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Molecular spin qudits to test generalized Bell inequalities

Silvia Macedonio1,2, Luca Lepori1,2, Alessandro Chiesa1,2,3, Simone Chicco1, Laura Bersani1, Marcos Rubin-Osanz1, Lukas Bradley Woodcock4, Athanasios Mavromagoulos4, Giuseppe Allodi1 et al.

Elena Garlatti1,2,3, Stergios Piligkos4, Augusto Smerzi5, and Stefano Carretta1,2,3,*

  • *Contact author: stefano.carretta@unipr.it

Phys. Rev. Research 8, 013138 – Published 9 February, 2026

DOI: https://doi.org/10.1103/sn37-b9rr

Abstract

We show that molecular nanomagnets provide an ideal platform to investigate entanglement in qubit-qudit and qudit-qudit pairs. A prototypical example is Yb173 (trensal), embedding an electronic spin qubit coupled to a nuclear spin qudit. We demonstrate violation of generalized Bell inequalities on this molecule by developing an optimized pulse sequence and by performing realistic numerical simulations using experimentally measured Hamiltonian parameters and coherence times. We find that the inequalities are safely violated in a wide range of parameters, proving the robustness of entanglement in the investigated system. Furthermore, we propose a scheme to study qudit-qudit entanglement on a molecular spin trimer, in which two spins 3/2 are linked via an interposed switch to turn on and off their mutual interaction.

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References (83)

