Entanglement-Swapping Network (S₁ & S₂ → Alice, Bob, Charlie)
Independent Dual Emitters
Observer Alice (A)
A₀, A₁
Observable: σ_z cos(θ) + σ_x sin(θ)
Station Bob (B)
Bell State Measurement
Projects into 4 Bell states |Φ±⟩, |Ψ±⟩
Observer Charlie (C)
C₀, C₁
Target: (σ_z + σ_x)/√2 rotation
Renou-Gisin-Navascués Inequality Test
P_real ≤ 2.000
2.8284
Complex Quantum Regime (Exceeds Real Bound)
Real Bound (2.0)
0.00 (Uncorrelated)
2.00 (Max Real)
2.828 (2√2 Tsirelson)
Complex Correlation P
2.8284
Max Real Correlation Bound
2.0000
Inequality Margin Δ
+0.8284
Hilbert Space Required
Complex ℂ² ⊗ ℂ²
Proved Mathematical Theorems vs. Conjectured Physical Ontologies
Grounded in Nature 2021 & Experimental Realizations 2022
While standard real-number mechanics can mimic isolated quantum particles via Stueckelberg's theorem (by simply doubling vector dimensions), Renou, Gisin, and Navascués proved that independent multi-party networks with independent sources strictly require complex numbers.
| Domain / Setup | Mathematically Rigorous Proof | Conjectured Reality & Interpretation |
|---|---|---|
| Single Qubit Systems Isolated spin-1/2 or photon |
Proved Theorem Stueckelberg Theorem (1960): A complex 2D Hilbert space ℂ² is isomorphic to a 4D real vector space ℝ⁴ with an antisymmetric operator J satisfying J² = -I. Single systems cannot rule out real mechanics. |
Interpretation Imaginary i might simply be a mathematical shortcut (like phase in classical electrical engineering) rather than a true physical reality for single isolated particles. |
| Dual Independent Network S₁ and S₂ independent sources |
Proved Theorem Renou et al. (Nature 2021): If physical state spaces satisfy tensor-product composition for independent sources (S₁ ⊗ S₂), real quantum mechanics enforces P ≤ 2. Complex quantum mechanics yields P = 2√2 ≈ 2.828. |
Physical Reality Nature operates intrinsically on complex amplitudes. Universal real simulators would require unphysical non-local superselection sectors or classical communication between distant emitters. |
| 2022 Experimental Tests Pan et al. & Fan et al. (PRL) |
Experimental Fact Photonic and superconducting qudit circuits violated the real-amplitude bound by > 4.5 standard deviations, demonstrating correlations exceeding 2.0 without shared past history between sources. |
Loophole Discussions Locality and detection loopholes in network scenarios are still being tightened, but mathematically real Hilbert space is decisively ruled out unless postulating global hidden correlations. |
The Key Insight: Why does independent entanglement swapping break real quantum mechanics? In complex mechanics, a tensor product carries phase information across parties naturally ($e^{i(\phi_1 + \phi_2)}$). In real Hilbert space, an artificial generator $J = \left(\begin{smallmatrix} 0 & -1 \\ 1 & 0 \end{smallmatrix}\right)$ cannot be split independently across two uncorrelated sources without missing critical cross-phase interference terms when Bob executes a joint Bell measurement.