Cat states and violation of the Bell-CHSH inequality in relativistic Quantum Field Theory

TL;DR

This work achieves an explicit Bell-CHSH violation within a relativistic quantum field theory by constructing a right-wedge cat state and using bounded Hermitian sign operators $A(f)=sign(\varphi(f))$ together with Summers-Werner type test functions derived from Tomita-Takesaki modular theory. The authors obtain a closed-form expression for the Bell-CHSH correlator $\langle \psi| {\cal C} |\psi\rangle$ that comprises a vacuum term $\langle {\cal C} \rangle_0$ and an interference term ${\cal R}$, with the total reducing to $2 + \frac{2}{\pi} \frac{\cos(\sigma) e^{-2\alpha^2}}{1+\cos(\sigma) e^{-2\alpha^2}} {\cal J}(\alpha)$ in the $\lambda=1$ limit. For suitable choices of parameters, notably $\sigma=\pi$, the correlator exceeds the classical bound, achieving a modest violation around $2.012$, thereby realizing Summers-Werner type nonlocality in an explicit QFT setting. The result highlights the role of interference from cat states and the utility of modular theory-inspired constructions in enabling analytic QFT-level Bell tests with wedge-local observables.

Abstract

A cat state localized in the right Rindler wedge is employed to study the violation of the Bell-CHSH inequality in a relativistic scalar free Quantum Field Theory. By means of the bounded Hermitian operator $sign(\varphi(f))$, where $\varphi(f)$ stands for the smeared scalar field, it turns out that the Bell-CHSH correlator can be evaluated in closed analytic form in terms of the imaginary error function. Being the superposition of two coherent states, cat states allow for the existence of interference terms which give rise to a violation of the Bell-CHSH inequality. As such, the present setup can be considered as an explicit realization of the results obtained by Summers-Werner.

Figures (2)

• Figure 1: Behavior of the correlator$\langle {\cal C}\rangle_0$ as a function of the spectral parameter $\lambda$. Although $\lambda$ belongs to the spectral interval $[0,1]$, the plot has been extended over a larger interval to show that $\langle {\cal C} \rangle_0$ is bounded by 2.
• Figure 2: Behavior of the correlator $\langle {\cal C}\rangle_{\psi}$ as function of the parameter $\alpha$, showing the violation of the Bell-CHSH inequality. The value 2 is asymptotically recovered for very large values of $\alpha$.