  1. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  2. L. Pezzè, A. Smerzi, M. K. Oberthaler, R. Schmied, and P. Treutlein, Quantum metrology with nonclassical states of atomic ensembles, Rev. Mod. Phys. 90, 035005 (2018).
  3. J. Yago Malo, L. Lepori, L. Gentini, and M. L. M. Chiofalo, Atomic quantum technologies for quantum matter and fundamental physics applications, Technologies 12, 64 (2024).
  4. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, UK, 2011).
  5. A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities, Nat. Phys. 7, 677 (2011).
  6. H.-P. Lo, C.-M. Li, A. Yabushita, Y.-N. Chen, C.-W. Luo, and T. Kobayashi, Experimental violation of Bell inequalities for multi-dimensional systems, Sci. Rep. 6, 22088 (2016).
  7. Y. Bao, S. S. Yu, L. Anderegg, E. Chae, W. Ketterle, K.-K. Ni, and J. M. Doyle, Dipolar spin-exchange and entanglement between molecules in an optical tweezer array, Science 382, 1138 (2023).
  8. C. M. Holland, Y. Lu, and L. W. Cheuk, On-demand entanglement of molecules in a reconfigurable optical tweezer array, Science 382, 1143 (2023).
  9. A. Smerzi, Entanglement with tweezed molecules, Science 382, 1118 (2023).
  10. A. Chiesa, P. Santini, E. Garlatti, F. Luis, and S. Carretta, Molecular nanomagnets: A viable path toward quantum information processing?, Rep. Prog. Phys. 87, 034501 (2024).
  11. Z. Wang, R. W. Parker, E. Champion, and M. S. Blok, High-EJ/EC transmon qudits with up to 12 levels, Phys. Rev. Appl. 23, 034046 (2025).
  12. Y. Wang, Z. Hu, B. C. Sanders, and S. Kais, Qudits and high-dimensional quantum computing, Front. Phys. 8, 589504 (2020).
  13. P. Imany, J. A. Jaramillo-Villegas, M. S. Alshaykh, J. M. Lukens, O. D. Odele, A. J. Moore, D. E. Leaird, M. Qi, and A. M. Weiner, High-dimensional optical quantum logic in large operational spaces, npj Quantum Inf 5, 59 (2019).
  14. B. Lanyon, M. Barbieri, M. Almeida, T. Jennewein, T. Ralph, K. Resch, G. Pryde, J. O’Brien, A. Gilchrist, and A. White, Simplifying quantum logic using higher-dimensional Hilbert spaces, Nat. Phys. 5, 134 (2009).
  15. F. Tacchino, A. Chiesa, R. Sessoli, I. Tavernelli, and S. Carretta, A proposal for using molecular spin qudits as quantum simulators of light–matter interactions, J. Mater. Chem. C 9, 10266 (2021).
  16. S. Roca-Jerat, E. Macaluso, A. Chiesa, P. Santini, and S. Carretta, Simulating open quantum systems with molecular spin qudits, Mater. Horiz. 12, 3918 (2025).
  17. M. Meth, J. Zhang, J. F. Haase, et al., Simulating two-dimensional lattice Gauge theories on a qudit quantum computer, Nat. Phys. 21, 570 (2025).
  18. A. N. Ciavarella and C. W. Bauer, Quantum simulation of SU(3) lattice Yang-Mills theory at leading order in large-Nc expansion, Phys. Rev. Lett. 133, 111901 (2024).
  19. M. H. Michael, M. Silveri, R. T. Brierley, V. V. Albert, J. Salmilehto, L. Jiang, and S. M. Girvin, New class of quantum error-correcting codes for a bosonic mode, Phys. Rev. X 6, 031006 (2016).
  20. V. V. Albert, J. P. Covey, and J. Preskill, Robust encoding of a qubit in a molecule, Phys. Rev. X 10, 031050 (2020).
  21. J. A. Gross, Designing codes around interactions: The case of a spin, Phys. Rev. Lett. 127, 010504 (2021).
  22. M. Mezzadri, A. Chiesa, L. Lepori, and S. Carretta, Fault-tolerant computing with single-qudit encoding in a molecular spin, Mater. Horiz. 11, 4961 (2024).
  23. D. Cozzolino, B. Da Lio, D. Bacco, and L. K. Oxenløwe, High‐dimensional quantum communication: Benefits, progress, and future challenges, Adv. Quantum Technol. 2, 1900038 (2019).
