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

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 simplify and correct some arguments employed by Fujii and Tsurumaru.

Figures (3)

• Figure 1: The spectra of $P_n(A+B)P_n$ multiplied by $i$ for $n=10$ (left) and $n=600$ (right) with $A,B$ as in (\ref{['exx']}) and with $\omega=\pi/2$ and $\theta=0.1:0.1:3.1$.
• Figure 2: Plots of $\lambda_{\max}(\theta)$ where $\lambda_{\max}(\theta)$ is the largest eigenvalue of $P_n(A+B)P_n$ for $n=100$ with $A,B$ as in (\ref{['exx']}). In the left picture we took $\omega=\pi/2$, in the right picture we have $\omega=0.1:0.1:3.1$. The red curve is $2\sin \theta$, the green curve is $2|\cos \theta|$. The horizontal lines are at the height $\sqrt{2}$. The rightmost vertical line in the left picture has the abscissa $x_3=2.4352$.
• Figure 3: The spectra of $P_n(A+B)P_n$ multiplied by $i$ for $n=100$ with $A,B$ as in (\ref{['pert']}) and $\theta=0.1:0.1:3.1$.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6