Real Geometric Quantum Mechanics Exactly Reproduces the 2022 USTC 43σ Entanglement-Swapping Experiment

Real Geometric Quantum Mechanics Exactly Reproduces the 2022 USTC 43σ Entanglement-Swapping Experiment Se Kyun Kwon Department of Physics, Pohang University of Science and Technology, Pohang 37673, Republic of Korea (Correspondence: [email protected]) Abstract 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 entanglementswapping 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 ℝ2𝑁 and containing no imaginary unit—reproduces the experimental result, 𝑇exp =8.09, with matching precision, 𝑇RGQM =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 ℝ8. 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 𝑇max =6√2≅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. 1. Introduction Quantum mechanics is conventionally formulated on complex Hilbert spaces, where the imaginary unit 𝑖 is treated as a primitive mathematical necessity. This viewpoint was recently reinforced by the experiment of Chen et al. (2022), which implemented the nonlocal game proposed by Renou et al. (2021). Their measured value, 𝑇exp =8.09, exceeds the real-number bound (≤7.66) by 43σ, and has been widely interpreted as experimental evidence that real Hilbert-space quantum theories are fundamentally insufficient. However, the “real-number theories” ruled out in these arguments assume only linear real vector spaces with no intrinsic geometric structure. In contrast, Real Geometric Quantum Mechanics (RGQM) provides a fully equivalent, yet far more structured, differential-geometric framework to the complex formalism: • states live in ℝ2𝑁; • dynamics are generated by real skew-symmetric matrices 𝐴(𝑡)∈𝔰𝔬(2𝑁); • phases arise from SO(𝟐) holonomy of local rotational planes; • superposition is encoded in real two-dimensional rotational subspaces; • entanglement corresponds to coupled torsion planes in the underlying manifold. Within this geometric framework, we show that the USTC result is not evidence for the necessity of complex numbers. Rather, the experiment supports RGQM by demonstrating that the observed nonlocal correlations—including the 43σ violation—are exactly reproducible through real geometric holonomy without invoking the imaginary unit. The quantity 𝑇 defined by Renou et al. is a nonlocal correlation functional whose value quantifies the strength of three-party quantum correlations beyond any classical or realHilbert-space model. Real theories satisfy the upper bound 𝑇≤7.66, while complex quantum mechanics allows a maximum of 𝑇max =8.485. Thus, the USTC value 8.09 lies deep in the fully quantum region and very close to the theoretical quantum maximum. 2. RGQM in a nutshell In RGQM, the state of an 𝑁-level system is a real vector Φ(𝑡)∈ℝ2𝑁, and its evolution is governed by the real moving-frame equation 𝑑Φ(𝑡) 𝑑𝑡 =𝐴(𝑡) Φ(𝑡), 𝐴(𝑡)∈𝔰𝔬(2𝑁). Every skew-symmetric generator 𝐴 decomposes uniquely into elementary planar rotations, 𝐴=∑𝜔𝑖𝑗 𝐽𝑖𝑗 𝑖<𝑗 , where 𝐽𝑖𝑗 is the generator of rotation in the (𝑥𝑖,𝑦𝑗) plane. The coefficients 𝜔𝑖𝑗 carry direct geometric and physical meaning. Diagonal components give curvature, 𝜔𝑖𝑖 =𝐸𝑖 ℏ, setting the intrinsic rotation rate associated with energy eigenvalues. Energy differences give torsion, 𝜔𝑖𝑗 =𝐸𝑖−𝐸𝑗 ℏ, which governs interference and entanglement between the corresponding real planes. The time-evolution operator lies in SO(𝟐𝑵), and is expressed as the path-ordered exponential, 𝑈(𝛾)=𝒫exp∮𝐴 𝛾. No imaginary unit appears anywhere in the formalism. All quantum phases—and all interference and holonomy effects traditionally attributed to the complex number 𝑖—arise naturally from real planar rotations and the holonomy of closed loops in ℝ2𝑁. 3. Geometry of the four-qubit entanglement-swapping network The four superconducting qubits Q1, Q2, Q3, and Q4 embed in Φ∈ℝ8, with coordinates (𝑥1,𝑦1),…,(𝑥4,𝑦4). 3.1 Two independent torsion planes The two entanglement sources create two independent SO(2) torsion planes: • P1 (A–B plane): torsion amplitude 𝜏1 • P2 (C–D plane): torsion amplitude 𝜏2 When an entangled pair is perfect, parallel transport around each torsion plane yields holonomy 2π. Thus, perfect two-source symmetry gives: 𝜗1=2𝜋, 𝜗2=2𝜋. Figure 1 provides a geometric visualization of this mechanism. The two entangled sources generate two orthogonal SO(2) torsion planes P1 and P2 in ℝ8, and the Bell-state measurement corresponds to a closed parallel-transport loop 𝛾 linking them. This loop accumulates the holonomies of both planes, making the origin of the total geometric phase 𝜗total =𝜗1+𝜗2=4𝜋 immediately clear. 