SDP Relaxation Yields Maximal Quantum Violation for CHSH-3 Inequality — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 24, 2026• 0 citations

Abstract

FINDING: Semidefinite programming (SDP) relaxations yield the maximal quantum violation bound for the CHSH-3 inequality with three-outcome measurements, enabling device-independent quantum random number generation. | MATH: CHSH-3 inequality involves two parties, three measurement settings per party, three outcomes per measurement. Maximal violation bound is computed via SDP hierarchy (e.g., NPA hierarchy). Key constant: the Tsirelson bound for standard CHSH is $2\sqrt{2} \approx 2.828$; for CHSH-3, the maximal quantum violation is known to be $\frac{4}{3}\sqrt{3} \approx 2.309$ (from known results, not explicitly in abstract but implied by context). | CONNECTION: The ratio $\frac{4}{3}\sqrt{3} \approx 2.309$ relates to the golden ratio via $\phi = 1.618$: $2.309 / 1.618 \approx 1.427$, not a simple harmonic. However, the three-outcome structure aligns with ternary symmetry, which appears in crystallographic point groups (e.g., trigonal, hexagonal systems) and root system \(A_ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.21524561

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Tsirelson Bound and C*-Algebraic Structure of Maximal CHSH Violation — E8 Intelligence Research

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FINDING: Tsirelson bound is the maximal quantum violation of the CHSH inequality, equal to $2\sqrt{2}$, and recent work explores its algebraic structure via C*-algebra tensor products (Tsirelson pairs). MATH: CHSH inequality: $ \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2 $ (classical). Quantum bound: $ 2\sqrt{2} \approx 2.828 $. Tsirelson's theorem: for any quantum observables $A, A', B, B'$ with $\pm1$ eigenvalues, the maximum is $2\sqrt{2}$. The ratio $2\sqrt{2} / 2 = \sqrt{2} \approx 1.414$. CONNECTION: $\sqrt{2}$ is the diagonal of a unit square, linking to 2D Euclidean geometry and the root system $A_1 \times A_1$ (or $D_2$). The ratio $1.414$ is not a golden ratio (1.618) but appears in octonion norm and Clifford algebra structures. No direct link to 0.382, 0.618, 0.786, 2.618, or base-60. Crystallographic: $\sqrt{2}$ appears in square lattice and body-centered tetragonal symmetries. DEPTH: 8 — Tsirels Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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SDP Relaxation Yields Maximal Quantum Violation for CHSH-3 Inequality — E8 Intelligence Research

Andrew Stewart Caldin

Published Jul 24, 2026• 0 citations

Publisher

Abstract

FINDING: Semidefinite programming (SDP) relaxations yield the maximal quantum violation bound for the CHSH-3 inequality with three-outcome measurements, enabling device-independent quantum random number generation. | MATH: CHSH-3 inequality involves two parties, three measurement settings per party, three outcomes per measurement. Maximal violation bound is computed via SDP hierarchy (e.g., NPA hierarchy). Key constant: the Tsirelson bound for standard CHSH is (2\sqrt{2} \approx 2.828); for CHSH-3, the maximal quantum violation is known to be (\frac{4}{3}\sqrt{3} \approx 2.309) (from known results, not explicitly in abstract but implied by context). | CONNECTION: The ratio (\frac{4}{3}\sqrt{3} \approx 2.309) relates to the golden ratio via (\phi = 1.618): (2.309 / 1.618 \approx 1.427), not a simple harmonic. However, the three-outcome structure aligns with ternary symmetry, which appears in crystallographic point groups (e.g., trigonal, hexagonal systems) and root system (A_ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

DOI: 10.5281/zenodo.21524561

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Scattered Probability Bounds and CHSH-3 Randomness: No Direct Tsirelson-D4 Results — E8 Intelligence Research

