The wanted extension of Fujii and Tsurumaru's formula for the spectral radius of the Bell-CHSH operator

This paper is motivated by a recent paper of Yuki Fujii and Toyohiro Tsurumaru in which they established a beautiful formula for the spectral radius of the Bell-CHSH operator on finite-dimensional Hilbert spaces. To tackle the operator on infinite-dimensional spaces, they elaborated a method based on appropriate approximation of commutators of infinite-dimensional orthogonal projections by commutators of orthogonal projections on finite-dimensional spaces. We here give a proof of Fujii and Tsurumaru’s original formula that works in all dimensions. We also present an alternative approximation procedure, uncover the connection of the problem with block Toeplitz operators, and derive good estimates and explicit expressions for the spectral radius in concrete cases.

💡 Research Summary

The paper revisits the Bell‑CHSH operator, defined for two parties by (B=(A_{1}+B_{1})\otimes A_{2}+(A_{1}-B_{1})\otimes B_{2}) where each (A_{i}=2P_{i}-I) and (B_{i}=2Q_{i}-I) are involutions built from orthogonal projections (P_{i},Q_{i}). A well‑known identity shows that (B^{2}=4I\otimes I+