  24. H. Yu, S. Sciara, M. Chemnitz, N. Montaut, B. Crockett, B. Fischer, R. Helsten, B. Wetzel, T. Goebel, R. Krämer, B. Little, S. Chu, S. Nolte, Z. Wang, J. Azaña, W. Munro, D. Moss, and R. Morandotti, Quantum key distribution implemented with d-level time-bin entangled photons, Nat. Commun. 16, 171 (2025).
  25. W. Weiss, G. Benenti, G. Casati, I. Guarneri, T. Calarco, M. Paternostro, and S. Montangero, Violation of Bell inequalities in larger Hilbert spaces: Robustness and challenges, New J. Phys. 18, 013021 (2016).
  26. A. Fonseca, A. de Rosier, T. Vértesi, W. Laskowski, and F. Parisio, Survey on the Bell nonlocality of a pair of entangled qudits, Phys. Rev. A 98, 042105 (2018).
  27. A. Fine, Hidden variables, joint probability, and the Bell inequalities, Phys. Rev. Lett. 48, 291 (1982).
  28. M. Howard, J. Wallman, V. Veitch, and J. Emerson, Contextuality supplies the ‘magic’ for quantum computation, Nature (London) 510, 351 (2014).
  29. A. Karanjai, J. J. Wallman, and S. D. Bartlett, Contextuality bounds the efficiency of classical simulation of quantum processes, arXiv:1802.07744.
  30. J. Zadrozny, J. Niklas, O. Poluektov, and D. Freedman, Millisecond coherence time in a tunable molecular electronic spin qubit, ACS Cent. Sci. 1, 488 (2015).
  31. M. Atzori and R. Sessoli, The second quantum revolution: Role and challenges of molecular chemistry, J. Am. Chem. Soc. 141, 11339 (2019).
  32. M. Atzori, L. Tesi, E. Morra, M. Chiesa, L. Sorace, and R. Sessoli, Room-temperature quantum coherence and Rabi oscillations in vanadyl phthalocyanine: Toward multifunctional molecular spin qubits, J. Am. Chem. Soc. 138, 2154 (2016).
  33. E. Moreno-Pineda, C. Godfrin, F. Balestro, W. Wernsdorfer, and M. Ruben, Molecular spin qudits for quantum algorithms, Chem. Soc. Rev. 47, 501 (2018).
  34. A. Chiesa, G. Whitehead, S. Carretta, L. Carthy, G. Timco, S. Teat, G. Amoretti, E. Pavarini, R. Winpenny, and P. Santini, Molecular nanomagnets with switchable coupling for quantum simulation, Sci. Rep. 4, 7423 (2014).
  35. J. Ferrando-Soria, E. Pineda, A. Chiesa, A. Fernandez, S. Magee, S. Carretta, P. Santini, I. Yrezabal, F. Tuna, G. Timco, E. Mcinnes, and R. Winpenny, A modular design of molecular qubits to implement universal quantum gates, Nat. Commun. 7, 11377 (2016).
  36. T. Bennett, S. Nawaz, S. Lockyer, D. Asthana, G. Whitehead, I. Vitorica Yrezabal, G. Timco, N. Burton, R. Winpenny, and E. McInnes, A ring of rotaxanes: Studies of a large paramagnetic assembly in solution, Inorg. Chem. Front. 10, 6945 (2023).
  37. S. J. Lockyer, A. Chiesa, A. Brookfield, G. A. Timco, G. F. S. Whitehead, E. J. L. McInnes, S. Carretta, and R. E. P. Winpenny, Five-spin supramolecule for simulating quantum decoherence of Bell states, J. Am. Chem. Soc. 144, 16086 (2022).
  38. A. Chiesa, F. Petiziol, M. Chizzini, P. Santini, and S. Carretta, Theoretical design of optimal molecular qudits for quantum error correction, J. Phys. Chem. Lett. 13, 6468 (2022).
  39. M. Mezzadri, L. Lepori, A. Chiesa, and S. Carretta, Dephasing-tolerant quantum sensing for transverse magnetic fields with spin qudits, Quantum Sci. Technol. 10, 015045 (2025).
  40. S. Carretta, D. Zueco, A. Chiesa, A. Gómez-León, and F. Luis, A perspective on scaling up quantum computation with molecular spins, Appl. Phys. Lett. 118, 240501 (2021).
  41. S. Chicco, G. Allodi, A. Chiesa, E. Garlatti, C. D. Buch, P. Santini, R. De Renzi, S. Piligkos, and S. Carretta, Proof-of-concept quantum simulator based on molecular spin qudits, J. Am. Chem. Soc. 146, 1053 (2024).
  42. V. Gebhart and A. Smerzi, Extending the fair sampling assumption using causal diagrams, Quantum 7, 897 (2023).
