Abstract
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.
Highlights
Since its formulation in 1964, Bell’s theorem has played a crucial role in quantum theory, both for its role in understanding the foundations of the theory, and for its ubiquitous applications in quantum technologies
We show that the violation of the CHSH-Bell inequality is independent of the quantum reference frame (QRF) chosen and that, in particular, it can be maximally violated with a specific choice of the initial state and by transforming appropriately the observables from the rest frame to the laboratory frame
Key to the result is the introduction of a transformation to “jump” to the rest frame of a general quantum system, in the case where two Dirac particles are moving in a superposition of relativistic velocities
Summary
Since its formulation in 1964, Bell’s theorem has played a crucial role in quantum theory, both for its role in understanding the foundations of the theory, and for its ubiquitous applications in quantum technologies. We show that the CHSH-Bell inequality for massive particles in the special-relativistic regime can be maximally violated without postselecting on the momentum of the particles, thereby solving the open problem. Key to this result is the definition of a quantum reference frame (QRF) transformation to the rest frame of a particle moving in a quantum superposition of velocities (momenta). Thanks to the operational identification of the observables that maximally violate Bell’s inequalities in the relativistic regime, the range of application of the technologies utilising tools from the field of Bell nonlocality, among which Quantum Key Distribution, Quantum Communication Complexity, and deviceindependent protocols This involves a Wigner rotation under change of QRF
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