Meyer, Ivan Šupić, Frédéric Grosshans, and Damian Markham, "Robustly self-testing all maximally entangled states in every finite dimension", arXiv:2508.01071, (2025). [6] Huan Liu, Zu-wu Chen, Xue-feng Zhan, Hong-chun Yuan, and Xue-xiang Xu, "Exploring entanglement, Wigner negativity and Bell nonlocality for anisotropic two-qutrit states", Laser Physics Letters 22 11, 115206 (2025). The above citations are from SAO/NASA ADS (last updated successfully 2026-06-15 14:02:32). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-06-15 14:02:30: Could not fetch cited-by data for 10.22331/q-2026-06-15-2139 from Crossref. This is normal if the DOI was registered recently.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. AbstractNonlocality is an essential concept that distinguishes quantum from classical models and has been extensively studied in systems of qubits. For higher-dimensional systems, certain results for their two-level counterpart, like Bell violations with stabilizer states and Clifford operators, do not generalize. On the other hand, similar to continuous variable systems, Wigner negativity is necessary for nonlocality in qudit systems. We propose a new generalization of the CHSH inequality for qudits by inquiring correlations related to the Wigner negativity of stabilizer states under the adjoint action of a generalization of the qubit $\pi/8$-gate. A specified stabilizer state maximally violates the inequality among all qudit states based on its Wigner negativity. The Bell operator not only serves as a measure for the singlet fraction but also quantifies the volume of Wigner negativity. Additionally, we show how a bipartite entangled qudit state can serve as a witness for contextuality when it exhibits Wigner negativity. Furthermore, we identify rational-phase diagonal unitaries as the key resource that exactly reproduce the CGLMP and SATWAP violation with the maximally entangled state through simple phase-difference alignment.► BibTeX data@article{Meyer2026quditclauserhorne, doi = {10.22331/q-2026-06-15-2139}, url = {https://doi.org/10.22331/q-2026-06-15-2139}, title = {Qudit {C}lauser-{H}orne-{S}himony-{H}olt {I}nequality and {N}onlocality from {W}igner {N}egativity}, author = {Meyer, Uta Isabella and {\v{S}}upi{\'{c}}, Ivan and Markham, Damian and Grosshans, Fr{\'{e}}d{\'{e}}ric}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2139}, month = jun, year = {2026} }► References [1] J.S. Shaari, M.R.B. Wahiddin, and S. Mancini. Blind encoding into qudits. Physics Letters A, 372 (12): 1963–1967, 2008. ISSN 0375-9601. https://doi.org/10.1016/j.physleta.2007.08.076. https://doi.org/10.1016/j.physleta.2007.08.076 [2] Xie Chen, Hyeyoun Chung, Andrew W. Cross, Bei Zeng, and Isaac L. Chuang. Subsystem stabilizer codes cannot have a universal set of transversal gates for even one encoded qudit. Phys. Rev. A, 78: 012353, Jul 2008. 10.1103/PhysRevA.78.012353. https://doi.org/10.1103/PhysRevA.78.012353 [3] Julien Niset, Jaromír Fiurášek, and Nicolas J. Cerf. No-go theorem for Gaussian quantum error correction. Phys. Rev. Lett., 102: 120501, Mar 2009. 10.1103/PhysRevLett.102.120501. https://doi.org/10.1103/PhysRevLett.102.120501 [4] Matthias Fitzi, Nicolas Gisin, and Ueli Maurer. Quantum solution to the byzantine agreement problem. Phys. Rev. Lett., 87: 217901, Nov 2001. 10.1103/PhysRevLett.87.217901. https://doi.org/10.1103/PhysRevLett.87.217901 [5] Damian Markham and Barry C. Sanders. Graph states for quantum secret sharing. Phys. Rev. A, 78: 042309, Oct 2008. 10.1103/PhysRevA.78.042309. https://doi.org/10.1103/PhysRevA.78.042309 [6] Elham Kashefi, Damian Markham, Mehdi Mhalla, and Simon Perdrix. Information flow in secret sharing protocols. Electronic Proceedings in Theoretical Computer Science, 9: 87–97, November 2009. ISSN 2075-2180. 10.4204/eptcs.9.10. https://doi.org/10.4204/eptcs.9.10 [7] Adrian Keet, Ben Fortescue, Damian Markham, and Barry C. Sanders. Quantum secret sharing with qudit graph states. Phys. Rev. A, 82: 062315, Dec 2010. 10.1103/PhysRevA.82.062315. https://doi.org/10.1103/PhysRevA.82.062315 [8] Richard Cleve, Daniel Gottesman, and Hoi-Kwong Lo. How to share a quantum secret. Phys. Rev. Lett., 83: 648–651, Jul 1999. 10.1103/PhysRevLett.83.648. https://doi.org/10.1103/PhysRevLett.83.648 [9] J. S. Bell. On the Einstein Podolsky Rosen paradox.