  43. M. Pawłowski and N. Brunner, Semi-device-independent security of one-way quantum key distribution, Phys. Rev. A 84, 010302(R) (2011).
  44. N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014).
  45. A. S. Friedman, A. H. Guth, M. J. W. Hall, D. I. Kaiser, and J. Gallicchio, Relaxed Bell inequalities with arbitrary measurement dependence for each observer, Phys. Rev. A 99, 012121 (2019).
  46. B. Hensen, H. Bernien, A. Dréau, et al., Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres, Nature (London) 526, 682 (2015).
  47. M. Giustina, et al., Significant-loophole-free test of Bell's theorem with entangled photons, Phys. Rev. Lett. 115, 250401 (2015).
  48. L. K. Shalm, et al., Strong loophole-free test of local realism, Phys. Rev. Lett. 115, 250402 (2015).
  49. N. Lambert, E. Giguère, P. Menczel, B. Li, P. Hopf, G. Suárez, M. Gali, J. Lishman, R. Gadhvi, R. Agarwal, A. Galicia, N. Shammah, P. Nation, J. R. Johansson, S. Ahmed, S. Cross, A. Pitchford, and F. Nori, QuTiP 5: The quantum toolbox in Python, Phys. Rep. 1153, 1 (2026).
  50. D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, Bell inequalities for arbitrarily high-dimensional systems, Phys. Rev. Lett. 88, 040404 (2002).
  51. J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed experiment to test local hidden-variable theories, Phys. Rev. Lett. 23, 880 (1969).
  52. A. Bernal, J. A. Casas, and J. M. Moreno, Maximal Clauser-Horne-Shimony-Holt violation for qubit-qudit states, Phys. Rev. A 112, 042404 (2025).
  53. R. Horodecki, P. Horodecki, and M. Horodecki, Violating Bell inequality by mixed spin-12 states: Necessary and sufficient condition, Phys. Lett. A 200, 340 (1995).
  54. S. Pironio, All Clauser–Horne–Shimony–Holt polytopes, J. Phys. A: Math. Theor. 47, 424020 (2014).
  55. J. S. Bell, On the Einstein Podolsky Rosen paradox, Phys. Phys. Fiz. 1, 195 (1964).
  56. See Supplemental Material at http://link.aps.org/supplemental/10.1103/sn37-b9rr for details on pulse sequences, numerical simulations, and additional experimental information, which includes Refs. [73, 74, 75, 76, 77, 78, 79, 80, 81].
  57. S. Hansen, C. Buch, and S. Piligkos, Structural isomerism-tuned magnetization relaxation dynamics in lanthanide coordination complexes, Inorg. Chem. Front. 11, 2116 (2024).
  58. G. Lindblad, On the generators of quantum dynamical semigroups, Commun. Math. Phys. 48, 119 (1976).
  59. A. Chiesa, S. Roca, S. Chicco, M. C. de Ory, A. Gómez-León, A. Gomez, D. Zueco, F. Luis, and S. Carretta, Blueprint for a molecular-spin quantum processor, Phys. Rev. Appl. 19, 064060 (2023).
  60. A. Chiesa, E. Macaluso, F. Petiziol, S. Wimberger, P. Santini, and S. Carretta, Molecular nanomagnets as qubits with embedded quantum-error correction, J. Phys. Chem. Lett. 11, 8610 (2020).
  61. Note that any evolution due to the hyperfine coupling term in Eq. (5) is ineffective on the measured diagonal observables.
  62. J. Johansson, P. Nation, and F. Nori, QuTiP: An open-source Python framework for the dynamics of open quantum systems, Comput. Phys. Commun. 183, 1760 (2012).
  63. J. Johansson, P. Nation, and F. Nori, QuTiP 2: A Python framework for the dynamics of open quantum systems, Comput. Phys. Commun. 184, 1234 (2013).
  64. D. Stefanatos, N. Khaneja, and S. J. Glaser, Optimal control of coupled spins in the presence of longitudinal and transverse relaxation, Phys. Rev. A 69, 022319 (2004).
  65. P. de Fouquieres, S. Schirmer, S. Glaser, and I. Kuprov, Second order gradient ascent pulse engineering, J. Magn. Reson. 212, 412 (2011).
