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Quantum Non-Locality and Tsirelson Bound: Unifying Entanglement Tests via Semi-Device-Independent Frameworks — E8 Intelligence Research

Andrew Stewart Caldin

Published Oct 15, 2026• 0 citations
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Abstract

FINDING: Bell inequality tests confirm quantum non-locality; CHSH form bounds classical correlations at 2, quantum at 2√2 ≈ 2.828; new frameworks (Buscemi) unify entanglement tests via semi-device-independent rules. MATH: - CHSH inequality: \( \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2 \) (local hidden variables) - Quantum Tsirelson bound: \( 2\sqrt{2} \approx 2.828 \) — maximal violation for singlet state - Bell operator: \( \mathcal{B} = \sum_{a,b,x,y} (-1)^{a+b+xy} P(a,b|x,y) \) - Buscemi's unification: replaces measurement choices with quantum inputs, yielding *semi-device-independent* inequalities — no fixed ratio but generalizes to \( \mathcal{B} \leq \beta_{\text{classical}} < \beta_{\text{quantum}} \) - Ballistic electron proposal: 5 gates → entangled state; Bell test via Coulomb-coupled wires (no optical analog) CONNECTION: - Tsirelson bound \( 2\sqrt{2} = 2 \times \sqrt{2} \) — ratio \( \sqrt{2} \approx 1.414 \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI
10.5281/zenodo.23386205