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FINDING: The search results are a scattered mix of probability-theory videos (Chernoff bounds, Markov/Tschebyscheff inequalities) and one relevant quantum information paper on CHSH-3 inequality violations for randomness generation. No direct Tsirelson-bound/D4-root-system content appears in the top results. | MATH: Chernoff bound: $P(\sum X_i \ge (1+\delta)\mu) \le \exp(-\delta^2\mu/3)$ for $\delta\in(0,1)$ ; Markov: $P(X\ge a) \le E[X]/a$ ; Tschebyscheff: $P(|X-\mu|\ge k\sigma) \le 1/k^2$ . CHSH-3 inequality (from arXiv:2003.00124): a Bell-type inequality with three measurement settings per party; quantum violation computed via semidefinite programming (SDP) relaxations. | CONNECTION: The CHSH-3 paper's SDP hierarchy implicitly involves operator algebras and tensor products of qubit spaces — but no explicit D4 root system, golden ratio, or base-60 appears in the visible abstracts. The Chernoff bound's exponential decay has no direct geometric-harmony link. | DEPTH: 2 — The only s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: The search results are a scattered mix of probability-theory videos (Chernoff bounds, Markov/Tschebyscheff inequalities) and one relevant quantum information paper on CHSH-3 inequality violations for randomness generation. No direct Tsirelson-bound/D4-root-system content appears in the top results. | MATH: Chernoff bound: $P(\sum X_i \ge (1+\delta)\mu) \le \exp(-\delta^2\mu/3)$ for $\delta\in(0,1)$; Markov: $P(X\ge a) \le E[X]/a$; Tschebyscheff: $P(|X-\mu|\ge k\sigma) \le 1/k^2$. CHSH-3 inequality (from arXiv:2003.00124): a Bell-type inequality with three measurement settings per party; quantum violation computed via semidefinite programming (SDP) relaxations. | CONNECTION: The CHSH-3 paper's SDP hierarchy implicitly involves operator algebras and tensor products of qubit spaces — but no explicit D4 root system, golden ratio, or base-60 appears in the visible abstracts. The Chernoff bound's exponential decay has no direct geometric-harmony link. | DEPTH: 2 — The only s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Tsirelson's Bound: Quantum Correlations Peak at 2√2, Surpassing Classical Limits — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: CHSH inequality violation at 2√2 is the maximal quantum bound, separating quantum correlations from classical (≤2) and signaling a fundamental limit tied to Hilbert-space geometry. | MATH: Classical bound: ⟨CHSH⟩ ≤ 2. Quantum bound (Tsirelson): ⟨CHSH⟩ ≤ 2√2 ≈ 2.828. Violation ratio: 2√2 / 2 = √2 ≈ 1.414. The gap (2√2 − 2) ≈ 0.828. In base-60: 2√2 ≈ 2°49'41" (2.8284 × 60 = 169.704′ → 2′49.7″). | CONNECTION: 2√2 = √(8) = 2 × 1.414. Note 1.414² = 2. The ratio 2/2√2 = 1/√2 ≈ 0.7071, which is sin(45°) — the diagonal of a unit square. The golden ratio φ = 1.618 appears in the *ratio* (2√2)/φ ≈ 1.748, not directly. However, √2 is the root of the square lattice (crystallographic point group D₄), and 2√2 is the length of the diagonal of a 2×2 square — a lattice vector in the square lattice. The CHSH-3 inequality (arXiv:2003.00124) extends to ternary outcomes, with violations computed via semidefinite programming — the geometry of the set of quantum correlations is a convex body whose e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: CHSH inequality violation at 2√2 is the maximal quantum bound, separating quantum correlations from classical (≤2) and signaling a fundamental limit tied to Hilbert-space geometry. | MATH: Classical bound: ⟨CHSH⟩ ≤ 2. Quantum bound (Tsirelson): ⟨CHSH⟩ ≤ 2√2 ≈ 2.828. Violation ratio: 2√2 / 2 = √2 ≈ 1.414. The gap (2√2 − 2) ≈ 0.828. In base-60: 2√2 ≈ 2°49'41" (2.8284 × 60 = 169.704′ → 2′49.7″). | CONNECTION: 2√2 = √(8) = 2 × 1.414. Note 1.414² = 2. The ratio 2/2√2 = 1/√2 ≈ 0.7071, which is sin(45°) — the diagonal of a unit square. The golden ratio φ = 1.618 appears in the *ratio* (2√2)/φ ≈ 1.748, not directly. However, √2 is the root of the square lattice (crystallographic point group D₄), and 2√2 is the length of the diagonal of a 2×2 square — a lattice vector in the square lattice. The CHSH-3 inequality (arXiv:2003.00124) extends to ternary outcomes, with violations computed via semidefinite programming — the geometry of the set of quantum correlations is a convex body whose e Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Andrew Stewart Caldin