  66. S. J. Glaser, U. Boscain, T. Calarco, C. P. Koch, W. Köckenberger, R. Kosloff, I. Kuprov, B. Luy, S. Schirmer, T. Schulte-Herbrüggen, D. Sugny, and F. K. Wilhelm, Training Schrödinger's cat: Quantum optimal control: Strategic report on current status, visions and goals for research in Europe, Eur. Phys. J. D 69, 279 (2015).
  67. A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions (Clarendon Press, Oxford, 1970).
  68. R. Hussain, G. Allodi, A. Chiesa, E. Garlatti, D. Mitcov, A. Konstantatos, K. S. Pedersen, R. De Renzi, S. Piligkos, and S. Carretta, Coherent manipulation of a molecular Ln-based nuclear qudit coupled to an electron qubit, J. Am. Chem. Soc. 140, 9814 (2018).
  69. The weak exchange interaction values between Cr and Yb used in this study are consistent with those reported in the literature for analogous Cr-Lanthanide complexes, where magnetic interactions are generally considered very weak. For example, for the Cr-Gd coupling, a J=2.5×10−3meV was determined in Ref. [82].
  70. E. Polozova and F. W. Strauch, Higher-dimensional Bell inequalities with noisy qudits, Phys. Rev. A 93, 032130 (2016).
  71. J. Ferrando-Soria, S. A. Magee, A. Chiesa, S. Carretta, P. Santini, I.J. Vitorica-Yrezabal, F. Tuna, G. F. Whitehead, S. Sproules, K. M. Lancaster, A.-L. Barra, G. A. Timco, E. J. McInnes, and R. E. Winpenny, Switchable interaction in molecular double qubits, Chem 1, 727 (2016).
  72. P. Santini, S. Carretta, F. Troiani, and G. Amoretti, Molecular nanomagnets as quantum simulators, Phys. Rev. Lett. 107, 230502 (2011).
  73. H. Georgi, Lie Algebras in Particle Physics: From Isospin to Unified Theories, Frontiers in Physics (Avalon Publishing, Boulder, CO, USA, 1999).
  74. P. C. K. Vesborg, I. Chorkendorff, T. Brock-Nannestad, J. R. Dethlefsen, and J. Bendix, Note: Simple means for selective removal of the 365 nm line from the Hg spectrum using Dy, Rev. Sci. Instrum. 82, 096102 (2011).
  75. K. S. Pedersen, L. Ungur, M. Sigrist, A. Sundt, M. Schau-Magnussen, V. Vieru, H. Mutka, S. Rols, H. Weihe, O. Waldmann, L. F. Chibotaru, J. Bendix, and J. Dreiser, Modifying the properties of 4f single-ion magnets by peripheral ligand functionalisation, Chem. Sci. 5, 1650 (2014).
  76. G. Allodi, A. Banderini, R. De Renzi, and C. Vignali, HyReSpect: A broadband fast-averaging spectrometer for nuclear magnetic resonance of magnetic materials, Rev. Sci. Instrum. 76, 083911 (2005).
  77. E. L. Hahn, Spin echoes, Phys. Rev. 80, 580 (1950).
  78. S. Carretta, A. Chiesa, F. Troiani, D. Gerace, G. Amoretti, and P. Santini, Quantum information processing with hybrid spin-photon qubit encoding, Phys. Rev. Lett. 111, 110501 (2013).
  79. M. Atzori, A. Chiesa, E. Morra, M. Chiesa, L. Sorace, S. Carretta, and R. Sessoli, A two-qubit molecular architecture for electron-mediated nuclear quantum simulation, Chem. Sci. 9, 6183 (2018).
  80. M. Chizzini, L. Crippa, A. Chiesa, F. Tacchino, F. Petiziol, I. Tavernelli, P. Santini, and S. Carretta, Molecular nanomagnets with competing interactions as optimal units for qudit-based quantum computation, Phys. Rev. Res. 4, 043135 (2022).
  81. D. D'Alessandro, Introduction to Quantum Control and Dynamics, Chapman and Hall/CRC Applied Mathematics and Nonlinear Science (Taylor and Francis Ltd, Hoboken, NJ, 2007).
  82. T. Sanada, T. Suzuki, T. Yoshida, and S. Kaizaki, Heterodinuclear complexes containing d- and f-block elements: Synthesis, structural characterization, and metal–metal interactions of novel chromium(III)–lanthanide(III) compounds bridged by oxalate, Inorg. Chem. 37, 4712 (1998).
  83. A. Chiesa, S. Macedonio, Molecular spin qudits to test generalized Bell inequalities (2026), 10.5281/zenodo.17079527.

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