Relativistic Bell Test within Quantum Reference Frames

A still widely debated question in the field of relativistic quantum information is whether entanglement and the degree of violation of Bell’s inequalities for massive relativistic particles are frame independent or not. At the core of this question is the effect that spin gets entangled with the momentum degree of freedom at relativistic velocities. Here, we show that Bell’s inequalities for a pair of particles can be maximally violated in a special-relativistic regime, even without any postselection of the momentum of the particles. To this end, we use the methodology of quantum reference frames, which allows us to transform the problem to the rest frame of a particle, whose state can be in a superposition of relativistic momenta from the viewpoint of the laboratory frame. We show that, when the relative motion of two particles is noncollinear, the optimal measurements for violation of Bell’s inequalities in the laboratory frame involve “coherent Wigner rotations.” Moreover, the degree of violation of Bell’s inequalities is independent of the choice of the quantum reference frame. Our results open up the possibility of extending entanglement-based quantum communication protocols to relativistic regimes.

Figure 1

In $A$’s perspective (above), the spin $s_B$ of the Dirac particle $B$ depends on its momentum since $B$ is moving in a superposition of two sharp relativistic velocities $v_{B,1}$ and $v_{B,2}$. Moreover, the state of the laboratory $C$ is in a superposition of two relativistic velocities $-v_1$ and $-v_2$ relative to $A$. In the initial QRF $A$, the spin $s_A$ and the Dirac particle $B$ are entangled (similarly to the singlet state $|{\mathrm{\Psi }}^{-}⟩=(|\uparrow \downarrow ⟩-|\downarrow \uparrow ⟩)/\sqrt{2}$) which is illustrated by the correlation between the dashed and between the solid arrows. The QRF transformation ${\widehat{S}}_2$ from $A$ to $C$ coherently boosts the two Dirac particles by the velocity of $C$ and outputs the perspective of the laboratory (below). In the laboratory frame $C$, the two Dirac particles $A$ and $B$ are entangled and both spin $s_A$ and $s_B$ depend on the corresponding momentum d.o.f. $A$ and $B$.