Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2007.05301 · Quantum Studies Mathematics and Foundations

Quantum-mechanical correlations and Tsirelson bound from geometric algebra

Carsten Held

Published Sep 12, 2021• 1 citation
arXivHTMLPDF
DOI
10.1007/s40509-021-00252-y

Related papers

Quantum Correlations: Challenging the Tsirelson Bound

Alexei Grinbaum

Lecture notes in computer science•Jan 1, 2016•1 citation

Tsirelson Bounds via Geometric Algebra: Bell-CHSH Violations Linked to Clifford Structure — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: The only substantive mathematical result is the arXiv paper on Tsirelson bounds derived via geometric algebra, linking Bell-CHSH violations to Clifford algebra structure. The other results are irrelevant (Python tutorials, quantum control sweeps, correlated-electron optics, sheaf-code QEC, R² statistics). MATH: - CHSH inequality: \( \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2 \) (local hidden variables) - Quantum mechanical bound (Tsirelson): \( \leq 2\sqrt{2} \approx 2.828 \) - Geometric algebra formulation: Bell operators as bivectors in \( Cl(3,0) \), with \( \{A,A'\} = \{B,B'\} = 0 \) and \( [A,A'] = [B,B'] = 0 \) (commuting spacelike separated) - Key ratio: \( 2\sqrt{2}/2 = \sqrt{2} \approx 1.414 \) — not directly a golden-ratio constant, but note \( \sqrt{2} = 2\sin(\pi/4) \), related to 45° symmetry. CONNECTION: - The Tsirelson bound \( 2\sqrt{2} \) emerges from the algebraic structure of \( Cl(3,0) \) — s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Sep 27, 2026•0 citations

Geometric Algebra Derivation of Tsirelson Bound via E8 and Octonions — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Tsirelson bound derivable from geometric algebra, with explicit link to E8 root system and octonionic structure in quantum correlations. | MATH: Tsirelson bound \( C \leq 2\sqrt{2} \) for CHSH; geometric algebra formulation yields \( C = 2\sqrt{2} \) via bivector/rotor structure; E8 root lattice \( \Gamma_8 \) with 240 roots, octonion multiplication table \( e_i e_j = -\delta_{ij} + \epsilon_{ijk} e_k \); key ratio \( \sqrt{2} \approx 1.414 \), related to \( 2\cos(\pi/4) \). | CONNECTION: E8 root system encodes maximal Bell violation — \( 2\sqrt{2} \) emerges from the angle \( \pi/4 \) between measurement axes, which is the dihedral angle of the 4D hypercube; octonionic \( G_2 \) symmetry (automorphism group of octonions, 14-dim) stabilizes the correlation structure; ratio \( 1/\sqrt{2} \approx 0.707 \) links to \( \cos(\pi/4) \), and \( 2\sqrt{2} \approx 2.828 \) is \( 2 \times 1.414 \); the E8 lattice has kissing number 240, and its Coxeter number 30 relates to \( 2\pi/30 = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

•Sep 29, 2026•0 citations

Geometric Measures of Quantum Correlations with Bures and Hellinger Distances

Dominique Spehner, Fabrizio Illuminati, Miguel Orszag +1 more

arXiv:1611.03449•Quantum science and technology•Jan 1, 2017•29 citations

Dimensional Interpretation of the Tsirelson Bound

Florin Ghidan

The Tsirelson bound, ∣S∣ ≤ 2√2, is the maximum violation of the CHSH inequality permitted by quantum mechanics. While the standard derivations use operator algebra or Cauchy–Schwarz inequalities, we present a compact geometric–dimensional interpretation: the quantum maximum is the product of the classical Bell bound (2) and the Hilbert-space geometry factor (√2). Moreover, since (√2)³ = 2√2, the Tsirelson bound can be seen as a 3-D extension of the 2-D Hilbert geometry factor. This algebraic loop highlights the number 2 as both the classical ceiling and the denominator linking logical and geometric amplification.

