Single particle entanglement in the mid-and ultra-relativistic regime

TL;DR

This paper investigates how relativistic effects, specifically Lorentz boosts and Wigner rotations, influence the entanglement between spin and momentum in a single massive spin-1/2 particle, revealing complex behavior in different regimes.

Contribution

It provides a detailed analysis of entanglement behavior under Lorentz transformations, highlighting the contrasting monotonic trends in mid- and ultra-relativistic regimes for helicity states.

Findings

Entanglement decreases monotonically in the mid-relativistic regime for equal helicity states.

Entanglement increases in the ultra-relativistic regime for equal helicity states.

Superpositions of unequal helicity states invert the monotonic behavior of entanglement.

Abstract

In this work we analyze the amount of entanglement associated with the spin and momentum degrees of freedom of a single massive spin-$\frac{1}{2}$ particle from a relativistic perspective. The effect of a Lorentz boost introduces a Wigner rotation that correlates the spin and momentum degrees of freedom. We show that the natural basis to discuss the geometrical effects of the boost are the helicity eigenstates in the rest frame. In the mid-relativistic regime (where the Wigner rotation angle is limited by $\delta < \frac{\pi}{2}$) we prove for states with equal helicity that the entanglement with respect to the Wigner rotation angle is monotonically decreasing, however, in the ultra-relativistic regime ($\delta > \frac{\pi}{2}$) the entanglement is increasing. If the states are prepared as a superposition of unequal helicity eigenstates, the monotonic behaviour is inverted. This implies that in the ultra-relativistic regime a geometrical setup can be found such that the amount of entanglement exhibits local maxima or minima. This shows a counter-intuitive behaviour of the relative amount of entanglement, an effect due to the internal and external geometrical configuration space, and points towards the difficulties in achieving a Lorentz invariant formulation of entanglement in general.