Tsirelson Bound Rooted in Singular Values of Correlation Matrix — E8 Intelligence Research

Abstract

FINDING: The CHSH inequality's maximal quantum violation (Tsirelson bound) is exactly \(2\sqrt{2}\), a value rooted in the singular-value structure of a 4×4 correlation matrix, not an arbitrary limit. MATH: - CHSH operator: \(S = A_1 \otimes B_1 + A_1 \otimes B_2 + A_2 \otimes B_1 - A_2 \otimes B_2\), with \(A_i, B_j \in \{\pm1\}\) observables. - Classical bound: \(\langle S \rangle \leq 2\). - Quantum bound (Tsirelson): \(\langle S \rangle_{\max} = 2\sqrt{2} \approx 2.828\). - Singular value decomposition of the 4×4 correlation matrix \(C_{ij} = \langle A_i B_j \rangle\): the maximum of \(\sum_{ij} M_{ij} C_{ij}\) over all quantum correlations equals \(2\sqrt{2} \cdot \sigma_{\max}(M)\), where \(M\) is the CHSH coefficient matrix \(\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\). Here \(\sigma_{\max}(M) = \sqrt{2}\), so the bound is \(2\sqrt{2}\). - Equivalently: \(\sqrt{2} = \frac{2}{\sqrt{2}} = \frac{1}{\sin(\pi/4)} = \frac{1}{\cos(\pi/4)}\). - The ratio \(2\sqrt{2}/2 = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com