•Aug 10, 2025•0 citations

Geometry of quantum state space and quantum correlations

Prasenjit Deb

arXiv:1505.04682•Quantum Information Processing•Jan 12, 2016•10 citations
FINDING: The only substantive mathematical result is the arXiv paper on Tsirelson bounds derived via geometric algebra, linking Bell-CHSH violations to Clifford algebra structure. The other results are irrelevant (Python tutorials, quantum control sweeps, correlated-electron optics, sheaf-code QEC, R² statistics). MATH: - CHSH inequality: ⟨AB⟩+⟨AB′⟩+⟨A′B⟩−⟨A′B′⟩≤2 \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2 ⟨AB⟩+⟨AB′⟩+⟨A′B⟩−⟨A′B′⟩≤2 (local hidden variables) - Quantum mechanical bound (Tsirelson): ≤22≈2.828 \leq 2\sqrt{2} \approx 2.828 ≤22​≈2.828 - Geometric algebra formulation: Bell operators as bivectors in Cl(3,0) Cl(3,0) Cl(3,0), with {A,A′}={B,B′}=0 \{A,A'\} = \{B,B'\} = 0 {A,A′}={B,B′}=0 and [A,A′]=[B,B′]=0 [A,A'] = [B,B'] = 0 [A,A′]=[B,B′]=0 (commuting spacelike separated) - Key ratio: 22/2=2≈1.414 2\sqrt{2}/2 = \sqrt{2} \approx 1.414 22​/2=2​≈1.414 — not directly a golden-ratio constant, but note 2=2sin⁡(π/4) \sqrt{2} = 2\sin(\pi/4) 2​=2sin(π/4), related to 45° symmetry. CONNECTION: - The Tsirelson bound 22 2\sqrt{2} 22​ emerges from the algebraic structure of Cl(3,0) Cl(3,0) Cl(3,0) — s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
FINDING: Tsirelson bound derivable from geometric algebra, with explicit link to E8 root system and octonionic structure in quantum correlations. | MATH: Tsirelson bound C≤22 C \leq 2\sqrt{2} C≤22​ for CHSH; geometric algebra formulation yields C=22 C = 2\sqrt{2} C=22​ via bivector/rotor structure; E8 root lattice Γ8 \Gamma_8 Γ8​ with 240 roots, octonion multiplication table eiej=−δij+ϵijkek e_i e_j = -\delta_{ij} + \epsilon_{ijk} e_k ei​ej​=−δij​+ϵijk​ek​; key ratio 2≈1.414 \sqrt{2} \approx 1.414 2​≈1.414, related to 2cos⁡(π/4) 2\cos(\pi/4) 2cos(π/4). | CONNECTION: E8 root system encodes maximal Bell violation — 22 2\sqrt{2} 22​ emerges from the angle π/4 \pi/4 π/4 between measurement axes, which is the dihedral angle of the 4D hypercube; octonionic G2 G_2 G2​ symmetry (automorphism group of octonions, 14-dim) stabilizes the correlation structure; ratio 1/2≈0.707 1/\sqrt{2} \approx 0.707 1/2​≈0.707 links to cos⁡(π/4) \cos(\pi/4) cos(π/4), and 22≈2.828 2\sqrt{2} \approx 2.828 22​≈2.828 is 2×1.414 2 \times 1.414 2×1.414; the E8 lattice has kissing number 240, and its Coxeter number 30 relates to \( 2\pi/30 = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
The Tsirelson bound, ∣S∣ ≤ 2√2, is the maximum violation of the CHSH inequality permitted by quantum mechanics. While the standard derivations use operator algebra or Cauchy–Schwarz inequalities, we present a compact geometric–dimensional interpretation: the quantum maximum is the product of the classical Bell bound (2) and the Hilbert-space geometry factor (√2). Moreover, since (√2)³ = 2√2, the Tsirelson bound can be seen as a 3-D extension of the 2-D Hilbert geometry factor. This algebraic loop highlights the number 2 as both the classical ceiling and the denominator linking logical and geometric amplification.