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

Abstract

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.com