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 four joint measurements turns out to give a universal value, $〈\^a\ensuremath{\bigotimes}\mathrm{b\ifmmode \hat{}\else \^{}\fi{}}〉+〈\^a\ensuremath{\bigotimes}{b}^{\ensuremath{'}}〉+〈{a}^{\ensuremath{'}}\ensuremath{\bigotimes}\mathrm{b\ifmmode \hat{}\else \^{}\fi{}}〉\ensuremath{-}〈{a}^{\ensuremath{'}}\ensuremath{\bigotimes}{b}^{\ensuremath{'}}〉=(2/\sqrt{2\ensuremath{-}{\ensuremath{\beta}}^{2}})(1+\sqrt{1\ensuremath{-}{\ensuremath{\beta}}^{2}}),$ where $\^a,\mathrm{b\ifmmode \hat{}\else \^{}\fi{}}$ are the relativistic spin observables derived from the Pauli-Lubanski pseudovector and $\ensuremath{\beta}=(v/c).$ We found that the degree of violation of the Bell's inequality is decreasing with increasing velocity of the observer and Bell's inequality is satisfied in the ultrarelativistic limit where the boost speed reaches the speed of light.