Remarks on the Bell-Clauser-Horne-Shimony-Holt inequality

We discuss the relationship between the Bogoliubov transformations, squeezed states, entanglement, and maximum violation of the Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality. In particular, we point out that the construction of the four bounded operators entering the Bell-CHSH inequality can be worked out in a simple and general way, covering a large variety of models, ranging from quantum mechanism to relativistic quantum field theories. Various examples are employed to illustrate the above mentioned framework. We start by considering a pair of entangled spin-one particles and a squeezed oscillator in quantum mechanics, moving then to the relativistic complex quantum scalar field and to the analysis of the vacuum state in Minkowski space-time in terms of the left and right Rindler modes. In the latter case, the Bell-CHSH inequality turns out to be parametrized by the Unruh temperature.

Automated summary

We discuss the relationship between Bogoliubov transformations, squeezed states, entanglement, and the maximum violation of the Bell-CHSH inequality. We point out a simple and general method to construct the bounded operators involved in the Bell-CHSH inequality, which is applicable to a wide range of models from quantum mechanics to relativistic quantum field theories. Various examples are provided to illustrate this framework, including entangled spin-one particles, squeezed oscillators, and the vacuum state in Minkowski spacetime parametrized by the Unruh temperature.