Efficient discrimination between real and complex quantum theories

Josep Batle; Tomasz Białecki; Tomasz Rybotycki; Jakub Tworzydło; Adam Bednorz

doi:10.22331/q-2025-01-15-1595

Efficient discrimination between real and complex quantum theories

Josep Batle^1,2, Tomasz Białecki³, Tomasz Rybotycki^4,5,6, Jakub Tworzydło³, and Adam Bednorz³

¹Departament de Física UIB i Institut d'Aplicacions Computacionals de Codi Comunitari (IAC3), Campus UIB, E-07122 Palma de Mallorca, Balearic Islands, Spain
²CRISP - Centre de Recerca Independent de sa Pobla, 07420 sa Pobla, Balearic Islands, Spain
³Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL02-093 Warsaw, Poland
⁴Systems Research Institute, Polish Academy of Sciences, 6 Newelska Street, PL01-447 Warsaw, Poland
⁵Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, 18 Bartycka Street, PL00-716 Warsaw, Poland
⁶Center of Excellence in Artificial Intelligence, AGH University, 30 Mickiewicza Lane, PL30-059 Cracow, Poland

Published:	2025-01-15, volume 9, page 1595
Editor:	Paul Skrzypczyk
Eprint:	arXiv:2405.03013v2
Doi:	https://doi.org/10.22331/q-2025-01-15-1595
Citation:	Quantum 9, 1595 (2025).

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Abstract

We improve the test to show the impossibility of a quantum theory based on real numbers by a larger ratio of complex-to-real bound on a Bell-type parameter. In contrast to previous theoretical and experimental proposals the test requires three settings for the parties $A$ and $C$, but also six settings for the middle party $B$, assuming separability of the sources. The bound we found for this symmetric configuration imposed on a real theory is $14.69$ while the complex maximum is $18$. This large theoretical difference enables us to demonstrate the concomitant experimental violation on IBM quantum computer via a designed quantum network, without resorting to error mitigation, obtaining as a result $15.44$ at more than $100$ standard deviations above the found real bound.

Featured image: The combination of correlations $\mathcal F$, with the classical bound $12$, the quantum real bound $\mathcal F_{\mathrm{r}}=6\sqrt{6}$, the experimental value $\mathcal F_{\mathrm{exp}}$, and the complex quantum maximum $18$.

Popular summary

In quantum mechanics, real numbers are insufficient to describe independent, separable systems. To demonstrate the necessity of complex numbers, one can perform a special experiment, involving three parties and testing bounds of correlations, constructed originally with the help of semidefinite programming.

We develop the analytic derivation of an improved test, an inequality for a combination of correlations $\mathcal F$, based on special sums of squares, assuming real-valued separability. We have run this experiment on the public IBM quantum computer and ruled out the real-valued quantum mechanics by more than $100$ standard deviations.

► BibTeX data

@article{Batle2025efficient,
  doi = {10.22331/q-2025-01-15-1595},
  url = {https://doi.org/10.22331/q-2025-01-15-1595},
  title = {Efficient discrimination between real and complex quantum theories},
  author = {Batle, Josep and Bia{\l{}}ecki, Tomasz and Rybotycki, Tomasz and Tworzyd{\l{}}o, Jakub and Bednorz, Adam},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1595},
  month = jan,
  year = {2025}
}

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Cited by

[1] Mao-Sheng Li and Yi-Xi Tan, "Bargmann invariants for quantum imaginarity", Physical Review A 111 2, 022409 (2025).

[2] Sergio Giardino, "Expectation Value Dynamics Within Real Hilbert Space Quantum Mechanics", International Journal of Theoretical Physics 64 10, 257 (2025).

[3] Sergio Giardino, "Self-Interacting Quantum Particles and the Dirac Delta Potential", Brazilian Journal of Physics 56 1, 46 (2026).

[4] Shreya Vardhan, Bowen Shi, Isaac H. Kim, and Yijian Zou, "Chirality, magic, and quantum correlations in multipartite quantum states", SciPost Physics 20 3, 066 (2026).

[5] Shubhayan Sarkar, David Trillo, Marc Olivier Renou, and Remigiusz Augusiak, "Gap between quantum theory based on real and complex numbers is arbitrarily large", Reports on Progress in Physics 89 7, 070503 (2026).

[6] Chuanfa Wu and Zhaoqi Wu, "Quantifying imaginarity of quantum operations", Science China Physics, Mechanics & Astronomy 69 3, 230316 (2026).

[7] Gábor Drótos, Károly F. Pál, Abdelmalek Taoutioui, and Tamás Vértesi, "Towards minimal self-testing of qubit states and measurements in prepare-and-measure scenarios", New Journal of Physics 26 6, 063012 (2024).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-09 13:33:46) and SAO/NASA ADS (last updated successfully 2026-08-09 13:33:53). The list may be incomplete as not all publishers provide suitable and complete citation data.

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