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Quantum Mechanics Based on Real Numbers: A Consistent Description

Pedro Barrios Hita1,2,*, Anton Trushechkin2, Hermann Kampermann2, Michael Epping1, and Dagmar Bruß2

  • *Contact author: Pedro.Barrios@dlr.de
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Phys. Rev. Lett. 136, 240202 – Published 18 June, 2026

DOI: https://doi.org/10.1103/4k13-sdjh

Abstract

Complex numbers play a crucial role in quantum mechanics. However, their necessity remains debated: whether they are fundamental or merely convenient. Recently, it was shown that any real-number quantum theory satisfying certain postulates can be falsified with multipartite experiments. In this Letter we show that a physically motivated postulate about composite quantum systems allows us to construct quantum mechanics based on real numbers that reproduces predictions for all multipartite quantum experiments. Thus, we argue that real-valued quantum mechanics cannot be falsified, and therefore the use of complex numbers is a matter of convenience.

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A New Perspective on Real-Valued Quantum Theory

Published 18 June, 2026

By exploiting a physically motivated principle, rather than a mathematical postulate, researchers offer a new perspective on how a real-valued quantum theory can be constructed.

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Article Text

Supplemental Material

References (36)

  1. F. J. Dyson, J. Math. Phys. (N.Y.) 3, 1199 (1962).
  2. G. Birkhoff and J. V. Neumann, Ann. Math. 37, 823 (1936).
  3. J. Jauch, Foundation of Quantum Mechanics (Addison Wesley, Reading, Massachusetts, 1968).
  4. V. Varadarajan, Geometry of Quantum Theory, 2nd ed. (Springer, New York, 1985).
  5. I. V. Volovich, Ultrametric Anal. Appl. 2, 77 (2010), first publication: preprint CERN-TH.4781/87, Geneva, 1987.
  6. V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, P-Adic Analysis and Mathematical Physics (World Scientific, Singapore, 1994).
  7. B. Dragovich, A. Y. Khrennikov, S. V. Kozyrev, I. V. Volovich, and E. I. Zelenov, Ultrametric Anal. Appl. 9, 87 (2017).
  8. E. Stueckelberg, Helv. Phys. Acta 33, 727 (1960).
  9. J. Myrheim, arXiv:quant-ph/9905037.
  10. M. McKague, M. Mosca, and N. Gisin, Phys. Rev. Lett. 102, 020505 (2009).
  11. D. E. Koh, M. Y. Niu, and T. J. Yoder, J. Phys. A 51, 195302 (2018).
  12. A. Aleksandrova, V. Borish, and W. K. Wootters, Phys. Rev. A 87, 052106 (2013).
  13. W. Wootters, Found. Phys. 42, 19 (2012).
  14. J. Barrett, arXiv:quant-ph/0508211.
  15. L. Hardy, arXiv:quant-ph/0101012.
  16. G. M. D’Ariano, F. Manessi, P. Perinotti, and A. Tosini, Europhys. Lett. 107, 20009 (2014).
  17. M.-O. Renou, D. Trillo, M. Weilenmann, T. P. Le, A. Tavakoli, N. Gisin, A. Acín, and M. Navascués, Nature (London) 600, 625 (2021).
  18. S. Sarkar, D. Trillo, M. O. Renou, and R. Augusiak, arXiv:2503.09724.
  19. M. Weilenmann, N. Gisin, and P. Sekatski, Phys. Rev. Lett. 135, 180201 (2025).
  20. T. J. Elliott, Phys. Rev. A 111, 062401 (2025).
  21. M.-C. Chen, C. Wang, F.-M. Liu, J.-W. Wang, C. Ying, Z.-X. Shang, Y. Wu, M. Gong, H. Deng, F.-T. Liang, Q. Zhang, C.-Z. Peng, X. Zhu, A. CaBello, C.-Y. Lu, and J.-W. Pan, Phys. Rev. Lett. 128, 040403 (2022).
  22. D. Wu, Y.-F. Jiang, X.-M. Gu, L. Huang, B. Bai, Q.-C. Sun, X. Zhang, S.-Q. Gong, Y. Mao, H.-S. Zhong, M.-C. Chen, J. Zhang, Q. Zhang, C.-Y. Lu, and J.-W. Pan, Phys. Rev. Lett. 129, 140401 (2022).
  23. Z.-D. Li, Y.-L. Mao, M. Weilenmann, A. Tavakoli, H. Chen, L. Feng, S.-J. Yang, M.-O. Renou, D. Trillo, T. P. Le, N. Gisin, A. Acín, M. Navascués, Z. Wang, and J. Fan, Phys. Rev. Lett. 128, 040402 (2022).
  24. See Supplemental Material at http://link.aps.org/supplemental/10.1103/4k13-sdjh for the proofs of the statements that appear in the main text are provided. Furthermore, the consistency of the model proposed is checked and extensions to said model are studied, which includes Refs. [5,7–9].
  25. A. Peres, Phys. Rev. A 61, 022117 (2000).
  26. C. Hindlycke, N. Johansson, and J. Larsson, Entropy 27, 596 (2025).
  27. S. Simon, R. Santagati, M. Degroote, N. Moll, M. Streif, and N. Wiebe, PRX Quantum 5, 010343 (2024).
  28. N. C. Jones, arXiv:1310.7290.
  29. R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, Proc. R. Soc. A 454, 339 (1997).
  30. K. Eckert, J. Schliemann, D. Bruß, and M. Lewenstein, Ann. Phys. (Amsterdam) 299, 88 (2002).

  31. Also these four product operators map canonical representatives into canonical representatives.

  32. N. D. Elkies, Linear algebra and tensors: Lecture notes (2010), available online: https://people.math.harvard.edu/ elkies/M55a.10/tensor.pdf.
  33. A. S. Holevo, Introduction to Quantum Information Theory (MCNMO, Moscow, 2002) (in Russian).
  34. I. Volovich, arXiv:2504.16838.
  35. T. Hoffreumon and M. P. Woods, arXiv:2504.02808.
  36. Y. Yīng, M. C. Alañón, D. Centeno, J. Surace, M. M. Ansanelli, R. Liu, D. Schmid, and R. W. Spekkens, arXiv:2506.08091.

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Phys. Rev. Lett. 136, 240202– Published 18 June, 2026
  • Received 5 June 2025
  • Revised 11 October 2025
  • Accepted 11 March 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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