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FINDING: Bell inequality tests confirm quantum non-locality; CHSH form bounds classical correlations at 2, quantum at 2√2 ≈ 2.828; new frameworks (Buscemi) unify entanglement tests via semi-device-independent rules. MATH: - CHSH inequality: ( \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2 ) (local hidden variables) - Quantum Tsirelson bound: ( 2\sqrt{2} \approx 2.828 ) — maximal violation for singlet state - Bell operator: ( \mathcal{B} = \sum_{a,b,x,y} (-1)^{a+b+xy} P(a,b|x,y) ) - Buscemi's unification: replaces measurement choices with quantum inputs, yielding semi-device-independent inequalities — no fixed ratio but generalizes to ( \mathcal{B} \leq \beta_{\text{classical}} < \beta_{\text{quantum}} ) - Ballistic electron proposal: 5 gates → entangled state; Bell test via Coulomb-coupled wires (no optical analog) CONNECTION: - Tsirelson bound ( 2\sqrt{2} = 2 \times \sqrt{2} ) — ratio ( \sqrt{2} \approx 1.414 \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Bell inequality tests confirm quantum non-locality; new generalized rules (Buscemi) unify entanglement tests via device-independent frameworks. MATH: CHSH inequality: S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2 (local hidden variables); quantum violation max S=22≈2.828S = 2\sqrt{2} \approx 2.828S=22​≈2.828. Bell's original inequality: P(a⃗,b⃗)−P(a⃗,c⃗)≤1+P(b⃗,c⃗)P(\vec{a},\vec{b}) - P(\vec{a},\vec{c}) \leq 1 + P(\vec{b},\vec{c})P(a,b)−P(a,c)≤1+P(b,c). Buscemi's unification: replaces measurement settings with quantum inputs, yielding semi-device-independent bounds—generalizes to Sq≤22S_q \leq 2\sqrt{2}Sq​≤22​ for qubits, but for higher-dimensional systems, Tsirelson bounds scale as 2d2\sqrt{d}2d​ for maximally entangled states. CONNECTION: The maximal quantum violation 222\sqrt{2}22​ relates to the golden ratio via 22=2×1.4142\sqrt{2} = 2 \times 1.41422​=2×1.414, and 1.4142=21.414^2 = 21.4142=2—a Pythagorean doubling. More directly, the CHSH bound 222\sqrt{2}22​ appears in the geometry of the square (diagonal/side ratio), linking to 45° rotations in 2D lattice symme Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Bell inequality tests confirm quantum non-locality; new generalized rules (Buscemi) unify entanglement frameworks; condensed-matter proposal extends tests to ballistic electrons. MATH: - Bell inequality (CHSH form): S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2 S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2 S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2 (local hidden variables); quantum mechanics predicts S=22≈2.828 S = 2\sqrt{2} \approx 2.828 S=22​≈2.828. - Violation ratio: 22/2=2≈1.414 2\sqrt{2}/2 = \sqrt{2} \approx 1.414 22​/2=2​≈1.414 — not a golden ratio, but a Pythagorean root. - Buscemi's unification: generalizes Bell inequalities via *semidefinite programming* hierarchies (NPA hierarchy), replacing fixed measurement settings with *quantum channels* — key constant: Tsirelson bound 22 2\sqrt{2} 22​ remains the supremum for CHSH. - Ballistic electron proposal: uses Coulomb-coupled quantum wires; entanglement generation via *beam-splitter-like* gates — no new constants, but maps to spin-1/2 Pauli algebra σx,σy,σz \sigma_x, \sigma_y, \sigma_z σx​,σy​,σz​. CONNECTION: - The 2 \sqrt{2} 2​ ratio app Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Bell inequality tests confirm quantum non-locality; new Buscemi framework unifies entanglement tests via "unsteerability" conditions; condensed-matter proposal uses ballistic electrons in Coulomb-coupled quantum wires. | MATH: Bell's original inequality: P(a,b)−P(a,b′)+P(a′,b)+P(a′,b′)≤2P(a,b) - P(a,b') + P(a',b) + P(a',b') \leq 2P(a,b)−P(a,b′)+P(a′,b)+P(a′,b′)≤2 (CHSH form: S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2S = E(a,b)+E(a,b')+E(a',b)-E(a',b') \leq 2S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2); quantum violation SQM=22≈2.828S_{\text{QM}} = 2\sqrt{2} \approx 2.828SQM​=22​≈2.828 (Tsirelson bound). Buscemi's generalization replaces local hidden variables with "unsteerable" assemblages, yielding a hierarchy of inequalities with tighter bounds (e.g., SBuscemi≤2S_{\text{Buscemi}} \leq 2SBuscemi​≤2 for classical, 222\sqrt{2}22​ for quantum, but extended to non-projective measurements). Ballistic electron proposal: entanglement generation via Coulomb interaction in quantum wires, requiring only 5 gates; Bell test via spin/charge correlations with predicted violation S≈2.7S \approx 2.7S≈2.7 (finite temperature corrections). | CONNECTION: Tsirelson bound \(2\sqrt{ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Bell/CHSH inequality tests reveal quantum non-locality; new Buscemi framework unifies entanglement tests via "semiquantum" games; ballistic-electron proposal extends tests to condensed matter. MATH: CHSH inequality: S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2 S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2 S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2 (local hidden variables); quantum bound SQM=22≈2.828 S_{\text{QM}} = 2\sqrt{2} \approx 2.828 SQM​=22​≈2.828. Bell's original: ∣P(a,b)−P(a,b′)∣+∣P(a′,b)+P(a′,b′)∣≤2 |P(a,b) - P(a,b')| + |P(a',b) + P(a',b')| \leq 2 ∣P(a,b)−P(a,b′)∣+∣P(a′,b)+P(a′,b′)∣≤2. Buscemi's unification: replaces classical inputs with quantum states, yielding "semiquantum" Bell inequalities with tighter bounds (e.g., ≤2 \leq 2 ≤2 classical, 22 2\sqrt{2} 22​ quantum, but extended to arbitrary dimensions via 2d 2\sqrt{d} 2d​ for d d d-level systems). Ballistic electron proposal: entanglement via Coulomb coupling, requires only 5 gates — no new constants, but uses spin/charge degrees of freedom. CONNECTION: The quantum bound 22 2\sqrt{2} 22​ is the ratio 22≈2.828 2\sqrt{2} \approx 2.828 22​≈2.828, which relates to the diagonal of a unit square Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Bell inequality tests reveal quantum non-locality; new generalized frameworks (Buscemi) unify entanglement tests via information-theoretic constraints, while condensed-matter proposals (ballistic electrons) extend violation to solid-state systems. | MATH: CHSH inequality: S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2S = E(a,b) + E(a,b') + E(a',b) - E(a',b') \leq 2S=E(a,b)+E(a,b′)+E(a′,b)−E(a′,b′)≤2 (local hidden variables); quantum violation Smax=22≈2.828S_{\text{max}} = 2\sqrt{2} \approx 2.828Smax​=22​≈2.828. Bell's original: P(a⃗,b⃗)−P(a⃗,b⃗′)≤1+P(b⃗,b⃗′)P(\vec{a},\vec{b}) - P(\vec{a},\vec{b}') \leq 1 + P(\vec{b},\vec{b}')P(a,b)−P(a,b′)≤1+P(b,b′). Buscemi's unification: replaces measurement settings with quantum inputs, yielding semi-device-independent bounds via teleportation-replication constraints. | CONNECTION: The CHSH maximum 222\sqrt{2}22​ is the ratio of the diagonal of a unit square to its side — a direct geometric expression of the square's symmetry (dihedral group D4D_4D4​). The violation threshold 222 vs 222\sqrt{2}22​ maps to the ratio 1:21:\sqrt{2}1:2​, which is the aspect ratio of A-series paper (ISO 216) and t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Bell inequality tests confirm quantum non-locality; new generalized rules unify entanglement tests; condensed-matter proposal uses ballistic electrons in quantum wires. MATH: Bell inequality: S=E(a,b)−E(a,b′)+E(a′,b)+E(a′,b′)≤2S = E(a,b) - E(a,b') + E(a',b) + E(a',b') \leq 2S=E(a,b)−E(a,b′)+E(a′,b)+E(a′,b′)≤2 (local hidden variables); quantum violation: SQM=22≈2.828S_{\text{QM}} = 2\sqrt{2} \approx 2.828SQM​=22​≈2.828 (CHSH). Buscemi's unification: replaces measurement settings with quantum inputs, yielding "nonlocal" bounds via semidefinite programming — generalized Bell functionals B(ρ)≤βlocalB(\rho) \leq \beta_{\text{local}}B(ρ)≤βlocal​ with βlocal\beta_{\text{local}}βlocal​ derived from tensor product structures. Condensed-matter: entangled electron pairs via Coulomb coupling in quantum wires; Bell test with five gates — spin/charge degrees of freedom, violation expected at S≈2.828S \approx 2.828S≈2.828 for singlet state. CONNECTION: The CHSH maximum 222\sqrt{2}22​ relates to the golden ratio via 22=8≈2.8282\sqrt{2} = \sqrt{8} \approx 2.82822​=8​≈2.828, and 2\sqrt{2}2​ appears in root systems of D4D_4D4​ (cryst Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com