Relativistic entanglement and Bell's inequality

Abstract

In this paper, the Lorentz transformation of entangled Bell states seen by a moving observer is studied. The calculated Bell observable for 4 joint measurements turns out to give a universal value, $<\hat{a}\otimes\vec{b}>+<\hat{a}\otimes\vec{b}'>+\lang le\hat{a}'\otimes\vec{b}> -<\hat{a}'\otimes\vec{b}'>=\frac{2}{\sqrt{2-\beta^2}}(1+\sqrt{1-\be ta^2})$, where $\hat{a}, \hat{b}$ are the relativistic spin observables derived from the Pauli-Lubanski pseudo vector and $\beta=\frac{v}{c}$. We found that the degree of violation of the Bell's inequality is decreasing with increasing velocity of the observer and the Bell's inequality is satisfied in the ultra-relativistic limit where the boost speed reaches the speed of light.