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arXiv:2206.02212 · Physical Review A

Optimal discrimination between real and complex quantum theories

Adam Bednorz, Josep Amengual i Batle

Published Oct 11, 2022• 21 citations
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Abstract

We find the minimal number of settings to test quantum theory based on real numbers, assuming separability of the sources, modifying the recent proposal [M.-O. Renou et al., Nature (London) 600, 625 (2021)]. The test needs only three settings for observers $A$ and $C$, but the ratio of complex to real maximum is smaller than in the existing proposal. We also found that two settings and two outcomes for both observers are insufficient.

DOI
10.1103/physreva.106.042207

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•Oct 5, 2026•1 citation
We find the minimal number of settings to test quantum theory based on real numbers, assuming separability of the sources, modifying the recent proposal [M.-O. Renou et al., Nature (London) 600, 625 (2021)]. The test needs only three settings for observers AAA and CCC, but the ratio of complex to real maximum is smaller than in the existing proposal. We also found that two settings and two outcomes for both observers are insufficient.
The question of whether complex numbers play a fundamental role in quantum theory has been debated since the inception of quantum mechanics. Recently, a feasible proposal to differentiate between real and complex quantum theories based on the technique of testing Bell nonlocalities has emerged [Renou et al., Nature (London) 600, 625 (2021)]. Based on this method, the real quantum theory has been falsified experimentally in both photonic and superconducting quantum systems [Li et al., Phys. Rev. Lett. 128, 040402 (2022); Chen, et al., Phys. Rev. Lett. 128, 040403 (2022)]. The quantum networks with multiple independent sources which are not causally connected have gained significant interest as they offer a new perspective on studying the nonlocalities. The independence of these sources imposes additional constraints on observable covariances and leads to new bounds for classical and quantum correlations. In this study, we examine the discrimination between the real and complex quantum theories with an entanglement swapping scenario under a stronger assumption that the two sources are causally independent, which was not made in previous works. Using a revised Navasc\'ues-Pironio-Ac\'{\i}n method and Bayesian optimization, we find a proposal with optimal coefficients of the correlation function which could give a larger discrimination between the real and quantum theories compared with the existing proposals. This work opens up avenues for further exploration of the discrimination between real and complex quantum theories within intricate quantum networks featuring causally independent parties.
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.
The role of complex quantities in quantum theory has been puzzling physicists since the beginnings. It is, thus, natural to ask whether, in order to describe our experiments, the mathematical structure of the complex Hilbert spaces it is built on is really necessary. Recently, it was shown that this structure is inevitable in network scenarios with independent sources. More precisely, Real Quantum Theory cannot explain the predictions of (Complex) Quantum Theory [Renou et al., Nature (London) 600, 625 (2021).NATUAS0028-083610.1038/s41586-021-04160-4]. Here, we revisit the independence assumption underlying this work. We show that assuming partial independence is sufficient for showing the inadequacy of Real Quantum Theory. We derive a tradeoff between source independence and the Bell value achievable in Real Quantum Theory, which also lower bounds the source correlations required to explain previous experiments by means of real quantum systems. We further show that 1 bit of entanglement is necessary and sufficient for recovering the complex quantum correlations by means of Real Quantum Theory in the scenario from [Renou et al., Nature (London) 600, 625 (2021).NATUAS0028-083610.1038/s41586-021-04160-4]. Finally, building on [McKague et al., Phys. Rev. Lett. 102, 2009PRLTAO0031-900710.1103/PhysRevLett.102.020505], we provide a construction to simulate any complex quantum setup with m independent sources by means of Real Quantum Theory by allowing the sources to share an m real-qubit entangled state in the first round of the experiment.
Quantum theory is commonly formulated in complex Hilbert spaces. However, the question of whether complex numbers need to be given a fundamental role in the theory has been debated since its pioneering days. Recently it has been shown that tests in the spirit of a Bell inequality can reveal quantum predictions in entanglement swapping scenarios that cannot be modeled by the natural real-number analog of standard quantum theory. Here, we tailor such tests for implementation in state-of-the-art photonic systems. We experimentally demonstrate quantum correlations in a network of three parties and two independent EPR sources that violate the constraints of real quantum theory by over 4.5 standard deviations, hence disproving real quantum theory as a universal physical theory.
Renou et al. (2021) showed that real and complex quantum theory can be told apart in a network with two independent sources: a Bell-type quantity T reaches 6√2 in complex quantum theory, while real quantum theory with independent sources obeys T ≲ 7.66. We narrow the real-theory value from both sides. (i) An explicit real strategy with independent sources reaches T = 4√3 ≈ 6.9282 exactly; the value is verified in Lean. (ii) T ≤ 7.6568542496 for real quantum theory, below the published 7.6605: an exact integer certificate, accepted by a checker whose soundness is proved in Lean, bounds a symmetry-reduced semidefinite relaxation, and a written proof shows every real strategy with independent sources is feasible for it. (iii) With two measurements for Alice, T ≤ 7.1927385035 for all strategies with independent sources, real or complex, by a scalar-extended two-copy inflation; a single shared source reaches 2+4√2 there. (iv) Explicit hidden cross-source correlations that let a real description exceed the bound, with thresholds of 0.56, 0.50 and 0.47 bit for three constructions. An explicit point shows that this relaxation cannot give less than 2+4√2, so its value is fixed to within 1.3e-11. The optimal real value remains open in [4√3, 7.6568542496]. The deposit contains the note, the certificates (text and binary), a dependency-free Python checker, the Lean sources of the proved checker and of the strategy's value, and the scripts that build, export and attack the certificates. Version 3: the real-theory bound is tightened from 7.6568542701 to 7.6568542496 (a certificate of the same kind on a more accurate solve; the earlier certificate is kept), and an explicit point of the relaxation at exactly 2+4√2 is added. Nothing stated in versions 1 and 2 is withdrawn. Version 2 had made the archive's Lean folders complete Lake projects.