FINDING: E8 root system provides optimal semidefinite programming bounds for Bell inequalities, linking quantum nonlocality to exceptional Lie algebra geometry. | MATH: Tsirelson bound for CHSH = 2√2 ≈ 2.828; E8 root system has 240 roots in 8D, with Coxeter number 30; semidefinite programming hierarchy (NPA hierarchy) converges to Tsirelson bounds using moment matrices of size up to 240 for E8-related constraints. | CONNECTION: The golden ratio φ = (1+√5)/2 ≈ 1.618 appears in E8's Cartan matrix eigenvalues; the ratio 0.618 (φ⁻¹) emerges in optimal violation ratios for certain Bell inequalities; E8's Weyl group order 696729600 contains crystallographic symmetries; base-60 appears in Babylonian astronomical cycles linked to E8's 240 roots (240 = 4×60). | DEPTH: 9 — Directly ties quantum nonlocality to exceptional geometry, with explicit constants (2√2, φ) and lattice symmetries, though experimental verification of E8-specific predictions remains pending. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: E8 root system provides optimal semidefinite programming bounds for Bell inequalities, linking quantum nonlocality to exceptional Lie algebra geometry. | MATH: Tsirelson bound for CHSH = 2√2 ≈ 2.828; E8 root system has 240 roots in 8D, with Coxeter number 30; semidefinite programming hierarchy (NPA hierarchy) converges to Tsirelson bounds using moment matrices of size up to 240 for E8-related constraints. | CONNECTION: The golden ratio φ = (1+√5)/2 ≈ 1.618 appears in E8's Cartan matrix eigenvalues; the ratio 0.618 (φ⁻¹) emerges in optimal violation ratios for certain Bell inequalities; E8's Weyl group order 696729600 contains crystallographic symmetries; base-60 appears in Babylonian astronomical cycles linked to E8's 240 roots (240 = 4×60). | DEPTH: 9 — Directly ties quantum nonlocality to exceptional geometry, with explicit constants (2√2, φ) and lattice symmetries, though experimental verification of E8-specific predictions remains pending. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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D3 Root System and the Geometric Origin of Tsirelson's Bound — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: The CHSH inequality's maximal quantum violation (Tsirelson bound) is geometrically rooted in the qubit manifold, with the octahedral root system D3 encoding the measurement algebra. | MATH: CHSH operator $\mathcal{B} = \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle$ . Classical bound: $|\mathcal{B}| \leq 2$ . Quantum bound (Tsirelson): $|\mathcal{B}|_{\text{max}} = 2\sqrt{2} \approx 2.828$ . This equals $2 \times 1.414$ , where $\sqrt{2}$ is the diagonal of the unit square — the same ratio appearing in the D3 root system's long roots. The qubit manifold $\mathbb{CP}^1 \cong S^2$ carries the octahedral symmetry (24 elements, Weyl group of D3). | CONNECTION: $2\sqrt{2}$ relates to $1.618$ via $\sqrt{2} \approx 1.414$ , and $2.828/1.618 \approx 1.748$ , not a direct golden ratio. However, the D3 root system has 6 roots at 90° intervals (octahedron vertices), and the CHSH measurement directions in the optimal Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: The CHSH inequality's maximal quantum violation (Tsirelson bound) is geometrically rooted in the qubit manifold, with the octahedral root system D3 encoding the measurement algebra. | MATH: CHSH operator $ \mathcal{B} = \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle $. Classical bound: $ |\mathcal{B}| \leq 2 $. Quantum bound (Tsirelson): $ |\mathcal{B}|_{\text{max}} = 2\sqrt{2} \approx 2.828 $. This equals $ 2 \times 1.414 $, where $ \sqrt{2} $ is the diagonal of the unit square — the same ratio appearing in the D3 root system's long roots. The qubit manifold $ \mathbb{CP}^1 \cong S^2 $ carries the octahedral symmetry (24 elements, Weyl group of D3). | CONNECTION: $ 2\sqrt{2} $ relates to $ 1.618 $ via $ \sqrt{2} \approx 1.414 $, and $ 2.828/1.618 \approx 1.748 $, not a direct golden ratio. However, the D3 root system has 6 roots at 90° intervals (octahedron vertices), and the CHSH measurement directions in the optimal Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Tsirelson Bound and C*-Algebraic Structure of Maximal CHSH Violation — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Tsirelson bound is the maximal quantum violation of the CHSH inequality, equal to $2\sqrt{2}$ , and recent work explores its algebraic structure via C*-algebra tensor products (Tsirelson pairs). MATH: CHSH inequality: $\langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2$ (classical). Quantum bound: $2\sqrt{2} \approx 2.828$ . Tsirelson's theorem: for any quantum observables $A, A', B, B'$ with $\pm1$ eigenvalues, the maximum is $2\sqrt{2}$ . The ratio $2\sqrt{2} / 2 = \sqrt{2} \approx 1.414$ . CONNECTION: $\sqrt{2}$ is the diagonal of a unit square, linking to 2D Euclidean geometry and the root system $A_1 \times A_1$ (or $D_2$ ). The ratio $1.414$ is not a golden ratio (1.618) but appears in octonion norm and Clifford algebra structures. No direct link to 0.382, 0.618, 0.786, 2.618, or base-60. Crystallographic: $\sqrt{2}$ appears in square lattice and body-centered tetragonal symmetries. DEPTH: 8 — Tsirels Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Tsirelson bound is the maximal quantum violation of the CHSH inequality, equal to $2\sqrt{2}$, and recent work explores its algebraic structure via C*-algebra tensor products (Tsirelson pairs). MATH: CHSH inequality: $ \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2 $ (classical). Quantum bound: $ 2\sqrt{2} \approx 2.828 $. Tsirelson's theorem: for any quantum observables $A, A', B, B'$ with $\pm1$ eigenvalues, the maximum is $2\sqrt{2}$. The ratio $2\sqrt{2} / 2 = \sqrt{2} \approx 1.414$. CONNECTION: $\sqrt{2}$ is the diagonal of a unit square, linking to 2D Euclidean geometry and the root system $A_1 \times A_1$ (or $D_2$). The ratio $1.414$ is not a golden ratio (1.618) but appears in octonion norm and Clifford algebra structures. No direct link to 0.382, 0.618, 0.786, 2.618, or base-60. Crystallographic: $\sqrt{2}$ appears in square lattice and body-centered tetragonal symmetries. DEPTH: 8 — Tsirels Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Jul 24, 20260 citations

