The 2022 superconducting-qubit experiment by USTC (Chen et al., PRL 128, 040403) reported a 43σ violation of the “real-number bound” in a three-party entanglement-swapping network, widely interpreted as decisive evidence that complex numbers are indispensable in quantum mechanics.Here we show that Real Geometric Quantum Mechanics (RGQM)—a formulation defined purely on real differential geometry (curvature, torsion, holonomy) of ℝ^{2𝑁} and containing no imaginary unit—reproduces the experimental result, T_exp = 8.09, with matching precision, 𝑇_RGQM = 8.08 without introducing any free parameters.The sole input is the independently measured Bell-state-measurement (BSM) fidelity 𝑓 = 0.952. In RGQM, each entanglement source creates an SO(2) torsion plane in ℝ^8.The BSM corresponds to a parallel-transport loop 𝛾 that couples these planes, and the observed nonlocal correlations arise from the resulting holonomy 𝑈(𝛾) ∈ SO(8).The two torsion planes contribute independent 2𝜋 holonomies, giving a total geometric phase of 4𝜋, which is the geometric origin of the theoretical maximum 𝑇_max = 8.485. The remarkable agreement between the experiment and the parameter-free RGQM prediction shows that complex numbers are not fundamental physical ingredients, but compact symbols for real geometric rotations. Version 3 description: Correction of numerical values. The values for the nonlocal correlation functional (T) and the Bell-state measurement fidelity (f) have been corrected to align with the experimental data reported by Chen et al. (Phys. Rev. Lett. 128, 040403). The previous version contained incorrect scaling factors (𝑇_max = 13.93 instead of 8.485). The theoretical conclusion remains unchanged.
The 2022 superconducting-qubit experiment by USTC (Chen et al., PRL 128, 040403) reported a 43σ violation of the “real-number bound” in a three-party entanglement-swapping network, widely interpreted as decisive evidence that complex numbers are indispensable in quantum mechanics.Here we show that Real Geometric Quantum Mechanics (RGQM)—a formulation defined purely on real differential geometry (curvature, torsion, holonomy) of ℝ^{2𝑁} and containing no imaginary unit—reproduces the experimental result, T_exp = 8.09, with matching precision, 𝑇_RGQM = 8.08 without introducing any free parameters.The sole input is the independently measured Bell-state-measurement (BSM) fidelity 𝑓 = 0.952. In RGQM, each entanglement source creates an SO(2) torsion plane in ℝ^8.The BSM corresponds to a parallel-transport loop 𝛾 that couples these planes, and the observed nonlocal correlations arise from the resulting holonomy 𝑈(𝛾) ∈ SO(8).The two torsion planes contribute independent 2𝜋 holonomies, giving a total geometric phase of 4𝜋, which is the geometric origin of the theoretical maximum 𝑇_max = 8.485. The remarkable agreement between the experiment and the parameter-free RGQM prediction shows that complex numbers are not fundamental physical ingredients, but compact symbols for real geometric rotations. Version 3 description: Correction of numerical values. The values for the nonlocal correlation functional (T) and the Bell-state measurement fidelity (f) have been corrected to align with the experimental data reported by Chen et al. (Phys. Rev. Lett. 128, 040403). The previous version contained incorrect scaling factors (𝑇_max = 13.93 instead of 8.485). The theoretical conclusion remains unchanged.
While the complex formalism of quantum mechanics has been extraordinarily successful, the fundamental geometric origin of its core elements—the imaginary unit, operator non-commutativity, quantum phase, and the intrinsic geometry with quantum spin—has remained unresolved for more than a century. Here we resolve these foundational questions by deriving an exact and physically equivalent formulation of the Schrödinger equation as a Real Geometric Flow—a moving-frame evolution on a 2𝑁-dimensional real state space. In this restored framework, the global imaginary unit 𝑖 fully decomposes into a set of local skew-symmetric generators that drive planar rotations in real space, revealing that the traditional complex formalism is a compressed representation of an intrinsic rotational geometry.Within the real geometric flow, energy eigenvalues appear as the Curvatures of the state trajectory, energy gaps correspond to Torsions that couple internal planes—naturally generating the geometry of quantum spin—and quantum phase emerges naturally as the Holonomy of the real moving frame. This formulation preserves the full empirical content of quantum mechanics while exposing the geometric mechanisms hidden beneath its notation, providing the first coherent explanation of curvature, torsion, non-commutativity, spin behavior, and quantum phase within a single framework.By demonstrating that the algebraic postulates of quantum mechanics arise from a deeper geometric order—one that stands alongside curvature-based theories such as General Relativity—this work establishes a unified differential-geometric foundation for quantum dynamics.It opens new directions for quantum control, quantum information, and fundamental physics.
