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

TL;DR

This paper extends Fujii and Tsurumaru's formula for the spectral radius of the Bell-CHSH operator to infinite-dimensional spaces, introduces new approximation methods, and provides explicit spectral radius estimates.

Contribution

It provides a proof of the spectral radius formula valid in all dimensions and introduces an alternative approximation approach linked to block Toeplitz operators.

Findings

Proof of the spectral radius formula in all dimensions

New approximation method for infinite-dimensional operators

Explicit spectral radius estimates in specific cases

Abstract

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.