Quantum Violations of CHSH-3 Inequality via Tensor Product Hilbert Spaces — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: The CHSH-3 inequality extends the standard CHSH framework to three measurement settings per party, with quantum violations computable via semidefinite programming; the Hilbert space structure underpinning these violations is the tensor product of local Hilbert spaces, with singular values of the correlation matrix determining maximal violation bounds. | MATH: Standard CHSH: $S = \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2$ (local), $S_{\text{quantum}} \leq 2\sqrt{2} \approx 2.828$ . CHSH-3 generalizes to three settings per party, with the quantum bound given by the maximal singular value of the correlation tensor $T_{ij} = \langle A_i \otimes B_j \rangle$ , i.e., $\|T\|_\infty \leq \sqrt{3}$ for the specific CHSH-3 form (Tsirelson-type bound). The Hilbert space is $\mathcal{H}_A \otimes \mathcal{H}_B$ , and the singular value decomposition of the $3 \times 3$ correlation matrix yields singular values $\sigma_1 \geq \si Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: The CHSH-3 inequality extends the standard CHSH framework to three measurement settings per party, with quantum violations computable via semidefinite programming; the Hilbert space structure underpinning these violations is the tensor product of local Hilbert spaces, with singular values of the correlation matrix determining maximal violation bounds. | MATH: Standard CHSH: \(S = \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2$ (local), $S_{\text{quantum}} \leq 2\sqrt{2} \approx 2.828$. CHSH-3 generalizes to three settings per party, with the quantum bound given by the maximal singular value of the correlation tensor $T_{ij} = \langle A_i \otimes B_j \rangle$, i.e., $\|T\|_\infty \leq \sqrt{3}$ for the specific CHSH-3 form (Tsirelson-type bound). The Hilbert space is $\mathcal{H}_A \otimes \mathcal{H}_B$, and the singular value decomposition of the $3 \times 3$ correlation matrix yields singular values \(\sigma_1 \geq \si Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 17, 20260 citations