Physics Physique Fizika, 1: 195–200, Nov 1964. 10.1103/PhysicsPhysiqueFizika.1.195. https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195 [10] Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner. Bell nonlocality. Rev. Mod. Phys., 86: 419–478, Apr 2014. 10.1103/RevModPhys.86.419. https://doi.org/10.1103/RevModPhys.86.419 [11] Daniel Collins, Nicolas Gisin, Noah Linden, Serge Massar, and Sandu Popescu. Bell inequalities for arbitrarily high-dimensional systems. Phys. Rev. Lett., 88: 040404, Jan 2002. 10.1103/PhysRevLett.88.040404. https://doi.org/10.1103/PhysRevLett.88.040404 [12] Yeong-Cherng Liang, Chu-Wee Lim, and Dong-Ling Deng. Reexamination of a multisetting Bell inequality for qudits. Phys. Rev. A, 80: 052116, Nov 2009. 10.1103/PhysRevA.80.052116. https://doi.org/10.1103/PhysRevA.80.052116 [13] Se-Wan Ji, Jinhyoung Lee, James Lim, Koji Nagata, and Hai-Woong Lee. Multisetting Bell inequality for qudits. Phys. Rev. A, 78: 052103, Nov 2008. 10.1103/PhysRevA.78.052103. https://doi.org/10.1103/PhysRevA.78.052103 [14] Alexia Salavrakos, Remigiusz Augusiak, Jordi Tura, Peter Wittek, Antonio Acín, and Stefano Pironio. Bell inequalities tailored to maximally entangled states. Phys. Rev. Lett., 119: 040402, Jul 2017. 10.1103/PhysRevLett.119.040402. https://doi.org/10.1103/PhysRevLett.119.040402 [15] Nicolas J. Cerf, Serge Massar, and Stefano Pironio. Greenberger-Horne-Zeilinger paradoxes for many qudits. Phys. Rev. Lett., 89: 080402, Aug 2002. 10.1103/PhysRevLett.89.080402. https://doi.org/10.1103/PhysRevLett.89.080402 [16] Weidong Tang, Sixia Yu, and C. H. Oh. Greenberger-Horne-Zeilinger paradoxes from qudit graph states. Phys. Rev. Lett., 110: 100403, Mar 2013. 10.1103/PhysRevLett.110.100403. https://doi.org/10.1103/PhysRevLett.110.100403 [17] Jay Lawrence. Mermin inequalities for perfect correlations in many-qutrit systems. Phys. Rev. A, 95: 042123, Apr 2017. 10.1103/PhysRevA.95.042123. https://doi.org/10.1103/PhysRevA.95.042123 [18] Dagomir Kaszlikowski, L. C. Kwek, Jing-Ling Chen, Marek Żukowski, and C. H. Oh. Clauser-Horne inequality for three-state systems. Phys. Rev. A, 65: 032118, Feb 2002a. 10.1103/PhysRevA.65.032118. https://doi.org/10.1103/PhysRevA.65.032118 [19] Dagomir Kaszlikowski, Darwin Gosal, E. J. Ling, L. C. Kwek, Marek Żukowski, and C. H. Oh. Three-qutrit correlations violate local realism more strongly than those of three qubits. Phys. Rev. A, 66: 032103, Sep 2002b. 10.1103/PhysRevA.66.032103. https://doi.org/10.1103/PhysRevA.66.032103 [20] A. Acín, J. L. Chen, N. Gisin, D. Kaszlikowski, L. C. Kwek, C. H. Oh, and M. Żukowski. Coincidence Bell inequality for three three-dimensional systems. Phys. Rev. Lett., 92: 250404, Jun 2004. 10.1103/PhysRevLett.92.250404. https://doi.org/10.1103/PhysRevLett.92.250404 [21] Jacek Gruca, Wiesław Laskowski, and Marek Żukowski. Nonclassicality of pure two-qutrit entangled states. Phys. Rev. A, 85: 022118, Feb 2012. 10.1103/PhysRevA.85.022118. https://doi.org/10.1103/PhysRevA.85.022118 [22] Hai Tao Li and Xiao Yu Chen. Entanglement of graph qutrit states. In 2011 International Conference on Intelligence Science and Information Engineering, pages 61–64, 2011. ISBN 9780769544809. 10.1109/ISIE.2011.10. https://doi.org/10.1109/ISIE.2011.10 [23] Jelena Mackeprang, Daniel Bhatti, Matty J. Hoban, and Stefanie Barz. The power of qutrits for non-adaptive measurement-based quantum computing. New Journal of Physics, 25 (7): 073007, jul 2023. 10.1088/1367-2630/acdf77. https://doi.org/10.1088/1367-2630/acdf77 [24] Jędrzej Kaniewski, Ivan Šupić, Jordi Tura, Flavio Baccari, Alexia Salavrakos, and Remigiusz Augusiak. Maximal nonlocality from maximal entanglement and mutually unbiased bases, and self-testing of two-qutrit quantum systems. Quantum, 3: 198, October 2019. ISSN 2521-327X. 10.22331/q-2019-10-24-198. https://doi.org/10.22331/q-2019-10-24-198 [25] Mark Howard. Maximum nonlocality and minimum uncertainty using magic states. Phys. Rev. A, 91: 042103, Apr 2015. 10.1103/PhysRevA.91.042103. https://doi.org/10.1103/PhysRevA.91.042103 [26] Mark Howard and Jiri Vala. Qudit versions of the qubit ${\pi}/8$ gate. Phys. Rev. A, 86: 022316, Aug 2012. 10.1103/PhysRevA.86.022316. https://doi.org/10.1103/PhysRevA.86.022316 [27] D. Gross. Hudson's theorem for finite-dimensional quantum systems. Journal of Mathematical Physics, 47, 2006. ISSN 00222488. 10.1063/1.2393152. https://doi.org/10.1063/1.2393152 [28] Mark Howard, Joel Wallman, Victor Veitch, and Joseph Emerson. Contextuality supplies the 'magic' for quantum computation. Nature, 510, 2014. ISSN 14764687. 10.1038/nature13460. https://doi.org/10.1038/nature13460 [29] Nicolas Delfosse, Cihan Okay, Juan Bermejo-Vega, Dan E. Browne, and Robert Raussendorf. 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