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.
Standard quantum mechanics employs complex Hilbert spaces, but whether complex numbers are fundamental or merely convenient has long been debated. For decades, real-valued equivalents were considered mathematically possible but cumbersome. However, a highly cited 2021 result claimed that any quantum theory based on real numbers is experimentally falsifiable via network Bell experiments. Yet, it remains an open question whether this falsification applies to all real-valued theories. Here we show that this conclusion rests on an incomplete real formulation, and we present a rigorous real-valued framework that perfectly reproduces all predictions of standard quantum mechanics. We demonstrate that the standard real tensor product (⊗R) used in previous no-go theorems is algebraically incompatible with the rich structure of conventional quantum mechanics. We present a real framework based on Kähler space and prove that it is exactly isomorphic to established quantum mechanics via an explicit bijection γ. The isomorphism extends to composite systems through a symplectic composition rule $\otimes^{\ks}$ that replaces the Kronecker product. Consequently, our formulation achieves the maximal CHSH3 violation of 62 using purely real variables, demonstrating that the no-go theorem is specific to a particular real representation of states and operators and to the composition rule ⊗R built upon it, neither of which extends to the present Kähler framework. These results demonstrate that complex numbers are not fundamentally required by nature; rather, they encode a deeper real geometric structure that governs quantum interference and entanglement, settling this long debate.
Standard quantum mechanics has been formulated with complex-valued Schrodinger equations, wave functions, operators, and Hilbert spaces. However, previous work has shown possible to simulate quantum systems using only real numbers by adding extra qubits and exploiting an enlarged Hilbert space. A fundamental question arises: are the complex numbers really necessary for the quantum mechanical description of nature? To answer this question, a non-local game has been developed to reveal a contradiction between a multiqubit quantum experiment and a player using only real numbers. Here, based on deterministic and high-fidelity entanglement swapping with superconducting qubits, we experimentally implement the Bell-like game and observe a quantum score of 8.09(1), which beats the real number bound of 7.66 by 43 standard deviations. Our results disprove the real-number description of nature and establish the indispensable role of complex numbers in quantum mechanics.
Quantum mechanics is commonly formulated in a complex, rather than real, Hilbert space. However, whether quantum theory really needs the participation of complex numbers has been debated ever since its birth. Recently, a Bell-like test in an entanglement-swapping scenario has been proposed to distinguish standard quantum mechanics from its real-valued analog. Previous experiments have conceptually demonstrated, yet not satisfied, the central requirement of independent state preparation and measurements and leave several loopholes. Here, we implement such a Bell-like test with two separated independent sources delivering entangled photons to three separated parties under strict locality conditions that are enforced by spacelike separation of the relevant events, rapid random setting generation, and fast measurement. With the fair-sampling assumption and closed loopholes of independent source, locality, and measurement independence simultaneously, we violate the constraints of real-valued quantum mechanics by 5.30 standard deviations. Our results disprove the real-valued quantum theory to describe nature and ensure the indispensable role of complex numbers in quantum mechanics.
Experimental Equivalence of Real and Complex Quantum Mechanics: Resolving the CHSH3 No-Go Result
A recent study by Renou et al. (Nature 600, 625, 2021) reported that quantum theory formulated over real numbers can be experimentally falsified via a gap in the maximal Clauser–Horne–Shimony–Holt CHSH3 violation. Here, we show that this apparent distinction is not fundamental but arises from an incomplete composition rule for composite systems. Using a structure‑preserving real representation of quantum mechanics (Singh, Ann. Phys. 487, 170368, 2026), we demonstrate that the maximal violation 62 is achieved using only real arithmetic. Our explicit construction for the tripartite entanglement‑swapping scenario reproduces exactly the complex quantum prediction. We further establish an algebraic isomorphism between real and complex quantum mechanics that extends to a structure-preserving strict monoidal functor. This establishes an operational equivalence between real and complex quantum mechanics, demonstrating that all quantum correlations can be realized without invoking complex numbers.Keywords: Quantum foundations, Real quantum mechanics, Symplectic geometry, Tensor product structure, Real Hilbert space, Real-complex isomorphism, Bell nonlocality, Entanglement, Quantum correlations.