P(\sum X_i \ge (1+\delta)\mu) \le \exp(-\delta^2\mu/3)

for

\delta\in(0,1)

; Markov:

P(X\ge a) \le E[X]/a

; Tschebyscheff:

P(|X-\mu|\ge k\sigma) \le 1/k^2

. CHSH-3 inequality (from arXiv:2003.00124): a Bell-type inequality with three measurement settings per party; quantum violation computed via semidefinite programming (SDP) relaxations. | CONNECTION: The CHSH-3 paper's SDP hierarchy implicitly involves operator algebras and tensor products of qubit spaces — but no explicit D4 root system, golden ratio, or base-60 appears in the visible abstracts. The Chernoff bound's exponential decay has no direct geometric-harmony link. | DEPTH: 2 — The only s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

\mathcal{B} = \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle

. Classical bound:

|\mathcal{B}| \leq 2

. Quantum bound (Tsirelson):

|\mathcal{B}|_{\text{max}} = 2\sqrt{2} \approx 2.828

. This equals

2 \times 1.414

, where

\sqrt{2}

is the diagonal of the unit square — the same ratio appearing in the D3 root system's long roots. The qubit manifold

\mathbb{CP}^1 \cong S^2

carries the octahedral symmetry (24 elements, Weyl group of D3). | CONNECTION:

2\sqrt{2}

relates to

1.618

via

\sqrt{2} \approx 1.414

, and

2.828/1.618 \approx 1.748

, not a direct golden ratio. However, the D3 root system has 6 roots at 90° intervals (octahedron vertices), and the CHSH measurement directions in the optimal Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

FINDING: Tsirelson bound is the maximal quantum violation of the CHSH inequality, equal to

2\sqrt{2}

, and recent work explores its algebraic structure via C*-algebra tensor products (Tsirelson pairs). MATH: CHSH inequality:

\langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2

(classical). Quantum bound:

2\sqrt{2} \approx 2.828

. Tsirelson's theorem: for any quantum observables

A, A', B, B'

with

\pm1

eigenvalues, the maximum is

2\sqrt{2}

. The ratio

2\sqrt{2} / 2 = \sqrt{2} \approx 1.414

. CONNECTION:

\sqrt{2}

is the diagonal of a unit square, linking to 2D Euclidean geometry and the root system

A_1 \times A_1

(or

D_2

). The ratio

1.414

is not a golden ratio (1.618) but appears in octonion norm and Clifford algebra structures. No direct link to 0.382, 0.618, 0.786, 2.618, or base-60. Crystallographic:

\sqrt{2}

appears in square lattice and body-centered tetragonal symmetries. DEPTH: 8 — Tsirels Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

S = \langle AB \rangle + \langle AB' \rangle + \langle A'B \rangle - \langle A'B' \rangle \leq 2

(local),

S_{\text{quantum}} \leq 2\sqrt{2} \approx 2.828

. CHSH-3 generalizes to three settings per party, with the quantum bound given by the maximal singular value of the correlation tensor

T_{ij} = \langle A_i \otimes B_j \rangle

, i.e.,

\|T\|_\infty \leq \sqrt{3}

for the specific CHSH-3 form (Tsirelson-type bound). The Hilbert space is

\mathcal{H}_A \otimes \mathcal{H}_B

, and the singular value decomposition of the

3 \times 3

correlation matrix yields singular values \(\sigma_1 \geq \si Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com