3.2 BSM as a geometric loop The Bell-state measurement executed at the middle node couples the two torsion planes. In RGQM, this is a parallel transport loop: 𝛾: (P1)→(BSM coupling)→(P2)→(P1), and produces holonomy 𝑈(𝛾)=𝒫exp∮𝐴 𝛾 ∈SO(8). 3.3 Total holonomy = 4π Because the two entanglement sources generate two orthogonal SO(2) torsion planes in ℝ8, the holonomy acquired under BSM-induced parallel transport is simply the sum of the two independent planar rotations: 𝜗total =𝜗1+𝜗2=2𝜋+2𝜋=4𝜋. This 4π geometric phase is the real-space origin of the theoretical maximum of the Renou functional: 𝑇max =8.485. In other words, the complex phases responsible for the maximal quantum score in standard quantum mechanics correspond, in RGQM, to the holonomies of the two torsion planes generated by the entanglement sources. This combined holonomy of 4𝜋 is precisely the geometric origin of the maximal quantum value 𝑇max =8.485. in the Renou scenario: the functional 𝑇 increases monotonically with the net SO(2) rotation generated by the entanglement-swapping loop. Thus, the theoretical maximum of the experiment corresponds to the full holonomy accumulated from two perfectly coherent torsion planes. 4. Fidelity scaling and the USTC result The Bell-state measurement fidelity 𝑓=0.952 acts as a direct geometric scaling of the torsion amplitude in RGQM, 𝜏→𝑓𝜏. Because the nonlocal correlation functional in the Renou–USTC scenario depends linearly on the torsion amplitude for small deviations from perfect coherence, the Bell functional satisfies the simple relation 𝑇(𝑓) = 𝑓𝑇max. Substituting the independently measured fidelity yields 𝑇RGQM =𝑓𝑇max =0.952×8.485=8.08, in close agreement with the experimental result, 𝑇exp =8.09. No adjustable parameters were introduced: the single measured quantity 𝑓 fully determines the reduction of torsion and therefore the magnitude of the observed nonlocal correlation. This parameter-free match strongly supports the RGQM description of the underlying geometric mechanism. 5. Significance: Why this agreement matters Although the RGQM computation looks simple, that simplicity emerges from the deep geometric structure of entangled states in ℝ8. Curvature generates local phase evolution, torsion produces entanglement, holonomy replaces complex phase, and parallel transport replaces unitary evolution. The entire 43σ violation is explained by the geometric identity 𝑇=𝑓𝑇max, 𝑇max attained at 𝜗total =4𝜋 In contrast, no such structural connection exists in complex quantum mechanics, where fidelity, nonlocal phase, and visibility are algebraically independent objects with no unifying geometric interpretation. Thus, the USTC experiment—intended as a test against real-number quantum theories—instead supports a richer real geometric formulation in which complex phases are revealed to be nothing more than rotations and holonomies ℝ2𝑁. 6. Conclusion Real Geometric Quantum Mechanics reproduces the strongest experimental evidence ever cited against real-number quantum theory—the USTC 43σ entanglement-swapping violation—with parameter-free precision. RGQM demonstrates that complex numbers are not fundamental; quantum phases are geometric rotations. Entanglement swapping is parallel transport across torsion planes. The 4π holonomy of two entangled sources generates the theoretical maximum of nonlocal correlation. The measured fidelity 𝑓 plays the role of geometric torsion amplitude. The USTC experiment therefore supports—rather than contradicts—a fully real, fully geometric foundation for quantum mechanics. Figure 1 | Entanglement swapping in Real Geometric Quantum Mechanics. Two independent entanglement sources generate two orthogonal SO(2) torsion planes in ℝ8: P1 (A–B plane, blue) and P2 (C–D plane, green). A Bell-state measurement at the central node corresponds to a parallel-transport loop 𝛾 (red), which couples the two planes. Each torsion plane contributes a 2𝜋 holonomy when the corresponding source is perfectly coherent, yielding a total holonomy of 𝟒𝝅 in the combined loop. This 4𝜋 geometric phase is the origin of the theoretical maximum of the Renou functional, 𝑇max =8.485. The experimentally measured fidelity 𝑓 simply rescales the torsion amplitudes, producing the observed value 𝑇exp =8.09. Thus, the entire entanglement-swapping process—and its nonlocal correlations—is described purely as real-geometry holonomy in ℝ𝟖. References [1] M.-C. Chen et al., Phys. Rev. Lett. 128, 040403 (2022). [2] M.-O. Renou et al., Nature 600, 580 (2021). [3] S. K. Kwon, Quantum Mechanics as Real Geometry: Curvature, Torsion, and Holonomy, https://doi.org/10.5281/zenodo.17699147 (2025).