Entanglement and nonlocality of a single relativistic particle

Recent work has argued that the concepts of entanglement and nonlocality must be taken seriously even in systems consisting of only a single particle. These treatments, however, are nonrelativistic, and, if single-particle entanglement is fundamental, it should also persist in a relativistic description. Here, we consider a spin-1/2 particle in a superposition of two different velocities as viewed by an observer in a relativistically boosted inertial frame and show that the entanglement between the two velocity modes survives right up to the speed of light. We also discuss how quantum gates could be implemented in this way and apply our results to the case of a superconductor. In particular, we show that an $s$-wave superconductor would have $p$-wave components for a boosted observer.

Figure 1

A single spin-1/2 particle initially has its spin pointing up in the $z$ direction. It is boosted to some velocity $v_1$ in the $y$ direction (or a superposition of velocities $+v_1$ and $-v_1$ along the $y$ axis). The observer of the system is moving at velocity $v_2$ perpendicular to the velocity of the particle, i.e., in the $x\text{−}z$ plane. The direction of the observer’s boost is given by the angle $\varphi$ from the $z$ axis.

Figure 2

(Color online) The relative entropy of entanglement for ${\rho }^{\prime }$ given by Eq. (8) as a function of the velocities $v_1$ and $v_2$.

Figure 3

Scheme for measuring the Wigner rotation. Two relativistically boosted spin-1/2 particles (in the same spin state) are scattered or passed through a beam splitter. The probability that both particles are detected at the same output port is related to the angle $\omega$ of the Wigner rotation.